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 Material Information Title: Collaborative Statistics Physical Description: Book Language: en-US Creator: Illowsky, BarbaraDean, Susan Publisher: Connexions, Rice University Place of Publication: Houston, Texas Publication Date: 2008-08-25
 Subjects Subjects / Keywords: Mathematics, Statistics, anova, binomial, bivariate, central, chi, confidence, continuous, data, descriptive, discrete, distribution, elementary, exponential, geometric, hypergeometric, hypothesis, intervals, limit, linear, mean, normal, poisson, probability, random, regression, sampling, square, statistics, student-t, testing, theorem, uniform, univariate, â€¦Mathematics, Statistical Analysis, Statistical Distributions, Statistics
 Notes Abstract: This textbook is intended for introductory statistics courses being taken by students at twoâ€“ and fourâ€“year colleges who are majoring in fields other than math or engineering. Intermediate algebra is the only prerequisite. The book focuses on applications of statistical knowledge rather than the theory behind it. The textbook was developed over several years and has been used in regular and honors-level classroom settings and in distance learning classes. Chapter titles: 1. Sampling and Data, 2. Descriptive Statistics, 3. Probability Topics, 4. Discrete Random Variables, 5. Continuous Random Variables, 6. The Normal Distribution, 7. The Central Limit Theorem, 8. Confidence Intervals, 9. Hypothesis Testing: Single Mean and Single Proportion, 10. Hypothesis Testing: Two Means, Paired Data, Two Proportions, 11. The Chi-Square Distribution, 12. Linear Regression and Correlation, 13. F Distribution and ANOVA, 14. Appendix. The textbook contains full materials for course offerings, including expository text, examples, labs, homework, and projects. A Teacherâ€™s Guide is currently available in print form and on the Connexions site at http://cnx.org/content/col10547/latest10. The on-line text meets the Section 508 standards for accessibility. *** WebAssign ancillaries are available.*** (http://www.webassign.net/collaborativestatistics). WebAssign Content for the Collaborative Statistics project includes: 1. Approximately two thirds of the Homework problems from the text 2. Approximately one Lab from each chapter 3. Applicable sections of the text linked to individual questions 4. Video and web resources linked from each question 5. Tutorial items that interactively step students through problem-solving strategies General Note: Expositive General Note: Community College, Higher Education General Note: http://www.ogtp-cart.com/product.aspx?ISBN=9781616100193 General Note: Adobe Reader General Note: Dr. Barbara Illowsky, Susan Dean General Note: Graph, Narrative text, Table, Textbook General Note: illowskybarbara@deanza.edu, deansusan@deanza.edu General Note: STA 022 - BASIC STATISTICS, STA 023 - STATISTICAL METHODS I General Note: http://florida.theorangegrove.org/og/file/687aa918-8ada-78a1-e889-f532825afd6b/1/col10522.pdf
 Record Information Source Institution: University Press of Florida Holding Location: University Press of Florida Rights Management: Creative Commons Attribution License (CC-BY 2.0). Under terms of this license, you are free: 1) to share - to copy, distribute and transmit the work ; 2) to remix - to adapt the work; under the following conditions: 1) Attribution. You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse). Resource Identifier: isbn - 9781616100193 System ID: AA00011706:00001

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CollaborativeStatistics By: BarbaraIllowsky SusanDean

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CollaborativeStatistics By: BarbaraIllowsky SusanDean Online: < http://cnx.org/content/col10522/1.25/ > CONNEXIONS RiceUniversity,Houston,Texas

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TableofContents Preface ................................................................................................1 AuthorAckowledgements ............................................................................5 StudentWelcomeLetter ...............................................................................6 1SamplingandData 1.1 SamplingandData...........................................................................7 1.2 Statistics......................................................................................7 1.3 Probability...................................................................................9 1.4 KeyTerms....................................................................................9 1.5 Data........................................................................................11 1.6 Sampling....................................................................................12 1.7 Variation....................................................................................17 1.8 AnswersandRoundingOff.................................................................18 1.9 Frequency...................................................................................18 1.10 Summary..................................................................................23 1.11 Practice:SamplingandData................................................................25 1.12 Homework.................................................................................28 1.13 Lab1:DataCollection......................................................................36 1.14 Lab2:SamplingExperiment...............................................................38 Solutions........................................................................................41 2DescriptiveStatistics 2.1 DescriptiveStatistics........................................................................43 2.2 DisplayingData.............................................................................43 2.3 StemandLeafGraphsStemplots...........................................................44 2.4 Histograms.................................................................................45 2.5 BoxPlots....................................................................................48 2.6 MeasuresoftheLocationoftheData........................................................51 2.7 MeasuresoftheCenteroftheData..........................................................54 2.8 SkewnessandtheMean,Median,andMode.................................................56 2.9 MeasuresoftheSpreadoftheData..........................................................57 2.10 SummaryofFormulas......................................................................64 2.11 Practice1:CenteroftheData...............................................................65 2.12 Practice2:SpreadoftheData...............................................................68 2.13 Homework.................................................................................69 2.14 Lab:DescriptiveStatistics..................................................................84 Solutions........................................................................................86 3ProbabilityTopics 3.1 ProbabilityTopics...........................................................................93 3.2 Terminology................................................................................94 3.3 IndependentandMutuallyExclusiveEvents................................................95 3.4 TwoBasicRulesofProbability...............................................................97 3.5 ContingencyTables........................................................................100 3.6 VennDiagramsoptional..................................................................103 3.7 TreeDiagramsoptional...................................................................105 3.8 SummaryofFormulas......................................................................109 3.9 Practice1:ContingencyTables.............................................................110 3.10 Practice2:CalculatingProbabilities.......................................................112

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iv 3.11 Homework...............................................................................113 3.12 Review...................................................................................121 3.13 Lab:ProbabilityTopics....................................................................124 Solutions.......................................................................................127 4DiscreteRandomVariables 4.1 DiscreteRandomVariables.................................................................133 4.2 ProbabilityDistributionFunctionPDFforaDiscreteRandomVariable....................134 4.3 MeanorExpectedValueandStandardDeviation...........................................135 4.4 CommonDiscreteProbabilityDistributionFunctions.......................................138 4.5 Binomial...................................................................................138 4.6 Geometricoptional.......................................................................141 4.7 Hypergeometricoptional.................................................................143 4.8 Poisson....................................................................................146 4.9 SummaryofFunctions.....................................................................148 4.10 Practice1:DiscreteDistribution...........................................................150 4.11 Practice2:BinomialDistribution..........................................................151 4.12 Practice3:PoissonDistribution...........................................................153 4.13 Practice4:GeometricDistribution.........................................................154 4.14 Practice5:HypergeometricDistribution...................................................156 4.15 Homework...............................................................................157 4.16 Review...................................................................................165 4.17 Lab1:DiscreteDistributionPlayingCardExperiment....................................168 4.18 Lab2:DiscreteDistributionLuckyDiceExperiment.....................................172 Solutions.......................................................................................176 5ContinuousRandomVariables 5.1 ContinuousRandomVariables.............................................................183 5.2 ContinuousProbabilityFunctions..........................................................184 5.3 TheUniformDistribution..................................................................186 5.4 TheExponentialDistribution...............................................................190 5.5 SummaryoftheUniformandExponentialProbabilityDistributions........................196 5.6 Practice1:UniformDistribution............................................................197 5.7 Practice2:ExponentialDistribution........................................................200 5.8 Homework.................................................................................202 5.9 Review.....................................................................................207 5.10 Lab:ContinuousDistribution.............................................................210 Solutions.......................................................................................213 6TheNormalDistribution 6.1 TheNormalDistribution...................................................................217 6.2 TheStandardNormalDistribution.........................................................218 6.3 Z-scores....................................................................................218 6.4 AreastotheLeftandRightofx.............................................................220 6.5 CalculationsofProbabilities................................................................220 6.6 SummaryofFormulas......................................................................224 6.7 Practice:TheNormalDistribution..........................................................225 6.8 Homework.................................................................................227 6.9 Review.....................................................................................232 6.10 Lab1:NormalDistributionLapTimes...................................................234 6.11 Lab2:NormalDistributionPinkieLength...............................................236 Solutions.......................................................................................240 7TheCentralLimitTheorem

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v 7.1 TheCentralLimitTheorem.................................................................243 7.2 TheCentralLimitTheoremforSampleMeansAverages...................................244 7.3 TheCentralLimitTheoremforSums.......................................................246 7.4 UsingtheCentralLimitTheorem...........................................................248 7.5 SummaryofFormulas......................................................................253 7.6 Practice:TheCentralLimitTheorem.......................................................254 7.7 Homework.................................................................................257 7.8 Review.....................................................................................263 7.9 Lab1:CentralLimitTheoremPocketChange.............................................265 7.10 Lab2:CentralLimitTheoremCookieRecipes............................................269 Solutions.......................................................................................274 8CondenceIntervals 8.1 CondenceIntervals.......................................................................277 8.2 CondenceInterval,SinglePopulationMean,PopulationStandardDeviationKnown,Normal 279 8.3 CondenceInterval,SinglePopulationMean,StandardDeviationUnknown,Student-T....282 8.4 CondenceIntervalforaPopulationProportion............................................284 8.5 SummaryofFormulas......................................................................288 8.6 Practice1:CondenceIntervalsforAverages,KnownPopulationStandardDeviation.......289 8.7 Practice2:CondenceIntervalsforAverages,UnknownPopulationStandardDeviation....291 8.8 Practice3:CondenceIntervalsforProportions............................................293 8.9 Homework.................................................................................295 8.10 Review...................................................................................304 8.11 Lab1:CondenceIntervalHomeCosts..................................................308 8.12 Lab2:CondenceIntervalPlaceofBirth.................................................310 8.13 Lab3:CondenceIntervalWomens'Heights............................................312 Solutions.......................................................................................314 9HypothesisTesting:SingleMeanandSingleProportion 9.1 HypothesisTesting:SingleMeanandSingleProportion....................................321 9.2 NullandAlternateHypotheses.............................................................322 9.3 OutcomesandtheTypeIandTypeIIErrors................................................323 9.4 DistributionNeededforHypothesisTesting................................................324 9.5 Assumption................................................................................325 9.6 RareEvents................................................................................325 9.7 UsingtheSampletoSupportOneoftheHypotheses.......................................325 9.8 DecisionandConclusion...................................................................327 9.9 AdditionalInformation....................................................................327 9.10 SummaryoftheHypothesisTest..........................................................328 9.11 Examples.................................................................................329 9.12 SummaryofFormulas....................................................................340 9.13 Practice1:SingleMean,KnownPopulationStandardDeviation...........................341 9.14 Practice2:SingleMean,UnknownPopulationStandardDeviation........................343 9.15 Practice3:SingleProportion..............................................................345 9.16 Homework...............................................................................347 9.17 Review...................................................................................359 9.18 Lab:HypothesisTestingofaSingleMeanandSingleProportion...........................362 Solutions.......................................................................................366 10HypothesisTesting:TwoMeans,PairedData,TwoProportions 10.1 HypothesisTesting:TwoPopulationMeansandTwoPopulationProportions..............373

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vi 10.2 ComparingTwoIndependentPopulationMeanswithUnknownPopulationStandardDeviations.......................................................................................374 10.3 ComparingTwoIndependentPopulationMeanswithKnownPopulationStandardDeviations 377 10.4 ComparingTwoIndependentPopulationProportions.....................................379 10.5 MatchedorPairedSamples...............................................................381 10.6 SummaryofTypesofHypothesisTests....................................................386 10.7 Practice1:HypothesisTestingforTwoProportions........................................387 10.8 Practice2:HypothesisTestingforTwoAverages...........................................389 10.9 Homework...............................................................................391 10.10 Review..................................................................................401 10.11 Lab:HypothesisTestingforTwoMeansandTwoProportions............................404 Solutions.......................................................................................409 11TheChi-SquareDistribution 11.1 TheChi-SquareDistribution..............................................................415 11.2 Notation..................................................................................416 11.3 FactsAbouttheChi-SquareDistribution..................................................416 11.4 Goodness-of-FitTest......................................................................417 11.5 TestofIndependence......................................................................424 11.6 TestofaSingleVarianceOptional........................................................427 11.7 SummaryofFormulas....................................................................430 11.8 Practice1:Goodness-of-FitTest...........................................................431 11.9 Practice2:ContingencyTables............................................................433 11.10 Practice3:TestofaSingleVariance.......................................................435 11.11 Homework..............................................................................437 11.12 Review..................................................................................443 11.13 Lab1:Chi-SquareGoodness-of-Fit.......................................................448 11.14 Lab2:Chi-SquareTestforIndependence.................................................453 Solutions.......................................................................................455 12LinearRegressionandCorrelation 12.1 LinearRegressionandCorrelation........................................................461 12.2 LinearEquations..........................................................................461 12.3 SlopeandY-InterceptofaLinearEquation................................................463 12.4 ScatterPlots...............................................................................463 12.5 TheRegressionEquation..................................................................466 12.6 TheCorrelationCoefcient................................................................469 12.7 FactsAbouttheCorrelationCoefcientforLinearRegression..............................470 12.8 Prediction.................................................................................472 12.9 Outliers...................................................................................473 12.10 95%CriticalValuesoftheSampleCorrelationCoefcientTable...........................478 12.11 Summary................................................................................480 12.12 Practice:LinearRegression...............................................................481 12.13 Homework..............................................................................484 12.14 Lab1:RegressionDistancefromSchool.................................................498 12.15 Lab2:RegressionTextbookCost........................................................501 12.16 Lab3:RegressionFuelEfciency.......................................................504 Solutions.......................................................................................508 13FDistributionandANOVA 13.1 FDistributionandANOVA...............................................................511 13.2 ANOVA..................................................................................511

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vii 13.3 TheFDistributionandtheFRatio.........................................................512 13.4 FactsAbouttheFDistribution............................................................513 13.5 TestofTwoVariances.....................................................................517 13.6 Summary.................................................................................519 13.7 Practice:ANOVA.........................................................................520 13.8 Homework...............................................................................522 13.9 Review...................................................................................523 13.10 Lab:ANOVA............................................................................527 Solutions.......................................................................................529 14Appendix 14.1 PracticeFinalExam1.....................................................................531 14.2 PracticeFinalExam2.....................................................................540 14.3 DataSets..................................................................................549 14.4 GroupProjects............................................................................552 14.5 SolutionSheets............................................................................563 14.6 EnglishPhrasesWrittenMathematically...................................................567 14.7 SymbolsandtheirMeanings..............................................................568 14.8 Formulas.................................................................................573 Solutions.......................................................................................575 TI-83+andTI-84CalculatorInstructions ............................................................579 Glossary .............................................................................................589 Index ................................................................................................597

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3 thinkingstatistically incorporatingtechnology workingcollaboratively writingthoughtfully Theseconceptsareintegraltoourcourse.Studentslearnthebestbyactivelyparticipating,notbyjust watchingandlistening.Teachingshouldbehighlyinteractive.Studentsneedtobethoroughlyengaged inthelearningprocessinordertomakesenseofstatisticalconcepts. CollaborativeStatistics provides techniquesforstudentstowriteacrossthecurriculum,tocollaboratewiththeirpeers,tothinkstatistically, andtoincorporatetechnology. Thisbooktakesstudentsstepbystep.Thetextisinteractive.Therefore,studentscanimmediatelyapply whattheyread.Oncestudentshavecompletedtheprocessofproblemsolving,theycantackleinteresting andchallengingproblemsrelevanttotoday'sworld.Theproblemsrequirethestudentstoapplytheir newlyfoundskills.Inaddition,technologyTI-83graphingcalculatorsarehighlightedisincorporated throughoutthetextandtheproblems,aswellasinthespecialgroupactivitiesandprojects.Thebookalso containslabsthatuserealdataandpracticesthatleadstudentsstepbystepthroughtheproblemsolving process. AtDeAnza,alongwithhundredsofothercollegesacrossthecountry,thecollegeaudienceinvolvesa largenumberofESLstudentsaswellasstudentsfrommanydisciplines.TheESLstudents,aswellas thenon-ESLstudents,havebeenespeciallyappreciativeofthistext.Theynditextremelyreadableand understandable. CollaborativeStatistics hasbeenusedinclassesthatrangefrom20to120students,andin regular,honor,anddistancelearningclasses. SusanDean BarbaraIllowsky

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8 CHAPTER1.SAMPLINGANDDATA 1.2.1OptionalCollaborativeClassroomExercise Inyourclassroom,trythisexercise.Haveclassmemberswritedowntheaveragetimeinhours,tothe nearesthalf-hourtheysleeppernight.Yourinstructorwillrecordthedata.Thencreateasimplegraph calleda dotplot ofthedata.Adotplotconsistsofanumberlineanddotsorpointspositionedabove thenumberline.Forexample,considerthefollowingdata: 5;5.5;6;6;6;6.5;6.5;6.5;6.5;7;7;8;8;9 Thedotplotforthisdatawouldbeasfollows: FrequencyofAverageTimeinHoursSpentSleepingperNight Figure1.1 Doesyourdotplotlookthesameasordifferentfromtheexample?Why?Ifyoudidthesameexamplein anEnglishclasswiththesamenumberofstudents,doyouthinktheresultswouldbethesame?Whyor whynot? Wheredoyourdataappeartocluster?Howcouldyouinterprettheclustering? Thequestionsaboveaskyoutoanalyzeandinterpretyourdata.Withthisexample,youhavebegunyour studyofstatistics. Inthiscourse,youwilllearnhowtoorganizeandsummarizedata.Organizingandsummarizingdatais called descriptivestatistics .Twowaystosummarizedataarebygraphingandbynumbersforexample, ndinganaverage.Afteryouhavestudiedprobabilityandprobabilitydistributions,youwilluseformal methodsfordrawingconclusionsfrom"good"data.Theformalmethodsarecalled inferentialstatistics Statisticalinferenceusesprobabilitytodetermineifconclusionsdrawnarereliableornot. Effectiveinterpretationofdatainferenceisbasedongoodproceduresforproducingdataandthoughtful examinationofthedata.Youwillencounterwhatwillseemtobetoomanymathematicalformulasfor interpretingdata.Thegoalofstatisticsisnottoperformnumerouscalculationsusingtheformulas,butto gainanunderstandingofyourdata.Thecalculationscanbedoneusingacalculatororacomputer.The understandingmustcomefromyou.Ifyoucanthoroughlygraspthebasicsofstatistics,youcanbemore condentinthedecisionsyoumakeinlife.

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10 CHAPTER1.SAMPLINGANDDATA isanumericalvariable.Ifwelet Y beaperson'spartyafliation,thenexamplesof Y includeRepublican, Democrat,andIndependent. Y isacategoricalvariable.Wecoulddosomemathwithvaluesof X calculate theaveragenumberofpointsearned,forexample,butitmakesnosensetodomathwithvaluesof Y calculatinganaveragepartyafliationmakesnosense. Data aretheactualvaluesofthevariable.Theymaybenumbersortheymaybewords.Datumisasingle value. Twowordsthatcomeupofteninstatisticsare average and proportion .Ifyouweretotakethreeexams inyourmathclassesandobtainedscoresof86,75,and92,youcalculateyouraveragescorebyaddingthe threeexamscoresanddividingbythreeyouraveragescorewouldbe84.3toonedecimalplace.If,inyour mathclass,thereare40studentsand22aremenand18arewomen,thentheproportionofmenstudents is 22 40 andtheproportionofwomenstudentsis 18 40 .Averageandproportionarediscussedinmoredetailin laterchapters. Example1.1 Denethekeytermsfromthefollowingstudy:Wewanttoknowtheaverageamountofmoney rstyearcollegestudentsspendatABCCollegeonschoolsuppliesthatdonotincludebooks. Threestudentsspent\$150,\$200,and\$225,respectively. Solution The population isallrstyearstudentsattendingABCCollegethisterm. The sample couldbeallstudentsenrolledinonesectionofabeginningstatisticscourseatABC Collegealthoughthissamplemaynotrepresenttheentirepopulation. The parameter istheaverageamountofmoneyspentexcludingbooksbyrstyearcollegestudentsatABCCollegethisterm. The statistic istheaverageamountofmoneyspentexcludingbooksbyrstyearcollegestudents inthesample. The variable couldbetheamountofmoneyspentexcludingbooksbyonerstyearstudent. Let X =theamountofmoneyspentexcludingbooksbyonerstyearstudentattendingABC College. The data arethedollaramountsspentbytherstyearstudents.Examplesofthedataare\$150, \$200,and\$225. 1.4.1OptionalCollaborativeClassroomExercise Dothefollowingexercisecollaborativelywithuptofourpeoplepergroup.Findapopulation,asample, theparameter,thestatistic,avariable,anddataforthefollowingstudy:Youwanttodeterminetheaverage numberofglassesofmilkcollegestudentsdrinkperday.Supposeyesterday,inyourEnglishclass,you askedvestudentshowmanyglassesofmilktheydrankthedaybefore.Theanswerswere1,0,1,3,and4 glassesofmilk.

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11 1.5Data 5 Datamaycomefromapopulationorfromasample.Smallletterslike x or y generallyareusedtorepresent datavalues.Mostdatacanbeputintothefollowingcategories: Qualitative Quantitative Qualitativedata aretheresultofcategorizingordescribingattributesofapopulation.Haircolor,blood type,ethnicgroup,thecarapersondrives,andthestreetapersonlivesonareexamplesofqualitativedata. Qualitativedataaregenerallydescribedbywordsorletters.Forinstance,haircolormightbeblack,dark brown,lightbrown,blonde,gray,orred.BloodtypemightbeAB+,O-,orB+.Qualitativedataarenotas widelyusedasquantitativedatabecausemanynumericaltechniquesdonotapplytothequalitativedata. Forexample,itdoesnotmakesensetondanaveragehaircolororbloodtype. Quantitativedata arealwaysnumbersandareusuallythedataofchoicebecausetherearemanymethods availableforanalyzingthedata.Quantitativedataaretheresultof counting or measuring attributesofa population.Amountofmoney,pulserate,weight,numberofpeoplelivinginyourtown,andthenumber ofstudentswhotakestatisticsareexamplesofquantitativedata.Quantitativedatamaybeeither discrete or continuous Alldatathataretheresultofcountingarecalled quantitativediscretedata .Thesedatatakeononlycertain numericalvalues.Ifyoucountthenumberofphonecallsyoureceiveforeachdayoftheweek,youmight get0,1,2,3,etc. Alldatathataretheresultofmeasuringare quantitativecontinuousdata assumingthatwecanmeasure accurately.Measuringanglesinradiansmightresultinthenumbers p 6 p 3 p 2 p 3 p 4 ,etc.Ifyouandyour friendscarrybackpackswithbooksinthemtoschool,thenumbersofbooksinthebackpacksarediscrete dataandtheweightsofthebackpacksarecontinuousdata. Example1.2:DataSampleofQuantitativeDiscreteData Thedataarethenumberofbooksstudentscarryintheirbackpacks.Yousamplevestudents. Twostudentscarry3books,onestudentcarries4books,onestudentcarries2books,andone studentcarries1book.Thenumbersofbooks,4,2,and1arethequantitativediscretedata. Example1.3:DataSampleofQuantitativeContinuousData Thedataaretheweightsofthebackpackswiththebooksinit.Yousamplethesamevestudents. Theweightsinpoundsoftheirbackpacksare6.2,7,6.8,9.1,4.3.Noticethatbackpackscarrying threebookscanhavedifferentweights.Weightsarequantitativecontinuousdatabecauseweights aremeasured. Example1.4:DataSampleofQualitativeData Thedataarethecolorsofbackpacks.Again,yousamplethesamevestudents.Onestudenthas aredbackpack,twostudentshaveblackbackpacks,onestudenthasagreenbackpack,andone studenthasagraybackpack.Thecolorsred,black,black,green,andgrayarequalitativedata. N OTE :Youmaycollectdataasnumbersandreportitcategorically.Forexample,thequizscores foreachstudentarerecordedthroughouttheterm.Attheendoftheterm,thequizscoresare reportedasA,B,C,D,orF. Example1.5 Workcollaborativelytodeterminethecorrectdatatypequantitativeorqualitative.Indicate whetherquantitativedataarecontinuousordiscrete.Hint:Datathatarediscreteoftenstartwith thewords"thenumberof." 5 Thiscontentisavailableonlineat.

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12 CHAPTER1.SAMPLINGANDDATA 1.Thenumberofpairsofshoesyouown. 2.Thetypeofcaryoudrive. 3.Whereyougoonvacation. 4.Thedistanceitisfromyourhometothenearestgrocerystore. 5.Thenumberofclassesyoutakeperschoolyear. 6.Thetuitionforyourclasses 7.Thetypeofcalculatoryouuse. 8.Movieratings. 9.Politicalpartypreferences. 10.Weightofsumowrestlers. 11.Amountofmoneyindollarswonplayingpoker. 12.Numberofcorrectanswersonaquiz. 13.Peoples'attitudestowardthegovernment. 14.IQscores.Thismaycausesomediscussion. 1.6Sampling 6 Gatheringinformationaboutanentirepopulationoftencoststoomuchorisvirtuallyimpossible.Instead, weuseasampleofthepopulation. Asampleshouldhavethesamecharacteristicsasthepopulationitis representing. Twocommonmethodsofsamplingare withreplacement and withoutreplacement .Ifeachmemberofa populationmaybechosenmorethanoncethenthesamplingiswithreplacement.Ifeachmembermaybe chosenonlyonce,thenthesamplingiswithoutreplacement. Oneofthemostimportantmethodsofobtainingsamplesiscalled randomsampling .Ifeachmemberofa populationhasanequalchanceofbeingselectedforthesample,thesampleiscalleda simplerandomsample .Twosimplerandomsampleswouldcontainmembersequallyrepresentativeoftheentirepopulation. Inotherwords,eachsampleofthesamesizehasanequalchanceofbeingselected.Forexample,suppose Lisawantstoformafour-personstudygroupherselfandthreeotherpeoplefromherpre-calculusclass, whichhas32membersincludingLisa.Tochooseasimplerandomsampleofsize3fromtheothermembers ofherclass,Lisarstliststhelastnamesofthemembersofherclasstogetherwithatwo-digitnumberas shownbelow. 6 Thiscontentisavailableonlineat.

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13 ClassRoster ID Name 00 Anselmo 01 Bautista 02 Bayani 03 Cheng 04 Cuarismo 05 Cuningham 06 Fontecha 07 Hong 08 Hoobler 09 Jiao 10 Khan 11 King 12 Legeny 13 Lundquist 14 Macierz 15 Motogawa 16 Okimoto 17 Patel 18 Price 19 Quizon 20 Reyes 21 Roquero 22 Roth 23 Rowell 24 Salangsang 25 Slade 26 Stracher 27 Tallai 28 Tran 29 Wai 30 Wood Lisacaneitheruseatableofrandomnumbersfoundinmanystatisticsbooksaswellasmathematical handbooksoracalculatororcomputertogeneraterandomnumbers.Forthisexample,supposeLisa choosestogeneraterandomnumbersfromacalculator.Thenumbersgeneratedare:

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14 CHAPTER1.SAMPLINGANDDATA .94360;.99832;.14669;.51470;.40581;.73381;.04399 Lisareadstwo-digitgroupsuntilshehaschosenthreeclassmembersthatis,shereads.94360asthegroups 94,43,36,60.Eachrandomnumbermayonlycontributeoneclassmember.Ifsheneededto,Lisacould havegeneratedmorerandomnumbers. Therandomnumbers.94360and.99832donotcontainappropriatetwodigitnumbers.Howeverthethird randomnumber,.14669,contains14thefourthrandomnumberalsocontains14,thefthrandomnumber contains05,andtheseventhrandomnumbercontains04.Thetwo-digitnumber14correspondstoMacierz, 05correspondstoCunningham,and04correspondstoCuarismo.Besidesherself,Lisa'sgroupwillconsist ofMarcierz,andCunningham,andCuarismo. Sometimes,itisdifcultorimpossibletoobtainasimplerandomsamplebecausepopulationsaretoo large.Thenwechooseotherformsofsamplingmethodsthatinvolveachanceprocessforgettingthe sample. Otherwell-knownrandomsamplingmethodsarethestratiedsample,theclustersample,andthe systematicsample. Tochoosea stratiedsample ,dividethepopulationintogroupscalledstrataandthentakeasamplefrom eachstratum.Forexample,youcouldstratifygroupyourcollegepopulationbydepartmentandthen chooseasimplerandomsamplefromeachstratumtogetastratiedrandomsample. Tochoosea clustersample ,dividethepopulationintosectionsandthenrandomlyselectsomeofthesections.Allthemembersfromthesesectionsareintheclustersample.Forexample,ifyourandomlysample fourdepartmentsfromyourstratiedcollegepopulationrandomlychoosefourdepartmentsfromallof thedepartments,thefourdepartmentsmakeuptheclustersample. Tochoosea systematicsample ,randomlyselectastartingpointandtakeeverynthpieceofdatafroma listingofthepopulation.Forexample,supposeyouhavetodoaphonesurvey.Yourphonebookcontains 20,000residencelistings.Youmustchoose400namesforthesample.Youstartbyrandomlypickingone oftherst50namesandthenchooseevery50thnamethereafter.Systematicsamplingisfrequentlychosen becauseitisasimplemethod. Atypeofsamplingthatisnonrandomisconveniencesampling. Conveniencesampling involvesusing resultsthatarereadilyavailable.Forexample,acomputersoftwarestoreconductsamarketingstudyby interviewingpotentialcustomerswhohappentobeinthestorebrowsingthroughtheavailablesoftware. Theresultsofconveniencesamplingmaybeverygoodinsomecasesandhighlybiasedfavorscertain outcomesinothers. Samplingdatashouldbedoneverycarefully.Collectingdatacarelesslycanhavedevastatingresults.Surveysmailedtohouseholdsandthenreturnedmaybeverybiasedforexample,theymayfavoracertain group.Itisbetterforthepersonconductingthesurveytoselectthesamplerespondents. Whenyouanalyzedata,itisimportanttobeawareof samplingerrors andnonsamplingerrors.Theactual processofsamplingcausessamplingerrors.Forexample,thesamplemaynotbelargeenoughorrepresentativeofthepopulation.Factorsnotrelatedtothesamplingprocesscause nonsamplingerrors .Adefective countingdevicecancauseanonsamplingerror. Example1.6 Determinethetypeofsamplingusedsimplerandom,stratied,systematic,cluster,orconvenience. 1.Asoccercoachselects6playersfromagroupofboysaged8to10,7playersfromagroupof boysaged11to12,and3playersfromagroupofboysaged13to14toformarecreational soccerteam. 2.Apollsterinterviewsallhumanresourcepersonnelinvedifferenthightechcompanies.

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15 3.Anengineeringresearcherinterviews50womenengineersand50menengineers. 4.Amedicalresearcherinterviewseverythirdcancerpatientfromalistofcancerpatientsata localhospital. 5.Ahighschoolcounselorusesacomputertogenerate50randomnumbersandthenpicks studentswhosenamescorrespondtothenumbers. 6.Astudentinterviewsclassmatesinhisalgebraclasstodeterminehowmanypairsofjeansa studentowns,ontheaverage. Solution 1.stratied 2.cluster 3.stratied 4.systematic 5.simplerandom 6.convenience Ifweweretoexaminetwosamplesrepresentingthesamepopulation,theywould,morethanlikely,not bethesame.Justasthereisvariationindata,thereisvariationinsamples.Asyoubecomeaccustomedto sampling,thevariabilitywillseemnatural. Example1.7 SupposeABCCollegehas10,000part-timestudentsthepopulation.Weareinterestedinthe averageamountofmoneyapart-timestudentspendsonbooksinthefallterm.Askingall10,000 studentsisanalmostimpossibletask. Supposewetaketwodifferentsamples. First,weuseconveniencesamplingandsurvey10studentsfromarsttermorganicchemistry class.Manyofthesestudentsaretakingrsttermcalculusinadditiontotheorganicchemistry class.Theamountofmoneytheyspendisasfollows: \$128;\$87;\$173;\$116;\$130;\$204;\$147;\$189;\$93;\$153 ThesecondsampleistakenbyusingalistfromtheP.E.departmentofseniorcitizenswhotake P.E.classesandtakingevery5thseniorcitizenonthelist,foratotalof10seniorcitizens.They spend: \$50;\$40;\$36;\$15;\$50;\$100;\$40;\$53;\$22;\$22 Problem1 Doyouthinkthateitherofthesesamplesisrepresentativeoforischaracteristicoftheentire 10,000part-timestudentpopulation? Solution No .Therstsampleprobablyconsistsofscience-orientedstudents.Besidesthechemistrycourse, someofthemaretakingrst-termcalculus.Booksfortheseclassestendtobeexpensive.Most ofthesestudentsare,morethanlikely,payingmorethantheaveragepart-timestudentfortheir books.Thesecondsampleisagroupofseniorcitizenswhoare,morethanlikely,takingcourses forhealthandinterest.Theamountofmoneytheyspendonbooksisprobablymuchlessthanthe averagepart-timestudent.Bothsamplesarebiased.Also,inbothcases,notallstudentshavea chancetobeineithersample.

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16 CHAPTER1.SAMPLINGANDDATA Problem2 Sincethesesamplesarenotrepresentativeoftheentirepopulation,isitwisetousetheresultsto describetheentirepopulation? Solution No. Neveruseasamplethatisnotrepresentativeordoesnothavethecharacteristicsofthe population. Now,supposewetakeathirdsample.Wechoosetendifferentpart-timestudentsfromthedisciplinesofchemistry,math,English,psychology,sociology,history,nursing,physicaleducation,art, andearlychildhooddevelopment.Eachstudentischosenusingsimplerandomsampling.Using acalculator,randomnumbersaregeneratedandastudentfromaparticulardisciplineisselected ifhe/shehasacorrespondingnumber.Thestudentsspend: \$180;\$50;\$150;\$85;\$260;\$75;\$180;\$200;\$200;\$150 Problem3 Doyouthinkthissampleisrepresentativeofthepopulation? Solution Yes. Itischosenfromdifferentdisciplinesacrossthepopulation. Studentsoftenaskifitis"goodenough"totakeasample,insteadofsurveyingtheentirepopulation.Ifthesurveyisdonewell,theanswerisyes. 1.6.1OptionalCollaborativeClassroomExercise Exercise1.6.1 Asaclass,determinewhetherornotthefollowingsamplesarerepresentative.Iftheyarenot, discussthereasons. 1.TondtheaverageGPAofallstudentsinauniversity,useallhonorstudentsattheuniversityasthesample. 2.Tondoutthemostpopularcerealamongyoungpeopleundertheageof10,standoutside alargesupermarketforthreehoursandspeaktoevery20thchildunderage10whoenters thesupermarket. 3.TondtheaverageannualincomeofalladultsintheUnitedStates,sampleU.S.congressmen.Createaclustersamplebyconsideringeachstateasastratumgroup.Byusingsimple randomsampling,selectstatestobepartofthecluster.ThensurveyeveryU.S.congressman inthecluster. 4.Todeterminetheproportionofpeopletakingpublictransportationtowork,survey20peopleinNewYorkCity.ConductthesurveybysittinginCentralParkonabenchandinterviewingeverypersonwhositsnexttoyou. 5.TodeterminetheaveragecostofatwodaystayinahospitalinMassachusetts,survey100 hospitalsacrossthestateusingsimplerandomsampling.

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17 1.7Variation 7 1.7.1VariationinData Variationispresentinanysetofdata.Forexample,16-ouncecansofbeveragemaycontainmoreorless than16ouncesofliquid.Inonestudy,eight16ouncecansweremeasuredandproducedthefollowing amountinouncesofbeverage: 15.8;16.1;15.2;14.8;15.8;15.9;16.0;15.5 Measurementsoftheamountofbeverageina16-ouncecanmayvarybecausedifferentpeoplemakethe measurementsorbecausetheexactamount,16ouncesofliquid,wasnotputintothecans.Manufacturers regularlyrunteststodetermineiftheamountofbeverageina16-ouncecanfallswithinthedesiredrange. Beawarethatasyoutakedata,yourdatamayvarysomewhatfromthedatasomeoneelseistakingforthe samepurpose.Thisiscompletelynatural.However,iftwoormoreofyouaretakingthesamedataand getverydifferentresults,itistimeforyouandtheotherstoreevaluateyourdata-takingmethodsandyour accuracy. 1.7.2VariationinSamples Itwasmentionedpreviouslythattwoormore samples fromthesame population andhavingthesame characteristicsasthepopulationmaybedifferentfromeachother.SupposeDoreenandJungbothdecide tostudytheaverageamountoftimestudentssleepeachnightanduseallstudentsattheircollegeasthe population.DoreenusessystematicsamplingandJungusesclustersampling.Doreen'ssamplewillbe differentfromJung'ssampleeventhoughbothsampleshavethecharacteristicsofthepopulation.Even ifDoreenandJungusedthesamesamplingmethod,inalllikelihoodtheirsampleswouldbedifferent. Neitherwouldbewrong,however. ThinkaboutwhatcontributestomakingDoreen'sandJung'ssamplesdifferent. IfDoreenandJungtooklargersamplesi.e.thenumberofdatavaluesisincreased,theirsampleresults theaverageamountoftimeastudentsleepswouldbeclosertotheactualpopulationaverage.Butstill, theirsampleswouldbe,inalllikelihood,differentfromeachother.This variabilityinsamples cannotbe stressedenough. 1.7.2.1SizeofaSample Thesizeofasampleoftencalledthenumberofobservationsisimportant.Theexamplesyouhaveseenin thisbooksofarhavebeensmall.Smallsamplescan"work"butthepersontakingthesamplemustbevery careful.Samplesthatarefrom1200to1500observationsareconsideredlargeenoughandgoodenoughif thesurveyisrandomandiswelldone.Youwilllearnwhywhenyoustudycondenceintervals. 1.7.2.2OptionalCollaborativeClassroomExercise Exercise1.7.1 Divideintogroupsoftwo,three,orfour.Yourinstructorwillgiveeachgroupone6-sideddie. Trythisexperimenttwice. Rollonefairdie-sided20times.Recordthenumberofones,twos, threes,fours,ves,andsixesyougetbelow"frequency"isthenumberoftimesaparticularface ofthedieoccurs: 7 Thiscontentisavailableonlineat.

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19 5;6;3;3;2;4;7;5;2;3;5;6;5;4;4;3;5;2;5;3 Belowisafrequencytablelistingthedifferentdatavaluesinascendingorderandtheirfrequencies. FrequencyTableofStudentWorkHours DATAVALUE FREQUENCY 2 3 3 5 4 3 5 6 6 2 7 1 A frequency isthenumberoftimesagivendatumoccursinadataset.Accordingtothetableabove, therearethreestudentswhowork2hours,vestudentswhowork3hours,etc.Thetotalofthefrequency column,20,representsthetotalnumberofstudentsincludedinthesample. A relativefrequency isthefractionoftimesanansweroccurs.Tondtherelativefrequencies,divide eachfrequencybythetotalnumberofstudentsinthesample-inthiscase,20.Relativefrequenciescanbe writtenasfractions,percents,ordecimals. FrequencyTableofStudentWorkHoursw/RealativeFrequency DATAVALUE FREQUENCY RELATIVEFREQUENCY 2 3 3 20 or0.15 3 5 5 20 or0.25 4 3 3 20 or0.15 5 6 6 20 or0.30 6 2 2 20 or0.10 7 1 1 20 or0.05 Thesumoftherelativefrequencycolumnis 20 20 ,or1. Cumulativerelativefrequency istheaccumulationofthepreviousrelativefrequencies.Tondthecumulativerelativefrequencies,addallthepreviousrelativefrequenciestotherelativefrequencyforthecurrent row.

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20 CHAPTER1.SAMPLINGANDDATA FrequencyTableofStudentWorkHoursw/RelativeandCumulativeFrequency DATAVALUE FREQUENCY RELATIVEFREQUENCY CUMULATIVERELATIVEFREQUENCY 2 3 3 20 or0.15 0.15 3 5 5 20 or0.25 0.15+0.25=0.40 4 3 3 20 or0.15 0.40+0.15=0.55 5 6 6 20 or0.10 0.55+0.30=0.85 6 2 2 20 or0.10 0.85+0.10=0.95 7 1 1 20 or0.05 0.95+0.05=1.00 Thelastentryofthecumulativerelativefrequencycolumnisone,indicatingthatonehundredpercentof thedatahasbeenaccumulated. N OTE :Becauseofrounding,therelativefrequencycolumnmaynotalwayssumtooneandthe lastentryinthecumulativerelativefrequencycolumnmaynotbeone.However,theyeachshould beclosetoone. Thefollowingtablerepresentstheheights,ininches,ofasampleof100malesemiprofessionalsoccerplayers. FrequencyTableofSoccerPlayerHeight HEIGHTSINCHES FREQUENCYOFSTUDENTS RELATIVEFREQUENCY CUMULATIVERELATIVEFREQUENCY 59.95-61.95 5 5 100 =0.05 0.05 61.95-63.95 3 3 100 =0.03 0.05+0.03=0.08 63.95-65.95 15 15 100 =0.15 0.08+0.15=0.23 65.95-67.95 40 40 100 =0.40 0.23+0.40=0.63 67.95-69.95 17 17 100 =0.17 0.63+0.17=0.80 69.95-71.95 12 12 100 =0.12 0.80+0.12=0.92 71.95-73.95 7 7 100 =0.07 0.92+0.07=0.99 73.95-75.95 1 1 100 =0.01 0.99+0.01=1.00 Total=100 Total=1.00 Thedatainthistablehasbeen grouped intothefollowingintervals: 59.95-61.95inches 61.95-63.95inches 63.95-65.95inches 65.95-67.95inches 67.95-69.95inches 69.95-71.95inches 71.95-73.95inches 73.95-75.95inches

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21 N OTE :ThisexampleisusedagainintheDescriptiveStatisticsSection2.1chapter,wherethe methodusedtocomputetheintervalswillbeexplained. Inthissample,thereare 5 playerswhoseheightsarebetween59.95-61.95inches, 3 playerswhoseheights fallwithintheinterval61.95-63.95inches, 15 playerswhoseheightsfallwithintheinterval63.95-65.95 inches, 40 playerswhoseheightsfallwithintheinterval65.95-67.95inches, 17 playerswhoseheights fallwithintheinterval67.95-69.95inches, 12 playerswhoseheightsfallwithintheinterval69.95-71.95, 7playerswhoseheightfallswithintheinterval71.95-73.95,and 1 playerwhoseheightfallswithinthe interval73.95-75.95.Allheightsfallbetweentheendpointsofanintervalandnotattheendpoints. Example1.8 Fromthetable,ndthepercentageofheightsthatarelessthan65.95inches. Solution Ifyoulookattherst,second,andthirdrows,theheightsarealllessthan65.95inches.Thereare 5+3+15=23maleswhoseheightsarelessthan65.95inches.Thepercentageofheightslessthan 65.95inchesisthen 23 100 or23%.Thispercentageisthecumulativerelativefrequencyentryinthe thirdrow. Example1.9 Fromthetable,ndthepercentageofheightsthatfallbetween61.95and65.95inches. Solution Addtherelativefrequenciesinthesecondandthirdrows:0.03+0.15=0.18or18%. Example1.10 Usethetableofheightsofthe100malesemiprofessionalsoccerplayers.Fillintheblanksand checkyouranswers. 1.Thepercentageofheightsthatarefrom67.95to71.95inchesis: 2.Thepercentageofheightsthatarefrom67.95to73.95inchesis: 3.Thepercentageofheightsthataremorethan65.95inchesis: 4.Thenumberofplayersinthesamplewhoarebetween61.95and71.95inchestallis: 5.Whatkindofdataaretheheights? 6.Describehowyoucouldgatherthisdatatheheightssothatthedataarecharacteristicofall malesemiprofessionalsoccerplayers. Remember,you countfrequencies .Tondtherelativefrequency,dividethefrequencybythetotal numberofdatavalues.Tondthecumulativerelativefrequency,addallofthepreviousrelative frequenciestotherelativefrequencyforthecurrentrow. 1.9.1OptionalCollaborativeClassroomExercise Exercise1.9.1 Inyourclass,havesomeoneconductasurveyofthenumberofsiblingsbrothersandsisterseach studenthas.Createafrequencytable.Addtoitarelativefrequencycolumnandacumulative relativefrequencycolumn.Answerthefollowingquestions: 1.Whatpercentageofthestudentsinyourclasshave0siblings? 2.Whatpercentageofthestudentshavefrom1to3siblings? 3.Whatpercentageofthestudentshavefewerthan3siblings?

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22 CHAPTER1.SAMPLINGANDDATA Example1.11 Nineteenpeoplewereaskedhowmanymiles,tothenearestmiletheycommutetoworkeach day.Thedataareasfollows: 2;5;7;3;2;10;18;15;20;7;10;18;5;12;13;12;4;5;10 Thefollowingtablewasproduced: FrequencyofCommutingDistances DATA FREQUENCY RELATIVEFREQUENCY CUMULATIVERELATIVEFREQUENCY 3 3 3 19 0.1579 4 1 1 19 0.2105 5 3 3 19 0.1579 7 2 2 19 0.2632 10 3 4 19 0.4737 12 2 2 19 0.7895 13 1 1 19 0.8421 15 1 1 19 0.8948 18 1 1 19 0.9474 20 1 1 19 1.0000 Problem 1.Isthetablecorrect?Ifitisnotcorrect,whatiswrong? 2.TrueorFalse:Threepercentofthepeoplesurveyedcommute3miles.Ifthestatementisnot correct,whatshoulditbe?Ifthetableisincorrect,makethecorrections. 3.Whatfractionofthepeoplesurveyedcommute5or7miles? 4.Whatfractionofthepeoplesurveyedcommute12milesormore?Lessthan12miles?Between5and13milesdoesnotinclude5and13miles?

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23 1.10Summary 10 Statistics Dealswiththecollection,analysis,interpretation,andpresentationofdata Probability Mathematicaltoolusedtostudyrandomness KeyTerms Population Parameter Sample Statistic Variable Data TypesofData QuantitativeDataanumber DiscreteYoucountit. ContinuousYoumeasureit. QualitativeDataacategory,words Sampling WithReplacement :Amemberofthepopulationmaybechosenmorethanonce WithoutReplacement :Amemberofthepopulationmaybechosenonlyonce RandomSampling Eachmemberofthepopulationhasanequalchanceofbeingselected SamplingMethods Random Simplerandomsample Stratiedsample Clustersample Systematicsample NotRandom Conveniencesample N OTE :Samplesmustberepresentativeofthepopulationfromwhichtheycome.Theymusthave thesamecharacteristics.However,theymayvarybutstillrepresentthesamepopulation. Frequencyfreq.orf Thenumberoftimesanansweroccurs 10 Thiscontentisavailableonlineat.

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24 CHAPTER1.SAMPLINGANDDATA RelativeFrequencyrel.freq.orRF Theproportionoftimesanansweroccurs Canbeinterpretedasafraction,decimal,orpercent CumulativeRelativeFrequenciescum.rel.freq.orcumRF Anaccumulationofthepreviousrelativefrequencies

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25 1.11Practice:SamplingandData 11 1.11.1StudentLearningOutcomes Thestudentwillpracticeconstructingfrequencytables. Thestudentwilldifferentiatebetweenkeyterms. Thestudentwillcomparesamplingtechniques. 1.11.2Given Studiesareoftendonebypharmaceuticalcompaniestodeterminetheeffectivenessofatreatmentprogram. SupposethatanewAIDSantibodydrugiscurrentlyunderstudy.ItisgiventopatientsoncetheAIDS symptomshaverevealedthemselves.Ofinterestistheaveragelengthoftimeinmonthspatientsliveonce startingthetreatment.Tworesearcherseachfollowadifferentsetof40AIDSpatientsfromthestartof treatmentuntiltheirdeaths.Thefollowingdatainmonthsarecollected. Researcher1 3;4;11;15;16;17;22;44;37;16;14;24;25;15;26;27;33;29;35;44;13;21;22;10;12;8;40;32; 26;27;31;34;29;17;8;24;18;47;33;34 Researcher2 3;14;11;5;16;17;28;41;31;18;14;14;26;25;21;22;31;2;35;44;23;21;21;16;12;18;41;22; 16;25;33;34;29;13;18;24;23;42;33;29 1.11.3OrganizetheData Completethetablesbelowusingthedataprovided. Researcher1 SurvivalLengthin months Frequency RelativeFrequency CumulativeRel.Frequency 0.5-6.5 6.5-12.5 12.5-18.5 18.5-24.5 24.5-30.5 30.5-36.5 36.5-42.5 42.5-48.5 11 Thiscontentisavailableonlineat.

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26 CHAPTER1.SAMPLINGANDDATA Researcher2 SurvivalLengthin months Frequency RelativeFrequency CumulativeRel.Frequency 0.5-6.5 6.5-12.5 12.5-18.5 18.5-24.5 24.5-30.5 30.5-36.5 36.5-42.5 42.5-48.5 1.11.4KeyTerms DenethekeytermsbasedupontheaboveexampleforResearcher1. Exercise1.11.1 Population Exercise1.11.2 Sample Exercise1.11.3 Parameter Exercise1.11.4 Statistic Exercise1.11.5 Variable Exercise1.11.6 Data 1.11.5DiscussionQuestions Discussthefollowingquestionsandthenanswerincompletesentences. Exercise1.11.7 Listtworeasonswhythedatamaydiffer. Exercise1.11.8 Canyoutellifoneresearcheriscorrectandtheotheroneisincorrect?Why? Exercise1.11.9 Wouldyouexpectthedatatobeidentical?Whyorwhynot? Exercise1.11.10 Howcouldtheresearchersgatherrandomdata?

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27 Exercise1.11.11 Supposethattherstresearcherconductedhissurveybyrandomlychoosingonestateinthe nationandthenrandomlypicking40patientsfromthatstate.Whatsamplingmethodwouldthat researcherhaveused? Exercise1.11.12 Supposethatthesecondresearcherconductedhissurveybychoosing40patientsheknew.What samplingmethodwouldthatresearcherhaveused?Whatconcernswouldyouhaveaboutthis dataset,baseduponthedatacollectionmethod?

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28 CHAPTER1.SAMPLINGANDDATA 1.12Homework 12 Exercise1.12.1 Solutiononp.41. Foreachitembelow: i. Identifythetypeofdataquantitative-discrete,quantitative-continuous,orqualitative thatwouldbeusedtodescribearesponse. ii. Giveanexampleofthedata. a. Numberofticketssoldtoaconcert b. Amountofbodyfat c. Favoritebaseballteam d. Timeinlinetobuygroceries e. NumberofstudentsenrolledatEvergreenValleyCollege f. Mostwatchedtelevisionshow g. Brandoftoothpaste h. Distancetotheclosestmovietheatre i. AgeofexecutivesinFortune500companies j. Numberofcompetingcomputerspreadsheetsoftwarepackages Exercise1.12.2 Fiftypart-timestudentswereaskedhowmanycoursestheyweretakingthisterm.Theincompleteresultsareshownbelow: Part-timeStudentCourseLoads #ofCourses Frequency RelativeFrequency CumulativeRelative Frequency 1 30 0.6 2 15 3 a. Fillintheblanksinthetableabove. b. Whatpercentofstudentstakeexactlytwocourses? c. Whatpercentofstudentstakeoneortwocourses? Exercise1.12.3 Solutiononp.41. Sixtyadultswithgumdiseasewereaskedthenumberoftimesperweektheyusedtoossbefore theirdiagnoses.Theincompleteresultsareshownbelow: FlossingFrequencyforAdultswithGumDisease #FlossingperWeek Frequency RelativeFrequency CumulativeRelativeFreq. 0 27 0.4500 1 18 3 0.9333 6 3 0.0500 7 1 0.0167 12 Thiscontentisavailableonlineat.

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29 a. Fillintheblanksinthetableabove. b. Whatpercentofadultsossedsixtimesperweek? c. Whatpercentossedatmostthreetimesperweek? Exercise1.12.4 Atnesscenterisinterestedintheaverageamountoftimeaclientexercisesinthecentereach week.Denethefollowingintermsofthestudy.Giveexampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data Exercise1.12.5 Solutiononp.41. Skiresortsareinterestedintheaverageagethatchildrentaketheirrstskiandsnowboard lessons.Theyneedthisinformationtooptimallyplantheirskiclasses.Denethefollowingin termsofthestudy.Giveexampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data Exercise1.12.6 Acardiologistisinterestedintheaveragerecoveryperiodforherpatientswhohavehadheart attacks.Denethefollowingintermsofthestudy.Giveexampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data Exercise1.12.7 Solutiononp.41. Insurancecompaniesareinterestedintheaveragehealthcostseachyearfortheirclients,sothat theycandeterminethecostsofhealthinsurance.Denethefollowingintermsofthestudy.Give exampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data

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30 CHAPTER1.SAMPLINGANDDATA Exercise1.12.8 Apoliticianisinterestedintheproportionofvotersinhisdistrictthatthinkheisdoingagood job.Denethefollowingintermsofthestudy.Giveexampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data Exercise1.12.9 Solutiononp.42. Amarriagecounselorisinterestedintheproportiontheclientsshecounselsthatstaymarried. Denethefollowingintermsofthestudy.Giveexampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data Exercise1.12.10 Politicalpollstersmaybeinterestedintheproportionofpeoplethatwillvoteforaparticular cause.Denethefollowingintermsofthestudy.Giveexampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data Exercise1.12.11 Solutiononp.42. Amarketingcompanyisinterestedintheproportionofpeoplethatwillbuyaparticularproduct. Denethefollowingintermsofthestudy.Giveexampleswhereappropriate. a. Population b. Sample c. Parameter d. Statistic e. Variable f. Data Exercise1.12.12 Airlinecompaniesareinterestedintheconsistencyofthenumberofbabiesoneachight,sothat theyhaveadequatesafetyequipment.Supposeanairlineconductsasurvey.OverThanksgiving weekend,itsurveys6ightsfromBostontoSaltLakeCitytodeterminethenumberofbabieson theights.Itdeterminestheamountofsafetyequipmentneededbytheresultofthatstudy. a. Usingcompletesentences,listthreethingswrongwiththewaythesurveywasconducted. b. Usingcompletesentences,listthreewaysthatyouwouldimprovethesurveyifitwereto berepeated.

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31 Exercise1.12.13 Supposeyouwanttodeterminetheaveragenumberofstudentsperstatisticsclassinyourstate. Describeapossiblesamplingmethodin35completesentences.Makethedescriptiondetailed. Exercise1.12.14 Supposeyouwanttodeterminetheaveragenumberofcansofsodadrunkeachmonthbypersons intheirtwenties.Describeapossiblesamplingmethodin3-5completesentences.Makethe descriptiondetailed. Exercise1.12.15 726distancelearningstudentsatLongBeachCityCollegeinthe2004-2005academicyearwere surveyedandaskedthereasonstheytookadistancelearningclass. Source:AmitSchitai,Director ofInstructionalTechnologyandDistanceLearning,LBCC .Theresultsofthissurveyarelistedin thetablebelow. ReasonsforTakingLBCCDistanceLearningCourses Convenience 87.6% Unabletocometocampus 85.1% Takingon-campuscoursesinadditiontomyDLcourse 71.7% Instructorhasagoodreputation 69.1% Tofulllrequirementsfortransfer 60.8% TofulllrequirementsforAssociateDegree 53.6% ThoughtDEwouldbemorevariedandinteresting 53.2% Ilikecomputertechnology 52.1% HadsuccesswithpreviousDLcourse 52.0% On-campussectionswerefull 42.1% Tofulllrequirementsforvocationalcertication 27.1% Becauseofdisability 20.5% Assumethatthesurveyallowedstudentstochoosefromtheresponseslistedinthetableabove. a. Whycanthepercentsadduptoover100%? b. Doesthatnecessarilyimplyamistakeinthereport? c. Howdoyouthinkthequestionwaswordedtogetresponsesthattotaledover100%? d. Howmightthequestionbewordedtogetresponsesthattotaled100%? Exercise1.12.16 NineteenimmigrantstotheU.Swereaskedhowmanyyears,tothenearestyear,theyhavelived intheU.S.Thedataareasfollows: 2;5;7;2;2;10;20;15;0;7;0;20;5;12;15;12;4;5;10

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32 CHAPTER1.SAMPLINGANDDATA Thefollowingtablewasproduced: FrequencyofImmigrantSurveyResponses Data Frequency RelativeFrequency CumulativeRelativeFrequency 0 2 2 19 0.1053 2 3 3 19 0.2632 4 1 1 19 0.3158 5 3 3 19 0.1579 7 2 2 19 0.5789 10 2 2 19 0.6842 12 2 2 19 0.7895 15 1 1 19 0.8421 20 1 1 19 1.0000 a. Fixtheerrorsonthetable.Also,explainhowsomeonemighthavearrivedattheincorrect numbers. b. Explainwhatiswrongwiththisstatement:percentofthepeoplesurveyedhavelived intheU.S.for5years. c. Fixthestatementabovetomakeitcorrect. d. WhatfractionofthepeoplesurveyedhavelivedintheU.S.5or7years? e. WhatfractionofthepeoplesurveyedhavelivedintheU.S.atmost12years? f. WhatfractionofthepeoplesurveyedhavelivedintheU.S.fewerthan12years? g. WhatfractionofthepeoplesurveyedhavelivedintheU.S.from5to20years,inclusive? Exercise1.12.17 Arandomsurveywasconductedof3274peopleofthemicroprocessorgenerationpeople bornsince1971,theyearthemicroprocessorwasinvented.Itwasreportedthat48%ofthose individualssurveyedstatedthatiftheyhad\$2000tospend,theywoulduseitforcomputer equipment.Also,66%ofthosesurveyedconsideredthemselvesrelativelysavvycomputerusers. Source:SanJoseMercuryNews a. Doyouconsiderthesamplesizelargeenoughforastudyofthistype?Whyorwhynot? b. Basedonyourgutfeeling,doyoubelievethepercentsaccuratelyreecttheU.S.populationforthoseindividualsbornsince1971?Ifnot,doyouthinkthepercentsofthe populationareactuallyhigherorlowerthanthesamplestatistics?Why? Additionalinformation:ThesurveywasreportedbyIntelCorporationofindividualswhovisited theLosAngelesConventionCentertoseetheSmithsonianInstiture'sroadshowcalledAmerica's Smithsonian. c. Withthisadditionalinformation,doyoufeelthatalldemographicandethnicgroupswere equallyrepresentedattheevent?Whyorwhynot? d. Withtheadditionalinformation,commentonhowaccuratelyyouthinkthesamplestatisticsreectthepopulationparameters.

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33 Exercise1.12.18 a. Listsomepracticaldifcultiesinvolvedingettingaccurateresultsfromatelephonesurvey. b. Listsomepracticaldifcultiesinvolvedingettingaccurateresultsfromamailedsurvey. c. Withyourclassmates,brainstormsomewaystoovercometheseproblemsifyouneeded toconductaphoneormailsurvey. 1.12.1Trythesemultiplechoicequestions Thenextfourquestionsrefertothefollowing: ALakeTahoeCommunityCollegeinstructorisinterestedin theaveragenumberofdaysLakeTahoeCommunityCollegemathstudentsareabsentfromclassduringa quarter. Exercise1.12.19 Solutiononp.42. Whatisthepopulationsheisinterestedin? A. AllLakeTahoeCommunityCollegestudents B. AllLakeTahoeCommunityCollegeEnglishstudents C. AllLakeTahoeCommunityCollegestudentsinherclasses D. AllLakeTahoeCommunityCollegemathstudents Exercise1.12.20 Solutiononp.42. Considerthefollowing: X =numberofdaysaLakeTahoeCommunityCollegemathstudentisabsent Inthiscase, X isanexampleofa: A. Variable B. Population C. Statistic D. Data Exercise1.12.21 Solutiononp.42. Theinstructortakeshersamplebygatheringdataon5randomlyselectedstudentsfromeach LakeTahoeCommunityCollegemathclass.Thetypeofsamplingsheusedis A. Clustersampling B. Stratiedsampling C. Simplerandomsampling D. Conveniencesampling Exercise1.12.22 Solutiononp.42. Theinstructor'ssampleproducesanaveragenumberofdaysabsentof3.5days.Thisvalueisan exampleofa A. Parameter B. Data C. Statistic D. Variable

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34 CHAPTER1.SAMPLINGANDDATA Thenexttwoquestions refertothefollowingrelativefrequencytableonhurricanesthathavemadedirect hitsontheU.Sbetween1851and2004.Hurricanesaregivenastrengthcategoryratingbasedonthe minimumwindspeedgeneratedbythestorm. http://www.nhc.noaa.gov/gifs/table5.gif [ ? ] FrequencyofHurricaneDirectHits Category NumberofDirectHits RelativeFrequency CumulativeFrequency 1 109 0.3993 0.3993 2 72 0.2637 0.6630 3 71 0.2601 4 18 0.9890 5 3 0.0110 1.0000 Total=273 Exercise1.12.23 Solutiononp.42. Whatistherelativefrequencyofdirecthitsthatwerecategory4hurricanes? A. 0.0768 B. 0.0659 C. 0.2601 D. Notenoughinformationtocalculate Exercise1.12.24 Solutiononp.42. WhatistherelativefrequencyofdirecthitsthatwereATMOSTacategory3storm? A. 0.3480 B. 0.9231 C. 0.2601 D. 0.3370 Thenextthreequestionsrefertothefollowing: Astudywasdonetodeterminetheage,numberoftimes perweekandthedurationamountoftimeofresidentuseofalocalparkinSanJose.Thersthousein theneighborhoodaroundtheparkwasselectedrandomlyandthenevery8thhouseintheneighborhood aroundtheparkwasinterviewed. Exercise1.12.25 Solutiononp.42. `Numberoftimesperweek'iswhattypeofdata? A. qualitative B. quantitative-discrete C. quantitative-continuous Exercise1.12.26 Solutiononp.42. Thesamplingmethodwas: A. simplerandom B. systematic C. stratied D. cluster

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35 Exercise1.12.27 Solutiononp.42. `Durationamountoftime'iswhattypeofdata? A. qualitative B. quantitative-discrete C. quantitative-continuous

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36 CHAPTER1.SAMPLINGANDDATA 1.13Lab1:DataCollection 13 ClassTime: Names: 1.13.1StudentLearningOutcomes Thestudentwilldemonstratethesystematicsamplingtechnique. ThestudentwillconstructRelativeFrequencyTables. Thestudentwillinterpretresultsandtheirdifferencesfromdifferentdatagroupings. 1.13.2MovieSurvey Askveclassmatesfromadifferentclasshowmanymoviestheysawlastmonthatthetheater.Donot includerentedmovies. 1.Recordthedata 2.Inclass,randomlypickoneperson.Ontheclasslist,markthatperson'sname.Movedownfour people'snamesontheclasslist.Markthatperson'sname.Continuedoingthisuntilyouhavemarked 12people'snames.Youmayneedtogobacktothestartofthelist.Foreachmarkednamerecord belowthevedatavalues.Younowhaveatotalof60datavalues. 3.Foreachnamemarked,recordthedata: ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ 1.13.3OrdertheData Completethetworelativefrequencytablesbelowusingyourclassdata. 13 Thiscontentisavailableonlineat.

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37 FrequencyofNumberofMoviesViewed NumberofMovies Frequency RelativeFrequency CumulativeRelativeFrequency 0 1 2 3 4 5 6 7+ FrequencyofNumberofMoviesViewed NumberofMovies Frequency RelativeFrequency CumulativeRelativeFrequency 0-1 2-3 4-5 6-7+ 1.Usingthetables,ndthepercentofdatathatisatmost2.Whichtabledidyouuseandwhy? 2.Usingthetables,ndthepercentofdatathatisatmost3.Whichtabledidyouuseandwhy? 3.Usingthetables,ndthepercentofdatathatismorethan2.Whichtabledidyouuseandwhy? 4.Usingthetables,ndthepercentofdatathatismorethan3.Whichtabledidyouuseandwhy? 1.13.4DiscussionQuestions 1.Isoneofthetablesabove"morecorrect"thantheother?Whyorwhynot? 2.Ingeneral,whywouldsomeonegroupthedataindifferentways?Arethereanyadvantagestoeither wayofgroupingthedata? 3.Whydidyouswitchbetweentables,ifyoudid,whenansweringthequestionabove?

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38 CHAPTER1.SAMPLINGANDDATA 1.14Lab2:SamplingExperiment 14 ClassTime: Names: 1.14.1StudentLearningOutcomes Thestudentwilldemonstratethesimplerandom,systematic,stratied,andclustersamplingtechniques. Thestudentwillexplaineachofthedetailsofeachprocedureused. Inthislab,youwillbeaskedtopickseveralrandomsamples.Ineachcase,describeyourprocedurebriey, includinghowyoumighthaveusedtherandomnumbergenerator,andthenlisttherestaurantsinthe sampleyouobtained N OTE :Thefollowingsectioncontainsrestaurantsstratiedbycityintocolumnsandgrouped horizontallybyentreecostclusters. 1.14.2ASimpleRandomSample Picka simplerandomsample of15restaurants. 1.Descibetheprocedure: 2. 1.__________ 6.__________ 11.__________ 2.__________ 7.__________ 12.__________ 3.__________ 8.__________ 13.__________ 4.__________ 9.__________ 14.__________ 5.__________ 10.__________ 15.__________ 1.14.3ASystematicSample Picka systematicsample of15restaurants. 1.Descibetheprocedure: 2. 1.__________ 6.__________ 11.__________ 2.__________ 7.__________ 12.__________ 3.__________ 8.__________ 13.__________ 4.__________ 9.__________ 14.__________ 5.__________ 10.__________ 15.__________ 14 Thiscontentisavailableonlineat.

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39 1.14.4AStratiedSample Picka stratiedsample ,byentreecost,of20restaurantswithequalrepresentationfromeachstratum. 1.Descibetheprocedure: 2. 1.__________ 6.__________ 11.__________ 16.__________ 2.__________ 7.__________ 12.__________ 17.__________ 3.__________ 8.__________ 13.__________ 18.__________ 4.__________ 9.__________ 14.__________ 19.__________ 5.__________ 10.__________ 15.__________ 20.__________ 1.14.5AStratiedSample Picka stratiedsample ,bycity,of21restaurantswithequalrepresentationfromeachstratum. 1.Descibetheprocedure: 2. 1.__________ 6.__________ 11.__________ 16.__________ 2.__________ 7.__________ 12.__________ 17.__________ 3.__________ 8.__________ 13.__________ 18.__________ 4.__________ 9.__________ 14.__________ 19.__________ 5.__________ 10.__________ 15.__________ 20.__________ 21.__________ 1.14.6AClusterSample Picka clustersample ofresturantsfromtwocities.Thenumberofrestaurantswillvary. 1.Descibetheprocedure: 2. 1.__________ 6.__________ 11.__________ 16.__________ 21.__________ 2.__________ 7.__________ 12.__________ 17.__________ 22.__________ 3.__________ 8.__________ 13.__________ 18.__________ 23.__________ 4.__________ 9.__________ 14.__________ 19.__________ 24.__________ 5.__________ 10.__________ 15.__________ 20.__________ 25.__________

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40 CHAPTER1.SAMPLINGANDDATA 1.14.7RestaurantsStratiedbyCityandEntreeCost RestaurantsUsedinSample EntreeCost Under\$10 \$10tounder\$15 \$15tounder\$20 Over\$20 SanJose ElAbueloTaq, PastaMia, Emma'sExpress, BambooHut Emperor'sGuard, CreeksideInn Agenda,Gervais, Miro's Blake's,Eulipia, HayesMansion, Germania PaloAlto SenorTaco,Olive Garden,Taxi's Ming's,P.A.Joe's, Stickney's Scott'sSeafood, PoolsideGrill, FishMarket SundanceMine, Maddalena's, Spago's LosGatos Mary'sPatio, MountEverest, SweetPea's, AndeleTaqueria Lindsey's,Willow Street TollHouse CharterHouse,La MaisonDuCafe MountainView Maharaja,New Ma's,Thai-Ric, GardenFresh AmberIndian,La Fiesta,Fiestadel Mar,Dawit Austin's,Shiva's, Mazeh LePetitBistro Cupertino Hobees,HungFu, Samrat,PandaExpress SantaBarb.Grill, Mand.Gourmet, BombayOven, KathmanduWest Fontana's,Blue Pheasant Hamasushi,Helios Sunnyvale Chekijababi,Taj India,FullThrottle,TiaJuana, LemonGrass PacicFresh, CharleyBrown's, CafeCameroon, Faz,Aruba's Lion&Compass, ThePalace,Beau Sejour SantaClara Rangoli,ArmadilloWilly's, ThaiPepper, Pasand Arthur's,Katie's Cafe,Pedro's,La Galleria Birk's,Truya Sushi,Valley Plaza Lakeside,Mariani's N OTE : TheoriginallabwasdesignedandcontributedbyCarolOlmstead.

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41 SolutionstoExercisesinChapter1 Example1.5p.11 Items1,5,11,and12arequantitativediscrete;items4,6,10,and14arequantitativecontinuous;anditems 2,3,7,8,9,and13arequalitative. Example1.10p.21 1.29% 2.36% 3.77% 4.87 5.quantitativecontinuous 6.getrostersfromeachteamandchooseasimplerandomsamplefromeach Example1.11p.22 1.No.Frequencycolumnsumsto18,not19.Notallcumulativerelativefrequenciesarecorrect. 2.False.Frequencyfor3milesshouldbe1;for2milesleftout,2.Cumulativerelativefrequency columnshouldread:0.1052,0.1579,0.2105,0.3684,0.4737,0.6316,0.7368,0.7895,0.8421,0.9474,1. 3. 5 19 4. 7 19 12 19 7 19 SolutionstoHomework SolutiontoExercise1.12.1p.28 a. quantitative-discrete b. quantitative-continuous c. qualitative d. quantitative-continuous e. quantitative-discrete f. qualitative g. qualitative h. quantitative-continuous i. quantitative-continuous j. quantitative-discrete SolutiontoExercise1.12.3p.28 b. 5.00% c. 93.33% SolutiontoExercise1.12.5p.29 a. Childrenwhotakeskiandsnowboardlessons b. Agroupofthesechildren c. Thepopulationaverage d. Thesampleaverage e. X =theageofonechildwhotakestherstskiorsnowboardlesson f. Avaluefor X ,suchas3,7,etc. SolutiontoExercise1.12.7p.29 a. Theclientsoftheinsurancecompanies

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42 CHAPTER1.SAMPLINGANDDATA b. Agroupoftheclients c. Theaverageageoftheclients d. Theaverageageofthesample e. X =theageofoneclient f. Avaluefor X ,suchas34,9,82,etc. SolutiontoExercise1.12.9p.30 a. Alltheclientsofthecounselor b. Agroupoftheclients c. Theproportionofallherclientswhostaymarried d. Theproportionofthesamplewhostaymarried e. X =thenumberofcoupleswhostaymarried f. yes,no SolutiontoExercise1.12.11p.30 a. Allpeoplemaybeinacertaingeographicarea,suchastheUnitedStates b. Agroupofthepeople c. Theproportionofallpeoplewhowillbuytheproduct d. Theproportionofthesamplewhowillbuytheproduct e. X =thenumberofpeoplewhowillbuyit f. buy,notbuy SolutiontoExercise1.12.19p.33 D SolutiontoExercise1.12.20p.33 A SolutiontoExercise1.12.21p.33 B SolutiontoExercise1.12.22p.33 C SolutiontoExercise1.12.23p.34 B SolutiontoExercise1.12.24p.34 B SolutiontoExercise1.12.25p.34 B SolutiontoExercise1.12.26p.34 B SolutiontoExercise1.12.27p.35 C

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Chapter2 DescriptiveStatistics 2.1DescriptiveStatistics 1 2.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Displaydatagraphicallyandinterpretgraphs:stemplots,histogramsandboxplots. Recognize,describe,andcalculatethemeasuresoflocationofdata:quartilesandpercentiles. Recognize,describe,andcalculatethemeasuresofthecenterofdata:mean,median,andmode. Recognize,describe,andcalculatethemeasuresofthespreadofdata:variance,standarddeviation, andrange. 2.1.2Introduction Onceyouhavecollecteddata,whatwillyoudowithit?Datacanbedescribedandpresentedinmany differentformats.Forexample,supposeyouareinterestedinbuyingahouseinaparticulararea.Youmay havenoclueaboutthehouseprices,soyoumightaskyourrealestateagenttogiveyouasampledataset ofprices.Lookingatallthepricesinthesampleoftenisoverwhelming.Abetterwaymightbetolook atthemedianpriceandthevariationofprices.Themedianandvariationarejusttwowaysthatyouwill learntodescribedata.Youragentmightalsoprovideyouwithagraphofthedata. Inthischapter,youwillstudynumericalandgraphicalwaystodescribeanddisplayyourdata.Thisarea ofstatisticsiscalled "DescriptiveStatistics" .Youwilllearntocalculate,andevenmoreimportantly,to interpretthesemeasurementsandgraphs. 2.2DisplayingData 2 Astatisticalgraphisatoolthathelpsyoulearnabouttheshapeordistributionofasample.Thegraphcan beamoreeffectivewayofpresentingdatathanamassofnumbersbecausewecanseewheredataclusters andwherethereareonlyafewdatavalues.NewspapersandtheInternetusegraphstoshowtrendsand toenablereaderstocomparefactsandguresquickly. Statisticiansoftengraphdatarstinordertogetapictureofthedata.Then,moreformaltoolsmaybe applied. 1 Thiscontentisavailableonlineat. 2 Thiscontentisavailableonlineat. 43

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44 CHAPTER2.DESCRIPTIVESTATISTICS Someofthetypesofgraphsthatareusedtosummarizeandorganizedataarethedotplot,thebarchart, thehistogram,thestem-and-leafplot,thefrequencypolygonatypeofbrokenlinegraph,piecharts,and theboxplot.Inthischapter,wewillbrieylookatstem-and-leafplots.Ouremphasiswillbeonhistograms andboxplots. 2.3StemandLeafGraphsStemplots 3 Onesimplegraph,the stem-and-leafgraph or stemplot ,comesfromtheeldofexploratorydataanalysis.It isagoodchoicewhenthedatasetsaresmall.Tocreatetheplot,divideeachobservationofdataintoastem andaleaf.Theleafconsistsof onedigit .Forexample,23hasstem2andleaf3.Fourhundredthirty-two hasstem43andleaf2.Fivethousandfourhundredthirty-two,432hasstem543andleaf2.The decimal9.3hasstem9andleaf3.Writethestemsinaverticallinefromsmallestthelargest.Drawavertical linetotherightofthestems.Thenwritetheleavesinincreasingordernexttotheircorrespondingstem. Example2.1 ForSusanDean'sspringpre-calculusclass,scoresfortherstexamwereasfollowssmallestto largest: 33;42;49;49;53;55;55;61;63;67;68;68;69;69;72;73;74;78;80;83;88;88;88;90;92;94;94;94;96; 100 Stem-and-LeafDiagram 3 3 4 299 5 355 6 1378899 7 2348 8 03888 9 0244446 10 0 Thestemplotshowsthatmostscoresfellinthe60s,70s,80s,and90s.Eightoutofthe31scoresor approximately26%ofthescoreswereinthe90'sor100,afairlyhighnumberofAs. Thestemplotisaquickwaytographandgivesanexactpictureofthedata.Youwanttolookforanoverall patternandanyoutliers.An outlier isanobservationofdatathatdoesnotttherestofthedata.Itis sometimescalledan extremevalue. Whenyougraphanoutlier,itwillappearnottotthepatternofthe graph.Someoutliersareduetomistakesforexample,writingdown50insteadof500whileothersmay indicatethatsomethingunusualishappening.Ittakessomebackgroundinformationtoexplainoutliers. Intheexampleabove,therewerenooutliers. Example2.2 Createastemplotusingthedata: 1.1;1.5;2.3;2.5;2.7;3.2;3.3;3.3;3.5;3.8;4.0;4.2;4.5;4.5;4.7;4.8;5.5;5.6;6.5;6.7;12.3 Thedataarethedistanceinkilometersfromahometothenearestsupermarket. 3 Thiscontentisavailableonlineat.

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45 Problem 1.Arethereanyoutliers? 2.Dothedataseemtohaveanyconcentrationofvalues? H INT :Theleavesaretotherightofthedecimal. N OTE :Thisbookcontainsinstructionsforconstructinga histogram anda boxplot fortheTI-83+ andTI-84calculators.YoucanndadditionalinstructionsforusingthesecalculatorsontheTexas InstrumentsTIwebsite 4 2.4Histograms 5 Formostoftheworkyoudointhisbook,youwilluseahistogramtodisplaythedata.Oneadvantageofa histogramisthatitcanreadilydisplaylargedatasets.Aruleofthumbistouseahistogramwhenthedata setconsistsof100valuesormore. A histogram consistsofcontiguousboxes.Ithasbothahorizontalaxisandaverticalaxis.Thehorizontal axisislabeledwithwhatthedatarepresentsforinstance,distancefromyourhometoschool.Thevertical axisislabeledeither"frequency"or"relativefrequency".Thegraphwillhavethesameshapewitheither label. Frequency iscommonlyusedwhenthedatasetissmalland relativefrequency isusedwhenthe datasetislargeorwhenwewanttocompareseveraldistributions.Thehistogramlikethestemplotcan giveyoutheshapeofthedata,thecenter,andthespreadofthedata.Thenextsectiontellsyouhowto calculatethecenterandthespread. Therelativefrequencyisequaltothefrequencyforanobservedvalueofthedatadividedbythetotal numberofdatavaluesinthesample.InthechapteronSamplingandDataSection1.1,wedened frequencyasthenumberoftimesanansweroccurs.If: f =frequency n =totalnumberofdatavaluesorthesumoftheindividualfrequencies,and RF =relativefrequency, then: RF = f n .1 Forexample,if3studentsinMr.Ahab'sEnglishclassof40studentsreceivedanA,then, f = 3, n = 40 ,and RF = f n = 3 40 = 0 .075 SevenandahalfpercentofthestudentsreceivedanA. Toconstructahistogram,rstdecidehowmany bar sor intervals representthedata.Manyhistograms consistoffrom5to15barsorclassesforclarity.Choosethestartingpointtobelessthanthesmallestdata value.A convenientstartingpoint isalowervaluecarriedouttoonemoredecimalplacethanthevalue withthemostdecimalplaces.Forexample,ifthevaluewiththemostdecimalplacesis6.1,aconvenient startingpointis6.05.Wesaythat6.05hasmoreprecision.Ifthevaluewiththemostdecimalplacesis2.23, 4 http://education.ti.com/educationportal/sites/US/sectionHome/support.html 5 Thiscontentisavailableonlineat.

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46 CHAPTER2.DESCRIPTIVESTATISTICS aconvenientstartingpointis2.225.Also,whenthestartingpointandotherboundariesarecarriedtoone additionaldecimalplace,nodatavalueislikelytofallonaboundary. Example2.3 Thefollowingdataaretheheightsininchestothenearesthalfinchof100malesemiprofessional soccerplayers.Theheightsare continuous datasinceheightismeasured. 60;60.5;61;61.5 63.5;63.5;63.5 64;64;64;64;64;64;64;64.5;64.5;64.5;64.5;64.5;64.5;64.5;64.5 66;66;66;66;66;66;66;66;66;66;66.5;66.5;66.5;66.5;66.5;66.5;66.5;66.5;66.5;66.5;66.5;67;67; 67;67;67;67;67;67;67;67;67;67;67.5;67.5;67.5;67.5;67.5;67.5;67.5 68;68;69;69;69;69;69;69;69;69;69;69;69.5;69.5;69.5;69.5;69.5 70;70;70;70;70;70;70.5;70.5;70.5;71;71;71 72;72;72;72.5;72.5;73;73.5 74 Thesmallestdatavalueis60.Sincethedatawiththemostdecimalplaceshasonedecimalfor instance,61.5,wewantourstartingpointtohavetwodecimalplaces.Sincethenumbers0.5, 0.05,0.005,etc.areconvenientnumbers,use0.05andsubtractitfrom60,thesmallestvalue,for theconvenientstartingpoint. 60-0.05=59.95whichismoreprecisethan,say,61.5byonedecimalplace.Thestartingpointis, then,59.95. Thelargestvalueis74.74+0.05=74.05istheendingvalue. Next,calculatethewidthofeachbarorclassinterval.Tocalculatethiswidth,subtractthestarting pointfromtheendingvalueanddividebythenumberofbarsyoumustchoosethenumberof barsyoudesire.Supposeyouchoose8bars. 74.05 )]TJ/F58 9.9626 Tf 10.132 0 Td [(59.95 8 = 1.76.2 N OTE :Wewillroundupto2andmakeeachbarorclassinterval2unitswide.Roundingupto 2isonewaytopreventavaluefromfallingonaboundary.Forthisexample,using1.76asthe widthwouldalsowork. Theboundariesare: 59.95 59.95+2=61.95 61.95+2=63.95 63.95+2=65.95 65.95+2=67.95 67.95+2=69.95 69.95+2=71.95 71.95+2=73.95 73.95+2=75.95

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48 CHAPTER2.DESCRIPTIVESTATISTICS isthemostconvenient.Sincethedataconsistofthenumbers1,2,3,4,5,6andthestartingpointis 0.5,awidthofoneplacesthe1inthemiddleoftheintervalfrom0.5to1.5,the2inthemiddleof theintervalfrom1.5to2.5,the3inthemiddleoftheintervalfrom2.5to3.5,the4inthemiddleof theintervalfrom_______to_______,the5inthemiddleoftheintervalfrom_______to_______, andthe_______inthemiddleoftheintervalfrom_______to_______. Calculatethenumberofbarsasfollows: 6.5 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.5 bars = 1.3 where1isthewidthofabar.Therefore, bars = 6. Thefollowinghistogramdisplaysthenumberofbooksonthex-axisandthefrequencyonthe y-axis. 2.4.1OptionalCollaborativeExercise Countthemoneybillsandchangeinyourpocketorpurse.Yourinstructorwillrecordtheamounts.Asa class,constructahistogramdisplayingthedata.Discusshowmanyintervalsyouthinkisappropriate.You maywanttoexperimentwiththenumberofintervals.Discuss,also,theshapeofthehistogram. Recordthedata,indollarsforexample,1.25dollars. Constructahistogram. 2.5BoxPlots 6 Boxplots or box-whiskerplots giveagoodgraphicalimageoftheconcentrationofthedata.Theyalso showhowfarfrommostofthedatatheextremevaluesare.Theboxplotisconstructedfromvevalues: thesmallestvalue,therstquartile,themedian,thethirdquartile,andthelargestvalue.Themedian,the rstquartile,andthethirdquartilewillbediscussedhere,andthenagaininthesectiononmeasuringdata inthischapter.Weusethesevaluestocomparehowcloseotherdatavaluesaretothem. 6 Thiscontentisavailableonlineat.

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49 The median ,anumber,isawayofmeasuringthe"center"ofthedata.Youcanthinkofthemedianasthe "middlevalue".Itisanumberthatseparatesordereddataintohalves.Halfthevaluesarethesamenumber orsmallerthanthemedianandhalfthevaluesarethesamenumberorlarger.Forexample,considerthe followingdata: 1;11.5;6;7.2;4;8;9;10;6.8;8.3;2;2;10;1 Orderedfromsmallesttolargest: 1;1;2;2;4;6; 6.8 ; 7.2 ;8;8.3;9;10;10;11.5 Themedianisbetweenthe7thvalue,6.8,andthe8thvalue7.2.Tondthemedian,addthetwovalues togetheranddivideby2. 6.8 + 7.2 2 = 7.4 Themedianis7.Halfofthevaluesaresmallerthan7andhalfofthevaluesarelargerthan7. Quartiles arenumbersthatseparatethedataintoquarters.Quartilesmayormaynotbepartofthedata. Tondthequartiles,rstndthemedianorsecondquartile.Therstquartileisthemiddlevalueofthe lowerhalfofthedataandthethirdquartileisthemiddlevalueoftheupperhalfofthedata.Togetthe idea,considerthesamedatasetshownabove: 1;1;2;2;4;6;6.8;7.2;8;8.3;9;10;10;11.5 Themedianor secondquartile is7.Thelowerhalfofthedatais1,1,2,2,4,6,6.8.Themiddlevalueofthe lowerhalfis2. 1;1;2; 2 ;4;6;6.8 Thenumber2,whichispartofthedata,isthe rstquartile .One-fourthofthevaluesarethesameorless than2andthree-fourthsofthevaluesaremorethan2. Theupperhalfofthedatais7.2,8,8.3,9,10,10,11.5.Themiddlevalueoftheupperhalfis9. 7.2;8;8.3; 9 ;10;10;11.5 Thenumber9,whichispartofthedata,isthe thirdquartile .Three-fourthsofthevaluesarelessthan9and one-fourthofthevaluesaremorethan9. Toconstructaboxplot,useahorizontalnumberlineandarectangularbox.Thesmallestandlargestdata valueslabeltheendpointsoftheaxis.Therstquartilemarksoneendoftheboxandthethirdquartile markstheotherendofthebox. Themiddleftypercentofthedatafallinsidethebox. The"whiskers" extendfromtheendsoftheboxtothesmallestandlargestdatavalues.Theboxplotgivesagoodquick pictureofthedata. Considerthefollowingdata: 1;1;2;2;4;6;6.8;7.2;8;8.3;9;10;10;11.5 Therstquartileis2,themedianis7,andthethirdquartileis9.Thesmallestvalueis1andthelargest valueis11.5.Theboxplotisconstructedasfollowsseecalculatorinstructionsinthebackofthisbookor ontheTIwebsite 7 : 7 http://education.ti.com/educationportal/sites/US/sectionHome/support.html

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50 CHAPTER2.DESCRIPTIVESTATISTICS Thetwowhiskersextendfromtherstquartiletothesmallestvalueandfromthethirdquartiletothe largestvalue.Themedianisshownwithadashedline. Example2.5 Thefollowingdataaretheheightsof40studentsinastatisticsclass. 59;60;61;62;62;63;63;64;64;64;65;65;65;65;65;65;65;65;65;66;66;67;67;68;68;69;70;70;70; 70;70;71;71;72;72;73;74;74;75;77 Constructaboxplotwiththefollowingproperties: Smallestvalue=59 Largestvalue=77 Q1:Firstquartile=64.5 Q2:Secondquartileormedian=66 Q3:Thirdquartile=70 a. Eachquarterhas25%ofthedata. b. Thespreadsofthefourquartersare64.5-59=5.5rstquarter,66-64.5=1.5second quarter,70-66=4rdquarter,and77-70=7fourthquarter.So,thesecondquarter hasthesmallestspreadandthefourthquarterhasthelargestspread. c. InterquartileRange: IQR = Q 3 )]TJ/F132 9.9626 Tf 10.355 0 Td [(Q 1 = 70 )]TJ/F58 9.9626 Tf 10.131 0 Td [(64.5 = 5.5. d. Theinterval59through65hasmorethan25%ofthedatasoithasmoredatainitthanthe interval66through70whichhas25%ofthedata. Forsomesetsofdata,someofthelargestvalue,smallestvalue,rstquartile,median,andthird quartilemaybethesame.Forinstance,youmighthaveadatasetinwhichthemedianandthe thirdquartilearethesame.Inthiscase,thediagramwouldnothaveadottedlineinsidethebox displayingthemedian.Therightsideoftheboxwoulddisplayboththethirdquartileandthe median.Forexample,ifthesmallestvalueandtherstquartilewereboth1,themedianandthe thirdquartilewereboth5,andthelargestvaluewas7,theboxplotwouldlookasfollows:

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51 Example2.6 Testscoresforacollegestatisticsclassheldduringthedayare: 99;56;78;55.5;32;90;80;81;56;59;45;77;84.5;84;70;72;68;32;79;90 Testscoresforacollegestatisticsclassheldduringtheeveningare: 98;78;68;83;81;89;88;76;65;45;98;90;80;84.5;85;79;78;98;90;79;81;25.5 Problem Whatarethesmallestandlargestdatavaluesforeachdataset? Whatisthemedian,therstquartile,andthethirdquartileforeachdataset? Createaboxplotforeachsetofdata. Whichboxplothasthewidestspreadforthemiddle50%ofthedatathedatabetweenthe rstandthirdquartiles?Whatdoesthismeanforthatsetofdataincomparisontotheother setofdata? Foreachdataset,whatpercentofthedataisbetweenthesmallestvalueandtherstquartile?Answer:25%therstquartileandthemedian?Answer:25%themedianandthe thirdquartile?thethirdquartileandthelargestvalue?Whatpercentofthedataisbetween therstquartileandthelargestvalue?Answer:75% Therstdatasetthetopboxplothasthewidestspreadforthemiddle50%ofthedata. IQR = Q 3 )]TJ/F132 9.9626 Tf 10.419 0 Td [(Q 1is82.5 )]TJ/F58 9.9626 Tf 10.195 0 Td [(56 = 26.5fortherstdatasetand89 )]TJ/F58 9.9626 Tf 10.195 0 Td [(78 = 11fortheseconddataset.So,the rstsetofdatahasitsmiddle50%ofscoresmorespreadout. 25%ofthedataisbetween M and Q 3and25%isbetween Q 3and Xmax 2.6MeasuresoftheLocationoftheData 8 Thecommonmeasuresoflocationare quartiles and percentiles %iles.Quartilesarespecialpercentiles. Therstquartile, Q 1 isthesameasthe25thpercentileth%ileandthethirdquartile, Q 3 ,isthesameas the75thpercentileth%ile.Themedian, M ,iscalledboththesecondquartileandthe50thpercentile th%ile. Tocalculatequartilesandpercentiles,thedatamustbeorderedfromsmallesttolargest.Recallthatquartiles divideordereddataintoquarters.Percentilesdivideordereddataintohundredths.Toscoreinthe90th percentileofanexamdoesnotmean,necessarily,thatyoureceived90%onatest.Itmeansthatyourscore washigherthan90%ofthepeoplewhotookthetestandlowerthanthescoresoftheremaining10%of thepeoplewhotookthetest.Percentilesareusefulforcomparingvalues.Forthisreason,universitiesand collegesusepercentilesextensively. 8 Thiscontentisavailableonlineat.

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52 CHAPTER2.DESCRIPTIVESTATISTICS The interquartilerange isanumberthatindicatesthespreadofthemiddlehalforthemiddle50%ofthe data.Itisthedifferencebetweenthethirdquartile Q 3 andtherstquartile Q 1 IQR = Q 3 )]TJ/F132 9.9626 Tf 10.355 0 Td [(Q 1 .5 TheIQRcanhelptodeterminepotential outliers Avalueissuspectedtobeapotentialoutlierifitismore than .5IQR belowtherstquartileormorethan .5IQR abovethethirdquartile .Potentialoutliers alwaysneedfurtherinvestigation. Example2.7 Forthefollowing13realestateprices,calculatethe IQR anddetermineifanypricesareoutliers. Pricesareindollars. Source:SanJoseMercuryNews 389,950;230,500;158,000;479,000;639,000;114,950;5,500,000;387,000;659,000;529,000;575,000; 488,000;1,095,000 Solution Orderthedatafromsmallesttolargest. 114,950;158,000;230,500;387,000;389,950;479,000;488,000;529,000;575,000;639,000;659,000; 1,095,000;5,500,000 M = 488,800 Q 1 = 230500 + 387000 2 = 308750 Q 3 = 639000 + 659000 2 = 649000 IQR = 649000 )]TJ/F58 9.9626 Tf 10.132 0 Td [(308750 = 340250 1.5 IQR = 1.5 340250 = 510375 Q 1 )]TJ/F142 10.3811 Tf 10.256 -0.105 Td [( 1.5 IQR = 308750 )]TJ/F58 9.9626 Tf 10.132 0 Td [(510375 = )]TJ/F58 9.9626 Tf 8.195 0 Td [(201625 Q 3 + 1.5 IQR = 649000 + 510375 = 1159375 Nohousepriceislessthan-201625.However,5,500,000ismorethan1,159,375.Therefore, 5,500,000isapotential outlier Example2.8 Forthetwodatasetsinthetestscoresexamplep.51,ndthefollowing: a. Theinterquartilerange.Comparethetwointerquartileranges. b. Anyoutliersineitherset. c. The30thpercentileandthe80thpercentileforeachset.Howmuchdatafallsbelowthe 30thpercentile?Abovethe80thpercentile?

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53 Example2.9:FindingQuartilesandPercentilesUsingaTable Fiftystatisticsstudentswereaskedhowmuchsleeptheygetperschoolnightroundedtothe nearesthour.Theresultswerestudentdata: AMOUNTOFSLEEPPERSCHOOLNIGHT HOURS FREQUENCY RELATIVEFREQUENCY CUMULATIVERELATIVEFREQUENCY 4 2 0.04 0.04 5 5 0.10 0.14 6 7 0.14 0.28 7 12 0.24 0.52 8 14 0.28 0.80 9 7 0.14 0.94 10 3 0.06 1.00 Findthe28thpercentile :Noticethe0.28inthe"cumulativerelativefrequency"column.28%of50 datavalues=14.Thereare14valueslessthanthe28th%ile.Theyincludethetwo4s,theve5s, andtheseven6s.The28th%ileisbetweenthelast6andtherst7. The28th%ileis6.5. Findthemedian :Lookagainatthe"cumulativerelativefrequency"columnandnd0.52.The medianisthe50th%ileorthesecondquartile.50%of50=25.Thereare25valueslessthanthe median.Theyincludethetwo4s,theve5s,theseven6s,andelevenofthe7s.Themedianor 50th%ileisbetweenthe25thand26thvalues. Themedianis7. Findthethirdquartile :Thethirdquartileisthesameasthe75thpercentile.Youcan"eyeball"this answer.Ifyoulookatthe"cumulativerelativefrequency"column,yound0.52and0.80.When youhaveallthe4s,5s,6sand7s,youhave52%ofthedata.Whenyouincludeallthe8s,youhave 80%ofthedata. The75th%ile,then,mustbean8 .Anotherwaytolookattheproblemistond 75%of50=37.5androundupto38.Thethirdquartile, Q 3 ,isthe38thvaluewhichisan8.You cancheckthisanswerbycountingthevalues.Thereare37valuesbelowthethirdquartileand12 valuesabove. Example2.10 Usingthetable: 1.Findthe80thpercentile. 2.Findthe90thpercentile. 3.Findtherstquartile.Whatisanothernamefortherstquartile? 4.Constructaboxplotofthedata. CollaborativeClassroomExercise :Yourinstructororamemberoftheclasswillaskeveryoneinclasshow manysweaterstheyown.Answerthefollowingquestions. 1.Howmanystudentsweresurveyed? 2.Whatkindofsamplingdidyoudo? 3.Findthemeanandstandarddeviation. 4.Findthemode. 5.Construct2differenthistograms.Foreach,startingvalue=_____endingvalue=____. 6.Findthemedian,rstquartile,andthirdquartile.

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54 CHAPTER2.DESCRIPTIVESTATISTICS 7.Constructaboxplot. 8.Constructatableofthedatatondthefollowing: The10thpercentile The70thpercentile Thepercentofstudentswhoownlessthan4sweaters 2.7MeasuresoftheCenteroftheData 9 Thetwomostwidelyusedmeasuresofthe"center"ofthedataarethe mean averageandthe median Tocalculatethe meanweight of50people,addthe50weightstogetheranddivideby50.Tondthe medianweight ofthe50people,orderthedataandndthenumberthatsplitsthedataintotwoequal partspreviouslydiscussedunderboxplotsinthischapter.Themedianisgenerallyabettermeasureof thecenterwhenthereareextremevaluesoroutliers.Themeanisthemostcommonmeasureofthecenter. Themeancanalsobecalculatedbymultiplyingeachdistinctvaluebyitsfrequencyandthendividingthe sumbythetotalnumberofdatavalues.Theletterusedtorepresentthesamplemeanisan x withabar overitpronounced" x bar": x TheGreekletter m pronounced"mew"representsthepopulationmean.Ifyoutakeatrulyrandomsample, thesamplemeanisagoodestimateofthepopulationmean. Toseethatbothwaysofcalculatingthemeanarethesame,considerthesample: 1;1;1;2;2;3;4;4;4;4;4 x = 1 + 1 + 1 + 2 + 2 + 3 + 4 + 4 + 4 + 4 + 4 11 = 2.7.6 x = 3 1 + 2 2 + 1 3 + 5 4 11 = 2.7.7 Inthesecondexample,thefrequenciesare3,2,1,and5. Youcanquicklyndthelocationofthemedianbyusingtheexpression n + 1 2 Theletter n isthetotalnumberofdatavaluesinthesample.If n isanoddnumber,themedianisthe middlevalueoftheordereddata.If n isanevennumber,themedianisequaltothetwomiddlevalues addedtogetheranddividedby2.Thelocationofthemedianandthemedianitselfare not thesame.The uppercaseletter M isoftenusedtorepresentthemedian.Thenextexampleillustratesthelocationofthe medianandthemedianitself. Example2.11 AIDSdataindicatingthenumberofmonthsanAIDSpatientlivesaftertakinganewantibody drugareasfollowssmallesttolargest: 3;4;8;8;10;11;12;13;14;15;15;16;16;17;17;18;21;22;22;24;24;25;26;26;27;27;29;29;31;32; 33;33;34;34;35;37;40;44;44;47 Calculatethemeanandthemedian. Solution Thecalculationforthemeanis: 9 Thiscontentisavailableonlineat.

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55 x = [ 3 + 4 + 8 2 + 10 + 11 + 12 + 13 + 14 + 15 2 + 16 2 + ... + 35 + 37 + 40 + 44 2 + 47 ] 40 = 23.6 Tondthemedian, M ,rstusetheformulaforthelocation.Thelocationis: n + 1 2 = 40 + 1 2 = 20.5 Startingatthesmallestvalue,themedianislocatedbetweenthe20thand21stvalueshighlighted below: 3;4;8;8;10;11;12;13;14;15;15;16;16;17;17;18;21;22;22; 24 ; 24 ;25;26;26;27;27;29;29;31;32; 33;33;34;34;35;37;40;44;44;47 M = 24 + 24 2 = 24 Themedianis24. Example2.12 Supposethat,inasmalltownof50people,onepersonearns\$5,000,000peryearandtheother49 eachearn\$30,000.Whichisthebettermeasureofthe"center,"themeanorthemedian? Solution x = 5000000 + 49 30000 50 = 129400 M = 30000 Thereare49peoplewhoearn\$30,000andonepersonwhoearns\$5,000,000. Themedianisabettermeasureofthe"center"thanthemeanbecause49ofthevaluesare30,000 andoneis5,000,000.The5,000,000isanoutlier.The30,000givesusabettersenseofthemiddleof thedata. Anothermeasureofthecenteristhemode.The mode isthemostfrequentvalue.Ifadatasethastwo valuesthatoccurthesamenumberoftimes,thenthesetisbimodal. Example2.13:Statisticsexamscoresfor20studentsareasfollows Statisticsexamscoresfor20studentsareasfollows: 50;53;59;59;63;63;72;72;72;72;72;76;78;81;83;84;84;84;90;93 Problem Findthemode. Solution Themostfrequentscoreis72,whichoccursvetimes.Mode=72. Example2.14 Fiverealestateexamscoresare430,430,480,480,495.Thedatasetisbimodalbecausethescores 430and480eachoccurtwice. Whenisthemodethebestmeasureofthe"center"?Consideraweightlossprogramthatadvertises anaverageweightlossofsixpoundstherstweekoftheprogram.Themodemightindicatethat mostpeoplelosetwopoundstherstweek,makingtheprogramlessappealing.

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56 CHAPTER2.DESCRIPTIVESTATISTICS Statisticalsoftwarewilleasilycalculatethemean,themedian,andthemode.Somegraphing calculatorscanalsomakethesecalculations.Intherealworld,peoplemakethesecalculations usingsoftware. 2.7.1TheLawofLargeNumbersandtheMean TheLawofLargeNumberssaysthatifyoutakesamplesoflargerandlargersizefromanypopulation, thenthemean x ofthesamplegetscloserandcloserto m .Thisisdiscussedinmoredetailinthesection The CentralLimitTheorem ofthiscourse. N OTE :TheformulaforthemeanislocatedintheSummaryofFormulasSection2.10section course. 2.8SkewnessandtheMean,Median,andMode 10 Considerthefollowingdataset: 4;5;6;6;6;7;7;7;7;7;7;8;8;8;9;10 Thisdataproducesthehistogramshownbelow.Eachintervalhaswidthoneandeachvalueislocatedin themiddleofaninterval. Thehistogramdisplaysasymmetricaldistributionofdata.Themean,themedian,andthemodeareeach 7forthesedata. Inaperfectlysymmetricaldistribution,themean,themedian,andthemodearethesame. Thehistogramforthedata: 4;5;6;6;6;7;7;7;7;7;7;8 is skewedtotheleft 10 Thiscontentisavailableonlineat.

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57 Themeanis6.3,themedianis6.5,andthemodeis7. Noticethatthemeanislessthanthemedianandthey arebothlessthanthemode. Themeanandthemedianbothreecttheskewingbutthemeanmoreso. Thehistogramforthedata: 6;7;7;7;7;7;7;8;8;8;9;10 is skewedtotheright Themeanis7.7,themedianis7.5,andthemodeis7. Noticethatthemeanisthelargeststatistic,whilethe modeisthesmallest .Again,themeanreectstheskewingthemost. Tosummarize,generallyifthedistributionofdataisskewedtotheleft,themeanislessthanthemedian, whichislessthanthemode.Ifthedistributionofdataisskewedtotheright,themodeislessthanthe median,whichislessthanthemean. Skewnessandsymmetrybecomeimportantwhenwediscussprobabilitydistributionsinlaterchapters. 2.9MeasuresoftheSpreadoftheData 11 Themostcommonmeasureofspreadisthestandarddeviation.The standarddeviation isanumberthat measureshowfardatavaluesarefromtheirmean.Forexample,ifthemeanofasetofdatacontaining7is 11 Thiscontentisavailableonlineat.

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58 CHAPTER2.DESCRIPTIVESTATISTICS 5andthe standarddeviation is2,thenthevalue7isonestandarddeviationfromitsmeanbecause5+ =7. Thenumberlinemayhelpyouunderstandstandarddeviation.Ifweweretoput5and7onanumberline, 7istotherightof5.Wesay,then,that7is one standarddeviationtothe right of5.If1werealsopartof thedataset,then1is two standarddeviationstothe left of5because5+-2=1. 1=5+-2;7=5+ Formula: value= x +#ofSTDEVss Generally,avalue=mean+#ofSTDEVsstandarddeviation,where#ofSTDEVs=thenumberofstandard deviations. If x isavalueand x isthesamplemean,then x )]TJETq1 0 0 1 295.449 414.301 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 295.743 408.089 Td [(x iscalledadeviation.Inadataset,thereareasmany deviationsastherearedatavalues.Deviationsareusedtocalculatethesamplestandarddeviation. Tocalculatethestandarddeviation,calculatethevariancerst.The variance istheaverageofthesquares ofthedeviations.Thestandarddeviationisthesquarerootofthevariance.Youcanthinkofthestandard deviationasaspecialaverageofthedeviationsthe x )]TJETq1 0 0 1 325.123 357.514 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 325.417 351.302 Td [(x values.Thelowercaseletter s representsthe samplestandarddeviationandtheGreekletter s sigmarepresentsthepopulationstandarddeviation. Weuse s 2 torepresentthesamplevarianceand s 2 torepresentthepopulationvariance.Ifthesamplehas thesamecharacteristicsasthepopulation,thensshouldbeagoodestimateof s N OTE :Inpractice,useeitheracalculatororcomputersoftwaretocalculatethestandarddeviation.However,pleasestudythefollowingstep-by-stepexample. Example2.15 Inafthgradeclass,theteacherwasinterestedintheaverageageandthestandarddeviationof theagesofherstudents.Whatfollowsaretheagesofherstudentstothenearesthalfyear: 9;9.5;9.5;10;10;10;10;10.5;10.5;10.5;10.5;11;11;11;11;11;11;11.5;11.5;11.5 x = 9 + 9.5 2 + 10 4 + 10.5 4 + 11 6 + 11.5 3 20 = 10.525.8 Theaverageageis10.53years,roundedto2places. Thevariancemaybecalculatedbyusingatable.Thenthestandarddeviationiscalculatedby takingthesquarerootofthevariance.Wewillexplainthepartsofthetableaftercalculating s .

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59 Data Freq. Deviations Deviations 2 Freq. Deviations 2 x f x )]TJETq1 0 0 1 199.173 657.841 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 199.466 651.629 Td [(x x )]TJETq1 0 0 1 313.139 657.841 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 313.433 651.629 Td [(x 2 f x )]TJETq1 0 0 1 436.14 657.841 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 436.433 651.629 Td [(x 2 9 1 9 )]TJ/F58 9.9626 Tf 10.132 0 Td [(10.525 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(1.525 )]TJ/F58 9.9626 Tf 8.194 0 Td [(1.525 2 = 2.325625 1 2.325625 = 2.325625 9.5 2 9.5 )]TJ/F58 9.9626 Tf 10.132 0 Td [(10.525 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(1.025 )]TJ/F58 9.9626 Tf 8.194 0 Td [(1.025 2 = 1.050625 2 1.050625 = 2.101250 10 4 10 )]TJ/F58 9.9626 Tf 10.131 0 Td [(10.525 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.525 )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.525 2 = 0.275625 4 .275625 = 1.1025 10.5 4 10.5 )]TJ/F58 9.9626 Tf 10.131 0 Td [(10.525 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.025 )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.025 2 = 0.000625 4 .000625 = .0025 11 6 11 )]TJ/F58 9.9626 Tf 10.131 0 Td [(10.525 = 0.475 0.475 2 = 0.225625 6 .225625 = 1.35375 11.5 3 11.5 )]TJ/F58 9.9626 Tf 10.131 0 Td [(10.525 = 0.975 0.975 2 = 0.950625 3 .950625 = 2.851875 Thesamplevariance, s 2 ,isequaltothelastsum.7375dividedbythetotalnumberofdatavalues minusone-1: s 2 = 9.7375 20 )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 = 0.5125 Thesamplestandarddeviation, s ,isequaltothesquarerootofthesamplevariance: s = p 0.5125 = .0715891Roundedtotwodecimalplaces, s = 0.72 Typically,youdothecalculationforthestandarddeviationonyourcalculatororcomputer.The intermediateresultsarenotrounded.Thisisdoneforaccuracy. Problem1 Verifythemeanandstandarddeviationcalculatedaboveonyourcalculatororcomputer.Find themedianandmode. Solution Median=10.5 Mode=11 Problem2 Findthevaluethatis1standarddeviationabovethemean.Find x + 1 s Solution x + 1 s = 10.53 + 1 0.72 = 11.25 Problem3 Findthevaluethatistwostandarddeviationsbelowthemean.Find x )]TJ/F58 9.9626 Tf 10.131 0 Td [(2 s Solution x )]TJ/F58 9.9626 Tf 10.131 0 Td [(2 s = 10.53 )]TJ/F142 10.3811 Tf 10.256 -0.104 Td [( 2 0.72 = 9.09 Problem4 Findthevaluesthatare1.5standarddeviations from belowandabovethemean. Solution x )]TJ/F58 9.9626 Tf 10.131 0 Td [(1.5 s = 10.53 )]TJ/F142 10.3811 Tf 10.256 -0.105 Td [( 1.5 0.72 = 9.45 x + 1.5 s = 10.53 + 1.5 0.72 = 11.61 Explanationofthetable: Thedeviationsshowhowspreadoutthedataareaboutthemean.Thevalue11.5 isfartherfromthemeanthan11.Thedeviations0.97and0.47indicatethat. Ifyouaddthedeviations,the

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61 Solution a. Data Frequency RelativeFrequency CumulativeRelativeFrequency 33 1 0.032 0.032 42 1 0.032 0.064 49 2 0.065 0.129 53 1 0.032 0.161 55 2 0.065 0.226 61 1 0.032 0.258 63 1 0.032 0.29 67 1 0.032 0.322 68 2 0.065 0.387 69 2 0.065 0.452 72 1 0.032 0.484 73 1 0.032 0.516 74 1 0.032 0.548 78 1 0.032 0.580 80 1 0.032 0.612 83 1 0.032 0.644 88 3 0.097 0.741 90 1 0.032 0.773 92 1 0.032 0.805 94 4 0.129 0.934 96 1 0.032 0.966 100 1 0.032 0.998 Whyisn'tthisvalue1? b.i. Thesamplemean=73.5 ii. Thesamplestandarddeviation=17.9 iii. Themedian=73 iv. Therstquartile=61 v. Thethirdquartile=90 vi. IQR=90-61=29 c. Thex-axisgoesfrom32.5to100.5;y-axisgoesfrom-2.4to15forthehistogram;number ofintervalsis5forthehistogramsothewidthofanintervalis.5-32.5dividedby 5whichisequalto13.6.Endpointsoftheintervals:startingpointis32.5,32.5+13.6= 46.1,46.1+13.6=59.7,59.7+13.6=73.3,73.3+13.6=86.9,86.9+13.6=100.5=theending value;Nodatavaluesfallonanintervalboundary.

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62 CHAPTER2.DESCRIPTIVESTATISTICS Figure2.1 Thelongleftwhiskerintheboxplotisreectedintheleftsideofthehistogram.Thespreadof theexamscoresinthelower50%isgreater-33=40thanthespreadintheupper50%73=27.Thehistogram,boxplot,andchartallreectthis.ThereareasubstantialnumberofA andBgradess,90s,and100.Thehistogramclearlyshowsthis.Theboxplotshowsusthatthe middle50%oftheexamscoresIQR=29areDs,Cs,andBs.Theboxplotalsoshowsusthatthe lower25%oftheexamscoresareDsandFs. Example2.17 Twostudents,JohnandAli,fromdifferenthighschools,wantedtondoutwhohadthehighest G.P.A.whencomparedtohisschool.WhichstudenthadthehighestG.P.A.whencomparedtohis school? Student GPA SchoolMeanGPA SchoolStandardDeviation John 2.85 3.0 0.7 Ali 77 80 10 Solution Usetheformula value=mean+#ofSTDEVsstdev andsolvefor#ofSTDEVsforeachstudent stdev=standarddeviation: # ofSTDEVs = value )]TJ/F132 7.5716 Tf 6.323 0 Td [(mean stdev : ForJohn,# ofSTDEVs = 2.85 )]TJ/F58 7.5716 Tf 6.228 0 Td [(3.0 0.7 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.21 ForAli,# ofSTDEVs = 77 )]TJ/F58 7.5716 Tf 6.228 0 Td [(80 10 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.3

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63 JohnhasthebetterG.P.A.whencomparedtohisschoolbecausehisG.P.A.is0.21standarddeviations belowhis meanwhileAli'sG.P.A.is0.3standarddeviations belowhis mean.

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64 CHAPTER2.DESCRIPTIVESTATISTICS 2.10SummaryofFormulas 12 CommonlyUsedSymbols Thesymbol S meanstoaddortondthesum. n =thenumberofdatavaluesinasample N =thenumberofpeople,things,etc.inthepopulation x =thesamplemean s =thesamplestandarddeviation m =thepopulationmean s =thepopulationstandarddeviation f =frequency x =numericalvalue CommonlyUsedExpressions x f =Avaluemultipliedbyitsrespectivefrequency x =Thesumofthevalues x f =Thesumofvaluesmultipliedbytheirrespectivefrequencies x )]TJETq1 0 0 1 118.873 465.504 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 119.167 459.292 Td [(x or x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m =Deviationsfromthemeanhowfaravalueisfromthemean x )]TJETq1 0 0 1 118.873 451.818 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 119.167 445.607 Td [(x 2 or x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m 2 =Deviationssquared f x )]TJETq1 0 0 1 126.192 438.078 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 126.486 431.866 Td [(x 2 or f x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m 2 =Thedeviationssquaredandmultipliedbytheirfrequencies MeanFormulas: x = x n or x = f x n m = x N or m = f x N StandardDeviationFormulas: s = q S x )]TJETq1 0 0 1 145.004 343.427 cm[]0 d 0 J 0.303 w 0 0 m 4.179 0 l SQBT/F132 7.5716 Tf 145.228 338.706 Td [(x 2 n )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 or s = q S f x )]TJETq1 0 0 1 226.247 343.465 cm[]0 d 0 J 0.303 w 0 0 m 4.179 0 l SQBT/F132 7.5716 Tf 226.47 338.744 Td [(x 2 n )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 s = q S x )]TJETq1 0 0 1 146.788 323.849 cm[]0 d 0 J 0.303 w 0 0 m 4.603 0 l SQBT/F134 7.5716 Tf 146.882 319.128 Td [(m 2 N or s = q S f x )]TJETq1 0 0 1 230.237 323.849 cm[]0 d 0 J 0.303 w 0 0 m 4.603 0 l SQBT/F134 7.5716 Tf 230.332 319.128 Td [(m 2 N FormulasRelatingaValue,theMean,andtheStandardDeviation: value=mean+#ofSTDEVsstandarddeviation,where#ofSTDEVs=thenumberofstandarddeviations x = x +#ofSTDEVs s x = m +#ofSTDEVs s 12 Thiscontentisavailableonlineat.

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65 2.11Practice1:CenteroftheData 13 2.11.1StudentLearningOutcomes Thestudentwillcalculateandinterpretthecenter,spread,andlocationofthedata. Thestudentwillconstructandinterprethistogramsanboxplots. 2.11.2Given Sixty-verandomlyselectedcarsalespersonswereaskedthenumberofcarstheygenerallysellinone week.Fourteenpeopleansweredthattheygenerallysellthreecars;nineteengenerallysellfourcars;twelve generallysellvecars;ninegenerallysellsixcars;elevengenerallysellsevencars. 2.11.3CompletetheTable DataValue#cars Frequency RelativeFrequency CumulativeRelativeFrequency 2.11.4DiscussionQuestions Exercise2.11.1 Solutiononp.88. Whatdoesthefrequencycolumnsumto?Why? Exercise2.11.2 Solutiononp.88. Whatdoestherelativefrequencycolumnsumto?Why? Exercise2.11.3 Whatisthedifferencebetweenrelativefrequencyandfrequencyforeachdatavalue? Exercise2.11.4 Whatisthedifferencebetweencumulativerelativefrequencyandrelativefrequencyforeachdata value? 2.11.5EntertheData Enteryourdataintoyourcalculatororcomputer. 13 Thiscontentisavailableonlineat.

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66 CHAPTER2.DESCRIPTIVESTATISTICS 2.11.6ConstructaHistogram Determineappropriateminimumandmaximumxandyvaluesandthescaling.Sketchthehistogram below.Labelthehorizontalandverticalaxeswithwords.Includenumericalscaling. 2.11.7DataStatistics Calculatethefollowingvalues: Exercise2.11.5 Solutiononp.88. Samplemean= x = Exercise2.11.6 Solutiononp.88. Samplestandarddeviation= s x = Exercise2.11.7 Solutiononp.88. Samplesize= n = 2.11.8Calculations Usethetableinsection2.11.3tocalculatethefollowingvalues: Exercise2.11.8 Solutiononp.88. Median= Exercise2.11.9 Solutiononp.88. Mode= Exercise2.11.10 Solutiononp.88. Firstquartile= Exercise2.11.11 Solutiononp.88. Secondquartile=median=50thpercentile= Exercise2.11.12 Solutiononp.88. Thirdquartile= Exercise2.11.13 Solutiononp.88. Interquartilerange IQR =_____-_____=_____ Exercise2.11.14 Solutiononp.88. 10thpercentile= Exercise2.11.15 Solutiononp.88. 70thpercentile=

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67 Exercise2.11.16 Solutiononp.88. Findthevaluethatis3standarddeviations: a. Abovethemean b. Belowthemean 2.11.9BoxPlot Constructaboxplotbelow.Usearulertomeasureandscaleaccurately. 2.11.10Interpretation Lookingatyourboxplot,doesitappearthatthedataareconcentratedtogether,spreadoutevenly,or concentratedinsomeareas,butnotinothers?Howcanyoutell?

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68 CHAPTER2.DESCRIPTIVESTATISTICS 2.12Practice2:SpreadoftheData 14 2.12.1StudentLearningObjectives Thestudentwillcalculatemeasuresofthecenterofthedata. Thestudentwillcalculatethespreadofthedata. 2.12.2Given Thepopulationparametersbelowdescribethefull-timeequivalentnumberofstudentsFTESeachyear atLakeTahoeCommunityCollegefrom1976-77through2004-2005. Source:GraphicallySpeakingbyBill King,LTCCInstitutionalResearch,December2005 Usethesevaluestoanswerthefollowingquestions: m =1000FTES Median-1014FTES s =474FTES Firstquartile=528.5FTES Thirdquartile=1447.5FTES n =29years 2.12.3CalculatetheValues Exercise2.12.1 Solutiononp.88. Asampleof11yearsistaken.AbouthowmanyareexpectedtohaveaFTESof1014orabove? Explainhowyoudeterminedyouranswer. Exercise2.12.2 Solutiononp.88. 75%ofallyearshaveaFTES: a. Atorbelow: b. Atorabove: Exercise2.12.3 Solutiononp.88. Thepopulationstandarddeviation= Exercise2.12.4 Solutiononp.88. WhatpercentoftheFTESwerefrom528.5to1447.5?Howdoyouknow? Exercise2.12.5 Solutiononp.89. Whatisthe IQR ?Whatdoesthe IQR represent? Exercise2.12.6 Solutiononp.89. Howmanystandarddeviationsawayfromthemeanisthemedian? 14 Thiscontentisavailableonlineat.

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69 2.13Homework 15 Exercise2.13.1 Solutiononp.89. Twenty-verandomlyselectedstudentswereaskedthenumberofmoviestheywatchedthepreviousweek.Theresultsareasfollows: #ofmovies Frequency RelativeFrequency CumulativeRelativeFrequency 0 5 1 9 2 6 3 4 4 1 a. Findthesamplemean x b. Findthesamplestandarddeviation, s c. Constructahistogramofthedata. d. Completethecolumnsofthechart. e. Findtherstquartile. f. Findthemedian. g. Findthethirdquartile. h. Constructaboxplotofthedata. i. Whatpercentofthestudentssawfewerthanthreemovies? j. Findthe40thpercentile. k. Findthe90thpercentile. Exercise2.13.2 ThemedianageforU.S.blackscurrentlyis30.1years;forU.S.whitesitis36.6years.Source:U.S. Census a. Baseduponthisinformation,givetworeasonswhytheblackmedianagecouldbelower thanthewhitemedianage. b. Doesthelowermedianageforblacksnecessarilymeanthatblacksdieyoungerthan whites?Whyorwhynot? c. Howmightitbepossibleforblacksandwhitestodieatapproximatelythesameage,but forthemedianageforwhitestobehigher? Exercise2.13.3 Solutiononp.89. Fortyrandomlyselectedstudentswereaskedthenumberofpairsofsneakerstheyowned.LetX =thenumberofpairsofsneakersowned.Theresultsareasfollows: 15 Thiscontentisavailableonlineat.

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70 CHAPTER2.DESCRIPTIVESTATISTICS X Frequency RelativeFrequency CumulativeRelativeFrequency 1 2 2 5 3 8 4 12 5 12 7 1 a. Findthesamplemean x b. Findthesamplestandarddeviation, s c. Constructahistogramofthedata. d. Completethecolumnsofthechart. e. Findtherstquartile. f. Findthemedian. g. Findthethirdquartile. h. Constructaboxplotofthedata. i. Whatpercentofthestudentsownedatleastvepairs? j. Findthe40thpercentile. k. Findthe90thpercentile. Exercise2.13.4 600adultAmericanswereaskedbytelephonepoll,Whatdoyouthinkconstitutesamiddle-class income?Theresultsarebelow.Also,includeleftendpoint,butnottherightendpoint. Source: Timemagazine;surveybyYankelovichPartners,Inc. N OTE :"Notsure"answerswereomittedfromtheresults. Salary\$ RelativeFrequency < 20,000 0.02 20,000-25,000 0.09 25,000-30,000 0.19 30,000-40,000 0.26 40,000-50,000 0.18 50,000-75,000 0.17 75,000-99,999 0.02 100,000+ 0.01 a. Whatpercentofthesurveyanswered"notsure"? b. Whatpercentthinkthatmiddle-classisfrom\$25,000-\$50,000? c. Constructahistogramofthedata 1. i -Shouldallbarshavethesamewidth,basedonthedata?Whyorwhynot? 2. ii -Howshouldthe < 20,000andthe100,000+intervalsbehandled?Why?

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71 d. Findthe40thand80thpercentiles Exercise2.13.5 Solutiononp.89. FollowingarethepublishedweightsinpoundsofalloftheteammembersoftheSanFrancisco 49ersfromapreviousyearSource:SanJoseMercuryNews. 177;205;210;210;232;205;185;185;178;210;206;212;184;174;185;242;188;212;215;247;241; 223;220;260;245;259;278;270;280;295;275;285;290;272;273;280;285;286;200;215;185;230; 250;241;190;260;250;302;265;290;276;228;265 a. Organizethedatafromsmallesttolargestvalue. b. Findthemedian. c. Findtherstquartile. d. Findthethirdquartile. e. Constructaboxplotofthedata. f. Themiddle50%oftheweightsarefrom_______to_______. g. Ifourpopulationwereallprofessionalfootballplayers,wouldtheabovedatabeasample ofweightsorthepopulationofweights?Why? h. IfourpopulationweretheSanFrancisco49ers,wouldtheabovedatabeasampleof weightsorthepopulationofweights?Why? i. AssumethepopulationwastheSanFrancisco49ers.Find: i. thepopulationmean, m ii. thepopulationstandarddeviation, s iii. theweightthatis2standarddeviationsbelowthemean. iv. WhenSteveYoung,quarterback,playedfootball,heweighed205pounds.How manystandarddeviationsaboveorbelowthemeanwashe? j. Thatsameyear,theaverageweightfortheDallasCowboyswas240.08poundswitha standarddeviationof44.38pounds.EmmitSmithweighedinat209pounds.With respecttohisteam,whowaslighter,SmithorYoung?Howdidyoudetermineyour answer? Exercise2.13.6 Anelementaryschoolclassran1mileinanaverageof11minuteswithastandarddeviationof 3minutes.Rachel,astudentintheclass,ran1milein8minutes.Ajuniorhighschoolclassran 1mileinanaverageof9minutes,withastandarddeviationof2minutes.Kenji,astudentinthe class,ran1milein8.5minutes.Ahighschoolclassran1mileinanaverageof7minuteswitha standarddeviationof4minutes.Nedda,astudentintheclass,ran1milein8minutes. a. WhyisKenjiconsideredabetterrunnerthanNedda,eventhoughNeddaranfasterthan he? b. Whoisthefastestrunnerwithrespecttohisorherclass?Explainwhy. Exercise2.13.7 Inasurveyof20yearoldsinChina,GermanyandAmerica,peoplewereaskedthenumberof foreigncountriestheyhadvisitedintheirlifetime.Thefollowingboxplotsdisplaytheresults.

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72 CHAPTER2.DESCRIPTIVESTATISTICS a. Incompletesentences,describewhattheshapeofeachboxplotimpliesaboutthedistributionofthedatacollected. b. ExplainhowitispossiblethatmoreAmericansthanGermanssurveyedhavebeentoover eightforeigncountries. c. Comparethethreeboxplots.Whatdotheyimplyabouttheforeigntraveloftwentyyear oldresidentsofthethreecountrieswhencomparedtoeachother? Exercise2.13.8 Twelveteachersattendedaseminaronmathematicalproblemsolving.Theirattitudesweremeasuredbeforeandaftertheseminar.Apositivenumberchangeattitudeindicatesthatateacher's attitudetowardmathbecamemorepositive.Thetwelvechangescoresareasfollows: 3;8;-1;2;0;5;-3;1;-1;6;5;-2 a. Whatistheaveragechangescore? b. Whatisthestandarddeviationforthispopulation? c. Whatisthemedianchangescore? d. Findthechangescorethatis2.2standarddeviationsbelowthemean. Exercise2.13.9 Solutiononp.90. Threestudentswereapplyingtothesamegraduateschool.Theycamefromschoolswithdifferent gradingsystems.WhichstudenthadthebestG.P.A.whencomparedtohisschool?Explainhow youdeterminedyouranswer. Student G.P.A. SchoolAve.G.P.A. SchoolStandardDeviation Thuy 2.7 3.2 0.8 Vichet 87 75 20 Kamala 8.6 8 0.4

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73 Exercise2.13.10 Giventhefollowingboxplot: a. Whichquarterhasthesmallestspreadofdata?Whatisthatspread? b. Whichquarterhasthelargestspreadofdata?Whatisthatspread? c. FindtheInterQuartileRangeIQR. d. Aretheremoredataintheinterval5-10orintheinterval10-13?Howdoyouknow this? e. Whichintervalhasthefewestdatainit?Howdoyouknowthis? I. 0-2 II. 2-4 III. 10-12 IV. 12-13 Exercise2.13.11 Giventhefollowingboxplot: a. Thinkofanexampleinwordswherethedatamighttintotheaboveboxplot.In2-5 sentences,writedowntheexample. b. Whatdoesitmeantohavetherstandsecondquartilessoclosetogether,whilethe secondtofourthquartilesarefarapart? Exercise2.13.12 SantaClaraCounty,CA,hasapproximately27,873Japanese-Americans.Theiragesareasfollows. Source:Westmagazine AgeGroup PercentofCommunity 0-17 18.9 18-24 8.0 25-34 22.8 35-44 15.0 45-54 13.1 55-64 11.9 65+ 10.3 a. ConstructahistogramoftheJapanese-AmericancommunityinSantaClaraCounty,CA. Thebarswill not bethesamewidthforthisexample.Whynot? b. Whatpercentofthecommunityisunderage35?

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74 CHAPTER2.DESCRIPTIVESTATISTICS c. Whichboxplotmostresemblestheinformationabove? Exercise2.13.13 Supposethatthreebookpublisherswereinterestedinthenumberofctionpaperbacksadult consumerspurchasepermonth.Eachpublisherconductedasurvey.Inthesurvey,eachasked adultconsumersthenumberofctionpaperbackstheyhadpurchasedthepreviousmonth.The resultsarebelow. PublisherA #ofbooks Freq. Rel.Freq. 0 10 1 12 2 16 3 12 4 8 5 6 6 2 8 2

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75 PublisherB #ofbooks Freq. Rel.Freq. 0 18 1 24 2 24 3 22 4 15 5 10 7 5 9 1 PublisherC #ofbooks Freq. Rel.Freq. 0-1 20 2-3 35 4-5 12 6-7 2 8-9 1 a. Findtherelativefrequenciesforeachsurvey.Writetheminthecharts. b. Usingeitheragraphingcalculator,computer,orbyhand,usethefrequencycolumnto constructahistogramforeachpublisher'ssurvey.ForPublishersAandB,makebar widthsof1.ForPublisherC,makebarwidthsof2. c. Incompletesentences,givetworeasonswhythegraphsforPublishersAandBarenot identical. d. WouldyouhaveexpectedthegraphforPublisherCtolookliketheothertwographs? Whyorwhynot? e. MakenewhistogramsforPublisherAandPublisherB.Thistime,makebarwidthsof2. f. Now,comparethegraphforPublisherCtothenewgraphsforPublishersAandB.Are thegraphsmoresimilarormoredifferent?Explainyouranswer. Exercise2.13.14 Often,cruiseshipsconductallon-boardtransactions,withtheexceptionofgambling,onacashlessbasis.Attheendofthecruise,guestspayonebillthatcoversallon-boardtransactions.Supposethat60singletravelersand70couplesweresurveyedastotheiron-boardbillsforaseven-day cruisefromLosAngelestotheMexicanRiviera.Belowisasummaryofthebillsforeachgroup.

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76 CHAPTER2.DESCRIPTIVESTATISTICS Singles Amount\$ Frequency Rel.Frequency 51-100 5 101-150 10 151-200 15 201-250 15 251-300 10 301-350 5 Couples Amount\$ Frequency Rel.Frequency 100-150 5 201-250 5 251-300 5 301-350 5 351-400 10 401-450 10 451-500 10 501-550 10 551-600 5 601-650 5 a. Fillintherelativefrequencyforeachgroup. b. ConstructahistogramfortheSinglesgroup.Scalethex-axisby\$50.widths.Userelative frequencyonthey-axis. c. ConstructahistogramfortheCouplesgroup.Scalethex-axisby\$50.Userelativefrequencyonthey-axis. d. Comparethetwographs: i. Listtwosimilaritiesbetweenthegraphs. ii. Listtwodifferencesbetweenthegraphs. iii. Overall,arethegraphsmoresimilarordifferent? e. ConstructanewgraphfortheCouplesbyhand.Sinceeachcoupleispayingfortwo individuals,insteadofscalingthex-axisby\$50,scaleitby\$100.Userelativefrequency onthey-axis. f. ComparethegraphfortheSingleswiththenewgraphfortheCouples: i. Listtwosimilaritiesbetweenthegraphs. ii. Overall,arethegraphsmoresimilarordifferent? i. ByscalingtheCouplesgraphdifferently,howdiditchangethewayyoucompareditto theSingles?

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77 j. Basedonthegraphs,doyouthinkthatindividualsspendthesameamount,moreorless, assinglesastheydopersonbypersoninacouple?Explainwhyinoneortwocomplete sentences. Exercise2.13.15 Solutiononp.90. Refertothefollowinghistogramsandboxplot.Determinewhichofthefollowingaretrueand whicharefalse.Explainyoursolutiontoeachpartincompletesentences. a. Themediansforallthreegraphsarethesame. b. Wecannotdetermineifanyofthemeansforthethreegraphsisdifferent. c. Thestandarddeviationforbislargerthanthestandarddeviationfora. d. Wecannotdetermineifanyofthethirdquartilesforthethreegraphsisdifferent.

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78 CHAPTER2.DESCRIPTIVESTATISTICS Exercise2.13.16 Refertothefollowingboxplots. a. Incompletesentences,explainwhyeachstatementisfalse. i. Data1 hasmoredatavaluesabove2than Data2 hasabove2. ii. Thedatasetscannothavethesamemode. iii. For Data1 ,therearemoredatavaluesbelow4thanthereareabove4. b. Forwhichgroup,Data1orData2,isthevalueofmorelikelytobeanoutlier?Explain whyincompletesentences Exercise2.13.17 Solutiononp.90. Inarecentissueofthe IEEESpectrum ,84engineeringconferenceswereannounced.Fourconferenceslastedtwodays.Thirty-sixlastedthreedays.Eighteenlastedfourdays.Nineteenlasted vedays.Fourlastedsixdays.Onelastedsevendays.Onelastedeightdays.Onelastednine days.LetX=thelengthindaysofanengineeringconference. a. Organizethedatainachart. b. Findthemedian,therstquartile,andthethirdquartile. c. Findthe65thpercentile. d. Findthe10thpercentile. e. Constructaboxplotofthedata. f. Themiddle50%oftheconferenceslastfrom_______daysto_______days. g. Calculatethesamplemeanofdaysofengineeringconferences. h. Calculatethesamplestandarddeviationofdaysofengineeringconferences. i. Findthemode. j. Ifyouwereplanninganengineeringconference,whichwouldyouchooseasthelengthof theconference:mean;median;ormode?Explainwhyyoumadethatchoice. k. Givetworeasonswhyyouthinkthat3-5daysseemtobepopularlengthsofengineering conferences. Exercise2.13.18 Asurveyofenrollmentat35communitycollegesacrosstheUnitedStatesyieldedthefollowing gures source:MicrosoftBookshelf : 6414;1550;2109;9350;21828;4300;5944;5722;2825;2044;5481;5200;5853;2750;10012;6357; 27000;9414;7681;3200;17500;9200;7380;18314;6557;13713;17768;7493;2771;2861;1263;7285; 28165;5080;11622 a. Organizethedataintoachartwithveintervalsofequalwidth.Labelthetwocolumns "Enrollment"and"Frequency."

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79 b. Constructahistogramofthedata. c. Ifyouweretobuildanewcommunitycollege,whichpieceofinformationwouldbemore valuable:themodeortheaveragesize? d. Calculatethesampleaverage. e. Calculatethesamplestandarddeviation. f. Aschoolwithanenrollmentof8000wouldbehowmanystandarddeviationsawayfrom themean? Exercise2.13.19 Solutiononp.90. ThemedianageoftheU.S.populationin1980was30.0years.In1991,themedianagewas33.1 years. Source:BureauoftheCensus a. Whatdoesitmeanforthemedianagetorise? b. Givetworeasonswhythemedianagecouldrise. c. Forthemedianagetorise,istheactualnumberofchildrenlessin1991thanitwasin 1980?Whyorwhynot? Exercise2.13.20 Asurveywasconductedof130purchasersofnewBMW3seriescars,130purchasersofnew BMW5seriescars,and130purchasersofnewBMW7seriescars.Init,peoplewereaskedtheage theywerewhentheypurchasedtheircar.Thefollowingboxplotsdisplaytheresults. a. Incompletesentences,describewhattheshapeofeachboxplotimpliesaboutthedistributionofthedatacollectedforthatcarseries. b. Whichgroupismostlikelytohaveanoutlier?Explainhowyoudeterminedthat. c. Comparethethreeboxplots.WhatdotheyimplyabouttheageofpurchasingaBMW fromtheserieswhencomparedtoeachother? d. LookattheBMW5series.Whichquarterhasthesmallestspreadofdata?Whatisthat spread? e. LookattheBMW5series.Whichquarterhasthelargestspreadofdata?Whatisthat spread? f. LookattheBMW5series.FindtheInterQuartileRangeIQR. g. LookattheBMW5series.Aretheremoredataintheinterval31-38orintheinterval 45-55?Howdoyouknowthis?

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80 CHAPTER2.DESCRIPTIVESTATISTICS h. LookattheBMW5series.Whichintervalhasthefewestdatainit?Howdoyouknow this? i. 31-35 ii. 38-41 iii. 41-64 Exercise2.13.21 Solutiononp.90. ThefollowingboxplotshowstheU.S.populationfor1990,thelatestavailableyear.Source: BureauoftheCensus,1990Census a. Aretherefewerormorechildrenage17andunderthanseniorcitizensage65andover? Howdoyouknow? b. 12.6%areage65andover.Approximatelywhatpercentofthepopulationareofworking ageadultsaboveage17toage65? Exercise2.13.22 JavierandErciliaaresupervisorsatashoppingmall.Eachwasgiventhetaskofestimatingthe meandistancethatshopperslivefromthemall.Theyeachrandomlysurveyed100shoppers.The samplesyieldedthefollowinginformation: Javier Ercilla x 6.0miles 6.0miles s 4.0miles 7.0miles a. Howcanyoudeterminewhichsurveywascorrect? b. Explainwhatthedifferenceintheresultsofthesurveysimpliesaboutthedata. c. Ifthetwohistogramsdepictthedistributionofvaluesforeachsupervisor,whichone depictsErcilia'ssample?Howdoyouknow? Figure2.2 d. Ifthetwoboxplotsdepictthedistributionofvaluesforeachsupervisor,whichonedepictsErcilia'ssample?Howdoyouknow?

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81 Figure2.3 Exercise2.13.23 Solutiononp.90. Studentgradesonachemistryexamwere: 77,78,76,81,86,51,79,82,84,99 a. Constructastem-and-leafplotofthedata. b. Arethereanypotentialoutliers?Ifso,whichscoresarethey?Whydoyouconsiderthem outliers? 2.13.1Trythesemultiplechoicequestions. Thenextthreequestionsrefertothefollowinginformation. Weareinterestedinthenumberofyears studentsinaparticularelementarystatisticsclasshavelivedinCalifornia.Theinformationinthefollowing tableisfromtheentiresection. Numberofyears Frequency 7 1 14 3 15 1 18 1 19 4 20 3 22 1 23 1 26 1 40 2 42 2 Total=20 Exercise2.13.24 Solutiononp.90. WhatistheIQR? A. 8 B. 11 C. 15 D. 35

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82 CHAPTER2.DESCRIPTIVESTATISTICS Exercise2.13.25 Solutiononp.90. Whatisthemode? A. 19 B. 19.5 C. 14and20 D. 22.65 Exercise2.13.26 Solutiononp.90. Isthisasampleortheentirepopulation? A. sample B. entirepopulation C. neither Thenexttwoquestionsrefertothefollowingtable. X =thenumberofdaysperweekthat100clientsusea particularexercisefacility. X Frequency 0 3 1 12 2 33 3 28 4 11 5 9 6 4 Exercise2.13.27 Solutiononp.91. The80thpercentileis: A. 5 B. 80 C. 3 D. 4 Exercise2.13.28 Solutiononp.91. Thenumberthatis1.5standarddeviationsBELOWthemeanisapproximately: A. 0.7 B. 4.8 C. -2.8 D. Cannotbedetermined Thenexttwoquestionsrefertothefollowinghistogram. Supposeonehundredelevenpeoplewhoshopped inaspecialT-shirtstorewereaskedthenumberofT-shirtstheyowncostingmorethan\$19each.

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83 Exercise2.13.29 Solutiononp.91. ThepercentofpeoplethatownatmostthreeT-shirtscostingmorethan\$19eachisapproximately: A. 21 B. 59 C. 41 D. Cannotbedetermined Exercise2.13.30 Solutiononp.91. Ifthedatawerecollectedbyaskingtherst111peoplewhoenteredthestore,thenthetypeof samplingis: A. cluster B. simplerandom C. stratied D. convenience

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84 CHAPTER2.DESCRIPTIVESTATISTICS 2.14Lab:DescriptiveStatistics 16 ClassTime: Names: 2.14.1StudentLearningObjectives Thestudentwillconstructahistogramandaboxplot. Thestudentwillcalculateunivariatestatistics. Thestudentwillexaminethegraphstointerpretwhatthedataimplies. 2.14.2CollecttheData Recordthenumberofpairsofshoesyouown: 1.Randomlysurvey30classmates.Recordtheirvalues. SurveyResults _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ 2.Constructahistogram.Make5-6intervals.Sketchthegraphusingarulerandpencil.Scaletheaxes. 16 Thiscontentisavailableonlineat.

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85 Figure2.4 3.Calculatethefollowing: x = s = 4.Arethedatadiscreteorcontinuous?Howdoyouknow? 5.Describetheshapeofthehistogram.Usecompletesentences. 6.Arethereanypotentialoutliers?Whichvaluesisareitthey?Useaformulatochecktheend valuestodetermineiftheyarepotentialoutliers. 2.14.3AnalyzetheData 1.Determinethefollowing: Minimumvalue= Median= Maximumvalue= Firstquartile= Thirdquartile= IQR= 2.Constructaboxplotofdata 3.Whatdoestheshapeoftheboxplotimplyabouttheconcentrationofdata?Usecompletesentences. 4.Usingtheboxplot,howcanyoudetermineiftherearepotentialoutliers? 5.Howdoesthestandarddeviationhelpyoutodetermineconcentrationofthedataandwhetherornot therearepotentialoutliers? 6.WhatdoestheIQRrepresentinthisproblem? 7.Showyourworktondthevaluethatis1.5standarddeviations: a. Abovethemean: b. Belowthemean:

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86 CHAPTER2.DESCRIPTIVESTATISTICS SolutionstoExercisesinChapter2 Example2.2p.45 Thevalue12.3maybeanoutlier.Valuesappeartoconcentrateat3and4miles. Stem Leaf 1 15 2 357 3 33358 4 025578 5 566 6 57 7 8 9 10 11 12 3 Example2.4p.47 3.5to4.5 4.5to5.5 6 5.5to6.5 Example2.6p.51 FirstDataSet Xmin = 32 Q 1 = 56 M = 74.5 Q 3 = 82.5 Xmax = 99 SecondDataSet Xmin = 25.5 Q 1 = 78 M = 81 Q 3 = 89 Xmax = 98

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87 Example2.8p.52 FortheIQRs,seetheanswertothetestscoresexampleFirstDataSet,p.86SecondDataSet,p.86p.511 .TherstdatasethasthelargerIQR,sothescoresbetween Q 3and Q 1middle50%fortherstdataset aremorespreadoutandnotclusteredaboutthemedian. FirstDataSet )]TJ/F58 7.5716 Tf 5.883 -4.15 Td [(3 2 IQR = )]TJ/F58 7.5716 Tf 5.883 -4.15 Td [(3 2 26.5 = 39.75 Xmax )]TJ/F132 9.9626 Tf 12.846 0 Td [(Q 3 = 99 )]TJ/F58 9.9626 Tf 12.622 0 Td [(82.5 = 16.5 Q 1 )]TJ/F132 9.9626 Tf 12.896 0 Td [(Xmin = 56 )]TJ/F58 9.9626 Tf 12.622 0 Td [(32 = 24 )]TJ/F58 7.5716 Tf 5.882 -4.149 Td [(3 2 IQR = 39.75islargerthan16.5andlargerthan24,sotherstsethasnooutliers. SecondDataSet )]TJ/F58 7.5716 Tf 5.883 -4.149 Td [(3 2 IQR = )]TJ/F58 7.5716 Tf 5.882 -4.149 Td [(3 2 11 = 16.5 Xmax )]TJ/F132 9.9626 Tf 10.356 0 Td [(Q 3 = 98 )]TJ/F58 9.9626 Tf 10.131 0 Td [(89 = 9 Q 1 )]TJ/F132 9.9626 Tf 10.405 0 Td [(Xmin = 78 )]TJ/F58 9.9626 Tf 10.131 0 Td [(25.5 = 52.5 )]TJ/F58 7.5716 Tf 5.882 -4.15 Td [(3 2 IQR = 16.5islargerthan9butsmallerthan52.5,soforthesecondset45and25.5areoutliers. Tondthepercentiles,createafrequency,relativefrequency,andcumulativerelativefrequencychartsee "Frequency"fromtheSamplingandDataChapterSection1.9.Getthepercentilesfromthatchart. FirstDataSet 30th%ilebetweenthe6thand7thvalues = 56 + 59 2 = 57.5 80th%ilebetweenthe16thand17thvalues = 84 + 84.5 2 = 84.25 SecondDataSet 30th%ilethvalue = 78 80th%ilethvalue = 90 30%ofthedatafallsbelowthe30th%ile,and20%fallsabovethe80th%ile. Example2.10p.53 1. 8 + 9 2 = 8.5 2.9 3.6 4.FirstQuartile=25th%ile

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88 CHAPTER2.DESCRIPTIVESTATISTICS SolutionstoPractice1:CenteroftheData SolutiontoExercise2.11.1p.65 65 SolutiontoExercise2.11.2p.65 1 SolutiontoExercise2.11.5p.66 4.75 SolutiontoExercise2.11.6p.66 1.39 SolutiontoExercise2.11.7p.66 65 SolutiontoExercise2.11.8p.66 4 SolutiontoExercise2.11.9p.66 4 SolutiontoExercise2.11.10p.66 4 SolutiontoExercise2.11.11p.66 4 SolutiontoExercise2.11.12p.66 6 SolutiontoExercise2.11.13p.66 6 )]TJ/F58 9.9626 Tf 10.131 0 Td [(4 = 2 SolutiontoExercise2.11.14p.66 3 SolutiontoExercise2.11.15p.66 6 SolutiontoExercise2.11.16p.67 a. 8.93 b. 0.58 SolutionstoPractice2:SpreadoftheData SolutiontoExercise2.12.1p.68 6 SolutiontoExercise2.12.2p.68 a. 1447.5 b. 528.5 SolutiontoExercise2.12.3p.68 474FTES SolutiontoExercise2.12.4p.68 50%

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89 SolutiontoExercise2.12.5p.68 919 SolutiontoExercise2.12.6p.68 0.03 SolutionstoHomework SolutiontoExercise2.13.1p.69 a. 1.48 b. 1.12 e. 1 f. 1 g. 2 h. i. 80% j. 1 k. 3 SolutiontoExercise2.13.3p.69 a. 3.78 b. 1.29 e. 3 f. 4 g. 5 h. i. 32.5% j. 4 k. 5 SolutiontoExercise2.13.5p.71 b. 241 c. 205.5 d. 272.5 e.

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90 CHAPTER2.DESCRIPTIVESTATISTICS f. 205.5,272.5 g. sample h. population i.i. 236.34 ii. 37.50 iii. 161.34 iv. 0.84std.dev.belowthemean j. Young SolutiontoExercise2.13.9p.72 Kamala SolutiontoExercise2.13.15p.77 a. True b. True c. True d. False SolutiontoExercise2.13.17p.78 b. 4,3,5 c. 4 d. 3 e. f. 3,5 g. 3.94 h. 1.28 i. 3 j. mode SolutiontoExercise2.13.19p.79 c. Maybe SolutiontoExercise2.13.21p.80 a. morechildren b. 62.4% SolutiontoExercise2.13.23p.81 b. 51,99 SolutiontoExercise2.13.24p.81 A SolutiontoExercise2.13.25p.82 A SolutiontoExercise2.13.26p.82 B

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91 SolutiontoExercise2.13.27p.82 D SolutiontoExercise2.13.28p.82 A SolutiontoExercise2.13.29p.83 C SolutiontoExercise2.13.30p.83 D

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92 CHAPTER2.DESCRIPTIVESTATISTICS

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Chapter3 ProbabilityTopics 3.1ProbabilityTopics 1 3.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Understandandusetheterminologyofprobability. Determinewhethertwoeventsaremutuallyexclusiveorindependent. CalculateprobabilitiesusingtheAdditionRulesandMultiplicationRules. ConstructandinterpretContingencyTables. ConstructandinterpretVennDiagramsoptional. ConstructandinterpretTreeDiagramsoptional. 3.1.2Introduction Itisoftennecessaryto"guess"abouttheoutcomeofaneventinordertomakeadecision.Politiciansstudy pollstoguesstheirlikelihoodofwinninganelection.Teacherschooseaparticularcourseofstudybased onwhattheythinkstudentscancomprehend.Doctorsestimatethetreatmentsneededforvariousdiseases. Youmayhavevisitedacasinowherepeopleplaygameschosenbecauseofthebeliefthatthelikelihoodof winningisgood.Youmayhavechosenyourcourseofstudybasedontheprobableavailabilityofjobs. Youhave,morethanlikely,usedprobability.Infact,youprobablyhaveanintuitivesenseofprobability. Probabilitydealswiththechanceofaneventoccurring.Wheneveryouweightheoddsofwhetherornot todoyourhomeworkortostudyforanexam,youareusingprobability.Inthischapter,youwilllearnto solveprobabilityproblemsusingasystematicapproach. 3.1.3OptionalCollaborativeClassroomExercise Yourinstructorwillsurveyyourclass.Countthenumberofstudentsintheclasstoday. Raiseyourhandifyouhaveanychangeinyourpocketorpurse.Recordthenumberofraisedhands. Raiseyourhandifyourodeabuswithinthepastmonth.Recordthenumberofraisedhands. Raiseyourhandifyouanswered"yes"toBOTHofthersttwoquestions.Recordthenumberof raisedhands. 1 Thiscontentisavailableonlineat. 93

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94 CHAPTER3.PROBABILITYTOPICS Usetheclassdataasestimatesofthefollowingprobabilities. Pchange meanstheprobabilitythatarandomlychosenpersoninyourclasshaschangeinhis/herpocketorpurse. Pbus meanstheprobabilitythat arandomlychosenpersoninyourclassrodeabuswithinthelastmonthandsoon.Discussyouranswers. Find Pchange Find Pbus Find Pchangeandbus Findtheprobabilitythatarandomlychosenstudentinyourclasshaschange inhis/herpocketorpurseandrodeabuswithinthelastmonth. Find Pchange|bus Findtheprobabilitythatarandomlychosenstudenthaschangegiventhathe/she rodeabuswithinthelastmonth.Countallthestudentsthatrodeabus.Fromthegroupofstudents whorodeabus,countthosewhohavechange.Theprobabilityisequaltothosewhohavechange androdeabusdividedbythosewhorodeabus. 3.2Terminology 2 Probabilitymeasurestheuncertaintythatisassociatedwiththeoutcomesofaparticularexperimentor activity.An experiment isaplannedoperationcarriedoutundercontrolledconditions.Iftheresultisnot predetermined,thentheexperimentissaidtobea chance experiment.Flippingonefaircoinisanexample ofanexperiment. Theresultofanexperimentiscalledan outcome .A samplespace isasetofallpossibleoutcomes.Three waystorepresentasamplespacearetolistthepossibleoutcomes,tocreateatreediagram,ortocreatea Venndiagram.Theuppercaseletter S isusedtodenotethesamplespace.Forexample,ifyouiponefair coin, S = f H T g where H =headsand T =tailsaretheoutcomes. An event isanycombinationofoutcomes.Uppercaseletterslike A and B representevents.Forexample, iftheexperimentistoiponefaircoin,event A mightbegettingatmostonehead.Theprobabilityofan event A iswritten P A The probability ofanyoutcomeisthe long-termrelativefrequency ofthatoutcome.Forexample,ifyou iponefaircoinfrom20to2,000times,therelativefrequencyofheadsapproaches0.5theprobability ofheads.Probabilitiesarebetween0and1, inclusive includes0and1andallnumbersbetweenthese values. P A = 0meanstheevent A canneverhappen. P A = 1meanstheevent A alwayshappens. Tocalculatethe probabilityofanevent A ,counttheoutcomesforeventAanddividebythetotaloutcomesinthesamplespace.Forexample,ifyoutossafairdimeandafairnickel,thesamplespaceis f HH TH HT TT g where T =tailsand H =heads.Thesamplespacehasfouroutcomes. A =gettingone head.Therearetwooutcomes f HT TH g P A = 2 4 Equallylikely meansthateachoutcomeofanexperimentoccurswithequalprobability.Forexample,if youtossafair,six-sideddie,eachface,2,3,4,5,or6isaslikelytooccurasanyotherface. Anoutcomeisintheevent AORB iftheoutcomeisin A orisin B orisinboth A and B .Forexample,let A = f 1,2,3,4,5 g and B = f 4,5,6,7,8 g AORB = f 1,2,3,4,5,6,7,8 g .Noticethat4and5areNOT listedtwice. Anoutcomeisintheevent AANDB iftheoutcomeisinboth A and B atthesametime.Forexample,let A and B be f 1,2,3,4,5 g and f 4,5,6,7,8 g ,respectively.Then AANDB = f 4,5 g 2 Thiscontentisavailableonlineat.

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95 The complement ofevent A isdenoted A 'read"Aprime". A 'consistsofalloutcomesthatare NOT in A .Noticethat P A + P A = 1.Forexample,let S = f 1,2,3,4,5,6 g andlet A = f 1,2,3,4 g .Then, A = f 5,6 g P A = 4 6 P A = 2 6 andP A + P A = 4 6 + 2 6 = 1 The conditionalprobability of A given B iswritten P A j B .Theprobabilityof A iscalculatedknowing that B hasalreadyoccurred. Aconditionalreducesthesamplespace .Wecalculatetheprobabilityof A fromthereducedsamplespace B .Theformulatocalculate P A j B is P A j B = P AANDB P B where P B isgreaterthan0. Forexample,supposewetossonefair,six-sideddie.Thesamplespace S = f 1,2,3,4,5,6 g .Let A =faceis 2or3and B =faceiseven,4,6.Tocalculate P A j B ,wecountthenumberofoutcomes2or3inthe samplespace B = f 2,4,6 g .Thenwedividethatbythenumberofoutcomesin B andnot S Wegetthesameresultbyusingtheformula.Rememberthat S has6outcomes. P A j B = PAandB PB = thenumberofoutcomesthatare2or3andeveninS/6 thenumberofoutcomesthatareeveninS/6 = 1/6 3/6 = 1 3 3.3IndependentandMutuallyExclusiveEvents 3 3.3.1IndependentEvents Twoeventsareindependentifthefollowingaretrue: P A j B = P A P B j A = P B P AANDB = P A P B If A and B are independent ,thenthechanceof A occurringdoesnotaffectthechanceof B occurringand viceversa.Forexample,tworolesofafairdieareindependentevents.Theoutcomeoftherstrolldoes notchangetheprobabilityfortheoutcomeofthesecondroll.Toshowtwoeventsareindependent,you mustshow onlyone oftheaboveconditions. 3.3.2MutuallyExclusiveEvents A and B are mutuallyexclusive eventsiftheycannotoccuratthesametime.Thismeansthat A and B do notshareanyoutcomesand PAANDB = 0. Forexample,supposethesamplespace S = f 1,2,3,4,5,6,7,8,9,10 g .Let A = f 1,2,3,4,5 g B = f 4,5,6,7,8 g ,and C = f 7,9 g AANDB = f 4,5 g PAANDB= 2 10 andisnotequaltozero.Therefore, A and B arenotmutuallyexclusive. A and C donothaveanynumbersincommonso PAANDC=0 Therefore, A and C aremutuallyexclusive. N OTE :Independentandmutuallyexclusivedo not meanthesamething. Youmustshowthatanytwoeventsareindependentormutuallyexclusive. Youcannotassumeeitherof theseconditions. Ifitisnotknownwhether A and B areindependentordependent, assumetheyaredependentuntilyou canshowotherwise 3 Thiscontentisavailableonlineat.

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96 CHAPTER3.PROBABILITYTOPICS Thefollowingexamplesillustratethesedenitionsandterms. Example3.1 Fliptwofaircoins.Thisisanexperiment. Thesamplespaceis f HH HT TH TT g where T =tailsand H =heads.Theoutcomesare HH HT TH ,and TT .Theoutcomes HT and TH aredifferent.The HT meansthattherstcoinshowed headsandthesecondcoinshowedtails.The TH meansthattherstcoinshowedtailsandthe secondcoinshowedheads. Let A =theeventofgetting atmostonetail .Atmostonetailmeans0or1tail.Then A can bewrittenas f HH HT TH g .Theoutcome HH shows0tails. HT and TH eachshow1tail. Let B =theeventofgettingalltails. B canbewrittenas f TT g B isthe complement of A .So, B = A '.Also, P A + P B = P A + P A = 1. Theprobabilitiesfor A andfor B are P A = 3 4 and P B = 1 4 Let C =theeventofgettingallheads. C = f HH g .Since B = f TT g P BANDC = 0. B and C aremutuallyexclusive. B and C havenomembersincommonbecauseyoucannot havealltailsandallheadsatthesametime. Let D =eventofgetting morethanone tail. D = f TT g P D = 1 4 Let E =eventofgettingaheadontherstroll.Thisimpliesyoucangeteitheraheadortail onthesecondroll. E = f HT HH g P E = 2 4 Findtheprobabilityofgetting atleastone or2tailintwoips.Let F =eventofgetting atleastonetailintwoips. F = f HT TH TT g PF = 3 4 Example3.2 Rollonefair6-sideddie.Thesamplespaceis f 1,2,3,4,5,6 g .Letevent A =afaceisodd.Then A = f 1,3,5 g .Letevent B =afaceiseven.Then B = f 2,4,6 g Findthecomplementof A A '.Thecomplementof A A ',is B because A and B together makeupthesamplespace. PA+PB=PA+PA'=1 .Also, PA = 3 6 and PB = 3 6 Letevent C =oddfaceslargerthan2.Then C = f 3,5 g .Letevent D =allevenfacessmaller than5.Then D = f 2,4 g PCandD = 0becauseyoucannothaveanoddandevenfaceat thesametime.Therefore, C and D aremutuallyexclusiveevents. Letevent E =allfaceslessthan5. E = f 1,2,3,4 g Problem Are C and E mutuallyexclusiveevents?Answeryesorno.Whyorwhynot? FindPC|A .Thisisaconditional.Recallthattheevent C is f 3,5 g andevent A is f 1,3,5 g Tond PC|A ,ndtheprobabilityof C usingthesamplespace A .Youhavereducedthe samplespacefromtheoriginalsamplespace f 1,2,3,4,5,6 g to f 1,3,5 g .So, PC|A = 2 3 Example3.3 Letevent G =takingamathclass.Letevent H =takingascienceclass.Then, GANDH =taking amathclassandascienceclass.Suppose PG = 0.6, PH = 0.5,and PGANDH = 0.3.Are G and H independent? If G and H areindependent,thenyoumustshow ONE ofthefollowing: PG|H = PG PH|G = PH PGANDH = PG PH N OTE : Thechoiceyoumakedependsontheinformationyouhave. Youcouldchooseanyofthe methodsherebecauseyouhavethenecessaryinformation.

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97 Problem1 Showthat PG|H = PG Solution PG|H = PGANDH PH = 0.3 0.5 = 0.6 = PG Problem2 Show PGANDH = PG PH Solution P G P H = 0.6 0.5 = 0.3 = PGANDH Since G and H areindependent,then,knowingthatapersonistakingascienceclassdoesnot changethechancethathe/sheistakingmath.Ifthetwoeventshadnotbeenindependentthat is,theyaredependentthenknowingthatapersonistakingascienceclasswouldchangethe chancehe/sheistakingmath.Forpractice,showthat PH|G = PH toshowthat G and H are independentevents. Example3.4 Inaboxthereare3redcardsand5bluecards.Theredcardsaremarkedwiththenumbers1,2, and3,andthebluecardsaremarkedwiththenumbers1,2,3,4,and5.Thecardsarewell-shufed. Youreachintotheboxyoucannotseeintoitanddrawonecard. Let R =redcardisdrawn, B =bluecardisdrawn, E =even-numberedcardisdrawn. Thesamplespace S = R1,R2,R3,B1,B2,B3,B4,B5 S has8outcomes. PR = 3 8 PB = 5 8 PRANDB = 0.Youcannotdrawonecardthatisbothredandblue. PE = 3 8 .Thereare3even-numberedcards, R 2, B 2,and B 4. PE|B = 2 5 .Thereare5bluecards: B 1, B 2, B 3, B 4,and B 5.Outofthebluecards,thereare 2evencards: B 2and B 4. PB|E = 2 3 .Thereare3even-numberedcards: R 2, B 2,and B 4.Outoftheeven-numbered cards,2areblue: B 2and B 4. Theevents R and B aremutuallyexclusivebecause PRANDB = 0. Let G =cardwithanumbergreaterthan3. G = f B 4, B 5 g PG = 2 8 .Let H =bluecard numberedbetween1and4,inclusive. H = f B 1, B 2, B 3, B 4 g PG|H = 1 4 .Theonlycardin Hthathasanumbergreaterthan3is B 4.Since 2 8 = 1 4 PG = PG|H whichmeansthat G and H areindependent. 3.4TwoBasicRulesofProbability 4 3.4.1TheMultiplicationRule If A and B aretwoeventsdenedona samplespace ,then: PAANDB = PB PA|B Thisrulemayalsobewrittenas: P A j B = P AANDB P B Theprobabilityof A given B equalstheprobabilityof A and B dividedbytheprobabilityof B IfAandBare independent ,then PA|B = PA .Then PAANDB = PA|BPB becomes PAANDB = PAPB 4 Thiscontentisavailableonlineat.

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98 CHAPTER3.PROBABILITYTOPICS 3.4.2TheAdditionRule If A and B aredenedonasamplespace,then: PAORB = PA + PB )]TJ/F132 9.9626 Tf 10.131 0 Td [(PAANDB If A and B are mutuallyexclusive ,then PAANDB = 0.Then PAORB = PA + PB )]TJ/F132 9.9626 Tf 10.338 0 Td [(PAANDB becomes PAORB = PA + PB Example3.5 Klausistryingtochoosewheretogoonvacation.Histwochoicesare: A =NewZealandand B =Alaska Klauscanonlyaffordonevacation.Theprobabilitythathechooses A is PA = 0.6andthe probabilitythathechooses B is PB = 0.35. PAandB = 0becauseKlauscanonlyaffordtotakeonevacation Therefore,theprobabilitythathechooseseitherNewZealandorAlaskais PAORB = PA + PB = 0.6 + 0.35 = 0.95. Example3.6 Carlosplayscollegesoccer.Hemakesagoal65%ofthetimeheshoots.Carlosisgoingtoattempt twogoalsinarowinthenextgame. A =theeventCarlosissuccessfulonhisrstattempt. PA = 0.65. B =theeventCarlosis successfulonhissecondattempt. PB = 0.65.Carlostendstoshootinstreaks.Theprobability thathemakesthesecondgoal GIVEN thathemadetherstgoalis0.90. Problem1 Whatistheprobabilitythathemakesbothgoals? Solution Theproblemisaskingyoutond PAANDB = PBANDA .Since PB|A = 0.90: PBANDA = PB|APA = 0.90 0.65 = 0.585.1 Carlosmakestherstandsecondgoalswithprobability0.585. Problem2 WhatistheprobabilitythatCarlosmakeseithertherstgoalorthesecondgoal? Solution Theproblemisaskingyoutond PAORB PAORB = PA + PB )]TJ/F132 9.9626 Tf 10.132 0 Td [(PAANDB = 0.65 + 0.65 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.585 = 0.715.2 Carlosmakeseithertherstgoalorthesecondgoalwithprobability0.715. Problem3 Are A and B independent? Solution No,theyarenot,because PBANDA = 0.585. PB PA = 0.65 0.65 = 0.423.3 0.423 6 = 0.585 = PBANDA .4 So, PBANDA is not equalto PB PA Problem4 Are A and B mutuallyexclusive?

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99 Solution No,theyarenotbecause PAandB =0.585. Tobemutuallyexclusive, PAANDB mustequal0. Example3.7 Acommunityswimteamhas 150 members. Seventy-ve ofthemembersareadvancedswimmers. Forty-seven ofthemembersareintermediateswimmers.Theremainderarenoviceswimmers. Forty oftheadvancedswimmerspractice4timesaweek. Thirty oftheintermediateswimmerspractice4timesaweek. Ten ofthenoviceswimmerspractice4timesaweek.Supposeone memberoftheswimteamisrandomlychosen.AnswerthequestionsVerifytheanswers: Problem1 Whatistheprobabilitythatthememberisanoviceswimmer? Solution 28 150 Problem2 Whatistheprobabilitythatthememberpractices4timesaweek? Solution 80 150 Problem3 Whatistheprobabilitythatthememberisanadvancedswimmerandpractices4timesaweek? Solution 40 150 Problem4 Whatistheprobabilitythatamemberisanadvancedswimmerandanintermediateswimmer? Arebeinganadvancedswimmerandanintermediateswimmermutuallyexclusive?Whyorwhy not? Solution PadvancedANDintermediate = 0,sothesearemutuallyexclusiveevents.Aswimmercannotbe anadvancedswimmerandanintermediateswimmeratthesametime. Problem5 Arebeinganoviceswimmerandpracticing4timesaweekindependentevents?Whyorwhy not? Solution No,thesearenotindependentevents. PnoviceANDpractices4timesperweek = 0.0667.5 Pnovice Ppractices4timesperweek = 0.0996.6 0.0667 6 = 0.0996.7

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100 CHAPTER3.PROBABILITYTOPICS Example3.8 Studiesshowthat,ifshelivestobe90,about1womanin7approximately14.3%willdevelop breastcancer.Supposethatofthosewomenwhodevelopbreastcancer,atestisnegative2%ofthe time.Alsosupposethatinthegeneralpopulationofwomen,thetestforbreastcancerisbelieved tobenegativeabout85%ofthetime.Let B =womandevelopsbreastcancerandlet N =tests negative. Problem1 Whatistheprobabilitythatawomandevelopsbreastcancer?Whatistheprobabilitythatwoman testsnegative? Solution PB = 0.143; PN = 0.85 Problem2 Giventhatawomanhasbreastcancer,whatistheprobabilitythatshetestsnegative? Solution PN|B = 0.02 Problem3 WhatistheprobabilitythatawomanhasbreastcancerANDtestsnegative? Solution PBANDN = PB PN|B = 0.143 0.02 = 0.0029 Problem4 Whatistheprobabilitythatawomanhasbreastcancerortestsnegative? Solution PBORN = PB + PN )]TJ/F132 9.9626 Tf 10.131 0 Td [(PBANDN = 0.143 + 0.85 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.0029 = 0.9901 Problem5 Arehavingbreastcancerandtestingnegativeindependentevents? Solution No. PN = 0.85; PN|B = 0.02.So, PN|B doesnotequal PN Problem6 Arehavingbreastcancerandtestingnegativemutuallyexclusive? Solution No. PBANDN = 0.0020.For B and N tobemutuallyexclusive, PBANDN mustbe0. 3.5ContingencyTables 5 A contingencytable providesadifferentwayofcalculatingprobabilities.Thetablehelpsindetermining conditionalprobabilitiesquiteeasily.Thetabledisplayssamplevaluesinrelationtotwodifferentvariables thatmaybedependentorcontingentononeanother.Lateron,wewillusecontingencytablesagain,butin anothermanner. 5 Thiscontentisavailableonlineat.

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101 Example3.9 Supposeastudyofspeedingviolationsanddriverswhousecarphonesproducedthefollowing ctionaldata: Speedingviolationin thelastyear Nospeedingviolation inthelastyear Total Carphoneuser 25 280 305 Notacarphoneuser 45 405 450 Total 70 685 755 Thetotalnumberofpeopleinthesampleis755.Therowtotalsare305and450.Thecolumntotals are70and685.Noticethat305 + 450 = 755and70 + 685 = 755. Calculatethefollowingprobabilitiesusingthetable Problem1 Ppersonisacarphoneuser= Solution numberofcarphoneusers totalnumberinstudy = 305 755 Problem2 Ppersonhadnoviolationinthelastyear= Solution numberthathadnoviolation totalnumberinstudy = 685 755 Problem3 PpersonhadnoviolationinthelastyearANDwasacarphoneuser= Solution 280 755 Problem4 PpersonisacarphoneuserORpersonhadnoviolationinthelastyear= Solution )]TJ/F58 7.5716 Tf 5.883 -4.149 Td [(305 755 + 685 755 )]TJ/F58 7.5716 Tf 11.327 3.925 Td [(280 755 = 710 755 Problem5 PpersonisacarphoneuserGIVENpersonhadaviolationinthelastyear= Solution 25 70 Thesamplespaceisreducedtothenumberofpersonswhohadaviolation. Problem6 PpersonhadnoviolationlastyearGIVENpersonwasnotacarphoneuser= Solution 405 450 Thesamplespaceisreducedtothenumberofpersonswhowerenotcarphoneusers.

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102 CHAPTER3.PROBABILITYTOPICS Example3.10 Thefollowingtableshowsarandomsampleof100hikersandtheareasofhikingpreferred: HikingAreaPreference Sex TheCoastline NearLakesandStreams OnMountainPeaks Total Female 18 16 ___ 45 Male ___ ___ 14 55 Total ___ 41 ___ ___ Problem1 Completethetable. Problem2 Aretheevents"beingfemale"and"preferringthecoastline"independentevents? Let F =beingfemaleandlet C =preferringthecoastline. a. PFANDC = b. P F P C = Arethesetwonumbersthesame?Iftheyare,then F and C areindependent.Iftheyarenot,then F and C arenotindependent. Problem3 Findtheprobabilitythatapersonismalegiventhatthepersonprefershikingnearlakesand streams.Let M =beingmaleandlet L =prefershikingnearlakesandstreams. a. Whatwordtellsyouthisisaconditional? b. Fillintheblanksandcalculatetheprobability: P___|___ = ___. c. Isthesamplespaceforthisproblemall100hikers?Ifnot,whatisit? Problem4 Findtheprobabilitythatapersonisfemaleorprefershikingonmountainpeaks.Let F =being femaleandlet P =prefersmountainpeaks. a. PF = b. PP = c. PFANDP = d. Therefore, PFORP = Example3.11 MuddyMouselivesinacagewith3doors.IfMuddygoesouttherstdoor,theprobabilitythat hegetscaughtbyAlissathecatis 1 5 andtheprobabilityheisnotcaughtis 4 5 .Ifhegoesoutthe seconddoor,theprobabilityhegetscaughtbyAlissais 1 4 andtheprobabilityheisnotcaughtis 3 4 TheprobabilitythatAlissacatchesMuddycomingoutofthethirddooris 1 2 andtheprobability shedoesnotcatchMuddyis 1 2 .ItisequallylikelythatMuddywillchooseanyofthethreedoors sotheprobabilityofchoosingeachdooris 1 3 .

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103 DoorChoice CaughtorNot DoorOne DoorTwo DoorThree Total Caught 1 15 1 12 1 6 ____ NotCaught 4 15 3 12 1 6 ____ Total ____ ____ ____ 1 Therstentry 1 15 = 1 4 1 3 is PDoorOneANDCaught Theentry 4 15 = 4 5 1 3 is PDoorOneANDNotCaught Verifytheremainingentries. Problem1 Completetheprobabilitycontingencytable.Calculatetheentriesforthetotals.Verifythatthe lower-rightcornerentryis1. Problem2 WhatistheprobabilitythatAlissadoesnotcatchMuddy? Solution 41 60 Problem3 WhatistheprobabilitythatMuddychoosesDoorOne OR DoorTwogiventhatMuddyiscaught byAlissa? Solution 9 19 N OTE :Youcouldalsodothisproblembyusingaprobabilitytree.SeetheTreeDiagramsOptionalSection3.7sectionofthischapterforexamples. 3.6VennDiagramsoptional 6 A Venndiagram isapicturethatrepresentstheoutcomesofanexperiment.Itgenerallyconsistsofabox togetherwithcirclesorovals.Thecirclesorovalsrepresentevents. Example3.12 Supposeanexperimenthastheoutcomes1,2,3,...,12whereeachoutcomehasanequalchance ofoccurring.Letevent A = {1,2,3,4,5,6} andevent A = {6,7,8,9} .Then AANDB = {6} and AORB = {1,2,3,4,5,6,7,8,9} .TheVenndiagramisasfollows: 6 Thiscontentisavailableonlineat.

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104 CHAPTER3.PROBABILITYTOPICS Example3.13 Flip2faircoins.Let A =tailsontherstcoin.Let B =tailsonthesecondcoin.Then A = f TT TH g and B = f TT HT g .Therefore, AANDB = f TT g AORB = f TH TT HT g Thesamplespacewhenyouiptwofaircoinsis S = f HH HT TH TT g .Theoutcome HH isin neither A nor B .TheVenndiagramisasfollows: Example3.14 Fortypercent ofthestudentsatalocalcollegebelongtoacluband 50% workparttime. Five percent ofthestudentsworkparttimeandbelongtoaclub.DrawaVenndiagramshowingthe relationships.Let C =studentbelongstoacluband PT =studentworksparttime. Theprobabilitythatastudentsbelongstoaclubis PC = 0.40. Theprobabilitythatastudentworksparttimeis PPT = 0.50.

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105 TheprobabilitythatastudentbelongstoaclubANDworksparttimeis PCANDPT = 0.05. Theprobabilitythatastudentbelongstoaclub given thatthestudentworksparttimeis: PC|PT = PCANDPT PPT = 0.05 0.50 = 0.1.8 Theprobabilitythatastudentbelongstoaclub OR worksparttimeis: PCORPT = PC + PPT )]TJ/F132 9.9626 Tf 10.131 0 Td [(PCANDPT = 0.40 + 0.50 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0.05 = 0.85.9 3.7TreeDiagramsoptional 7 A treediagram isaspecialtypeofgraphusedtodeterminetheoutcomesofanexperiment.Itconsistsof "branches"thatarelabeledwitheitherfrequenciesorprobabilities.Treediagramscanmakesomeprobabilityproblemseasiertocalculate.Thefollowingexampleillustrateshowtouseatreediagram. Example3.15 Inanurn,thereare11balls.Threeballsarered R and8ballsareblue B .Drawtwoballs,one atatime, withreplacement ."Withreplacement"meansthatyouputtherstballbackintheurn beforeyouselectthesecondball.Thetreediagramusingfrequenciesthatshowallthepossible outcomesfollows. Figure3.1: Total = 64 + 24 + 24 + 9 = 121 Therstsetofbranchesrepresentstherstdraw.Thesecondsetofbranchesrepresentsthesecond draw.Eachoftheoutcomesisdistinct.Infact,wecanlisteachredballas R 1, R 2,and R 3andeach blueballas B 1, B 2, B 3, B 4, B 5, B 6, B 7,and B 8.Thenthe9 RR outcomescanbewrittenas: R 1 R 1; R 1 R 2; R 1 R 3; R 2 R 1; R 2 R 2; R 2 R 3; R 3 R 1; R 3 R 2; R 3 R 3 7 Thiscontentisavailableonlineat.

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106 CHAPTER3.PROBABILITYTOPICS Theotheroutcomesaresimilar. Thereareatotalof11ballsintheurn.Drawtwoballs,oneatatime,andwithreplacement.There are11 11 = 121outcomes,thesizeofthe samplespace Problem1 Listthe24 BR outcomes: B 1 R 1, B 1 R 2, B 1 R 3,... Problem2 Usingthetreediagram,calculate PRR Solution PRR = 3 11 3 11 = 9 121 Problem3 Usingthetreediagram,calculate PRBORBR Solution PRBORBR = 3 11 8 11 + 8 11 3 11 = 48 121 Problem4 Usingthetreediagram,calculate PRon1stdrawANDBon2nddraw Solution PRon1stdrawANDBon2nddraw = PRB = 3 11 8 11 = 24 121 Problem5 Usingthetreediagram,calculate PRon2nddrawgivenBon1stdraw Solution PRon2nddrawgivenBon1stdraw = PRon2nd|Bon1st = 24 88 = 3 11 Thisproblemisaconditional.Thesamplespacehasbeenreducedtothoseoutcomesthatalready haveablueontherstdraw.Thereare24 + 64 = 88possibleoutcomes BR and64 BB Twenty-fourofthe88possibleoutcomesare BR 24 88 = 3 11 Problem6 Usingthetreediagram,calculate PBB Problem7 Usingthetreediagram,calculate PBonthe2nddrawgivenRontherstdraw Example3.16 Anurnhas3redmarblesand8bluemarblesinit.Drawtwomarbles,oneatatime,thistime withoutreplacementfromtheurn. "Withoutreplacement" meansthatyoudonotputtherstball backbeforeyouselectthesecondball.Belowisatreediagram.Thebranchesarelabeledwith probabilitiesinsteadoffrequencies.Thenumbersattheendsofthebranchesarecalculatedby multiplyingthenumbersonthetwocorrespondingbranches,forexample, 3 11 2 10 = 6 110 .

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107 Figure3.2: Total = 56 + 24 + 24 + 6 110 = 110 110 = 1 N OTE :Ifyoudrawaredontherstdrawfromthe3redpossibilities,thereare2redlefttodraw ontheseconddraw.Youdonotputbackorreplacetherstballafteryouhavedrawnit. Calculatethefollowingprobabilitiesusingthetreediagram.Youdraw withoutreplacement ,so thatontheseconddrawthereare10marblesleftintheurn. Problem1 PRR = Solution PRR = 3 11 2 10 = 6 110 Problem2 Fillintheblanks: PRBORBR = 3 11 8 10 + ______ = 48 110 Problem3 PRon2d|Bon1st = Problem4 Fillintheblanks: PRon1standBon2nd = PRB = ______ = 24 110 Problem5 PBB = Problem6 PBon2nd|Ron1st = Solution Thereare6 + 24outcomesthathave R ontherstdraw RR and24 RB .The6andthe24 arefrequencies.Theyarealsothenumeratorsofthefractions 6 110 and 24 110 .Thesamplespaceisno

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108 CHAPTER3.PROBABILITYTOPICS longer110but6 + 24 = 30.Twenty-fourofthe30outcomeshave B ontheseconddraw.The probabilityisthen 24 30 .Didyougetthisanswer? Ifweareusingprobabilities,wecanlabelthetreeinthefollowinggeneralway. PR|R heremeans PRon2nd|Ron1st PB|R heremeans PBon2nd|Ron1st PR|B heremeans PRon2nd|Bon1st PB|B heremeans PBon2nd|Bon1st

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109 3.8SummaryofFormulas 8 Formula3.1: Compliment If A and A 'arecomplementsthen P A + PA' = 1 Formula3.2: AdditionRule PAORB = PA + PB )]TJ/F132 9.9626 Tf 10.131 0 Td [(PAANDB Formula3.3: MutuallyExclusive If A and B aremutuallyexclusivethen PAANDB = 0;so PAORB = PA + PB Formula3.4: MultiplicationRule PAANDB = PBPA|B PAANDB = PAPB|A Formula3.5: Independence If A and B areindependentthen: PA|B = PA PB|A = PB PAANDB = PAPB 8 Thiscontentisavailableonlineat.

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110 CHAPTER3.PROBABILITYTOPICS 3.9Practice1:ContingencyTables 9 3.9.1StudentLearningObjectives Thestudentwillpracticeconstructingandinterpretingcontingencytables. 3.9.2Given Anarticleinthe NewEnglandJournalofMedicine byHaiman,Stram,Wilkens,Pike,etal.,1/26/06 reportedaboutastudyofsmokersinCaliforniaandHawaii.Inonepartofthereport,theself-reported ethnicityandsmokinglevelsperdayweregiven.Ofthepeoplesmokingatmost10cigarettesperday, therewere9886AfricanAmericans,2745NativeHawaiians,12,831Latinos,8378JapaneseAmericans,and 7650Whites.Ofthepeoplesmoking11-20cigarettesperday,therewere6514AfricanAmericans,3062 NativeHawaiians,4932Latinos,10,680JapaneseAmericans,and9877Whites.Ofthepeoplesmoking 21-30cigarettesperday,therewere1671AfricanAmericans,1419NativeHawaiians,1406Latinos,4715 JapaneseAmericans,and6062Whites.Ofthepeoplesmokingatleast31cigarettesperday,therewere759 AfricanAmericans,788NativeHawaiians,800Latinos,2305JapaneseAmericans,and3970Whites. 3.9.3CompletetheTable Completethetablebelowusingthedataprovided. SmokingLevelsbyEthnicity Smoking Level African American Native Hawaiian Latino Japanese Americans White TOTALS 1-10 11-20 21-30 31+ TOTALS 3.9.4AnalyzetheData Supposethatonepersonfromthestudyisrandomlyselected. Exercise3.9.1 Solutiononp.128. Findtheprobabilitythatpersonsmoked11-20cigarettesperday. Exercise3.9.2 Solutiononp.128. FindtheprobabilitythatpersonwasLatino. 9 Thiscontentisavailableonlineat.

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111 3.9.5DiscussionQuestions Exercise3.9.3 Solutiononp.128. Inwords,explainwhatitmeanstopickonepersonfromthestudyandthatpersonisJapanese American AND smokes21-30cigarettesperday.Also,ndtheprobability. Exercise3.9.4 Solutiononp.128. Inwords,explainwhatitmeanstopickonepersonfromthestudyandthatpersonisJapanese American OR smokes21-30cigarettesperday.Also,ndtheprobability. Exercise3.9.5 Solutiononp.128. Inwords,explainwhatitmeanstopickonepersonfromthestudyandthatpersonisJapanese American GIVEN thatpersonsmokes21-30cigarettesperday.Also,ndtheprobability. Exercise3.9.6 Provethatsmokinglevel/dayandethnicityaredependentevents.

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112 CHAPTER3.PROBABILITYTOPICS 3.10Practice2:CalculatingProbabilities 10 3.10.1StudentLearningObjectives Studentswilldenebasicprobabilityterms. Studentswillpracticecalculatingprobabilities. Studentswilldifferentiatebetweenindependentandmutuallyexclusiveevents. N OTE :Useprobabilityrulestosolvetheproblemsbelow.Showyourwork. 3.10.2Given 68%ofCalifornianssupportthedeathpenalty.AmajorityofallracialgroupsinCaliforniasupportthe deathpenalty,exceptforblackCalifornians,ofwhom45%supportthedeathpenalty Source:SanJose MercuryNews,12/2005 .6%ofallCaliforniansareblack Source:U.S.CensusBureau Inthisproblem,let: C =Californianssupportingthedeathpenalty B =BlackCalifornians SupposethatoneCalifornianisrandomlyselected. 3.10.3AnalyzetheData Exercise3.10.1 Solutiononp.128. P C = Exercise3.10.2 Solutiononp.128. P B = Exercise3.10.3 Solutiononp.128. P C j B = Exercise3.10.4 Inwords,whatis" C j B "? Exercise3.10.5 Solutiononp.128. P BANDC = Exercise3.10.6 Inwords,whatis B and C ? Exercise3.10.7 Solutiononp.128. Are B and C independentevents?Showwhyorwhynot. Exercise3.10.8 Solutiononp.128. P BORC = Exercise3.10.9 Inwords,whatis B or C ? Exercise3.10.10 Solutiononp.128. Are B and C mutuallyexclusiveevents?Showwhyorwhynot. 10 Thiscontentisavailableonlineat.

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113 3.11Homework 11 Exercise3.11.1 Solutiononp.129. Supposethatyouhave8cards.5aregreenand3areyellow.The5greencardsarenumbered 1,2,3,4,and5.The3yellowcardsarenumbered1,2,and3.Thecardsarewellshufed.You randomlydrawonecard. G =carddrawnisgreen E =carddrawniseven-numbered a. Listthesamplespace. b. P G = c. P G j E = d. P GANDE = e. P GORE = f. Are G and E mutuallyexclusive?Justifyyouranswernumerically. Exercise3.11.2 Refertothepreviousproblem.Supposethatthistimeyourandomlydrawtwocards,oneata time,and withreplacement G 1 =rstcardisgreen G 2 =secondcardisgreen a. Drawatreediagramofthesituation. b. P G 1 ANDG 2 = c. P atleastonegreen = d. P G 2 j G 1 = e. Are G 2 and G 1 independentevents?Explainwhyorwhynot. Exercise3.11.3 Solutiononp.129. Refertothepreviousproblems.Supposethatthistimeyourandomlydrawtwocards,oneata time,and withoutreplacement G 1 =rstcardisgreen G 2 =secondcardisgreen a. Drawatreediagramofthesituation. b > P G 1 ANDG 2 = c. Patleastonegreen= d. P G 2 j G 1 = e. Are G 2 and G 1 independentevents?Explainwhyorwhynot. Exercise3.11.4 Rolltwofairdice.Eachdiehas6faces. a. Listthesamplespace. b. Let A betheeventthateithera3or4isrolledrst,followedbyanevennumber.Find P A 11 Thiscontentisavailableonlineat.

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115 d. Find PCANDD e. Find PD|C Exercise3.11.9 Solutiononp.129. E and F mutuallyexclusiveevents. P E = 0 4; P F = 0 5.Find P E j F Exercise3.11.10 J and K areindependentevents. PJ|K = 0.3.Find P J Exercise3.11.11 Solutiononp.129. U and V aremutuallyexclusiveevents. P U = 0 .26 ; P V = 0 .37 .Find: a. PUANDV = b. PU|V = c. PUORV = Exercise3.11.12 Q and R areindependentevents. P Q = 0.4; P QANDR = 0.1.Find P R Exercise3.11.13 Solutiononp.129. Y and Z areindependentevents. a. RewritethebasicAdditionRule PYORZ = P Y + P Z )]TJ/F132 9.9626 Tf 10.859 0 Td [(P YANDZ usingthe informationthatYandZareindependentevents. b. Usetherewrittenruletond P Z if P YORZ = 0.71and P Y = 0.42. Exercise3.11.14 G and H aremutuallyexclusiveevents. P G = 0 5; P H = 0 3 a. ExplainwhythefollowingstatementMUSTbefalse: P H j G = 0 4. b. Find: PHORG c. Are G and H independentordependentevents?Explaininacompletesentence. Exercise3.11.15 Solutiononp.129. ThefollowingarerealdatafromSantaClaraCounty,CA.AsofMarch31,2000,therewasatotal of3059documentedcasesofAIDSinthecounty.Theyweregroupedintothefollowingcategories Source:SantaClaraCountyPublicH.D. : Homosexual/Bisexual IVDrugUser* HeterosexualContact Other Totals Female 0 70 136 49 ____ Male 2146 463 60 135 ____ Totals ____ ____ ____ ____ ____ *includeshomosexual/bisexualIVdrugusers SupposeoneofthepersonswithAIDSinSantaClaraCountyisrandomlyselected.Computethe following: a. Ppersonisfemale = b. PpersonhasariskfactorHeterosexualContact = c. PpersonisfemaleORhasariskfactorofIVDrugUser = d. PpersonisfemaleANDhasariskfactorofHomosexual/Bisexual = e. PpersonismaleANDhasariskfactorofIVDrugUser = f. PfemaleGIVENpersongotthediseasefromheterosexualcontact =

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116 CHAPTER3.PROBABILITYTOPICS g. ConstructaVennDiagram.Makeonegroupfemalesandtheothergroupheterosexual contact. Exercise3.11.16 Solvethesequestionsusingprobabilityrules.DoNOTusethecontingencytableabove.3059 casesofAIDShadbeenreportedinSantaClaraCounty,CA,throughMarch31,2000.Thosecases willbeourpopulation.Ofthosecases,6.4%obtainedthediseasethroughheterosexualcontact and7.4%arefemale.Outofthefemaleswiththedisease,53.3%gotthediseasefromheterosexual contact. a. Ppersonisfemale= b. Ppersonobtainedthediseasethroughheterosexualcontact= c. PfemaleGIVENpersongotthediseasefromheterosexualcontact= d. ConstructaVennDiagram.Makeonegroupfemalesandtheothergroupheterosexual contact.Fillinallvaluesasprobabilities. Exercise3.11.17 Solutiononp.130. Thefollowingtableidentiesagroupofchildrenbyoneoffourhaircolors,andbytypeofhair. HairType Brown Blond Black Red Totals Wavy 20 15 3 43 Straight 80 15 12 Totals 20 215 a. Completethetableabove. b. Whatistheprobabilitythatarandomlyselectedchildwillhavewavyhair? c. Whatistheprobabilitythatarandomlyselectedchildwillhaveeitherbrownorblond hair? d. Whatistheprobabilitythatarandomlyselectedchildwillhavewavybrownhair? e. Whatistheprobabilitythatarandomlyselectedchildwillhaveredhair,giventhathehas straighthair? f. IfBistheeventofachildhavingbrownhair,ndtheprobabilityofthecomplementofB. g. Inwords,whatdoesthecomplementofBrepresent? Exercise3.11.18 Apreviousyear,theweightsofthemembersofthe SanFrancisco49ers andthe DallasCowboys werepublishedinthe SanJoseMercuryNews .Thefactualdataarecompiledintothefollowing table. Shirt# 210 211-250 251-290 290 1-33 21 5 0 0 34-66 6 18 7 4 66-99 6 12 22 5 Forthefollowing,supposethatyourandomlyselectoneplayerfromthe49ersorCowboys. a. Findtheprobabilitythathisshirtnumberisfrom1to33.

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117 b. Findtheprobabilitythatheweighsatmost210pounds. c. Findtheprobabilitythathisshirtnumberisfrom1to33ANDheweighsatmost210 pounds. d. Findtheprobabilitythathisshirtnumberisfrom1to33ORheweighsatmost210 pounds. e. Findtheprobabilitythathisshirtnumberisfrom1to33GIVENthatheweighsatmost 210pounds. f. Ifhavingashirtnumberfrom1to33andweighingatmost210poundswereindependent events,thenwhatshouldbetrueabout PShirt1-33| 210pounds ? Exercise3.11.19 Solutiononp.130. Approximately249,000,000peopleliveintheUnitedStates.Ofthesepeople,31,800,000speak alanguageotherthanEnglishathome.Ofthosewhospeakanotherlanguageathome,over50 percentspeakSpanish. Source:U.S.BureauoftheCensus,1990Census Let: E =speakEnglishathome; E '=speakanotherlanguageathome; S =speakSpanishathome Finisheachprobabilitystatementbymatchingthecorrectanswer. ProbabilityStatements Answers a.PE'= i.0.8723 b.PE= ii. > 0.50 c.PS= iii.0.1277 d.PS|E'= iv. > 0.0639 Exercise3.11.20 Theprobabilitythatamaledevelopssomeformofcancerinhislifetimeis0.4567Source:AmericanCancerSociety.Theprobabilitythatamalehasatleastonefalsepositivetestresultmeaning thetestcomesbackforcancerwhenthemandoesnothaveitis0.51Source:USAToday.Someof thequestionsbelowdonothaveenoughinformationforyoutoanswerthem.Writenotenough informationforthoseanswers. Let: C =amandevelopscancerinhislifetime; P =manhasatleastonefalsepositive a. Constructatreediagramofthesituation. b. P C = c. P P j C = d. P P j C = e. Ifatestcomesuppositive,baseduponnumericalvalues,canyouassumethatmanhas cancer?Justifynumericallyandexplainwhyorwhynot. Exercise3.11.21 Solutiononp.130. In1994,theU.S.governmentheldalotterytoissue55,000GreenCardspermitsfornon-citizens toworklegallyintheU.S..RenateDeutsch,fromGermany,wasoneofapproximately6.5million peoplewhoenteredthislottery.Let G=wonGreenCard a. WhatwasRenate'schanceofwinningaGreenCard?Writeyouranswerasaprobability statement.

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118 CHAPTER3.PROBABILITYTOPICS b. Inthesummerof1994,Renatereceivedaletterstatingshewasoneof110,000nalists chosen.Oncethenalistswerechosen,assumingthateachnalisthadanequalchance towin,whatwasRenate'schanceofwinningaGreenCard?Let F=wasanalist .Write youranswerasaconditionalprobabilitystatement. c. Are G and F independentordependentevents?Justifyyouranswernumericallyandalso explainwhy. d. Are G and F mutuallyexclusiveevents?Justifyyouranswernumericallyandalsoexplain why. N OTE :P.S.Amazingly,on2/1/95,RenatelearnedthatshewouldreceiveherGreenCardtrue story! Exercise3.11.22 ThreeprofessorsatGeorgeWashingtonUniversitydidanexperimenttodetermineifeconomists aremoreselshthanotherpeople.Theydropped64stamped,addressedenvelopeswith\$10cash indifferentclassroomsontheGeorgeWashingtoncampus.44%werereturnedoverall.Fromthe economicsclasses56%oftheenvelopeswerereturned.Fromthebusiness,psychology,andhistory classes31%werereturned. Source:WallStreetJournal Let: R =moneyreturned; E =economicsclasses; O =otherclasses a. Writeaprobabilitystatementfortheoverallpercentofmoneyreturned. b. Writeaprobabilitystatementforthepercentofmoneyreturnedoutoftheeconomics classes. c. Writeaprobabilitystatementforthepercentofmoneyreturnedoutoftheotherclasses. d. Ismoneybeingreturnedindependentoftheclass?Justifyyouranswernumericallyand explainit. e. Baseduponthisstudy,doyouthinkthateconomistsaremoreselshthanotherpeople? Explainwhyorwhynot.Includenumberstojustifyyouranswer. Exercise3.11.23 Solutiononp.130. ThechartbelowgivesthenumberofsuicidesestimatedintheU.S.forarecentyearbyage,race blackandwhite,andsex.Weareinterestedinpossiblerelationshipsbetweenage,race,andsex. Wewillletsuicidevictimsbeourpopulation. Source:TheNationalCenterforHealthStatistics, U.S.Dept.ofHealthandHumanServices RaceandSex 1-14 15-24 25-64 over64 TOTALS white,male 210 3360 13,610 22,050 white,female 80 580 3380 4930 black,male 10 460 1060 1670 black,female 0 40 270 330 allothers TOTALS 310 4650 18,780 29,760 N OTE :Donotinclude"allothers"forpartsf,g,andi. a. Fillinthecolumnforthesuicidesforindividualsoverage64. b. Fillintherowforallotherraces. c. Findtheprobabilitythatarandomlyselectedindividualwasawhitemale.

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119 d. Findtheprobabilitythatarandomlyselectedindividualwasablackfemale. e. Findtheprobabilitythatarandomlyselectedindividualwasblack f. ComparingRaceandSextoAge,whichtwogroupsaremutuallyexclusive?Howdo youknow? g. Findtheprobabilitythatarandomlyselectedindividualwasmale. h. Outoftheindividualsoverage64,ndtheprobabilitythatarandomlyselectedindividualwasablackorwhitemale. i. Arebeingmaleandcommittingsuicideoverage64independentevents?Howdoyou know? Thenexttwoquestionsrefertothefollowing: ThepercentoflicensedU.S.driversfromarecentyearthat arefemaleis48.60.Ofthefemales,5.03%areage19andunder;81.36%areage20-64;13.61%areage65or over.OfthelicensedU.S.maledrivers,5.04%areage19andunder;81.43%areage20-64;13.53%areage 65orover.Source:FederalHighwayAdministration,U.S.Dept.ofTransportation Exercise3.11.24 Completethefollowing: a. Constructatableoratreediagramofthesituation. b. Pdriverisfemale = c. Pdriverisage65orover|driverisfemale = d. Pdriverisage65oroverANDfemale = e. Inwords,explainthedifferencebetweentheprobabilitiesinpartcandpartd. f. Pdriverisage65orover = g. Arebeingage65oroverandbeingfemalemutuallyexclusiveevents?Howdoyouknow Exercise3.11.25 Solutiononp.130. Supposethat10,000U.S.licenseddriversarerandomlyselected. a. Howmanywouldyouexpecttobemale? b. Usingthetableortreediagramfromthepreviousexercise,constructacontingencytable ofgenderversusagegroup. c. Usingthecontingencytable,ndtheprobabilitythatoutoftheage20-64group,arandomlyselecteddriverisfemale. Exercise3.11.26 Approximately86.5%ofAmericanscommutetoworkbycar,truckorvan.Outofthatgroup, 84.6%drivealoneand15.4%driveinacarpool.Approximately3.9%walktoworkandapproximately5.3%takepublictransportation. Source:BureauoftheCensus,U.S.Dept.ofCommerce. Disregardroundingapproximations. a. Constructatableoratreediagramofthesituation.Includeabranchforallothermodes oftransportationtowork. b. Assumingthatthewalkerswalkalone,whatpercentofallcommuterstravelaloneto work? c. Supposethat1000workersarerandomlyselected.Howmanywouldyouexpecttotravel alonetowork? d. Supposethat1000workersarerandomlyselected.Howmanywouldyouexpecttodrive inacarpool?

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120 CHAPTER3.PROBABILITYTOPICS Exercise3.11.27 Explainwhatiswrongwiththefollowingstatements.Usecompletesentences. a. Ifthere'sa60%chanceofrainonSaturdayanda70%chanceofrainonSunday,then there'sa130%chanceofrainovertheweekend. b. Theprobabilitythatabaseballplayerhitsahomerunisgreaterthantheprobabilitythat hegetsasuccessfulhit. 3.11.1Trythesemultiplechoicequestions. Thenexttwoquestionsrefertothefollowingprobabilitytreediagram whichshowstossinganunfaircoin FOLLOWEDBY drawingonebeadfromacupcontaining3red R ,4yellow Y and5blue B beads.For thecoin, P H = 2 3 and P T = 1 3 where H="heads" and T="tails Figure3.3 Exercise3.11.28 Solutiononp.130. Find PtossingaHeadonthecoinANDaRedbead A. 2 3 B. 5 15 C. 6 36 D. 5 36 Exercise3.11.29 Solutiononp.130. Find PBluebead A. 15 36 B. 10 36 C. 10 12 D. 6 36 Thenextthreequestionsrefertothefollowingtable ofdataobtainedfrom www.baseball-almanac.com 12 12 http://cnx.org/content/m16836/latest/www.baseball-almanac.com

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122 CHAPTER3.PROBABILITYTOPICS Exercise3.12.2 Solutiononp.131. AlloftheNASDAQstocksarethe: A. Population B. Statistic C. Parameter D. Sample E. Variable Exercise3.12.3 Solutiononp.131. 9%isthe: A. Population B. Statistic C. Parameter D. Sample E. Variable Exercise3.12.4 Solutiononp.131. The100NASDAQstocksinthesurveyarethe: A. Population B. Statistic C. Parameter D. Sample E. Variable Exercise3.12.5 Solutiononp.131. Thepercentincreaseforonestockinthesurveyisthe: A. Population B. Statistic C. Parameter D. Sample E. Variable Exercise3.12.6 Solutiononp.131. Wouldthedatacollectedbequalitative,quantitativediscrete,orquantitativecontinuous? Thenexttwoquestionsrefertothefollowingstudy: ThirtypeoplespenttwoweeksaroundMardiGrasin NewOrleans.Theirtwo-weekweightgainisbelow.Note:alossisshownbyanegativeweightgain. WeightGain Frequency -2 3 -1 5 0 2 1 4 4 13 6 2 11 1

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123 Exercise3.12.7 Solutiononp.131. Calculatethefollowingvalues: a. Theaverageweightgainforthetwoweeks b. Thestandarddeviation c. Therst,second,andthirdquartiles Exercise3.12.8 Constructahistogramandaboxplotofthedata.

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124 CHAPTER3.PROBABILITYTOPICS 3.13Lab:ProbabilityTopics 14 Classtime: Names: 3.13.1StudentLearningOutcomes: Thestudentwillusetheoreticalandempiricalmethodstoestimateprobabilities. Thestudentwillappraisethedifferencesbetweenthetwoestimates. Thestudentwilldemonstrateanunderstandingoflong-termrelativefrequencies. 3.13.2DotheExperiment: Countout25or40mixed-colorM&M'ssmallbag'sworth.Recordthenumberofeachcolorinthe "Population"table.Usetheinformationfromthistabletocompletethetheoreticalprobabilityquestions. Next,puttheM&M'sinacup.Theexperimentistopick2M&M's,oneatatime.Do not lookatthemasyou pickthem.Thersttimethrough,replacetherstM&Mbeforepickingthesecondone.Recordtheresults intheWithReplacementcolumnoftheempiricaltable.Dothis24times.Thesecondtimethrough,after pickingtherstM&M,do not replaceitbeforepickingthesecondone.Then,pickthesecondone.Record theresultsintheWithoutReplacementcolumnsectionofthe"EmpiricalResults"table.Afteryourecord thepick,put both M&M'sback.Dothisatotalof24times,also.Usethedatafromthe"EmpiricalResults" tabletocalculatetheempiricalprobabilityquestions.Leaveyouranswersinunreducedfractionalform. Do not multiplyoutanyfractions. Population Color Quantity YellowY GreenG BlueBL BrownB OrangeO RedR TheoreticalProbabilities WithReplacement WithoutReplacement P 2reds P R 1 B 2 ORB 1 R 2 P R 1 ANDG 2 P G 2 |R 1 P noyellows P doubles P nodoubles 14 Thiscontentisavailableonlineat.

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125 Note: G 2 =greenonsecondpick; R 1 =redonrstpick;doubles=bothpicksarethesamecolour. EmpiricalResults WithReplacement WithoutReplacement __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ __,____,__ EmpiricalProbabilities WithReplacement WithoutReplacement P 2reds P R 1 B 2 ORB 1 R 2 P R 1 ANDG 2 P G 2 |R 1 P noyellows P doubles P nodoubles Note: 3.13.3DiscussionQuestions 1.WhyaretheWithReplacementandWithoutReplacementprobabilitiesdifferent? 2.Convert Pnoyellows todecimalformatforbothTheoreticalWithReplacementandforEmpirical WithReplacement.Roundto4decimalplaces. a. TheoreticalWithReplacement: Pnoyellows = b. EmpiricalWithReplacement: Pnoyellows = c. Arethedecimalvaluesclose?Didyouexpectthemtobeclosertogetherorfartherapart? Why? 3.Ifyouincreasedthenumberoftimesyoupicked2M&M'sto240times,whywouldempiricalprobabilityvalueschange?

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126 CHAPTER3.PROBABILITYTOPICS 4.Wouldthischangeseeabovecausetheempiricalprobabilitiesandtheoreticalprobabilitiestobe closertogetherorfartherapart?Howdoyouknow? 5.Explainthedifferencesinwhat P G 1 ANDR 2 and P R 1 |G 2 represent.

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127 SolutionstoExercisesinChapter3 Example3.2p.96 No. C = f 3,5 g and E = f 1,2,3,4 g P CANDE = 1 6 .Tobemutuallyexclusive, P CANDE mustbe0. Example3.10,Problem1p.102 HikingAreaPreference Sex TheCoastline NearLakesandStreams OnMountainPeaks Total Female 18 16 11 45 Male 16 25 14 55 Total 34 41 25 100 Example3.10,Problem2p.102 a. PFANDC = 18 100 = 0.18 b. P F P C = 45 100 45 100 = 0.45 0.45 = 0.153 PFANDC 6 = P F P C ,sotheevents F and C arenotindependent. Example3.10,Problem3p.102 a. Theword'given'tellsyouthatthisisaconditional. b. PM|L = 25 41 c. No,thesamplespaceforthisproblemis41. Example3.10,Problem4p.102 a. PF = 45 100 b. PP = 25 100 c. PFANDP = 11 100 d. PFORP = 45 100 + 25 100 )]TJ/F58 7.5716 Tf 13.22 3.925 Td [(11 100 = 59 100 Example3.11,Problem1p.103 DoorChoice CaughtorNot DoorOne DoorTwo DoorThree Total Caught 1 15 1 12 1 6 19 60 NotCaught 4 15 3 12 1 6 31 60 Total 5 15 4 12 2 6 1 Example3.15,Problem1p.106 B 1 R 1; B 1 R 2; B 1 R 3; B 2 R 1; B 2 R 2; B 2 R 3; B 3 R 1; B 3 R 2; B 3 R 3; B 4 R 1; B 4 R 2; B 4 R 3; B 5 R 1; B 5 R 2; B 5 R 3; B 6 R 1; B 6 R 2; B 6 R 3; B 7 R 1; B 7 R 2; B 7 R 3; B 8 R 1; B 8 R 2; B 8 R 3 Example3.15,Problem6p.106 PBB = 64 121

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128 CHAPTER3.PROBABILITYTOPICS Example3.15,Problem7p.106 PBon2nddraw|Ron1stdraw = 8 11 Thereare9 + 24outcomesthathave R ontherstdraw RR and24 RB .Thesamplespaceisthen 9 + 24 = 33.Twenty-fourofthe33outcomeshave B ontheseconddraw.Theprobabilityisthen 24 33 Example3.16,Problem2p.107 PRBorBR = 3 11 8 10 + )]TJ/F58 7.5716 Tf 7.776 -4.149 Td [(8 11 )]TJ/F58 7.5716 Tf 14.248 -4.149 Td [(3 10 = 48 110 Example3.16,Problem3p.107 PRon2d|Bon1st = 3 10 Example3.16,Problem4p.107 PRon1standBon2nd = PRB = )]TJ/F58 7.5716 Tf 7.776 -4.149 Td [(3 11 )]TJ/F58 7.5716 Tf 14.248 -4.149 Td [(8 10 = 24 110 Example3.16,Problem5p.107 PBB = 8 11 7 10 SolutionstoPractice1:ContingencyTables SolutiontoExercise3.9.1p.110 35,065 100,450 SolutiontoExercise3.9.2p.110 19,969 100,450 SolutiontoExercise3.9.3p.111 4,715 100,450 SolutiontoExercise3.9.4p.111 36,636 100,450 SolutiontoExercise3.9.5p.111 4715 15,273 SolutionstoPractice2:CalculatingProbabilities SolutiontoExercise3.10.1p.112 0.68 SolutiontoExercise3.10.2p.112 0.06 SolutiontoExercise3.10.3p.112 0.45 SolutiontoExercise3.10.5p.112 0.027 SolutiontoExercise3.10.7p.112 No SolutiontoExercise3.10.8p.112 0.713 SolutiontoExercise3.10.10p.112 No

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129 SolutionstoHomework SolutiontoExercise3.11.1p.113 a. f G 1, G 2, G 3, G 4, G 5, Y 1, Y 2, Y 3 g b. 5 8 c. 2 3 d. 2 8 e. 6 8 f. No SolutiontoExercise3.11.3p.113 b. )]TJ/F58 7.5716 Tf 5.883 -4.149 Td [(5 8 4 7 c. )]TJ/F58 7.5716 Tf 5.883 -4.149 Td [(5 8 )]TJ/F58 7.5716 Tf 12.355 -4.149 Td [(3 7 + )]TJ/F58 7.5716 Tf 5.883 -4.149 Td [(3 8 )]TJ/F58 7.5716 Tf 12.355 -4.149 Td [(5 7 + )]TJ/F58 7.5716 Tf 5.883 -4.149 Td [(5 8 4 7 d. 4 7 e. No SolutiontoExercise3.11.5p.114 a. f GH GT BH BT RH RT g b. 3 20 c. Yes d. No SolutiontoExercise3.11.7p.114 a. f HHH HHT HTH HTT THH THT TTH TTT g b. 4 8 c. Yes SolutiontoExercise3.11.9p.115 0 SolutiontoExercise3.11.11p.115 a. 0 b. 0 c. 0.63 SolutiontoExercise3.11.13p.115 b. 0.5 SolutiontoExercise3.11.15p.115 Thecompletedcontingencytableisasfollows: Homosexual/Bisexual IVDrugUser* HeterosexualContact Other Totals Female 0 70 136 49 255 Male 2146 463 60 135 2804 Totals 2146 533 196 174 3059 *includeshomosexual/bisexualIVdrugusers

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130 CHAPTER3.PROBABILITYTOPICS a. 255 3059 b. 196 3059 c. 718 3059 d. 0 e. 463 3059 f. 136 196 SolutiontoExercise3.11.17p.116 b. 43 215 c. 120 215 d. 20 215 e. 12 172 f. 115 215 SolutiontoExercise3.11.19p.117 a. iii b. i c. iv d. ii SolutiontoExercise3.11.21p.117 a. P G = 0 .008 b. 0.5 c. dependent d. No SolutiontoExercise3.11.23p.118 c. 22050 29760 d. 330 29760 e. 2000 29760 f. 23720 29760 g. 5010 6020 h. Blackfemalesandages1-14 i. No SolutiontoExercise3.11.25p.119 a. 5140 c. 0.49 SolutiontoExercise3.11.28p.120 C SolutiontoExercise3.11.29p.120 A SolutiontoExercise3.11.30p.121 B SolutiontoExercise3.11.31p.121 B SolutiontoExercise3.11.32p.121 C

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131 SolutionstoReview SolutiontoExercise3.12.1p.121 C. Parameter SolutiontoExercise3.12.2p.122 A. Population SolutiontoExercise3.12.3p.122 B. Statistic SolutiontoExercise3.12.4p.122 D. Sample SolutiontoExercise3.12.5p.122 E. Variable SolutiontoExercise3.12.6p.122 quantitative-continuous SolutiontoExercise3.12.7p.123 a. 2.27 b. 3.04 c. -1,4,4

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132 CHAPTER3.PROBABILITYTOPICS

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Chapter4 DiscreteRandomVariables 4.1DiscreteRandomVariables 1 4.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Recognizeandunderstanddiscreteprobabilitydistributionfunctions,ingeneral. Calculateandinterpretexpectedvalue. Recognizethebinomialprobabilitydistributionandapplyitappropriately. RecognizethePoissonprobabilitydistributionandapplyitappropriatelyoptional. Recognizethegeometricprobabilitydistributionandapplyitappropriatelyoptional. Recognizethehypergeometricprobabilitydistributionandapplyitappropriatelyoptional. Classifydiscretewordproblemsbytheirdistributions. 4.1.2Introduction Astudenttakesa10questiontrue-falsequiz.Becausethestudenthadsuchabusyschedule,heorshe couldnotstudyandrandomlyguessesateachanswer.Whatistheprobabilityofthestudentpassingthe testwithatleasta70%? Smallcompaniesmightbeinterestedinthenumberoflongdistancephonecallstheiremployeesmake duringthepeaktimeoftheday.Supposetheaverageis20calls.Whatistheprobabilitythattheemployees makemorethan20longdistancephonecallsduringthepeaktime? Thesetwoexamplesillustratetwodifferenttypesofprobabilityproblemsinvolvingdiscreterandomvariables.Recallthatdiscretedataisdatathatyoucancount.A randomvariable describestheoutcomesofa statisticalexperimentbothinwordsandnumerically.Thevaluesofarandomvariablecanvarywitheach repetitionofanexperiment. Inthischapter,youwillstudyprobabilityproblemsinvolvingdiscreterandomdistributions.Youwillalso studylong-termaveragesassociatedwiththem. 4.1.3RandomVariableNotation Uppercaseletterslike X or Y denotearandomvariable.Lowercaseletterslike x or y denotethevalueofa randomvariable.If X isarandomvariable,then X isdenedinwords. 1 Thiscontentisavailableonlineat. 133

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134 CHAPTER4.DISCRETERANDOMVARIABLES Forexample,let X =thenumberofheadsyougetwhenyoutossthreefaircoins.Thesamplespaceforthe tossofthreefaircoinsis TTT ; THH ; HTH ; HHT ; HTT ; THT ; TTH ; HHH .Then, x =0,1,2,3. X isin wordsand x isanumber.Noticethatforthisexample,the x valuesarecountableoutcomes.Becauseyou cancountthepossiblevaluesthat X cantakeonthe x values0,1,2,3, X isadiscreterandomvariable. 4.1.4OptionalCollaborativeClassroomActivity Tossacoin10timesandrecordthenumberofheads.Afterallmembersoftheclasshavecompletedthe experimenttossedacoin10timesandcountedthenumberofheads,llinthechartusingaheadinglike theonebelow.Let X =thenumberofheadsin10tossesofthecoin. X Frequencyof X RelativeFrequencyof X Whichvaluesof X occurredmostfrequently? Ifyoutossedthecoin1,000times,whatvalueswould X takeon?Whichvaluesof X doyouthink wouldoccurmostfrequently? Whatdoestherelativefrequencycolumnsumto? 4.2ProbabilityDistributionFunctionPDFforaDiscreteRandom Variable 2 Adiscrete probabilitydistributionfunction hastwocharacteristics: Eachprobabilityisbetween0and1,inclusive. Thesumoftheprobabilitiesis1. Example4.1 Achildpsychologistisinterestedinthenumberoftimesanewbornbaby'scryingwakesitsmother aftermidnight.Forarandomsampleof50mothers,thefollowinginformationwasobtained.Let X =thenumberoftimesanewbornwakesitsmotheraftermidnight.Forthisexample, x =0,1,2, 3,4,5. PX =probabilitythat X takesonavalue x 2 Thiscontentisavailableonlineat.

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135 X Px 0 PX=0 = 2 50 1 PX=1 = 11 50 2 PX=2 = 23 50 3 PX=3 = 9 50 4 PX=4 = 4 50 5 PX=5 = 1 50 X takesonthevalues0,1,2,3,4,5.Thisisadiscrete PDF because 1.Each PX isbetween0and1,inclusive. 2.Thesumoftheprobabilitiesis1,thatis, 2 50 + 11 50 + 23 50 + 9 50 + 4 50 + 1 50 = 1.1 Example4.2 SupposeNancyhasclasses 3days aweek.Sheattendsclasses3daysaweek 80% ofthetime, 2 days15% ofthetime, 1day4% ofthetime,and nodays1% ofthetime. Problem1 Let X =thenumberofdaysNancy____________________. Problem2 X takesonwhatvalues? Problem3 Constructaprobabilitydistributiontablecalleda PDF tableliketheoneinthepreviousexample. Thetableshouldhavetwocolumnslabeled X and PX .Whatdoesthe PX columnsumto? 4.3MeanorExpectedValueandStandardDeviation 3 The expectedvalue isoftenreferredtoasthe "long-term"averageormean .Thismeansthatoverthelong termofdoinganexperimentoverandover,youwould expect thisaverageeverytimeyouperforma particularexperiment. The mean ofarandomvariable X is m .Ifwedoanexperimentmanytimesforinstance,ipafaircoin,as KarlPearsondid,24,000timesandlet X =thenumberofheadsandrecordthevalueof X eachtime,the averagegetscloserandclosertomaswekeeprepeatingtheexperiment.Thisisknownasthe LawofLarge Numbers N OTE :Tondtheexpectedvalueorlongtermaverage, m ,simplymultiplyeachvalueofthe randomvariablebyitsprobabilityandaddtheproducts. 3 Thiscontentisavailableonlineat.

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136 CHAPTER4.DISCRETERANDOMVARIABLES AStep-by-StepExample Amen'ssoccerteamplayssoccer0,1,or2daysaweek.Theprobabilitythattheyplay0daysis0.2,the probabilitythattheyplay1dayis0.5,andtheprobabilitythattheyplay2daysis0.3.Findthelong-term average, m ,orexpectedvalueofthedaysperweekthemen'ssoccerteamplayssoccer. Todotheproblem,rstlettherandomvariable X =thenumberofdaysthemen'ssoccerteamplayssoccer perweek. X takesonthevalues0,1,2.Constructa PDF table,addingacolumn xP x .Inthiscolumn, youwillmultiplyeach X valuebyitsprobability. ExpectedValueTable X Px or PX=x xPx 0 0.2 .2=0 1 0.5 .5=0.5 2 0.3 .3=0.6 Thistableiscalledanexpectedvaluetable.Thetablehelpsyoucalculatetheexpectedvalueorlong-term average, m ,ofaprobabilitydistribution. Addthelastcolumntondthelongtermaverageorexpectedvalue: 0 0.2 + 1 0.5 + 2 0.3 = 0 + 0.5.0.6 = 1.1. Theexpectedvalueis1.1.Themen'ssoccerteamwould,ontheaverage,expecttoplaysoccer1.1days perweek.Thenumber1.1isthelongtermaverageorexpectedvalueifthemen'ssoccerteamplayssoccer weekafterweekafterweek.Wesay m = 1.1 Example4.3 Findtheexpectedvaluetheexpectednumberoftimesanewbornwakesitsmotheraftermidnightforexample4-1. X Px or PX=x xPx 0 PX=0 = 2 50 )]TJ/F58 7.5716 Tf 7.775 -4.149 Td [(2 50 =0 1 PX=1 = 11 50 11 50 = 11 50 2 PX=2 = 23 50 )]TJ/F58 7.5716 Tf 5.882 -4.15 Td [(23 50 = 46 50 3 PX=3 = 9 50 )]TJ/F58 7.5716 Tf 7.775 -4.15 Td [(9 50 = 27 50 4 PX=4 = 4 50 4 50 = 16 50 5 PX=5 = 1 50 1 50 = 5 50 The expectednumberoftimes anewbornwakesitsmotheraftermidnightis2.1times.Youexpect anewborntowakeitsmotheraftermidnight2.1times,onanaveragenight. N OTE : Addthelastcolumntondtheexpectedvalue. m =ExpectedValue= 105 50 = 2.1 Problem GobackandcalculatetheexpectedvalueforthenumberofdaysNancyattendsclassesaweek. Constructthethirdcolumntodoso. Solution 2.74daysaweek.

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137 Example4.4 Supposeyouplayagameofchanceinwhichyouchoose5numbersfrom0,1,2,3,4,5,6,7,8, 9.Youmaychooseanumbermorethanonce.Youpay\$2toplayandcouldprot\$100,000ifyou matchall5numbersinorderyougetyour\$2backplus\$100,000.Overthelongterm,whatis your expected protifyoumatchall5numbersinorder? Todothisproblem,setupanexpectedvaluetablefortheamountofmoneyyoucanprot. Let X =theamountofmoneyyouprot.Thevaluesof x arenot0,1,2,3,4,5,6,7,8,9.Sinceyou areinterestedinyourprotorloss,thevaluesof x are100,000dollarsand-2dollars. Towin,youmustgetall5numberscorrect,inorder.Theprobabilityofchoosingonecorrect numberis 1 10 becausethereare10numbers.Youmaychooseanumbermorethanonce.The probabilityofchoosingall5numberscorrectlyandinorderis: 1 10 1 10 1 10 1 10 1 10 = 1 10 )]TJ/F58 7.5716 Tf 6.227 0 Td [(5 = 0.00001.2 Therefore,theprobabilityofwinningis0.00001andtheprobabilityoflosingis 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.00001 = 0.99999.3 Theexpectedvaluetableisasfollows. X PX XPX Loss -2 0.99999 -2.99999=-1.99998 Prot 100,000 0.00001 .00001=1 Addthelastcolumn: m =ExpectedValue= )]TJ/F58 9.9626 Tf 8.195 0 Td [(1.99998+1= )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.99998 Youwould,ontheaverage,expecttoloseapproximatelyonedollarforeachgameyouplay.However,eachtimeyouplay,youeitherlose\$2orprot\$100,000.The\$1istheaverageorexpected LOSSpergameafterplayingthisgameoverandover. Example4.5 Supposeyouplayagamewithabiasedcoin.Youplayeachgamebytossingthecoinonce. Pheads = 2 3 and Ptails = 1 3 .Ifyoutossahead,youpay\$6.Ifyoutossatail,youwin\$10. Ifyouplaythisgamemanytimes,willyoucomeoutahead? Problem1 Denearandomvariable X Problem2 Completethefollowingexpectedvaluetable. X ____ ____ WIN 10 1 3 ____ LOSE ____ ____ )]TJ/F58 7.5716 Tf 6.228 0 Td [(12 3 Problem3 Whatistheexpectedvalue, m ?Doyoucomeoutahead?

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138 CHAPTER4.DISCRETERANDOMVARIABLES Likedata,probabilitydistributionshavestandarddeviations.Tocalculatethestandarddeviation, s ,ofa probabilitydistribution,ndeachdeviation,squareit,multiplyitbyitsprobability,andaddtheproducts. Tounderstandhowtodothecalculation,lookatthenumberofdaysperweekamen'ssoccerteamplays soccertableagain.Addthecolumn x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m 2 P x X Px or PX=x xPx xm 2 Px 0 0.2 .2=0 0 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1.1 2 .2 = 0.242 1 0.5 .5=0.5 1 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1.1 2 .5 = 0.005 2 0.3 .3=0.6 2 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1.1 2 .3 = 0.243 m =1.1Thesumofthelastcolumnisthevariance, s 2 s 2 =0.490Thestandarddeviationis s s = p 0.490= 0.7 Generallyforprobabilitydistributions,weuseacalculatororacomputertocalculate m and s toreduce roundofferror.Forsomeofthecommonprobabilitydistributions,thereareshort-cutformulasthatcalculate m and s 4.4CommonDiscreteProbabilityDistributionFunctions 4 Someofthemorecommondiscreteprobabilityfunctionsarebinomial,geometric,hypergeometric,and Poisson.Mostelementarycoursesdonotcoverthegeometric,hypergeometric,andPoisson.Yourinstructorwillletyouknowifheorshewishestocoverthesedistributions. Aprobabilitydistributionfunctionisapattern.Youtrytotaprobabilityproblemintoa pattern ordistributioninordertoperformthenecessarycalculations.Thesedistributionsaretoolstomakesolvingprobabilityproblemseasier.Eachdistributionhasitsownspecialcharacteristics.Learningthecharacteristics enablesyoutodistinguishamongthedifferentdistributions. 4.5Binomial 5 Thecharacteristicsofabinomialexperimentare: 1.Thereareaxednumberoftrials.Thinkoftrialsasrepetitionsofanexperiment.Theletter n denotes thenumberoftrials.The n trialsareindependentandarerepeatedusingidenticalconditions.Because the n trialsareindependent,theoutcomeofonetrialdoesnotaffecttheoutcomeofanyothertrial. 2.Thereareonly2possibleoutcomes,called"success"and,"failure"foreachtrial.Theletter p denotes theprobabilityofasuccessononetrialand q denotestheprobabilityofafailureononetrial. p + q = 1. 3.Foreachindividualtrial,theprobability, p ,ofasuccessandprobability, q ,ofafailureremainthe same.Forexample,randomlyguessingatatrue-falsestatisticsquestionhasonlytwooutcomes. Ifasuccessisguessingcorrectly,thenafailureisguessingincorrectly.SupposeJoealwaysguesses correctlyonanystatisticstrue-falsequestionwithprobability p = 0.6.Then, q = 0.4.Thismeans thatforeverytrue-falsestatisticsquestionJoeanswers,hisprobabilityofsuccess p = 0.6andhis probabilityoffailure q = 0.4remainthesame. Theoutcomesofabinomialexperimentta binomialprobabilitydistribution .Themean, m ,andvariance, s 2 ,forthebinomialprobabilitydistributionis m = np and s 2 = npq .Thestandarddeviation, s ,isthen s = p npq 4 Thiscontentisavailableonlineat. 5 Thiscontentisavailableonlineat.

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139 Anyexperimentthathascharacteristics2and3iscalleda BernoulliTrial namedafterJacobBernoulli who,inthelate1600s,studiedthemextensively.Abinomialexperimenttakesplacewhenthenumberof successesarecountedinoneormoreBernoulliTrials. Example4.6 AtABCCollege,thewithdrawalratefromanelementaryphysicscourseis30%foranygiven term.Thisimpliesthat,foranygiventerm,70%ofthestudentsstayintheclassfortheentire term.A"success"couldbedenedasanindividualwhowithdrew.Therandomvariableis X = thenumberofstudentswhowithdrawfromtheelementaryphysicscourseperterm. Example4.7 Supposeyouplayagamethatyoucanonlyeitherwinorlose.Theprobabilitythatyouwinany gameis55%andtheprobabilitythatyouloseis45%.Ifyouplaythegame20times,whatisthe probabilitythatyouwin15ofthe20games?Here,ifyoudene X =thenumberofwins,then X takesonthevalues X =0,1,2,3,...,20.Theprobabilityofasuccessis p = 0.55.Theprobability ofafailureis q = 0.45.Thenumberoftrialsis n = 20.Theprobabilityquestioncanbestated mathematicallyas P X = 15 Example4.8 Afaircoinisipped15times.Whatistheprobabilityofgettingmorethan10heads?Let X =the numberofheadsin15ipsofthefaircoin. X takesonthevalues x =0,1,2,3,...,15.Sincethecoin isfair, p =0.5and q =0.5.Thenumberoftrialsis n =15.Theprobabilityquestioncanbestated mathematicallyas P X > 10 Example4.9 Approximately70%ofstatisticsstudentsdotheirhomeworkintimeforittobecollectedand graded.Inastatisticsclassof50students,whatistheprobabilitythatatleast40willdotheir homeworkontime? Problem1 Thisisabinomialproblembecausethereisonlyasuccessora__________,thereareadenite numberoftrials,andtheprobabilityofasuccessis0.70foreachtrial. Problem2 Ifweareinterestedinthenumberofstudentswhodotheirhomework,thenhowdowedene X ? Problem3 Whatvaluesdoes X takeon? Problem4 Whatisa"failure",inwords? Theprobabilityofasuccessis p =0.70.Thenumberoftrialis n =50. Problem5 If p + q = 1,thenwhatis q ? Problem6 Thewords"atleast"translateaswhatkindininequality? Theprobabilityquestionis P X 40 .

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140 CHAPTER4.DISCRETERANDOMVARIABLES 4.5.1NotationfortheBinomial:B=BinomialProbabilityDistributionFunction X B n p Readthisas" X isarandomvariablewithabinomialdistribution."Theparametersare n and p n =number oftrials p =probabilityofasuccessoneachtrial Example4.10 Ithasbeenstatedthatabout41%ofadultworkershaveahighschooldiplomabutdonotpursue anyfurthereducation.If20adultworkersarerandomlyselected,ndtheprobabilitythatatmost 12ofthemhaveahighschooldiplomabutdonotpursueanyfurthereducation.Howmanyadult workersdoyouexpecttohaveahighschooldiplomabutdonotpursueanyfurthereducation? Let X =thenumberofworkerswhohaveahighschooldiplomabutdonotpursueanyfurther education. X takesonthevalues0,1,2,...,20where n =20and p =0.41. q =1-0.41=0.59. X B 20,0.41 Find P X 12 P X 12 = 0.9738.calculatororcomputer UsingtheTI-83+ortheTI-84calculators,thecalculationsareasfollows.Gointo2ndDISTR.The syntaxfortheinstructionsare Tocalculate X =value:binompdf n p ,number If"number"isleftout,theresultisthebinomial probabilitytable. Tocalculate P X value :binomcdf n p ,number If"number"isleftout,theresultisthecumulativebinomialprobabilitytable. Forthisproblem:Afteryouarein2ndDISTR,arrowdowntoA:binomcdf.PressENTER.Enter 20,.41,12.Theresultis P X 12 = 0.9738 N OTE :Ifyouwanttond P X = 12 ,usethepdf:binompdf.Ifyouwanttond P X > 12 use1-binomcdf,.41,12. Theprobabilityatmost12workershaveahighschooldiplomabutdonotpursueanyfurther educationis0.9738 Thegraphof X B 20,0.41 is: They-axiscontainstheprobabilityof X ,where X =thenumberofworkerswhohaveonlyahigh schooldiploma.

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141 Thenumberofadultworkersthatyouexpecttohaveahighschooldiplomabutnotpursueany furthereducationisthemean, m = np = 20 0.41 = 8.2. Theformulaforthevarianceis s 2 = npq .Thestandarddeviationis s = p npq s = p 20 0.41 0.59 = 2.20. Example4.11 Thefollowingexampleillustratesaproblemthatis not binomial.Itviolatestheconditionof independence.ABCCollegehasastudentadvisorycommitteemadeupof10staffmembersand 6students.Thecommitteewishestochooseachairpersonandarecorder.Whatistheprobability thatthechairpersonandrecorderarebothstudents?Allnamesofthecommitteeareputintoabox andtwonamesaredrawn withoutreplacement .Therstnamedrawndeterminesthechairperson andthesecondnametherecorder.Therearetwotrials.However,thetrialsarenotindependent becausetheoutcomeofthersttrialaffectstheoutcomeofthesecondtrial.Theprobabilityofa studentontherstdrawis 6 16 .Theprobabilityofastudentontheseconddrawis 5 15 ,whenthe rstdrawproducesastudent.Theprobabilityis 6 15 whentherstdrawproducesastaffmember. Theprobabilityofdrawingastudent'snamechangesforeachofthetrialsand,therefore,violates theconditionofindependence. 4.6Geometricoptional 6 Thecharacteristicsofageometricexperimentare: 1.ThereareoneormoreBernoullitrialswithallfailuresexceptthelastone,whichisasuccess.Inother words,youkeeprepeatingwhatyouaredoinguntiltherstsuccess.Thenyoustop.Forexample, youthrowadartatabull'seyeuntilyouhitthebull'seye.Thersttimeyouhitthebull'seyeisa "success"soyoustopthrowingthedart.Itmighttakeyou6triesuntilyouhitthebull'seye.Youcan thinkofthetrialsasfailure,failure,failure,failure,failure,success.STOP. 2.Intheory,thenumberoftrialscouldgoonforever.Theremustbeatleastonetrial. 3.Theprobability, p ,ofasuccessandtheprobability, q ,ofafailureisthesameforeachtrial. p + q = 1 and q = 1 )]TJ/F132 9.9626 Tf 10.982 0 Td [(p .Forexample,theprobabilityofrollinga3whenyouthrowonefairdieis 1 6 .Thisis truenomatterhowmanytimesyourollthedie.Supposeyouwanttoknowtheprobabilityofgetting therst3onthefthroll.Onrolls1,2,3,and4,youdonotgetafacewitha3.Theprobabilityfor eachofrolls1,2,3,and4is q = 5 6 ,theprobabilityofafailure.Theprobabilityofgettinga3onthe fthrollis 5 6 5 6 5 6 5 6 1 6 = 0.0804 Theoutcomesofageometricexperimenttageometricprobabilitydistribution.Themeanandvariance areinthesummaryinthischapter. Example4.12 Youplayagameofchancethatyoucaneitherwinorlosetherearenootherpossibilities until youlose.Yourprobabilityofwinningis p = 0.43.Whatistheprobabilitythatittakes5games untilyouwin?Let X =thenumberofgamesyouplayuntilyouwinincludesthewinninggame. Then X takesonthevalues1,2,3,...couldgoonindenitely.Since p = 0.43, q = 1 )]TJ/F58 9.9626 Tf 9.77 0 Td [(0.43 = 0.57. Theprobabilityquestionis P X = 5 Example4.13 Asafetyengineerfeelsthat35%ofallindustrialaccidentsinherplantarecausedbyfailureof employeestofollowinstructions.Shedecidestolookattheaccidentreports until shendsone thatshowsanaccidentcausedbyfailureofemployeestofollowinstructions.Ontheaverage, 6 Thiscontentisavailableonlineat.

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142 CHAPTER4.DISCRETERANDOMVARIABLES howmanyreportswouldthesafetyengineer expect tolookatuntilshendsareportshowing anaccidentcausedbyemployeefailuretofollowinstructions?Whatistheprobabilitythatthe safetyengineerwillhavetoexamineatleast3reportsuntilshendsareportshowinganaccident causedbyemployeefailuretofollowinstructions? Let X =thenumberofaccidentsthesafetyengineermustexamine until shendsareportshowing anaccidentcausedbyemployeefailuretofollowinstructions. X takesonthevalues1,2,3,....The rstquestionasksyoutondthe expectedvalue orthemean.Thesecondquestionasksyouto nd P X 3 ."Atleast"translatesasa"greaterthanorequalto"symbol. Example4.14 Supposethatyouarelookingforachemistrylabpartner.Theprobabilitythatsomeoneagrees tobeyourlabpartneris0.55.Sinceyouneedalabpartnerverysoon,youaskeverychemistry studentyouareacquaintedwith until onesaysthathe/shewillbeyourlabpartner.Whatisthe probabilitythatthefourthpersonsaysyes? Thisisageometricproblembecauseyoumayhaveanumberoffailuresbeforeyouhavetheone successyoudesire.Also,theprobabilityofasuccessstaysthesameeachtimeyouaskachemistry studenttobeyourlabpartner.Thereisnodenitenumberoftrialsnumberoftimesyouaska chemistrystudenttobeyourpartner. Problem1 Let X =thenumberof____________youmustask____________onesaysyes. Solution Let X =thenumberof chemistrystudents youmustask until onesaysyes. Problem2 Whatvaluesdoes X takeon? Problem3 Whatare p and q ? Problem4 TheprobabilityquestionisP_______. 4.6.1NotationfortheGeometric:G=GeometricProbabilityDistributionFunction X G p Readthisas" X isarandomvariablewithageometricdistribution."Theparameteris p p =theprobability ofasuccessforeachtrial. Example4.15 Assumethattheprobabilityofadefectivecomputercomponentis0.02.Findtheprobabilitythat therstdefectiscausedbythe7thcomponenttested.Howmanycomponentsdoyouexpectto testuntiloneisfoundtobedefective? Let X =thenumberofcomputercomponentstesteduntiltherstdefectisfound. X takesonthevalues1,2,3,...where p = 0.02. X G.02 Find P X = 7 P X = 7 = 0.0177.calculatororcomputer

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143 TI-83+andTI-84: Forageneraldiscussion,seethisexamplebinomial .Thesyntaxissimilar. Thegeometricparameterlistisp,numberIf"number"isleftout,theresultisthegeometric probabilitytable.Forthisproblem: Afteryouarein2ndDISTR,arrowdowntoD:geometpdf. PressENTER.Enter.02,7.Theresultis P X = 7 = 0.0177 Theprobabilitythatthe7thcomponentistherstdefectis0.0177. Thegraphof X G.02 is: The y -axiscontainstheprobabilityof X ,where X =thenumberofcomputercomponentstested. Thenumberofcomponentsthatyouwouldexpecttotestuntilyoundtherstdefectiveoneis themean, m =50. Theformulaforthemeanis m = 1 p = 1 0.02 = 50 Theformulaforthevarianceis s 2 = 1 p 1 p )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 1 0.02 1 0.02 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = 2450 Thestandarddeviationis s = r 1 p 1 p )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = r 1 0.02 1 0.02 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 49.5 4.7Hypergeometricoptional 7 Thecharacteristicsofahypergeometricexperimentare: 1.Youtakesamplesfrom 2 groups. 2.Youareconcernedwithagroupofinterest,calledtherstgroup. 3.Yousample withoutreplacement fromthecombinedgroups.Forexample,youwanttochoosea softballteamfromacombinedgroupof11menand13women.Theteamconsistsof10players. 4.Eachpickis not independent,sincesamplingiswithoutreplacement.Inthesoftballexample,the probabilityofpickingawomenrstis 13 24 .Theprobabilityofpickingamansecondis 11 23 ifawoman waspickedrst.Itis 10 23 ifamanwaspickedrst.Theprobabilityofthesecondpickdependsonwhat happenedintherstpick. 5.Youare not dealingwithBernoulliTrials. 7 Thiscontentisavailableonlineat.

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144 CHAPTER4.DISCRETERANDOMVARIABLES Theoutcomesofahypergeometricexperimentta hypergeometricprobability distribution.Themean andvariancearegiveninthesummaryinthischapter. Example4.16 Acandydishcontains100jellybeansand80gumdrops.Fiftycandiesarepickedatrandom.What istheprobabilitythat35ofthe50aregumdrops?Thetwogroupsarejellybeansandgumdrops. Sincetheprobabilityquestionasksfortheprobabilityofgumdrops,thegroupofinterestrst groupisgumdrops.Thesizeofthegroupofinterestrstgroupis80.Thesizeofthesecond groupis100.Thesizeofthesampleis50jellybeansorgumdrops.Let X =thenumberof gumdropsinthesampleof50. X takesonthevalues x =0,1,2,...,50.Theprobabilityquestionis P X = 35 Example4.17 Supposeashipmentof100VCRsisknowntohave10defectiveVCRs.Aninspectorchooses12 forinspection.Heisinterestedindeterminingtheprobabilitythat,amongthe12,atmost2are defective.Thetwogroupsarethe90non-defectiveVCRsandthe10defectiveVCRs.Thegroupof interestrstgroupisthedefectivegroupbecausetheprobabilityquestionasksfortheprobability ofatmost2defectiveVCRs.Thesizeofthesampleis12VCRs.Theymaybenon-defectiveor defective.Let X =thenumberofdefectiveVCRsinthesampleof12. X takesonthevalues0,1,2, ...,10. X maynottakeonthevalues11or12.Thesamplesizeis12,butthereareonly10defective VCRs.Theinspectorwantstoknow P X 2 "Atmost"means"lessthanorequalto". Example4.18 Youarepresidentofanon-campusspecialeventsorganization.Youneedacommitteeof7toplan aspecialbirthdaypartyforthepresidentofthecollege.Yourorganizationconsistsof18women and15men.Youareinterestedinthenumberofmenonyourcommittee.Whatistheprobability thatyourcommitteehasmorethan4men? Thisisahypergeometricproblembecauseyouarechoosingyourcommitteefromtwogroups menandwomen. Problem1 Areyouchoosingwithorwithoutreplacement? Problem2 Whatisthegroupofinterest? Problem3 Howmanyareinthegroupofinterest? Problem4 Howmanyareintheothergroup? Problem5 Let X =_________onthecommittee.Whatvaluesdoes X takeon? Problem6 Theprobabilityquestionis P_______ .

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145 4.7.1NotationfortheHypergeometric:H=HypergeometricProbabilityDistribution Function X H r b n Readthisas" X isarandomvariablewithahypergeometricdistribution."Theparametersare r b ,and n r =thesizeofthegroupofinterestrstgroup, b =thesizeofthesecondgroup, n =thesizeofthechosen sample Example4.19 Aschoolsitecommitteeistobechosenfrom6menand5women.Ifthecommitteeconsistsof4 members,whatistheprobabilitythat2ofthemaremen?Howmanymendoyouexpecttobeon thecommittee? Let X =thenumberofmenonthecommitteeof4.Themenarethegroupofinterestrstgroup. X takesonthevalues0,1,2,3,4,where r = 6, b = 5,and n = 4. X H 6,5,4 Find P X = 2 P X = 2 = 0.4545calculatororcomputer N OTE :Currently,theTI-83+andTI-84donothavehypergeometricprobabilityfunctions.There areanumberofcomputerpackages,includingMicrosoftExcel,thatdo. Theprobabilitythatthereare2menonthecommitteeisabout0.45. Thegraphof X H 6,5,4 is: The y -axiscontainstheprobabilityof X ,where X =thenumberofmenonthecommittee. Youwouldexpect m = 2.18about2menonthecommittee. Theformulaforthemeanis m = n r r + b = 4 6 6 + 5 = 2.18 Theformulaforthevarianceisfairlycomplex.YouwillnditintheSummaryoftheDiscrete ProbabilityFunctionsChapterSection4.9.

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147 4.8.1NotationforthePoisson:P=PoissonProbabilityDistributionFunction X P m Readthisas" X isarandomvariablewithaPoissondistribution."Theparameteris m or l m or l =the meanfortheintervalofinterest. Example4.23 Leah'sansweringmachinereceivesabout6telephonecallsbetween8a.m.and10a.m.Whatis theprobabilitythatLeahreceivesmorethan1call inthenext15minutes? Let X =thenumberofcallsLeahreceivesin15minutes.The intervalofinterest is15minutesor 1 4 hour. X takesonthevalues0,1,2,3,... IfLeahreceives,ontheaverage,6telephonecallsin2hours,andthereareeight15minutesintervalsin2hours,thenLeahreceives 1 8 6 = 0.75 callsin15minutes,ontheaverage.So, m =0.75forthisproblem. X P.75 Find P X > 1 P X > 1 = 0.1734calculatororcomputer TI-83+andTI-84:Forageneraldiscussion,see thisexampleBinomial .Thesyntaxissimilar.The Poissonparameterlistis m fortheintervalofinterest,number. Forthisproblem: Press1-andthenpress2ndDISTR.ArrowdowntoC:poissoncdf.PressENTER.Enter.75,1.The resultis P X > 1 = 0.1734 .NOTE:TheTIcalculatorsuse l lambdaforthemean. TheprobabilitythatLeahreceivesmorethan1telephonecallinthenextfteenminutesisabout 0.1734. Thegraphof X P.75 is: They-axiscontainstheprobabilityof X where X =thenumberofcallsin15minutes.

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148 CHAPTER4.DISCRETERANDOMVARIABLES 4.9SummaryofFunctions 9 Formula4.1: Binomial X B n p X =thenumberofsuccesses n =thenumberofindependenttrials X takesonthevalues x = 0,1,2,3,..., n p =theprobabilityofasuccess q =theprobabilityofafailure p + q = 1 q = 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p Themeanis m = np .Thevarianceis s 2 = npq Formula4.2: Geometric X G p X =thenumberoftrialsuntiltherstsuccesscountthefailuresandtherstsuccess X takesonthevalues x =1,2,3,... p =theprobabilityofasuccess q =theprobabilityofafailure p + q = 1 q = 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p Themeanis m = 1 p Thevarianceis s 2 = 1 p 1 p )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 Formula4.3: Hypergeometric X H r b n X =thenumberofitemsfromthegroupofinterestthatareinthechosensample. X maytakeonthevalues x =0,1,...,uptothesizeofthegroupofinterest.Theminimumvalue for X maybelargerthan0insomeinstances. r =thesizeofthegroupofinterestrstgroup b =thesizeofthesecondgroup n =thesizeofthechosensample. n r + b Themeanis: m = nr r + b 9 Thiscontentisavailableonlineat.

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149 Thevarianceis: s 2 = rbn r + b + n r + b 2 r + b )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 Formula4.4: Poisson X P m X =thenumberofoccurrencesintheintervalofinterest X takesonthevalues x =0,1,2,3,... Themean m istypicallygiven. l isoftenusedasthemeaninsteadof m .WhenthePoissonis usedtoapproximatethebinomial,weusethebinomialmean m = np n isthebinomialnumber oftrials. p =theprobabilityofasuccessforeachtrial.Thisformulaisvalidwhennis"large"and p "small". n "large"isabout100and p "small"maybesomethinglessthan0.1.If n islargeenough and p issmallenoughthenthePoissonapproximatesthebinomialverywell.

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150 CHAPTER4.DISCRETERANDOMVARIABLES 4.10Practice1:DiscreteDistribution 10 4.10.1StudentLearningObjectives Thestudentwillinvestigatethepropertiesofadiscretedistribution. 4.10.2Given: Aballetinstructorisinterestedinknowingwhatpercentofeachyear'sclasswillcontinueontothenext, sothatshecanplanwhatclassestooffer.Overtheyears,shehasestablishedthefollowingprobability distribution. Let X =thenumberofyearsastudentwillstudyballetwiththeteacher. Let P X = x =theprobabilitythatastudentwillstudyballetthatmanyyears. 4.10.3OrganizetheData Completethetablebelowusingthedataprovided. X PX=x X*PX=x 1 0.10 2 0.05 3 0.10 4 5 0.30 6 0.20 7 0.10 Exercise4.10.1 Inwords,denetheRandomVariable X Exercise4.10.2 P X = 4 = Exercise4.10.3 P X < 4 = Exercise4.10.4 Onaverage,howmanyyearswouldyouexpectachildtostudyballetwiththisteacher? 4.10.4DiscussionQuestion Exercise4.10.5 Whatdoesthecolumn" PX=x "sumtoandwhy? Exercise4.10.6 Whatdoesthecolumn" X PX=x "sumtoandwhy? 10 Thiscontentisavailableonlineat.

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151 4.11Practice2:BinomialDistribution 11 4.11.1StudentLearningOutcomes ThestudentwillpracticeconstructingBinomialDistributions. 4.11.2Given TheHigherEducationResearchInstituteatUCLAsurveyedmorethan263,000incomingfreshmenfrom 385colleges.36.7%ofrst-generationcollegestudentsexpectedtoworkfulltimewhileincollege. Source: EricHoover,TheChronicleofHigherEducation,2/3/2006 .Supposethatyourandomlypick8college freshmenfromthesurvey.Youareinterestedinthenumberthatexpecttoworkfull-timewhileincollege. 4.11.3InterprettheData Exercise4.11.1 Solutiononp.177. Inwords,denetherandomVariableX. Exercise4.11.2 Solutiononp.177. X ___________ Exercise4.11.3 Solutiononp.177. Whatvaluesdoes X takeon? Exercise4.11.4 ConstructtheprobabilitydistributionfunctionPDFfor X X PX=x Exercise4.11.5 Solutiononp.177. Onaverage u ,howmanywouldyouexpecttoansweryes? Exercise4.11.6 Solutiononp.177. Whatisthestandarddeviation s ? Exercise4.11.7 Solutiononp.177. Whatistheprobabilitywhatatmost5ofthefreshmenexpecttoworkfull-time? Exercise4.11.8 Solutiononp.178. Whatistheprobabilitythatatleast2ofthefreshmenexpecttoworkfull-time? 11 Thiscontentisavailableonlineat.

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152 CHAPTER4.DISCRETERANDOMVARIABLES Exercise4.11.9 Constructahistogramorplotalinegraph.Labelthehorizontalandverticalaxeswithwords. Includenumericalscaling.

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153 4.12Practice3:PoissonDistribution 12 4.12.1StudentLearningObjectives ThestudentwillinvestigatethepropertiesofaPoissondistribution. 4.12.2Given Onaverage,tenteensarekilledintheU.S.inteen-drivenautosperdayUSAToday,3/1/2005.Asaresult, statesacrossthecountryaredebatingraisingthedrivingage. 4.12.3InterprettheData Exercise4.12.1 Inwords,denetheRandomVariable X Exercise4.12.2 Solutiononp.178. X ______________ Exercise4.12.3 Solutiononp.178. Whatvaluesdoes X takeon? Exercise4.12.4 Forthegivenvaluesof X ,llinthecorrespondingprobabilities. X PX=x 0 4 8 10 11 15 Exercise4.12.5 Solutiononp.178. IsitlikelythattherewillbenoteenskilledintheU.S.inteen-drivenautosonanygivenday? Numerically,why? Exercise4.12.6 Solutiononp.178. Isitlikelythattherewillbemorethan20teenskilledintheU.S.inteen-drivenautosonanygiven day?Numerically,why? 12 Thiscontentisavailableonlineat.

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154 CHAPTER4.DISCRETERANDOMVARIABLES 4.13Practice4:GeometricDistribution 13 4.13.1StudentLearningObjectives Thestudentwillinvestigatethepropertiesofageometricdistribution. 4.13.2Given: UsetheinformationfromtheBinomialDistributionPracticeSection4.11.Supposethatyouwillrandomly selectonefreshmanfromthestudyuntilyoundonewhoexpectstoworkfull-timewhileincollege.You areinterestedinthenumberoffreshmenyoumustask. 4.13.3InterprettheData Exercise4.13.1 Inwords,denetheRandomVariable X Exercise4.13.2 Solutiononp.178. X Exercise4.13.3 Solutiononp.178. Whatvaluesdoes X takeon? Exercise4.13.4 ConstructtheprobabilitydistributionfunctionPDFfor X .Stopat X = 6. X PX=x 0 1 2 3 4 5 6 Exercise4.13.5 Solutiononp.178. Onaverage m ,howmanyfreshmenwouldyouexpecttohavetoaskuntilyoufoundonewho expectstoworkfull-timewhileincollege? Exercise4.13.6 Solutiononp.178. Whatistheprobabilitythatyouwillneedtoaskfewerthan3freshmen? 13 Thiscontentisavailableonlineat.

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155 Exercise4.13.7 Constructahistogramorplotalinegraph.Labelthehorizontalandverticalaxeswithwords. Includenumericalscaling.

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156 CHAPTER4.DISCRETERANDOMVARIABLES 4.14Practice5:HypergeometricDistribution 14 4.14.1StudentLearningObjectives Thestudentwillinvestigatethepropertiesofahypergeometricdistribution. 4.14.2Given Supposethatagroupofstatisticsstudentsisdividedintotwogroups:businessmajorsandnon-business majors.Thereare16businessmajorsinthegroupand7non-businessmajorsinthegroup.Arandom sampleof9studentsistaken.Weareinterestedinthenumberofbusinessmajorsinthegroup. 4.14.3InterprettheData Exercise4.14.1 Inwords,denetheRandomVariable X Exercise4.14.2 Solutiononp.178. X Exercise4.14.3 Solutiononp.178. Whatvaluesdoes X takeon? Exercise4.14.4 Constructtheprobabilitydistributionfunction PDF for X X PX=x Exercise4.14.5 Solutiononp.178. Onaverage m ,howmanywouldyouexpecttobebusinessmajors? 14 Thiscontentisavailableonlineat.

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157 4.15Homework 15 Exercise4.15.1 Solutiononp.178. 1.CompletethePDFandanswerthequestions. X P X = x X P X = x 0 0.3 1 0.2 2 3 0.4 a. Findtheprobabilitythat X = 2. b. Findtheexpectedvalue. Exercise4.15.2 Supposethatyouareofferedthefollowingdeal.Yourolladie.Ifyourolla6,youwin\$10.If yourolla4or5,youwin\$5.Ifyourolla1,2,or3,youpay\$6. a. Whatareyouultimatelyinterestedinherethevalueoftherollorthemoneyyouwin? b. Inwords,denetheRandomVariable X c. Listthevaluesthat X maytakeon. d. ConstructaPDF. e. Overthelongrunofplayingthisgame,whatareyourexpectedaveragewinningsper game? f. Basedonnumericalvalues,shouldyoutakethedeal?Explainyourdecisionincomplete sentences. Exercise4.15.3 Solutiononp.178. Aventurecapitalist,willingtoinvest\$1,000,000,hasthreeinvestmentstochoosefrom.Therst investment,asoftwarecompany,hasa10%chanceofreturning\$5,000,000prot,a30%chanceof returning\$1,000,000prot,anda60%chanceoflosingthemilliondollars.Thesecondcompany, ahardwarecompany,hasa20%chanceofreturning\$3,000,000prot,a40%chanceofreturning \$1,000,000prot,anda40%chanceoflosingthemilliondollars.Thethirdcompany,abiotech rm,hasa10%chanceofreturning\$6,000,000prot,a70%ofnoprotorloss,anda20%chance oflosingthemilliondollars. a. ConstructaPDFforeachinvestment. b. Findtheexpectedvalueforeachinvestment. c. Whichisthesafestinvestment?Whydoyouthinkso? d. Whichistheriskiestinvestment?Whydoyouthinkso? e. Whichinvestmenthasthehighestexpectedreturn,onaverage? Exercise4.15.4 Atheatergroupholdsafund-raiser.Itsells100rafeticketsfor\$5apiece.Supposeyoupurchase 4tickets.Theprizeis2passestoaBroadwayshow,worthatotalof\$150. a. Whatareyouinterestedinhere? b. Inwords,denetheRandomVariable X 15 Thiscontentisavailableonlineat.

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158 CHAPTER4.DISCRETERANDOMVARIABLES c. Listthevaluesthat X maytakeon. d. ConstructaPDF. e. Ifthisfund-raiserisrepeatedoftenandyoualwayspurchase4tickets,whatwouldbe yourexpectedaveragewinningspergame? Exercise4.15.5 Solutiononp.179. Supposethat20,000marriedadultsintheUnitedStateswererandomlysurveyedastothenumber ofchildrentheyhave.Theresultsarecompiledandareusedastheoreticalprobabilities.Let X = thenumberofchildren X P X = x X P X = x 0 0.10 1 0.20 2 0.30 3 4 0.10 5 0.05 6ormore 0.05 a. Findtheprobabilitythatamarriedadulthas3children. b. Inwords,whatdoestheexpectedvalueinthisexamplerepresent? c. Findtheexpectedvalue. d. Isitmorelikelythatamarriedadultwillhave23childrenor46children?Howdo youknow? Exercise4.15.6 SupposethatthePDFforthenumberofyearsittakestoearnaBachelorofScienceB.S.degree isgivenbelow. X P X = x 3 0.05 4 0.40 5 0.30 6 0.15 7 0.10 a. Inwords,denetheRandomVariable X b. Whatdoesitmeanthatthevalues0,1,and2arenotincludedfor X onthePDF? c. Onaverage,howmanyyearsdoyouexpectittotakeforanindividualtoearnaB.S.?

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161 d. Howmanyareexpectedtosufferfromanorexia? e. Findtheprobabilitythatnoonesuffersfromanorexia. f. Findtheprobabilitythatmorethanfoursufferfromanorexia. Exercise4.15.16 Theaveragenumberofchildrenofmiddle-agedJapanesecouplesis2.09 Source:TheYomiuri Shimbun,June28,2006. Supposethatonemiddle-agedJapanesecoupleisrandomlychosen. d. Findtheprobabilitythattheyhavenochildren. e. FindtheprobabilitythattheyhavefewerchildrenthantheJapaneseaverage. f. FindtheprobabilitythattheyhavemorechildrenthantheJapaneseaverage. Exercise4.15.17 Solutiononp.179. TheaveragenumberofchildrenperSpanishcoupleswas1.34in 2005.SupposethatoneSpanishcoupleisrandomlychosen. Source: http://www.typicallyspanish.com/news/publish/article_4897.shtml 18 ,June16,2006. d. Findtheprobabilitythattheyhavenochildren. e. FindtheprobabilitythattheyhavefewerchildrenthantheSpanishaverage. f. FindtheprobabilitythattheyhavemorechildrenthantheSpanishaverage. Exercise4.15.18 Fertilefemalecatsproduceanaverageof3littersperyear. Source:TheHumaneSocietyof theUnitedStates .Supposethatonefertile,femalecatisrandomlychosen.Inoneyear,ndthe probabilitysheproduces: d. Nolitters. e. Atleast2litters. f. Exactly3litters. Exercise4.15.19 Solutiononp.180. AconsumerlookingtobuyausedredMiatacarwillcalldealershipsuntilshendsadealership thatcarriesthecar.Sheestimatestheprobabilitythatanyindependentdealershipwillhavethe carwillbe28%.Weareinterestedinthenumberofdealershipsshemustcall. d. Onaverage,howmanydealershipswouldweexpecthertohavetocalluntilshends onethathasthecar? e. Findtheprobabilitythatshemustcallatmost4dealerships. f. Findtheprobabilitythatshemustcall3or4dealerships. Exercise4.15.20 SupposethattheprobabilitythatanadultinAmericawillwatchtheSuperBowlis40%.Each personisconsideredindependent.WeareinterestedinthenumberofadultsinAmericawemust surveyuntilwendonewhowillwatchtheSuperBowl. d. HowmanyadultsinAmericadoyouexpecttosurveyuntilyoundonewhowillwatch theSuperBowl? e. Findtheprobabilitythatyoumustask7people. f. Findtheprobabilitythatyoumustask3or4people. 18 http://www.typicallyspanish.com/news/publish/article_4897.shtml

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162 CHAPTER4.DISCRETERANDOMVARIABLES Exercise4.15.21 Solutiononp.180. AgroupofMartialArtsstudentsisplanningonparticipatinginanupcomingdemonstration. 6arestudentsofTaeKwonDo;7arestudentsofShotokanKarate.Supposethat8studentsare randomlypickedtobeintherstdemonstration.WeareinterestedinthenumberofShotokan Karatestudentsinthatrstdemonstration. d. HowmanyShotokanKaratestudentsdoweexpecttobeinthatrstdemonstration? e. Findtheprobabilitythat4studentsofShotokanKaratearepicked. f. Findtheprobabilitythatnomorethan6studentsofShotokanKaratearepicked. Exercise4.15.22 ThechanceofaIRSauditforataxreturnwithover\$25,000inincomeisabout2%peryear.We areinterestedintheexpectednumberofauditsapersonwiththatincomehasina20yearperiod. Assumeeachyearisindependent. d. Howmanyauditsareexpectedina20yearperiod? e. Findtheprobabilitythatapersonisnotauditedatall. f. Findtheprobabilitythatapersonisauditedmorethantwice. Exercise4.15.23 Solutiononp.180. Refertothepreviousproblem.Supposethat100peoplewithtaxreturnsover\$25,000arerandomlypicked.Weareinterestedinthenumberofpeopleauditedin1year.Onewaytosolvethis problemisbyusingtheBinomialDistribution.Since n islargeand p issmall,anotherdiscrete distributioncouldbeusedtosolvethefollowingproblems.Solvethefollowingquestionsd-f usingthatdistribution. d. Howmanyareexpectedtobeaudited? e. Findtheprobabilitythatnoonewasaudited. f. Findtheprobabilitythatmorethan2wereaudited. Exercise4.15.24 Supposethatatechnologytaskforceisbeingformedtostudytechnologyawarenessamonginstructors.Assumethat10peoplewillberandomlychosentobeonthecommitteefromagroup of28volunteers,20whoaretechnicallyprocientand8whoarenot.Weareinterestedinthe numberonthecommitteewhoare not technicallyprocient. d. Howmanyinstructorsdoyouexpectonthecommitteewhoare not technicallyprocient? e. Findtheprobabilitythatatleast5onthecommitteearenottechnicallyprocient. f. Findtheprobabilitythatatmost3onthecommitteearenottechnicallyprocient. Exercise4.15.25 Solutiononp.180. ReferbacktoExercise4.15.12.Solvethisproblemagain,usingadifferent,thoughstillacceptable, distribution. Exercise4.15.26 Supposethat9Massachusettsathletesarescheduledtoappearatacharitybenet.The9arerandomlychosenfrom8volunteersfromtheBostonCelticsand4volunteersfromtheNewEngland Patriots.WeareinterestedinthenumberofPatriotspicked. d. Isitmorelikelythattherewillbe2Patriotsor3Patriotspicked? e. WhatistheprobabilitythatallofthevolunteerswillbefromtheCeltics f. IsitmorelikelythatmoreofthevolunteerswillbefromthePatriotsorfromtheCeltics? Howdoyouknow?

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164 CHAPTER4.DISCRETERANDOMVARIABLES g. Supposethatapagemaybepickedmorethanonce.Weareinterestedinthenumberof pagesthatwemustrandomlysurveyuntilwendonethathasfootwearadvertised onit.DenetherandomvariableXandgiveitsdistribution. h. Doyouexpecttosurveymorethan10pagesinordertondonethatadvertisesfootwear onit?Why? i. Whatistheprobabilitythatyouonlyneedtosurveyatmost3pagesinordertondone thatadvertisesfootwearonit? j. Howmanypagesdoyouexpecttoneedtosurveyinordertondonethatadvertises footwear? Exercise4.15.32 Supposethatyourollafairdieuntileachfacehasappearedatleastonce.Itdoesnotmatterin whatorderthenumbersappear.Findtheexpectednumberofrollsyoumustmakeuntileachface hasappearedatleastonce. 4.15.2Trythesemultiplechoiceproblems. Forthenextthreeproblems :TheprobabilitythattheSanJoseSharkswillwinanygivengameis0.3694 basedontheir13yearwinhistoryof382winsoutof1034gamesplayedasofacertaindate.Their2005 scheduleforNovembercontains12games.Let X =numberofgameswoninNovember2005 Exercise4.15.33 Solutiononp.181. TheexpectednumberofwinsforthemonthofNovember2005is: A. 1.67 B. 12 C. 382 1043 D. 4.43 Exercise4.15.34 Solutiononp.181. WhatistheprobabilitythattheSanJoseSharkswin6gamesinNovember? A. 0.1476 B. 0.2336 C. 0.7664 D. 0.8903 Exercise4.15.35 Solutiononp.181. FindtheprobabilitythattheSanJoseSharkswinatleast5gamesinNovember. A. 0.3694 B. 0.5266 C. 0.4734 D. 0.2305 Forthenextthreequestions :TheaveragenumberoftimesperweekthatMrs.Plum'scatswakeherupat nightbecausetheywanttoplayis10.Weareinterestedinthenumberoftimeshercatswakeherupeach week. Exercise4.15.36 Solutiononp.181. Inwords,therandomvariable X = A. ThenumberoftimesMrs.Plum'scatswakeherupeachweek

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165 B. ThenumberoftimesMrs.Plum'scatswakeherupeachhour C. ThenumberoftimesMrs.Plum'scatswakeherupeachnight D. ThenumberoftimesMrs.Plum'scatswakeherup Exercise4.15.37 Solutiononp.181. Findtheprobabilitythathercatswillwakemeupnomorethan5timesnextweek. A. 0.5000 B. 0.9329 C. 0.0378 D. 0.0671 4.16Review 19 Thenexttwoquestionsrefertothefollowing: Arecentpollconcerningcreditcardsfoundthat35percentofrespondentsuseacreditcardthatgivesthem amileofairtravelforeverydollartheycharge.Thirtypercentoftherespondentschargemorethan\$2000 permonth.Ofthoserespondentswhochargemorethan\$2000,80percentuseacreditcardthatgivesthem amileofairtravelforeverydollartheycharge. Exercise4.16.1 Solutiononp.181. Whatistheprobabilitythatarandomlyselectedrespondentexpectedtospendmorethan\$2000 ANDuseacreditcardthatgivesthemamileofairtravelforeverydollartheycharge? A. 0 .30 0 .35 B. 0 .80 0 .35 C. 0 .80 0 .30 D. 0 .80 Exercise4.16.2 Solutiononp.181. Basedupontheaboveinformation,areusingacreditcardthatgivesamileofairtravelforeach dollarspentANDchargingmorethan\$2000permonthindependentevents? A. Yes B. No,andtheyarenotmutuallyexclusiveeither C. No,buttheyaremutuallyexclusive D. Notenoughinformationgiventodeterminetheanswer Exercise4.16.3 Solutiononp.181. Asociologistwantstoknowtheopinionsofemployedadultwomenaboutgovernmentfunding fordaycare.Sheobtainsalistof520membersofalocalbusinessandprofessionalwomen's clubandmailsaquestionnaireto100ofthesewomenselectedatrandom.68questionnairesare returned.Whatisthepopulationinthisstudy? A. Allemployedadultwomen B. Allthemembersofalocalbusinessandprofessionalwomen'sclub C. The100womenwhoreceivedthequestionnaire D. Allemployedwomenwithchildren 19 Thiscontentisavailableonlineat.

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166 CHAPTER4.DISCRETERANDOMVARIABLES Thenexttwoquestionsrefertothefollowing:AnarticlefromTheSanJoseMercuryNewswasconcerned withtheracialmixofthe1500studentsatProspectHighSchoolinSaratoga,CA.Thetablesummarizesthe results.Maleandfemalevaluesareapproximate. EthnicGroup Gender White Asian Hispanic Black AmericanIndian Male 400 168 115 35 16 Female 440 132 140 40 14 Exercise4.16.4 Solutiononp.181. FindtheprobabilitythatastudentisAsianorMale. Exercise4.16.5 Solutiononp.181. FindtheprobabilitythatastudentisBlackgiventhatthestudentisFemale. Exercise4.16.6 Solutiononp.181. Asampleofpoundslost,inacertainmonth,byindividualmembersofaweightreducingclinic producedthefollowingstatistics: Mean=5lbs. Median=4.5lbs. Mode=4lbs. Standarddeviation=3.8lbs. Firstquartile=2lbs. Thirdquartile=8.5lbs. Thecorrectstatementis: A. Onefourthofthememberslostexactly2pounds. B. Themiddleftypercentofthememberslostfrom2to8.5lbs. C. Mostpeoplelost3.5to4.5lbs. D. Allofthechoicesabovearecorrect. Exercise4.16.7 Solutiononp.181. Whatdoesitmeanwhenadatasethasastandarddeviationequaltozero? A. Allvaluesofthedataappearwiththesamefrequency. B. Themeanofthedataisalsozero. C. Allofthedatahavethesamevalue. D. Therearenodatatobeginwith. Exercise4.16.8 Solutiononp.181. Thestatementthatbestdescribestheillustrationbelowis: Figure4.1

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167 A. Themeanisequaltothemedian. B. Thereisnorstquartile. C. Thelowestdatavalueisthemedian. D. Themedianequals Q 1 + Q 3 2 Exercise4.16.9 Solutiononp.181. AccordingtoarecentarticleSanJoseMercuryNewstheaveragenumberofbabiesbornwith signicanthearinglossdeafnessisapproximately2per1000babiesinahealthybabynursery. Thenumberclimbstoanaverageof30per1000babiesinanintensivecarenursery. Supposethat1000babiesfromhealthynurserybabiesweresurveyed.Findtheprobabilitythat exactly2babieswereborndeaf. Exercise4.16.10 Solutiononp.181. Afriendoffersyouthefollowingdeal.Fora\$10fee,youmaypickanenvelopefromabox containing100seeminglyidenticalenvelopes.However,eachenvelopecontainsacouponfora freegift. 10ofthecouponsareforafreegiftworth\$6. 80ofthecouponsareforafreegiftworth\$8. 6ofthecouponsareforafreegiftworth\$12. 4ofthecouponsareforafreegiftworth\$40. Baseduponthenancialgainorlossoverthelongrun,shouldyouplaythegame? A. Yes,Iexpecttocomeoutaheadinmoney. B. No,Iexpecttocomeoutbehindinmoney. C. Itdoesn'tmatter.Iexpecttobreakeven. Thenextfourquestionsrefertothefollowing:Recently,anursecommentedthatwhenapatientcallsthe medicaladvicelineclaimingtohave theu ,thechancethathe/shetrulyhas theu andnotjustanasty coldisonlyabout4%.Ofthenext25patientscallinginclaimingtohave theu ,weareinterestedinhow manyactuallyhave theu Exercise4.16.11 Solutiononp.181. DenetheRandomVariableandlistitspossiblevalues. Exercise4.16.12 Solutiononp.181. Statethedistributionof X Exercise4.16.13 Solutiononp.181. Findtheprobabilitythatatleast4ofthe25patientsactuallyhave theu Exercise4.16.14 Solutiononp.181. Onaverage,forevery25patientscallingin,howmanydoyouexpecttohave theu ? Thenexttwoquestionsrefertothefollowing:Differenttypesofwritingcansometimesbedistinguished bythenumberoflettersinthewordsused.Astudentinterestedinthisfactwantstostudythenumberof lettersofwordsusedbyTomClancyinhisnovels.SheopensaClancynovelatrandomandrecordsthe numberoflettersoftherst250wordsonthepage. Exercise4.16.15 Solutiononp.182. Whatkindofdatawascollected? A. qualitative B. quantitative-continuous C. quantitativediscrete Exercise4.16.16 Solutiononp.182. Whatisthepopulationunderstudy?

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169 X Frequency RelativeFrequency 0 __________ __________ 1 __________ __________ 2 __________ __________ 3 __________ __________ 4 __________ __________ 5 __________ __________ 6 __________ __________ 7 __________ __________ 8 __________ __________ 9 __________ __________ 10 __________ __________ 2.Calculatethefollowing: a. x = b. s = 3.Constructahistogramoftheempiricaldata. Figure4.2

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170 CHAPTER4.DISCRETERANDOMVARIABLES 4.17.5TheorecticalDistribution 1.BuildthetheoreticalPDFchartforXbasedonthedistributioninthesectionabove. X P X = x 0 1 2 3 4 5 6 7 8 9 10 2.Calculatethefollowing: a. m = ____________ b. s = ____________ 3.Constuctahistogramofthetheoreticaldistribution. Figure4.3

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171 4.17.6UsingtheData Calculatethefollowing,roundingto4decimalplaces: N OTE :RF=relativefrequency Usethetablefromthesectiontitled"UsingtheData"here: P X = 3 = P 1 < X < 4 = P X 8 = Usethedatafromthesectiontitled"OrganizetheData"here: RF X = 3 = RF 1 < X < 4 = RF X 8 = 4.17.7DiscussionQuestions 1.Knowingthatdatavary,describethreesimilaritiesbetweenthegraphsanddistributionsofthetheoreticalandempiricaldistributions.Usecompletesentences.Note:Theseanswersmayvaryandstill becorrect. 2.Describethethreemostsignicantdifferencesbetweenthegraphsordistributionsofthetheoretical andempiricaldistributions.Note:Theseanswersmayvaryandstillbecorrect. 3.DoesitappearthatthedatatthedistributioninPartI?In1-3completesentences,explainwhyor whynot. 4.Supposethattheexperimenthadbeenrepeated500times.WhichchartfromPartIIorPartIIIwould youexpecttochange?Why?Whywouldn'ttheotherchartchange?Howmightthechartchange?

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172 CHAPTER4.DISCRETERANDOMVARIABLES 4.18Lab2:DiscreteDistributionLuckyDiceExperiment 21 ClassTime: Names: 4.18.1StudentLearningOutcomes: ThestudentwillcompareempiricaldataandatheoreticaldistributiontodetermineifaTetgambling gametsadiscretedistribution. Thestudentwilldemonstrateanunderstandingoflong-termprobabilities. 4.18.2Supplies: 1gameLuckyDiceor3regulardice N OTE :Foradetailedgamedescription,referhere. N OTE :Roundrelativefrequenciesandprobabilitiestofourdecimalplaces. 4.18.3TheProcedure 1.Theexperimentprocedureistobetononeobject.Then,roll3LuckyDiceandcountthenumberof matches.Thenumberofmatcheswilldecideyourprot. 2.Whatisthetheoreticalprobabilityof1diematchingtheobject? 3.Chooseoneobjecttoplaceabeton.Rollthe3LuckyDice.Countthenumberofmatches. 4.Let X =numberofmatches.Theoretically, X 5.Let Y =protpergame. 4.18.4OrganizetheData Inthechartbelow,llinthe Y valuethatcorrespondstoeach X value.Next,recordthenumberofmatches pickedforyourclass.Then,calculatetherelativefrequency. 1.Completethetable. X Y Frequency RelativeFrequency 0 1 2 3 2.CalculatetheFollowing: a. x = b. s x = c. y = 21 Thiscontentisavailableonlineat.

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173 d. s y = 3.Explainwhat x represents. 4.Explainwhat y represents. 5.Basedupontheexperiment: a. Whatwastheaverageprotpergame? b. Didthisrepresentanaveragewinorlosspergame? c. Howdoyouknow?Answerincompletesentences. 6.Constructahistogramoftheempiricaldata Figure4.4 4.18.5TheoreticalDistribution BuildthetheoreticalPDFchartfor X and Y basedonthedistributionfromthesectiontitled"TheProcedure". 1. X Y P X = x = P Y = y 0 1 2 3 2.Calculatethefollowing a. m x = b. s x = c. m y =

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174 CHAPTER4.DISCRETERANDOMVARIABLES 3.Explainwhat m x represents. 4.Explainwhat m y represents. 5.Basedupontheory: a. Whatwastheexpectedprotpergame? b. Didtheexpectedprotrepresentanaveragewinorlosspergame? c. Howdoyouknow?Answerincompletesentences. 6.Constructahistogramofthetheoreticaldistribution. Figure4.5 4.18.6UsetheData Calculatethefollowingroundedto4decimalplaces: N OTE : RF =relativefrequency Usethedatafromthesectiontitled"TheoreticalDistribution"here: 1. P X = 3 = ____________ 2. P 0 < X < 3 = ____________ 3. P X 2 = ____________ Usethedatafromthesectiontitled"OrganizetheData"here: 1. RF X = 3 = ____________ 2. RF 0 < X < 3 = ____________ 3. RF X 2 = ____________

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176 CHAPTER4.DISCRETERANDOMVARIABLES SolutionstoExercisesinChapter4 Example4.2,Problem1p.135 Let X =thenumberofdaysNancy attendsclassperweek Example4.2,Problem2p.135 0,1,2,and3 Example4.2,Problem3p.135 X P x 0 0.01 1 0.04 2 0.15 3 0.80 Example4.5,Problem1p.137 X =amountofprot Example4.5,Problem2p.137 X P x x P x WIN 10 1 3 10 3 LOSE -6 2 3 )]TJ/F58 7.5716 Tf 6.227 0 Td [(12 3 Example4.5,Problem3p.137 Theexpectedvalue m = )]TJ/F58 7.5716 Tf 6.228 0 Td [(2 3 .Youdonotcomeoutahead. Example4.9,Problem1p.139 failure Example4.9,Problem2p.139 X =thenumberofstatisticsstudentswhodotheirhomeworkontime Example4.9,Problem3p.139 0,1,2,...,50 Example4.9,Problem4p.139 Failureisastudentwhodoesnotdohisorherhomeworkontime. Example4.9,Problem5p.139 q =0.30 Example4.9,Problem6p.139 greaterthanorequalto Example4.14,Problem2p.142 0,1,2,...,totalnumberofchemistrystudents

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177 Example4.14,Problem3p.142 p =0.55 q =0.45 Example4.14,Problem4p.142 P X = 4 Example4.18,Problem1p.144 Without Example4.18,Problem2p.144 Themen Example4.18,Problem3p.144 15men Example4.18,Problem4p.144 18women Example4.18,Problem5p.144 Let X = thenumberofmen onthecommittee. X =0,1,2,...,7. Example4.18,Problem6p.144 PX > 4 Example4.22,Problem1p.146 The2pages Example4.22,Problem2p.146 2 Example4.22,Problem3p.146 Let X = thenumberofpagesofwrittenmathhomeworkyoudoperday Example4.22,Problem4p.146 PX > 2 SolutionstoPractice2:BinomialDistribution SolutiontoExercise4.11.1p.151 X =thenumberthatexpecttoworkfull-time. SolutiontoExercise4.11.2p.151 B,0.367 SolutiontoExercise4.11.3p.151 0,1,2,3,4,5,6,7,8 SolutiontoExercise4.11.5p.151 2.94 SolutiontoExercise4.11.6p.151 1.36 SolutiontoExercise4.11.7p.151 0.9677

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178 CHAPTER4.DISCRETERANDOMVARIABLES SolutiontoExercise4.11.8p.151 0.8547 SolutionstoPractice3:PoissonDistribution SolutiontoExercise4.12.2p.153 P SolutiontoExercise4.12.3p.153 0,1,2,3,4,... SolutiontoExercise4.12.5p.153 No SolutiontoExercise4.12.6p.153 No SolutionstoPractice4:GeometricDistribution SolutiontoExercise4.13.2p.154 G.367 SolutiontoExercise4.13.3p.154 0,1,2,... SolutiontoExercise4.13.5p.154 2.72 SolutiontoExercise4.13.6p.154 0.5993 SolutionstoPractice5:HypergeometricDistribution SolutiontoExercise4.14.2p.156 H,7,9 SolutiontoExercise4.14.3p.156 2,3,4,5,6,7,8,9 SolutiontoExercise4.14.5p.156 6.26 SolutionstoHomework SolutiontoExercise4.15.1p.157 a. 0.1 b. 1.6 SolutiontoExercise4.15.3p.157 b. \$200,000;\$600,000;\$400,000 c. thirdinvestment d. rstinvestment e. secondinvestment

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179 SolutiontoExercise4.15.5p.158 a. 0.2 c. 2.35 d. 2-3children SolutiontoExercise4.15.7p.159 a. X =thenumberofdicethatshowa1 b. 0,1,2,3,4,5,6 c. X B 6, 1 6 d. 1 e. 0.00002 f. 3dice SolutiontoExercise4.15.9p.159 a. X =thenumberofstudentsthatwillattendTet. b. 0,1,2,3,4,5,6,7,8,9,10,11,12 c. X B,0.18 d. 2.16 e. 0.9511 f. 0.3702 SolutiontoExercise4.15.11p.159 a. X =thenumberoffencersthatdo not usefoilastheirmainweapon b. 0,1,2,3,...25 c. X B,0.40 d. 10 e. 0.0442 f. Yes SolutiontoExercise4.15.13p.160 a. X =thenumberoffortunecookiesthathaveanextrafortune b. 0,1,2,3,...144 c. X B,0.40 or P.32 d. 4.32 e. 0.0124or0.0133 f. 0.6300or0.6264 SolutiontoExercise4.15.15p.160 a. X =thenumberofwomenthatsufferfromanorexia b. 0,1,2,3,...600canleaveoff600 c. X P d. 3 e. 0.0498 f. 0.1847 SolutiontoExercise4.15.17p.161 a. X =thenumberofchildrenforaSpanishcouple b. 0,1,2,3,... c. X P.34

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180 CHAPTER4.DISCRETERANDOMVARIABLES d. 0.2618 e. 0.6217 f. 0.3873 SolutiontoExercise4.15.19p.161 a. X =thenumberofdealersshecallsuntilshendsonewithausedredMiata b. 0,1,2,3,... c. X G.28 d. 3.57 e. 0.7313 f. 0.2497 SolutiontoExercise4.15.21p.162 d. 4.31 e. 0.4079 f. 0.9953 SolutiontoExercise4.15.23p.162 d. 2 e. 0.1353 f. 0.3233 SolutiontoExercise4.15.25p.162 a. X =thenumberofseniorsthatparticipatedinafter-schoolsportsall4yearsofhighschool b. 0,1,2,3,...60 c. X~P 4 8 d. 4.8 e. Yes f. 4 SolutiontoExercise4.15.27p.163 a. X =thenumberofshellpiecesinonecake b. 0,1,2,3,... c. X~P 1 5 d. 1.5 e. 0.2231 f. 0.0001 g. Yes SolutiontoExercise4.15.29p.163 d. 0.0043 e. none f. 3.3 SolutiontoExercise4.15.31p.163 d. 3.02 e. No f. 0.9997 h. 0.2291 i. 0.3881

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181 j. 6.6207pages SolutiontoExercise4.15.33p.164 D:4.43 SolutiontoExercise4.15.34p.164 A:0.1476 SolutiontoExercise4.15.35p.164 C:0.4734 SolutiontoExercise4.15.36p.164 A:ThenumberoftimesMrs.Plum'scatswakeherupeachweek SolutiontoExercise4.15.37p.165 D:0.0671 SolutionstoReview SolutiontoExercise4.16.1p.165 C SolutiontoExercise4.16.2p.165 B SolutiontoExercise4.16.3p.165 A SolutiontoExercise4.16.4p.166 0.5773 SolutiontoExercise4.16.5p.166 0.0522 SolutiontoExercise4.16.6p.166 B SolutiontoExercise4.16.7p.166 C SolutiontoExercise4.16.8p.166 C SolutiontoExercise4.16.9p.167 0.2709 SolutiontoExercise4.16.10p.167 B SolutiontoExercise4.16.11p.167 X =thenumberofpatientscallinginclaimingtohave theu ,whoactuallyhave theu X =0,1,2,...25 SolutiontoExercise4.16.12p.167 B 25 ,0 .04 SolutiontoExercise4.16.13p.167 0.0165 SolutiontoExercise4.16.14p.167 1

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182 CHAPTER4.DISCRETERANDOMVARIABLES SolutiontoExercise4.16.15p.167 C SolutiontoExercise4.16.16p.167 AllwordsusedbyTomClancyinhisnovels

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Chapter5 ContinuousRandomVariables 5.1ContinuousRandomVariables 1 5.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Recognizeandunderstandcontinuousprobabilitydistributionfunctionsingeneral. Recognizetheuniformprobabilitydistributionandapplyitappropriately. Recognizetheexponentialprobabilitydistributionandapplyitappropriately. 5.1.2Introduction Continuousrandomvariableshavemanyapplications.Baseballbattingaverages,IQscores,thelength oftimealongdistancetelephonecalllasts,theamountofmoneyapersoncarries,thelengthoftimea computerchiplasts,andSATscoresarejustafew.Theeldofreliabilitydependsonavarietyofcontinuous randomvariables. Thischaptergivesanintroductiontocontinuousrandomvariablesandthemanycontinuousdistributions. Wewillbestudyingthesecontinuousdistributionsforseveralchapters. Thecharacteristicsofcontinuousrandomvariablesare: Theoutcomesaremeasured,notcounted. Geometrically,theprobabilityofanoutcomeisequaltoanareaunderthedensitycurve, f x Eachindividualvaluehaszeroprobabilityofoccurring.Insteadwendtheprobabilitythatthevalue isbetweentwoendpoints. Wewillstartwiththetwosimplestcontinuousdistributions,the Uniform andthe Exponential NOTE:Thevaluesofdiscreteandcontinuousrandomvariablescanbeambiguous.Forexample, if X isequaltothenumberofmilestothenearestmileyoudrivetoworkthen X isadiscrete randomvariable.Youcountthemiles.If X isthedistanceyoudrivetowork,thenyoumeasure valuesof X and X isacontinuousrandomvariable.Howtherandomvariableisdenedisvery important. 1 Thiscontentisavailableonlineat. 183

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184 CHAPTER5.CONTINUOUSRANDOMVARIABLES 5.2ContinuousProbabilityFunctions 2 Webeginbydeningacontinuousprobabilitydensityfunction.Weusethefunctionnotation f X .Intermediatealgebramayhavebeenyourrstformalintroductiontofunctions.Inthestudyofprobability,the functionswestudyarespecial.Wedenethefunction f X sothattheareabetweenitandthex-axisis equaltoaprobability.Sincethemaximumprobabilityisone,themaximumareaisalsoone. Forcontinuousprobabilitydistributions,PROBABILITY=AREA. Example5.1 Considerthefunction f X = 1 20 where0 < X < 20. X =arealnumber.Thegraphof f X = 1 20 isahorizontalline.However,since0 < X < 20, f X isrestrictedtotheportionbetween X = 0 and X = 20. f X = 1 20 where 0 < X < 20 Thegraphof f X = 1 20 isahorizontallinesegmentwhen0 < X < 20. Theareabetween f X = 1 20 where0 < X < 20andthex-axisistheareaofarectanglewithbase =20andheight= 1 20 AREA = 20 1 20 = 1 Thisparticularfunction,wherewehaverestricted X sothattheareabetweenthefunctionand thex-axisis1,isanexampleofacontinuousprobabilitydensityfunction.Itisusedasatoolto calculateprobabilities. Supposewewanttondtheareabetween f X = 1 20 andthex-axiswhere 0 < X < 2 AREA = 2 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0 1 20 = 0.1 2 Thiscontentisavailableonlineat.

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185 2 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0 = 2 = baseofarectangle 1 20 =theheight. Theareacorrespondstoaprobability.Theprobabilitythat X isbetween0and2is0.1,whichcan bewrittenmathematicallyas P < X < 2 = PX < 2 = 0.1. Supposewewanttondtheareabetween f X = 1 20 andthex-axiswhere 4 < X < 15 AREA = 15 )]TJ/F58 9.9626 Tf 10.131 0 Td [(4 1 20 = 0.55 15 )]TJ/F58 9.9626 Tf 10.131 0 Td [(4 = 11 = thebaseofarectangle 1 20 =theheight. Theareacorrespondstotheprobability P 4 < X < 15 = 0.55. Supposewewanttond P X = 15 Onanx-ygraph, X = 15isaverticalline.Averticalline hasnowidthor0width.Therefore, PX=15 = baseheight = 0 1 20 = 0. P X x canbewrittenas P X < x forcontinuousdistributionsiscalledthecumulativedistributionfunctionor CDF .Noticethe"lessthanorequalto"symbol.Wecanusethe CDF to calculate P X > x .The CDF gives"areatotheleft"and P X > x gives"areatotheright."We calculate P X > x forcontinuousdistributionsasfollows: P X > x = 1 )]TJ/F132 9.9626 Tf 10.455 0 Td [(P X < x Labelthegraphwith f X and X .Scalethexandyaxeswiththemaximum x and y values. f X = 1 20 ,0 < X < 20.

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186 CHAPTER5.CONTINUOUSRANDOMVARIABLES P 2.3 < X < 12.7 = base height = 12.7 )]TJ/F58 9.9626 Tf 10.131 0 Td [(2.3 1 20 = 0.52 Thepreviousproblemisanexampleoftheuniformprobabilitydistribution. 5.3TheUniformDistribution 3 Example5.2 Illustratethe uniformdistribution Thedatathatfollowsare55smilingtimes,inseconds,ofan eight-weekoldbaby. 10.4 19.6 18.8 13.9 17.8 16.8 21.6 17.9 12.5 11.1 4.9 12.8 14.8 22.8 20.0 15.9 16.3 13.4 17.1 14.5 19.0 22.8 1.3 0.7 8.9 11.9 10.9 7.3 5.9 3.7 17.9 19.2 9.8 5.8 6.9 2.6 5.8 21.7 11.8 3.4 2.1 4.5 6.3 10.7 8.9 9.4 9.4 7.6 10.0 3.3 6.7 7.8 11.6 13.8 18.6 samplemean=11.49andsamplestandarddeviation=6.23 Wewillassumethatthesmilingtimes,inseconds,followauniformdistributionbetween0and23 seconds.Thismeansthatanysmilingtimebetween0and23secondsisequallylikely.Youcansee thisforyourselfbyconstructingthehistogramfromthedata. Let X =length,inseconds,ofaneight-weekold'ssmile. Thenotationfortheuniformdistributionis X U a b where a =thelowestvalueand b =thehighestvalue. Forthisexample, X U 0,23 .0 < X < 23. Formulasforthetheoreticalmeanandstandarddeviationare m = a + b 2 and s = q b )]TJ/F132 7.5716 Tf 6.435 0 Td [(a 2 12 Forthisproblem,thetheoreticalmeanandstandarddeviationare m = 0 + 23 2 = 11.50secondsand s = q 23 )]TJ/F58 7.5716 Tf 6.227 0 Td [(0 2 12 = 6.64seconds 3 Thiscontentisavailableonlineat.

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187 Noticethatthetheoreticalmeanandstandarddeviationareclosetothesamplemeanandstandard deviation. Example5.3 Problem1 Whatistheprobabilitythatarandomlychoseneight-weekoldsmilesbetween2and18seconds? Solution Find P 2 < X < 18 P 2 < X < 18 = base height = 18 )]TJ/F58 9.9626 Tf 10.131 0 Td [(2 1 23 = 16 23 Problem2 Findthe90thpercentileforaneightweekold'ssmilingtime. Solution Ninetypercentofthesmilingtimesfallbelowthe90thpercentile, k ,so P X < k = 0.90 P X < k = 0.90 base height = 0.90 k )]TJ/F58 9.9626 Tf 10.131 0 Td [(0 1 23 = 0.90 k = 23 0.90 = 20.7 Problem3 Findtheprobabilitythatarandomeightweekoldsmilesmorethan12seconds KNOWING that thebabysmiles MORETHAN8SECONDS Solution Find P X > 12 j X > 8 Therearetwowaystodotheproblem. Fortherstway ,usethefactthat thisisa conditional andchangesthesamplespace.Thegraphillustratesthenewsamplespace. Youalreadyknowthebabysmiledmorethan8seconds. Writeanew f X : f X = 1 23 )]TJ/F58 7.5716 Tf 6.227 0 Td [(8 = 1 15 for8 < X < 23

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188 CHAPTER5.CONTINUOUSRANDOMVARIABLES P X > 12 j X > 8 = 23 )]TJ/F58 9.9626 Tf 10.131 0 Td [(12 1 15 = 11 15 Forthesecondway,usetheconditionalformulafromChapter3withtheoriginaldistribution X U 0,23 : P A j B = P AANDB P B Forthisproblem, A is X > 12 and B is X > 8 So, P X > 12 j X > 8 = X > 12 ANDX > 8 P X > 8 = P X > 12 P X > 8 = 11 23 15 23 = 0.733 Example5.4 Uniform :Theamountoftime,inminutes,thatapersonmustwaitforabusisuniformlydistributedbetween0and15minutes. Problem1 Whatistheprobabilitythatapersonwaitsfewerthan12.5minutes? Solution Let X =thenumberofminutesapersonmustwaitforabus. a =0and b =15. X U 0,15 Writetheprobabilitydensityfunction. f X = 1 15 )]TJ/F58 7.5716 Tf 6.228 0 Td [(0 = 1 15 for0 < X < 15 Find P X < 12.5 .Drawagraph. P X < k = base height = 12.5 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0 1 15 = 0.8333 Theprobabilityapersonwaitslessthan12.5minutesis0.8333.

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189 Problem2 Ontheaverage,howlongmustapersonwait? Findthemean, m ,andthestandarddeviation, s Solution m = a + b 2 = 15 + 0 2 = 7.5.Ontheaverage,apersonmustwait7.5minutes. s = q b )]TJ/F132 7.5716 Tf 6.436 0 Td [(a 2 12 = q 15 )]TJ/F58 7.5716 Tf 6.228 0 Td [(0 2 12 = 4.3.TheStandarddeviationis4.3minutes. Problem3 Ninetypercentofthetime,thetimeapersonmustwaitfallsbelowwhatvalue? N OTE :Thisasksforthe90thpercentile. Solution Findthe90thpercentile.Drawagraph.Let k =the90thpercentile. P X < k = base height = k )]TJ/F58 9.9626 Tf 10.132 0 Td [(0 1 15 0.90 = k 1 15 k = 0.90 15 = 13.5 k issometimescalledacriticalvalue. The90thpercentileis13.5minutes.Ninetypercentofthetime,apersonmustwaitatmost13.5 minutes. Example5.5 Uniform :Theaveragenumberofdonutsanine-yearoldchildeatspermonthisuniformlydistributedfrom0.5to4.Let X =theaveragenumberofdonutsanine-yearoldchildeatspermonth. The X U 0.5,4 .

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190 CHAPTER5.CONTINUOUSRANDOMVARIABLES Problem1 Theprobabilitythatarandomlyselectednine-yearoldchildeatsanaverageofmorethantwo donutsis_______. Findtheprobabilitythatadifferentnine-yearoldchildeatsanaverageofmorethantwodonuts giventhathisorheramountismorethan1.5donuts. Thesecondprobabilityquestionhasa conditional referto"ProbabilityTopicsSection3.1".You areaskedtondtheprobabilitythatanine-yearoldeatsanaverageofmorethantwodonuts giventhathis/heramountismorethan1.5donuts.Solvetheproblemtwodifferentwaysseethe rstexampleExample5.2.Youmustreducethesamplespace. Firstway :Sinceyoualready knowthechildeatsmorethan1.5donuts,youarenolongerstartingat a = 0.5donut.Your startingpointis1.5donuts. WriteanewfX: f X = 1 4 )]TJ/F58 7.5716 Tf 6.228 0 Td [(1.5 = 2 5 for1.5 < x < 4 Find P X > 2 j X > 1.5 .Drawagraph. Problem2 P X > 2 j X > 1.5 = base newheight = 4 )]TJ/F58 9.9626 Tf 10.131 0 Td [(2 2/5 = ? Theprobabilitythatanine-yearoldchildeatsanaverageofmorethan2donutswhenhe/shehas alreadyeatenmorethan1.5donutsis__________. Secondway: Drawtheoriginalgraphfor X U 0.5,4 .Usetheconditionalformula P X > 2 j X > 1.5 = P X > 2 ANDX > 1.5 P X > 1.5 = P X > 2 P X > 1.5 = 2 3.5 2.5 3.5 = 0.8 = 4 5 N OTE :See"SummaryoftheUniformandExponentialProbabilityDistributionsSection5.5"for afullsummary. 5.4TheExponentialDistribution 4 The exponential distributionisoftenconcernedwiththeamountoftimeuntilsomespeciceventoccurs. Forexample,theamountoftimebeginningnowuntilanearthquakeoccurshasanexponentialdistribution.Otherexamplesincludethelength,inminutes,oflongdistancebusinesstelephonecalls,andthe 4 Thiscontentisavailableonlineat.

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191 amountoftime,inmonths,acarbatterylasts.Itcanbeshown,too,thattheamountofchangethatyou haveinyourpocketorpursefollowsanexponentialdistribution. Valuesforanexponentialrandomvariableoccurinthefollowingway.Therearefewerlargevaluesand moresmallvalues.Forexample,theamountofmoneycustomersspendinonetriptothesupermarket followsanexponentialdistribution.Therearemorepeoplethatspendlessmoneyandfewerpeoplethat spendlargeamountsofmoney. Theexponentialdistributioniswidelyusedintheeldofreliability.Reliabilitydealswiththeamountof timeaproductlasts. Example5.6 Illustratestheexponentialdistribution: Let X =amountoftimeinminutesapostalclerkspends withhis/hercustomer.Thetimeisknowntohaveanexponentialdistributionwiththeaverage amountoftimeequalto4minutes. X isa continuousrandomvariable sincetimeismeasured.Itisgiventhat m =4minutes.Todo anycalculations,youmustknow m ,thedecayparameter. m = 1 m .Therfore, m = 1 4 = 0.25 Thestandarddeviation, s ,isthesameasthemean. m = s Theprobabilitydensityfunctionis f X = m e )]TJ/F132 7.5716 Tf 6.323 0 Td [(m x Thenumber e =2.71828182846...Itisa numberthatisusedofteninmathematics.Scienticcalculatorshavethekey" e x ."Ifyouenter1 for x ,thecalculatorwilldisplaythevalue e Thecurveis: f X = 0.25 e )]TJ/F58 7.5716 Tf 6.227 0 Td [(0.25 X where X isatleast0and m =0.25. Forexample, f 5 = 0.25 e )]TJ/F58 7.5716 Tf 6.227 0 Td [(0.25 5 = 0.072 Thegraphisasfollows: Noticethegraphisadecliningcurve.When X =0, f X = 0.25 e )]TJ/F58 7.5716 Tf 6.227 0 Td [(0.25 0 = 0.25 1 = 0.25 = m Example5.7 Problem1 Findtheprobabilitythataclerkspendsfourtoveminuteswitharandomlyselectedcustomer.

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192 CHAPTER5.CONTINUOUSRANDOMVARIABLES Solution Find P 4 < X < 15 The cumulativedistributionfunctionCDF givestheareatotheleft. P X < x = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m x P X < 5 = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F58 7.5716 Tf 6.228 0 Td [(0.25 5 = 0.7135and P X < 4 = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F58 7.5716 Tf 6.228 0 Td [(0.25 4 = 0.6321 N OTE :Youcandothesecalculationseasilyonacalculator. Theprobabilitythatapostalclerkspendsfourtoveminuteswitharandomlyselectedcustomer is P 4 < X < 5 = P X < 5 )]TJ/F132 9.9626 Tf 10.455 0 Td [(P X < 4 = 0.7135 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.6321 = 0.0814 N OTE :TI-83+andTI-84:Onthehomescreen,enter-e-.25*5--e-.25*4orentere-.25*4e-.25*5. Problem2 Halfofallcustomersarenishedwithinhowlong?Findthe50thpercentile Solution Findthe50thpercentile. P X < k = 0.50, k =2.8minutescalculatororcomputer Halfofallcustomersarenishedwithin2.8minutes. Youcanalsodothecalculationasfollows:

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193 P X < k = 0.50and P X < k = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F58 7.5716 Tf 6.228 0 Td [(0.25 k Therefore,0.50 = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F58 7.5716 Tf 6.228 0 Td [(0.25 k and e )]TJ/F58 7.5716 Tf 6.227 0 Td [(0.25 k = 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.50 = 0.5 Takenaturallogs: ln e )]TJ/F58 7.5716 Tf 6.227 0 Td [(0.25 k = ln 0.50 .So, )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.25 k = ln 0.50 Solvefor k : k = ln .50 )]TJ/F58 7.5716 Tf 6.228 0 Td [(0.25 = 2.8minutes N OTE :Aformulaforthepercentile k is k = LN 1 )]TJ/F132 7.5716 Tf 6.701 0 Td [(AreaToTheLeft )]TJ/F132 7.5716 Tf 6.322 0 Td [(m whereLNisthenaturallog. N OTE :TI-83+andTI-84:Onthehomescreen,enterLN-.50/-.25.Pressthe-forthenegative. Problem3 Whichislarger,themeanorthemedian? Solution Isthemeanormedianlarger? Frompartb,themedianor50thpercentileis2.8minutes.Thetheoreticalmeanis4minutes.The meanislarger. 5.4.1OptionalCollaborativeClassroomActivity Haveeachclassmembercountthechangehe/shehasinhis/herpocketorpurse.Yourinstructorwill recordtheamountsindollarsandcents.Constructahistogramofthedatatakenbytheclass.Use5 intervals.Drawasmoothcurvethroughthebars.Thegraphshouldlookapproximatelyexponential.Then calculatethemean. Let X =theamountofmoneyastudentinyourclasshasinhis/herpocketorpurse. Thedistributionfor X isapproximatelyexponentialwithmean, m =_______and m =_______.Thestandard deviation, s =________. Drawtheappropriateexponentialgraph.Youshouldlabelthexandyaxes,thedecayrate,andthemean. Shadetheareathatrepresentstheprobabilitythatonestudenthaslessthan\$.40inhis/herpocketorpurse. Shade P X < 0.40 Example5.8 Ontheaverage,acertaincomputerpartlasts10years.Thelengthoftimethecomputerpartlasts isexponentiallydistributed. Problem1 Whatistheprobabilitythatacomputerpartlastsmorethan7years? Solution Let X =theamountoftimeinyearsacomputerpartlasts. m = 10so m = 1 m = 1 10 = 0.1 Find P X > 7 .Drawagraph.

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194 CHAPTER5.CONTINUOUSRANDOMVARIABLES P X > 7 = 1 )]TJ/F132 9.9626 Tf 10.455 0 Td [(P X < 7 Since P X < x = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.322 0 Td [(mx then P X > x = 1 )]TJ/F142 10.3811 Tf 10.256 -0.105 Td [( 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m x = e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m x P X > 7 = 1 )]TJ/F132 9.9626 Tf 10.37 0 Td [(e )]TJ/F58 7.5716 Tf 6.227 0 Td [(0.1 7 = 0.4966.Theprobabilitythatacomputerpartlastsmorethan7yearsis 0.4966. N OTE :TI-83+andTI-84:Onthehomescreen,entere-.1*7. Problem2 Ontheaverage,howlongwould5computerpartslastiftheyareusedoneafteranother? Solution Ontheaverage,1computerpartlasts10years.Therefore,5computerparts,iftheyareusedone rightaftertheotherwouldlast,ontheaverage, 5 10 = 50years. Problem3 Eightypercentofcomputerpartslastatmosthowlong? Solution Findthe80thpercentile.Drawagraph.Let k =the80thpercentile. P X < k = 0.80, k = 16.1calculatororcomputer Eightypercentofthecomputerpartslastatmost16.1years. N OTE :TI-83+andTI-84:Onthehomescreen,enterLN-.80/-.1

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195 Problem4 Whatistheprobabilitythatacomputerpartlastsbetween9and11years? Solution Find P 9 < X < 11 .Drawagraph. P 9 < X < 11 = P X < 11 )]TJ/F132 9.9626 Tf 10.812 0 Td [(P X < 9 = )]TJ/F58 9.9626 Tf 4.687 -8.074 Td [(1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F58 7.5716 Tf 6.227 0 Td [(0.1 11 )]TJ/F1 9.9626 Tf 10.613 8.074 Td [()]TJ/F58 9.9626 Tf 4.688 -8.074 Td [(1 )]TJ/F132 9.9626 Tf 10.255 0 Td [(e )]TJ/F58 7.5716 Tf 6.228 0 Td [(0.1 9 = 0.6671 )]TJ/F58 9.9626 Tf 10.488 0 Td [(0.5934 = 0.0737.calculatororcomputer Theprobabilitythatacomputerpartlastsbetween9and11yearsis0.0737. N OTE :TI-83+andTI-84:Onthehomescreen,entere-.1*9-e-.1*11. Example5.9 Supposethatthelengthofaphonecall,inminutes,isanexponentialrandomvariablewithdecay parameter= 1 12 .Ifanotherpersonarrivesatapublictelephonejustbeforeyou,ndtheprobability thatyouwillhavetowaitmorethan5minutes.Let X =thelengthofaphonecall,inminutes. Problem Whatis m m ,and s ?Theprobabilitythatyoumustwaitmorethan5minutesis_______. N OTE :Asummaryforexponentialdistributionisavailablein"SummaryofTheUniformand ExponentialProbabilityDistributionsSection5.5".

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196 CHAPTER5.CONTINUOUSRANDOMVARIABLES 5.5SummaryoftheUniformandExponentialProbabilityDistributions 5 Formula5.1: Uniform X =arealnumberbetween a and b insomeinstances, X cantakeonthevalues a and b a = smallest X ; b =largest X X U a b Themeanis m = a + b 2 Thestandarddeviationis s = q b )]TJ/F132 7.5716 Tf 6.436 0 Td [(a 2 12 Probabilitydensityfunction: f X = 1 b )]TJ/F132 7.5716 Tf 6.436 0 Td [(a for a < X < b or a X b AreatotheLeft,RightandInbetween: P X < x = abaseheight P X > x = abaseheight P c < X < d = abase height = d )]TJ/F132 9.9626 Tf 10.255 0 Td [(c height Formula5.2: Exponential X Exp m X =arealnumber,0orlarger. m =theparameterthatcontrolstherateofdecayordecline Themeanandstandarddeviation arethesame. m = s = 1 m and m = 1 m = 1 s Theprobabilitydensityfunction: f X = m e )]TJ/F132 7.5716 Tf 6.323 0 Td [(m X AreatotheLeft: P X < x = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m x AreatotheRight: P X > x = e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m x AreaInbetween: P c < X < d = P X < d )]TJ/F132 9.9626 Tf 10.533 0 Td [(P X < c = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m d )]TJ/F142 10.3811 Tf 10.334 -0.105 Td [( 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m c = e )]TJ/F132 7.5716 Tf 6.322 0 Td [(m c )]TJ/F132 9.9626 Tf -419.83 -16.596 Td [(e )]TJ/F132 7.5716 Tf 6.323 0 Td [(m d Percentile,k: k = LN-AreaToTheLeft )]TJ/F132 7.5716 Tf 6.322 0 Td [(m 5 Thiscontentisavailableonlineat.

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197 5.6Practice1:UniformDistribution 6 5.6.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofdatawithauniformdistribution. 5.6.2Given Theageofcarsinthestaffparkinglotofasuburbancollegeisuniformlydistributedfromsixmonths.5 yearsto9.5years. 5.6.3DescribetheData Exercise5.6.1 Solutiononp.213. Whatisbeingmeasuredhere? Exercise5.6.2 Solutiononp.213. Inwords,denetheRandomVariable X Exercise5.6.3 Solutiononp.213. Arethedatadiscreteorcontinuous? Exercise5.6.4 Solutiononp.213. Theintervalofvaluesfor X is: Exercise5.6.5 Solutiononp.213. Thedistributionfor X is: 5.6.4ProbabilityDistribution Exercise5.6.6 Solutiononp.213. Writetheprobabilitydensityfunction. Exercise5.6.7 Solutiononp.213. Graphtheprobabilitydistribution. a. Sketchthegraphoftheprobabilitydistribution. Figure5.1 6 Thiscontentisavailableonlineat.

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198 CHAPTER5.CONTINUOUSRANDOMVARIABLES b. Identifythefollowingvalues: i. Lowestvaluefor X : ii. Highestvaluefor X : iii. Heightoftherectangle: iv. Labelforx-axiswords: v. Labelfory-axiswords: 5.6.5RandomProbability Exercise5.6.8 Solutiononp.213. Findtheprobabilitythatarandomlychosencarinthelotwaslessthan4yearsold. a. Sketchthegraph.Shadetheareaofinterest. Figure5.2 b. Findtheprobability. P X < 4 = Exercise5.6.9 Solutiononp.213. Outofjustthecarslessthan7.5yearsold,ndtheprobabilitythatarandomlychosencarinthe lotwaslessthan4yearsold. a. Sketchthegraph.Shadetheareaofinterest. Figure5.3

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199 b. Findtheprobability. P X < 4 j X < 7 5 = Exercise5.6.10:DiscussionQuestion Whathaschangedintheprevioustwoproblemsthatmadethesolutionsdifferent? 5.6.6Quartiles Exercise5.6.11 Solutiononp.213. Findtheaverageageofthecarsinthelot. Exercise5.6.12 Solutiononp.213. Findthethirdquartileofagesofcarsinthelot.Thismeansyouwillhavetondthevaluesuch that 3 4 ,or75%,ofthecarsareatmostlessthanorequaltothatage. a. Sketchthegraph.Shadetheareaofinterest. Figure5.4 b. Findthevalue k suchthat P X < k = 0 .75 c. Thethirdquartileis:

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200 CHAPTER5.CONTINUOUSRANDOMVARIABLES 5.7Practice2:ExponentialDistribution 7 5.7.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofdatawithaexponentialdistribution. 5.7.2Given Carbon-14isaradioactiveelementwithahalf-lifeofabout5730years.Carbon-14issaidtodecayexponentially.Thedecayrateis0.000121.Westartwith1gramofcarbon-14.Weareinterestedinthetimeyears ittakestodecaycarbon-14. 5.7.3DescribetheData Exercise5.7.1 Whatisbeingmeasuredhere? Exercise5.7.2 Solutiononp.214. Arethedatadiscreteorcontinuous? Exercise5.7.3 Solutiononp.214. Inwords,denetheRandomVariable X Exercise5.7.4 Solutiononp.214. Whatisthedecayrate m ? Exercise5.7.5 Solutiononp.214. Thedistributionfor X is: 5.7.4Probability Exercise5.7.6 Solutiononp.214. Findtheamountpercentof1gramofcarbon-14lastinglessthan5730years.Thismeans,nd P X < 5730 a. Sketchthegraph.Shadetheareaofinterest. Figure5.5 7 Thiscontentisavailableonlineat.

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201 b. Findtheprobability. P X < 5730 = Exercise5.7.7 Solutiononp.214. Findthepercentageofcarbon-14lastinglongerthan10,000years. a. Sketchthegraph.Shadetheareaofinterest. Figure5.6 b. Findtheprobability. P X > 10000 = Exercise5.7.8 Solutiononp.214. Thirtypercent%ofcarbon-14willdecaywithinhowmanyyears? a. Sketchthegraph.Shadetheareaofinterest. Figure5.7 b. Findthevalue k suchthat P X < k = 0 .30 .

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202 CHAPTER5.CONTINUOUSRANDOMVARIABLES 5.8Homework 8 Foreachprobabilityandpercentileproblem,DRAWTHEPICTURE! Exercise5.8.1 Considerthefollowingexperiment.Youareoneof100peopleenlistedtotakepartinastudyto determinethepercentofnursesinAmericawithanR.N.registerednursedegree.Youasknurses iftheyhaveanR.N.degree.Thenursesansweryesorno.Youthencalculatethepercentage ofnurseswithanR.N.degree.Yougivethatpercentagetoyoursupervisor. a. Whatpartoftheexperimentwillyielddiscretedata? b. Whatpartoftheexperimentwillyieldcontinuousdata? Exercise5.8.2 Whenageisroundedtothenearestyear,dothedatastaycontinuous,ordotheybecomediscrete? Why? Exercise5.8.3 Solutiononp.214. Birthsareapproximatelyuniformlydistributedbetweenthe52weeksoftheyear.Theycanbe saidtofollowaUniformDistributionfrom153spreadof52weeks. a. X b. Graphtheprobabilitydistribution. c. f x = d. m = e. s = f. Findtheprobabilitythatapersonisbornattheexactmomentweek19starts.Thatis,nd P X = 19 g. P 2 < X < 31 = h. Findtheprobabilitythatapersonisbornafterweek40. i. P 12 < X j X < 28 = j. Findthe70thpercentile. k. Findtheminimumfortheupperquarter. Exercise5.8.4 Arandomnumbergeneratorpicksanumberfrom1to9inauniformmanner. a. X~ b. Graphtheprobabilitydistribution. c. f x = d. m = e. s = f. P 3 5 < X < 7 .25 = g. P X > 5 .67 = h. P X > 5 j X > 3 = i. Findthe90thpercentile. Exercise5.8.5 Solutiononp.214. ThespeedofcarspassingthroughtheintersectionofBlossomHillRoadandtheAlmadenExpresswayvariesfrom10to35mphandisuniformlydistributed.Noneofthecarstravelover35 mphthroughtheintersection. a. X = 8 Thiscontentisavailableonlineat.

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203 b. X~ c. Graphtheprobabilitydistribution. d. f x = e. m = f. s = g. Whatistheprobabilitythatthespeedofacarisatmost30mph? h. Whatistheprobabilitythatthespeedofacarisbetween16and22mph. i. P 20 < X < 53 = Statethisinaprobabilityquestionsimilarto g and h ,drawthe picture,andndtheprobability. j. Findthe90thpercentile.Thismeansthat90%ofthetime,thespeedislessthan_____mph whilepassingthroughtheintersectionperminute. k. Findthe75thpercentile.Inacompletesentence,statewhatthismeans.See j l. Findtheprobabilitythatthespeedismorethan24mphgivenorknowingthatitisat least15mph. Exercise5.8.6 AccordingtoastudybyDr.JohnMcDougallofhislive-inweightlossprogramatSt.Helena Hospital,thepeoplewhofollowhisprogramlosebetween6and15poundsamonthuntilthey approachtrimbodyweight.Let'ssupposethattheweightlossisuniformlydistributed.Weare interestedintheweightlossofarandomlyselectedindividualfollowingtheprogramforone month.Source: TheMcDougallProgramforMaximumWeightLoss byJohnA.McDougall, M.D. a. X = b. X~ c. Graphtheprobabilitydistribution. d. f x = e. m = f. s = g. Findtheprobabilitythattheindividuallostmorethan10poundsinamonth. h. Supposeitisknownthattheindividuallostmorethan10poundsinamonth.Findthe probabilitythathelostlessthan12poundsinthemonth. i. P 7 < X < 13 j X > 9 = Statethisinaprobabilityquestionsimilartogandh,drawthe picture,andndtheprobability. Exercise5.8.7 Solutiononp.214. AsubwaytrainontheRedLinearrivesevery8minutesduringrushhour.Weareinterestedinthe lengthoftimeacommutermustwaitforatraintoarrive.Thetimefollowsauniformdistribution. a. X = b. X~ c. Graphtheprobabilitydistribution. d. f x = e. m = f. s = g. Findtheprobabilitythatthecommuterwaitslessthanoneminute. h. Findtheprobabilitythatthecommuterwaitsbetweenthreeandfourminutes. i. 60%ofcommuterswaitmorethanhowlongforthetrain?Statethisinaprobabilityquestionsimilarto g and h ,drawthepicture,andndtheprobability. Exercise5.8.8 TheageofarstgraderonSeptember1atGardenElementarySchoolisuniformlydistributed from5.8to6.8years.Werandomlyselectonerstgraderfromtheclass.

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204 CHAPTER5.CONTINUOUSRANDOMVARIABLES a. X = b. X~ c. Graphtheprobabilitydistribution. d. f x = e. m = f. s = g. Findtheprobabilitythatsheisover6.5years. h. Findtheprobabilitythatsheisbetween4and6years. i. Findthe70thpercentilefortheageofrstgradersonSeptember1atGardenElementary School. Exercise5.8.9 Solutiononp.215. Let X~ Exp.1 a. decayrate= b. m = c. Graphtheprobabilitydistributionfunction. d. Ontheabovegraph,shadetheareacorrespondingto P X < 6 andndtheprobability. e. Sketchanewgraph,shadetheareacorrespondingto P 3 < X < 6 andndtheprobability. f. Sketchanewgraph,shadetheareacorrespondingto P X > 7 andndtheprobability. g. Sketchanewgraph,shadetheareacorrespondingtothe40thpercentileandndthe value. h. Findtheaveragevalueof X Exercise5.8.10 Supposethatthelengthoflongdistancephonecalls,measuredinminutes,isknowntohavean exponentialdistributionwiththeaveragelengthofacallequalto8minutes. a. X = b. Is X continuousordiscrete? c. X~ d. m = e. s = f. Drawagraphoftheprobabilitydistribution.Labeltheaxes. g. Findtheprobabilitythataphonecalllastslessthan9minutes. h. Findtheprobabilitythataphonecalllastsmorethan9minutes. i. Findtheprobabilitythataphonecalllastsbetween7and9minutes. j. If25phonecallsaremadeoneafteranother,onaverage,whatwouldyouexpectthetotal tobe?Why? Exercise5.8.11 Solutiononp.215. Supposethattheusefullifeofaparticularcarbattery,measuredinmonths,decayswithparameter 0.025.Weareinterestedinthelifeofthebattery. a. X = b. Is X continuousordiscrete? c. X~ d. Onaverage,howlongwouldyouexpect1carbatterytolast? e. Onaverage,howlongwouldyouexpect9carbatteriestolast,iftheyareusedoneafter another? f. Findtheprobabilitythatacarbatterylastsmorethan36months.

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205 g. 70%ofthebatterieslastatleasthowlong? Exercise5.8.12 Thepercentofpersonsages5andolderineachstatewhospeakalanguageathomeotherthan Englishisapproximatelyexponentiallydistributedwithameanof9.848.Supposewerandomly pickastate.Source:BureauoftheCensus,U.S.Dept.ofCommerce a. X = b. Is X continuousordiscrete? c. X~ d. m = e. s = f. Drawagraphoftheprobabilitydistribution.Labeltheaxes. g. Findtheprobabilitythatthepercentislessthan12. h. Findtheprobabilitythatthepercentisbetween8and14. i. ThepercentofallindividualslivingintheUnitedStateswhospeakalanguageathome otherthanEnglishis13.8. i. Whyisthisnumberdifferentfrom9.848%? ii. Whatwouldmakethisnumberhigherthan9.848%? Exercise5.8.13 Solutiononp.215. Thetimeinyears after reachingage60thatittakesanindividualtoretireisapproximately exponentiallydistributedwithameanofabout5years.Supposewerandomlypickoneretired individual.Weareinterestedinthetimeafterage60toretirement. a. X = b. Is X continuousordiscrete? c. X~ d. m = e. s = f. Drawagraphoftheprobabilitydistribution.Labeltheaxes. g. Findtheprobabilitythatthepersonretiredafterage70. h. Domorepeopleretirebeforeage65orafterage65? i. Inaroomof1000peopleoverage80,howmanydoyouexpectwillNOThaveretiredyet? Exercise5.8.14 Thecostofallmaintenanceforacarduringitsrstyearisapproximatelyexponentiallydistributedwithameanof\$150. a. X = b. X~ c. m = d. s = e. Drawagraphoftheprobabilitydistribution.Labeltheaxes. f. Findtheprobabilitythatacarrequiredover\$300formaintenanceduringitsrstyear.

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206 CHAPTER5.CONTINUOUSRANDOMVARIABLES 5.8.1Trythesemultiplechoiceproblems Thenextthreequestionsrefertothefollowinginformation. Theaveragelifetimeofacertainnewcellphone is3years.Themanufacturerwillreplaceanycellphonefailingwithin2yearsofthedateofpurchase.The lifetimeofthesecellphonesisknowntofollowanexponentialdistribution. Exercise5.8.15 Solutiononp.215. Thedecayrateis A. 0.3333 B. 0.5000 C. 2.0000 D. 3.0000 Exercise5.8.16 Solutiononp.215. Whatistheprobabilitythataphonewillfailwithin2yearsofthedateofpurchase? A. 0.8647 B. 0.4866 C. 0.2212 d. 0.9997 Exercise5.8.17 Solutiononp.215. Whatisthemedianlifetimeofthesephonesinyears? A. 0.1941 B. 1.3863 C. 2.0794 D. 5.5452 Thenextthreequestionsrefertothefollowinginformation. TheSkyTrainfromtheterminaltotherental carandlongtermparkingcenterissupposedtoarriveevery8minutes.Thewaitingtimesforthetrainare knowntofollowauniformdistribution. Exercise5.8.18 Solutiononp.215. Whatistheaveragewaitingtimeinminutes? A. 0.0000 B. 2.0000 C. 3.0000 D. 4.0000 Exercise5.8.19 Solutiononp.215. Findthe30thpercentileforthewaitingtimesinminutes. A. 2.0000 B. 2.4000 C. 2.750 D. 3.000 Exercise5.8.20 Solutiononp.215. Theprobabilityofwaitingmorethan7minutesgivenapersonhaswaitedmorethan4minutes is? A. 0.1250

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207 B. 0.2500 C. 0.5000 D. 0.7500 5.9Review 9 Exercise5.9.1Exercise5.9.7refertothefollowingstudy: Arecentstudyofmothersofjuniorhighschool childreninSantaClaraCountyreportedthat76%ofthemothersareemployedinpaidpositions.Ofthose motherswhoareemployed,64%workfull-timeover35hoursperweek,and36%workpart-time.However,outofallofthemothersinthepopulation,49%workfull-time.Thepopulationunderstudyismade upofmothersofjuniorhighschoolchildreninSantaClaraCounty. Let E = employed,Let F = full-timeemployment Exercise5.9.1 Solutiononp.215. a. FindthepercentofallmothersinthepopulationthatNOTemployed. b. Findthepercentofmothersinthepopulationthatareemployedpart-time. Exercise5.9.2 Solutiononp.216. Thetypeofemploymentisconsideredtobewhattypeofdata? Exercise5.9.3 Solutiononp.216. Insymbols,whatdoesthe36%represent? Exercise5.9.4 Solutiononp.216. FindtheprobabilitythatarandomlyselectedpersonfromthepopulationwillbeemployedOR workfull-time. Exercise5.9.5 Solutiononp.216. Basedupontheaboveinformation,arebeingemployedANDworkingpart-time: a. mutuallyexclusiveevents?Whyorwhynot? b. independentevents?Whyorwhynot? Exercise5.9.6-Exercise5.9.7refertothefollowing: Werandomlypick10mothersfromtheabovepopulation.Weareinterestedinthenumberofthemothersthatareemployed.Let X = numberofmothersthat areemployed. Exercise5.9.6 Solutiononp.216. Statethedistributionfor X Exercise5.9.7 Solutiononp.216. Findtheprobabilitythatatleast6areemployed. Exercise5.9.8 Solutiononp.216. WeexpecttheStatisticsDiscussionBoardtohave,onaverage,14questionspostedtoitperweek. Weareinterestedinthenumberofquestionspostedtoitperday. a. Dene X b. Whatarethevaluesthattherandomvariablemaytakeon? c. Statethedistributionfor X d. Findtheprobabilitythatfrom10to14inclusivequestionsarepostedtotheListservon arandomlypickedday. 9 Thiscontentisavailableonlineat.

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208 CHAPTER5.CONTINUOUSRANDOMVARIABLES Exercise5.9.9 Solutiononp.216. Apersoninvests\$1000instockofacompanythathopestogopublicin1year. Theprobabilitythatthepersonwillloseallhismoneyafter1yeari.e.hisstockwillbe worthlessis35%. Theprobabilitythattheperson'sstockwillstillhaveavalueof\$1000after1yeari.e.no protandnolossis60%. Theprobabilitythattheperson'sstockwillincreaseinvalueby\$10,000after1yeari.e.will beworth\$11,000is5%. FindtheexpectedPROFITafter1year. Exercise5.9.10 Solutiononp.216. Rachel'spianocost\$3000.Theaveragecostforapianois\$4000withastandarddeviationof \$2500.Becca'sguitarcost\$550.Theaveragecostforaguitaris\$500withastandarddeviation of\$200.Matt'sdrumscost\$600.Theaveragecostfordrumsis\$700withastandarddeviationof \$100.Whosecostwaslowestwhencomparedtohisorherowninstrument?Justifyyouranswer. Exercise5.9.11 Solutiononp.216. Forthefollowingdata,whichofthemeasuresofcentraltendencywouldbetheLEASTuseful: mean,median,mode?Explainwhy.WhichwouldbetheMOSTuseful?Explainwhy. 4,6,6, 12 18 18 18 200 Exercise5.9.12 Solutiononp.216. Foreachstatementbelow,explainwhyeachiseithertrueorfalse. a. 25%ofthedataareatmost5. b. Thereisthesameamountofdatafrom45asthereisfrom57. c. Therearenodatavaluesof3. d. 50%ofthedataare4. Exercise5.9.13Exercise5.9.14refertothefollowing: 64facultymemberswereaskedthenumberof carstheyownedincludingspouseandchildren'scars.Theresultsaregiveninthefollowinggraph: Exercise5.9.13 Solutiononp.216. Findtheapproximatenumberofresponsesthatwere. Exercise5.9.14 Solutiononp.216. Findtherst,secondandthirdquartiles.Usethemtoconstructaboxplotofthedata.

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209 Exercise5.9.15Exercise5.9.16refertothefollowingstudydoneoftheGirlssoccerteamSnowLeopards: HairStyle HairColor blond brown black ponytail 3 2 5 plain 2 2 1 SupposethatonegirlfromtheSnowLeopardsisrandomlyselected. Exercise5.9.15 Solutiononp.216. FindtheprobabilitythatthegirlhasblackhairGIVENthatshewearsaponytail. Exercise5.9.16 Solutiononp.216. FindtheprobabilitythatthegirlwearsherhairplainORhasbrownhair. Exercise5.9.17 Solutiononp.216. FindtheprobabilitythatthegirlhasblondhairANDthatshewearsherhairplain.

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210 CHAPTER5.CONTINUOUSRANDOMVARIABLES 5.10Lab:ContinuousDistribution 10 ClassTime: Names: 5.10.1StudentLearningOutcomes: Thestudentwillcompareandcontrastempiricaldatafromarandomnumbergeneratorwiththe UniformDistribution. 5.10.2CollecttheData Usearandomnumbergeneratortogenerate50valuesbetween0and1inclusive.Listthembelow.Round thenumbersto4decimalplacesorsetthecalculatorMODEto4places. 1.Completethetable: __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 2.Calculatethefollowing: a. x = b. s = c. 40thpercentile= d. 3rdquartile= e. Median= 5.10.3OrganizingtheData 1.Constructahistogramoftheempiricaldata.Make8bars. 10 Thiscontentisavailableonlineat.

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211 Figure5.8 2.Constructahistogramoftheempiricaldata.Make5bars. Figure5.9

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212 CHAPTER5.CONTINUOUSRANDOMVARIABLES 5.10.4DescribetheData 1.Describetheshapeofeachgraph.Use23completesentences.Keepitsimple.Doesthegraphgo straightacross,doesithaveaVshape,doesithaveahumpinthemiddleorateitherend,etc.?One waytohelpyoudetermineashape,istoroughlydrawasmoothcurvethroughthetopofthebars. 2.Describehowchangingthenumberofbarsmightchangetheshape. 5.10.5TheoreticalDistribution 1.Inwords, X = 2.Thetheoreticaldistributionof X is X U 0,1 .Useitforthispart. 3.Intheory,baseduponthedistributioninthesectiontitled"OrganizingtheData", a. m = b. s = c. 40thpercentile= d. 3rdquartile= e. median=__________ 4.Aretheempiricalvaluesthedatainthesectiontitled"CollecttheData"closetothecorresponding theoreticalvaluesabove?Whyorwhynot? 5.10.6PlottheData 1.Constructaboxplotofthedata.Besuretousearulertoscaleaccuratelyanddrawstraightedges. 2.Doyounoticeanypotentialoutliers?Ifso,whichvaluesarethey?Eitherway,numericallyjustifyyour answer.RecallthatanyDATAarelessthanQ11.5*IQRormorethanQ3+1.5*IQRarepotential outliers.IQRmeansinterquartilerange. 5.10.7ComparingtheData 1.Foreachpartbelow,useacompletesentencetocommentonhowthevalueobtainedfromthedata comparestothetheoreticalvalueyouexpectedfromthedistributioninthesectiontitled"Theoretical Data". a. minimumvalue: b. 40thpercentile: c. median: d. thirdquartile: e. maximumvalue: f. widthofIQR: g. overallshape: 2.Basedonyourcommentsinthesectiontitled"CollecttheData",howdoestheboxplottornott whatyouwouldexpectofthedistributioninthesectiontitled"TheoreticalDistribution"? 5.10.8DiscussionQuestion 1.Supposethatthenumberofvaluesgeneratedwas500,not50.Howwouldthataffectwhatyouwould expecttheempiricaldatatobeandtheshapeofitsgraphtolooklike?

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213 SolutionstoExercisesinChapter5 Example5.5,Problem1p.189 0.5714 Example5.5,Problem2p.190 4 5 Example5.9p.195 m = 1 12 m =12 s =12 P X > 5 = 0.6592 SolutionstoPractice1:UniformDistribution SolutiontoExercise5.6.1p.197 Theageofcarsinthestaffparkinglot SolutiontoExercise5.6.2p.197 X =Theageinyearsofcarsinthestaffparkinglot SolutiontoExercise5.6.3p.197 Continuous SolutiontoExercise5.6.4p.197 0.5-9.5 SolutiontoExercise5.6.5p.197 X U 0 5,9 5 SolutiontoExercise5.6.6p.197 f x = 1 9 SolutiontoExercise5.6.7p.197 b.i. 0.5 b.ii. 9.5 b.iii. AgeofCars b.iv. 1 9 b.v. f x SolutiontoExercise5.6.8p.198 b.3 5 9 SolutiontoExercise5.6.9p.198 b3 5 7 SolutiontoExercise5.6.11p.199 m =5 SolutiontoExercise5.6.12p.199 b. k =7.25

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214 CHAPTER5.CONTINUOUSRANDOMVARIABLES SolutionstoPractice2:ExponentialDistribution SolutiontoExercise5.7.2p.200 Continuous SolutiontoExercise5.7.3p.200 X =Timeyearstodecaycarbon-14 SolutiontoExercise5.7.4p.200 m =0.000121 SolutiontoExercise5.7.5p.200 X Exp.000121 SolutiontoExercise5.7.6p.200 b. P X < 5730 =0.5001 SolutiontoExercise5.7.7p.201 b. P X > 10000 =0.2982 SolutiontoExercise5.7.8p.201 b. k =2947.73 SolutionstoHomework SolutiontoExercise5.8.3p.202 a. X~U 1, 53 c. f x = 1 52 where1 x 53 d. 27 e. 15.01 f. 0 g. 29 52 h. 13 52 i. 16 27 j. 37.4 k. 40 SolutiontoExercise5.8.5p.202 b. X~U 10 35 d. f x = 1 25 where 10 X 35 e. 45 2 f. 7.22 g. 4 5 h. 6 25 i. 3 5 j. 32.5 k. 28.75 l. 11 20 SolutiontoExercise5.8.7p.203 b. X~U 0,8

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215 d. f x = 1 8 where 0 X 8 e. 4 f. 2.31 g. 1 8 h. 1 8 i. 3.2 SolutiontoExercise5.8.9p.204 a. 0.1 b. 10 d. 0.4512 e. 0.1920 f. 0.4966 g. 5.11 h. 10 SolutiontoExercise5.8.11p.204 c. X~Exp 0.025 d. 40months e. 360months f. 0.4066 g. 14.27 SolutiontoExercise5.8.13p.205 c. X~Exp 1 5 d. 5 e. 5 g. 0.1353 h. Before i. 18.3 SolutiontoExercise5.8.15p.206 A SolutiontoExercise5.8.16p.206 B SolutiontoExercise5.8.17p.206 C SolutiontoExercise5.8.18p.206 D SolutiontoExercise5.8.19p.206 B SolutiontoExercise5.8.20p.206 B SolutionstoReview SolutiontoExercise5.9.1p.207 a. 24%

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216 CHAPTER5.CONTINUOUSRANDOMVARIABLES b. 27% SolutiontoExercise5.9.2p.207 Qualitative SolutiontoExercise5.9.3p.207 P PT j E SolutiontoExercise5.9.4p.207 0.7336 SolutiontoExercise5.9.5p.207 a. No, b. No, SolutiontoExercise5.9.6p.207 B 10 ,0 .76 SolutiontoExercise5.9.7p.207 0.9330 SolutiontoExercise5.9.8p.207 a. X = thenumberofquestionspostedtotheStatisticsListservperday b. x = 0,1,2, ... c. X~P 2 d. 0 SolutiontoExercise5.9.9p.208 \$150 SolutiontoExercise5.9.10p.208 Matt SolutiontoExercise5.9.11p.208 Mean SolutiontoExercise5.9.12p.208 a. False b. True c. False d. False SolutiontoExercise5.9.13p.208 16 SolutiontoExercise5.9.14p.208 2,2,3 SolutiontoExercise5.9.15p.209 5 10 = 0 5 SolutiontoExercise5.9.16p.209 7 15 SolutiontoExercise5.9.17p.209 2 15

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Chapter6 TheNormalDistribution 6.1TheNormalDistribution 1 6.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Recognizethenormalprobabilitydistributionandapplyitappropriately. Recognizethestandardnormalprobabilitydistributionandapplyitappropriately. Comparenormalprobabilitiesbyconvertingtothestandardnormaldistribution. 6.1.2Introduction Thenormal,acontinuousdistribution,isthemostimportantofallthedistributions.Itiswidelyused andevenmorewidelyabused.Itsgraphisbell-shaped.Youseethebellcurveinalmostalldisciplines. Someoftheseincludepsychology,business,economics,thesciences,nursing,and,ofcourse,mathematics. Someofyourinstructorsmayusethenormaldistributiontohelpdetermineyourgrade.MostIQscoresare normallydistributed.Oftenrealestatepricestanormaldistribution.Thenormaldistributionisextremely importantbutitcannotbeappliedtoeverythingintherealworld. Inthischapter,youwillstudythenormaldistribution,thestandardnormal,andmanyapplicationassociatedwiththem. 6.1.3OptionalCollaborativeClassroomActivity Yourinstructorwillrecordtheheightsofbothmenandwomeninyourclass,separately.Drawhistograms ofyourdata.Thendrawasmoothcurvethrougheachhistogram.Iseachcurvesomewhatbell-shaped?Do youthinkthatifyouhadrecorded200datavaluesformenand200forwomenthatthecurveswouldlook bell-shaped?Calculatethemeanforeachdataset.Writethemeansonthex-axisoftheappropriategraph belowthepeak.Shadetheapproximateareathatrepresentstheprobabilitythatonerandomlychosen maleistallerthan72inches.Shadetheapproximateareathatrepresentstheprobabilitythatonerandomly chosenfemaleisshorterthan60inches.Ifthetotalareaundereachcurveisone,doeseitherprobability appeartobemorethan0.5? Thenormaldistributionhastwoparameterstwonumericaldescriptivemeasures,themean m andthe standarddeviation s 1 Thiscontentisavailableonlineat. 217

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218 CHAPTER6.THENORMALDISTRIBUTION NORMAL: X N m s X =aquantitytobemeasured.Theprobabilitydistributionfunctionisarathercomplicatedfunction. Do notmemorizeit .Itisnotnecessary. f x = 1 s p 2 p e )]TJ/F58 5.9776 Tf 7.423 2.983 Td [(1 2 x )]TJ/F134 5.9776 Tf 4.991 0 Td [(m s 2 Thecumulativedistributionfunctionis P X < x Itiscalculatedeitherbyacalculatororacomputerorit islookedupinatable Thecurveissymmetricalaboutaverticallinedrawnthroughthemean, m .Intheory,themeanisthesame asthemediansincethegraphissymmetricabout m .Asthenotationindicates,thenormaldistribution dependsonlyonthemeanandthestandarddeviation.Sincetheareaunderthecurvemustequalone,a changeinthestandarddeviation, s ,causesachangeintheshapeofthecurve;thecurvebecomesfatteror skinnierdependingon s .Achangein m causesthegraphtoshifttotheleftorright.Thismeansthereare aninnitenumberofnormalprobabilitydistributions.Oneofspecialinterestiscalledthestandard normal distribution 6.2TheStandardNormalDistribution 2 The standardnormaldistribution isanormaldistributionof standardizedvaluescalled z-scores Az-score ismeasuredinunitsofthestandarddeviation. Forexample,ifthemeanofanormaldistributionis5and thestandarddeviationis2,thevalue11is3standarddeviationsaboveortotherightofthemean.The calculationis: x = m + z s = 5 + 3 2 = 11.1 Thez-scoreis3. Themeanforthestandardnormaldistributionis0andthestandarddeviationis1.Thetransformation z = x )]TJ/F134 7.5716 Tf 6.322 0 Td [(m s producesthedistribution Z N 0,1 Thevalue x comesfromanormaldistributionwith mean m andstandarddeviation s 6.3Z-scores 3 If X isanormallydistributedrandomvariableand X N m s ,thenthez-scoreis: z = x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m s .2 2 Thiscontentisavailableonlineat. 3 Thiscontentisavailableonlineat.

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219 Thez-scoretellsyouhowmanystandarddeviationsthatthevalue x isabovetotherightoforbelowto theleftofthemean, m Valuesof x thatarelargerthanthemeanhavepositivez-scoresandvaluesof x that aresmallerthanthemeanhavenegativez-scores. Example6.1 Suppose X N 5,6 .Thissaysthat X isanormallydistributedrandomvariablewithmean m = 5 andstandarddeviation s = 6.Suppose x = 17.Then: z = x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m s = 17 )]TJ/F58 9.9626 Tf 10.132 0 Td [(5 6 = 2.3 Thismeansthat x = 17is 2standarddeviations 2 s aboveortotherightofthemean m = 5.The standarddeviationis s = 6. Noticethat: 5 + 2 6 = 17 Thepatternis m + z s = x .4 Nowsuppose x = 1.Then: z = x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m s = 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(5 6 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.67 roundedtotwodecimalplaces .5 Thismeansthat x = 1 is0.67standarddeviations )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.67 s belowortotheleftofthemean m = 5 Noticethat: 5 + )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.67 6 isapproximatelyequalto1Thishasthepattern m + )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.67 s = 1 Summarizing,when z ispositive, x isaboveortotherightof m andwhen z isnegative, x istothe leftoforbelow m Example6.2 Somedoctorsbelievethatapersoncanlose5pounds,ontheaverage,inamonthbyreducing his/herfatintakeandbyexercisingconsistently.Supposeweightlosshasanormaldistribution. Let X =theamountofweightlostinpoundsbyapersoninamonth.Useastandarddeviation of2pounds. X N 5,2 .Fillintheblanks. Problem1 Supposeaperson lost 10poundsinamonth.Thez-scorewhen x = 10poundsis z = 2.5verify. Thisz-scoretellsyouthat x = 10is________standarddeviationstothe________rightorleftof themean_____Whatisthemean?. Problem2 Supposeaperson gained 3poundsanegativeweightloss.Then z =__________.Thisz-score tellsyouthat x = )]TJ/F58 9.9626 Tf 8.194 0 Td [(3is________standarddeviationstothe__________rightorleftofthemean. Supposetherandomvariables X and Y havethefollowingnormaldistributions: X N 5,6 and Y N 2,1 .If x = 17,then z = 2.Thiswaspreviouslyshown.If y = 4,whatis z ? z = y )]TJ/F134 9.9626 Tf 10.256 0 Td [(m s = 4 )]TJ/F58 9.9626 Tf 10.131 0 Td [(2 1 = 2 where m =2and s =1. .6 Thez-scorefor y = 4is z = 2.Thismeansthat4is z = 2standarddeviationstotherightofthe mean.Therefore, x = 17and y = 4areboth2of their standarddeviationstotherightof their respectivemeans. Thez-scoreallowsustocomparedatathatarescaleddifferently. Tounderstandtheconcept, suppose X N 5,6 representsweightgainsforonegroupofpeoplewhoaretryingtogainweight ina6weekperiodand Y N 2,1 measuresthesameweightgainforasecondgroupofpeople.

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220 CHAPTER6.THENORMALDISTRIBUTION Anegativeweightgainwouldbeaweightloss.Since x = 17and y = 4areeach2standard deviationstotherightoftheirmeans,theyrepresentthesameweightgain inrelationshiptotheir means 6.4AreastotheLeftandRightofx 4 Thearrowinthegraphbelowpointstotheareatotheleftof x .Thisareaisrepresentedbytheprobability P X < x Normaltables,computers,andcalculatorsprovideorcalculatetheprobability P X < x Theareatotherightisthen P X > x = 1 )]TJ/F132 9.9626 Tf 10.455 0 Td [(P X < x Remember, P X < x = Areatotheleft oftheverticallinethrough x P X > x = 1 )]TJ/F132 9.9626 Tf 10.456 0 Td [(P X < x = Areatotheright oftheverticallinethrough x P X < x isthesameas P X x and P X > x isthesameas P X x forcontinuousdistributions. 6.5CalculationsofProbabilities 5 Probabilitiesarecalculatedbyusingtechnology.ThereareinstructionsinthechapterfortheTI-83+and TI-84calculators. Example6.3 Iftheareatotheleftis0.0228,thentheareatotherightis1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.0228 = 0.9772. Example6.4 Thenalexamscoresinastatisticsclasswerenormallydistributedwithameanof63anda standarddeviationof5. Problem1 Findtheprobabilitythatarandomlyselectedstudentscoredmorethan65ontheexam. Solution Let X = ascoreonthenalexam. X N 63,5 ,where m = 63and s = 5 Drawagraph. Then,nd P X > 65 P X > 65 = 0.3446calculatororcomputer 4 Thiscontentisavailableonlineat. 5 Thiscontentisavailableonlineat.

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221 Theprobabilitythatonestudentscoresmorethan65is0.3446. UsingtheTI-83+ortheTI-84calculators,thecalculationisasfollows.Gointo 2ndDISTR Afterpressing 2ndDISTR ,press 2:normalcdf Thesyntaxfortheinstructionsareshownbelow. normalcdflowervalue,uppervalue,mean,standarddeviationForthisproblem:normalcdf,1E99,63,5=0.3446.Youget1E99=10 99 bypressing 1 ,the EE keya2ndkeyandthen 99 Or,youcanenter 10 instead.Thenumber10 99 iswayoutintherighttailofthenormalcurve. Wearecalculatingtheareabetween65and10 99 .Insomeinstances,thelowernumberofthearea mightbe-1E99= )]TJ/F58 9.9626 Tf 8.194 0 Td [(10 99 .Thenumber )]TJ/F58 9.9626 Tf 8.194 0 Td [(10 99 iswayoutinthelefttailofthenormalcurve. H ISTORICAL N OTE :TheTIprobabilityprogramcalculatesaz-scoreandthentheprobabilityfrom thez-score.Beforetechnology,thez-scorewaslookedupinastandardnormalprobabilitytable becausethemathinvolvedistoocumbersometondtheprobability.Inthisexample,astandard normaltablewithareatotheleftofthez-scorewasused.Youcalculatethez-scoreandlookup theareatotheleft.Theprobabilityistheareatotheright. z = 65 )]TJ/F58 7.5716 Tf 6.227 0 Td [(63 5 = 0.4.Areatotheleftis0.6554. P X > 65 = P Z > 0.4 = 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.6554 = 0.3446 Problem2 Findtheprobabilitythatarandomlyselectedstudentscoredlessthan85. Solution Drawagraph. Thennd P X < 85 .Shadethegraph. P X < 85 = 1calculatororcomputer Theprobabilitythatonestudentscoreslessthan85isapproximately1or100%. TheTI-instructionsandanswerareasfollows: normalcdf,85,63,5=1roundsto1 Problem3 Findthe90thpercentilethatis,ndthescorekthathas90%ofthescoresbelowkand10%of thescoresabovek. Solution Findthe90thpercentile.Foreachproblemorpartofaproblem,drawanewgraph.Drawthe x-axis.Shadetheareathatcorrespondstothe90thpercentile. Let k =the90thpercentile. k islocatedonthex-axis. P X < k istheareatotheleftof k .The90th percentile k separatestheexamscoresintothosethatarethesameorlowerthan k andthosethat arethesameorhigher.Ninetypercentofthetestscoresarethesameorlowerthan k and10%are thesameorhigher. k isoftencalleda criticalvalue .

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222 CHAPTER6.THENORMALDISTRIBUTION k = 69.4calculatororcomputer The90thpercentileis69.4.Thismeansthat90%ofthetestscoresfallatorbelow69.4and10%fall atorabove.FortheTI-83+orTI-84calculators,use invNorm in 2ndDISTR .invNormareatothe left,mean,standarddeviationForthisproblem,invNorm.90,63,5=69.4 Problem4 Findthe70thpercentilethatis,ndthescoreksuchthat70%ofscoresarebelowkand30%of thescoresareabovek. Solution Findthe70thpercentile. Drawanewgraphandlabelitappropriately. k = 65.6 The70thpercentileis65.6.Thismeansthat70%ofthetestscoresfallatorbelow65.5and30%fall atorabove. invNorm.70,63,5=65.6 Example6.5 MoreandmorehouseholdsintheUnitedStateshaveatleastonecomputer.Thecomputeris usedforofceworkathome,research,communication,personalnances,education,entertainment,andamyriadofotherthings.Supposetheaveragenumberofhoursahouseholdpersonal computerisusedforentertainmentis2hoursperday.Assumethetimesforentertainmentare normallydistributedandthestandarddeviationforthetimesishalfanhour. Problem1 Findtheprobabilitythatahouseholdpersonalcomputerisusedbetween1.8and2.75hoursper day. Solution Let X =theamountoftimeinhoursahouseholdpersonalcomputerisusedforentertainment. X N 2,0.5 where m = 2and s = 0.5. Find P 1.8 < X < 2.75 Theprobabilityforwhichyouarelookingisthearea between x = 1.8and x = 2.75. P 1.8 < X < 2.75 = 0.5886

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223 normalcdf.8,2.75,2,.5=0.5886 Theprobabilitythatahouseholdpersonalcomputerisusedbetween1.8and2.75hoursperday forentertainmentis0.5886. Problem2 Findthemaximumnumberofhoursperdaythatthebottomquartileofhouseholdsuseapersonal computerforentertainment. Solution Tondthemaximumnumberofhoursperdaythatthebottomquartileofhouseholdsusesa personalcomputerforentertainment, ndthe25thpercentile, k ,where P X < k = 0.25. invNorm.25,2,.5=1.67 Themaximumnumberofhoursperdaythatthebottomquartileofhouseholdsusesapersonal computerforentertainmentis1.67hours.

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224 CHAPTER6.THENORMALDISTRIBUTION 6.6SummaryofFormulas 6 Formula6.1: NormalProbabilityDistribution X N m s m =themean s =thestandarddeviation Formula6.2: StandardNormalProbabilityDistribution Z N 0,1 Z =astandardizedvaluez-score mean=0standarddeviation=1 Formula6.3: FindingthekthPercentile Tondthe kth percentilewhenthez-scoreisknown: k = m + z s Formula6.4: z-score z = x )]TJ/F134 7.5716 Tf 6.323 0 Td [(m s Formula6.5: Findingtheareatotheleft Theareatotheleft: P X < x Formula6.6: Findingtheareatotheright Theareatotheright: P X > x = 1 )]TJ/F132 9.9626 Tf 10.455 0 Td [(P X < x 6 Thiscontentisavailableonlineat.

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225 6.7Practice:TheNormalDistribution 7 6.7.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofdatawithanormaldistribution. 6.7.2Given ThelifeofSunshineCDplayersisnormallydistributedwithameanof4.1yearsandastandarddeviation of1.3years.ACDplayerisguaranteedfor3years.WeareinterestedinthelengthoftimeaCDplayer lasts. 6.7.3NormalDistribution Exercise6.7.1 DenetheRandomVariable X inwords. X = Exercise6.7.2 X Exercise6.7.3 Solutiononp.240. FindtheprobabilitythataCDplayerwillbreakdownduringtheguaranteeperiod. a. Sketchthesituation.Labelandscaletheaxes.Shadetheregioncorrespondingtothe probability. Figure6.1 b. P 0 < X < _________ = _________ Exercise6.7.4 Solutiononp.240. FindtheprobabilitythataCDplayerwilllastbetween2.8and6years. a. Sketchthesituation.Labelandscaletheaxes.Shadetheregioncorrespondingtothe probability. 7 Thiscontentisavailableonlineat.

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226 CHAPTER6.THENORMALDISTRIBUTION Figure6.2 b. P _______ < X < _______ = _________ Exercise6.7.5 Solutiononp.240. Findthe70thpercentileofthedistributionforthetimeaCDplayerlasts. a. Sketchthesituation.Labelandscaletheaxes.Shadetheregioncorrespondingtothe lower70%. Figure6.3 b. P X < k = _________ .Therefore, k = __________ .

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227 6.8Homework 8 Exercise6.8.1 Solutiononp.240. AccordingtoastudydonebyDeAnzastudents,theheightforAsianadultmalesisnormally distributedwithanaverageof66inchesandastandarddeviationof2.5inches.Supposeone Asianadultmaleisrandomlychosen.Let X = heightoftheindividual. a. X ______________,_______ b. Findtheprobabilitythatthepersonisbetween65and69inches.Includeasketchofthe graphandwriteaprobabilitystatement. c. WouldyouexpecttomeetmanyAsianadultmalesover72inches?Explainwhyorwhy not,andjustifyyouranswernumerically. d. Themiddle40%ofheightsfallbetweenwhattwovalues?Sketchthegraphandwritethe probabilitystatement. Exercise6.8.2 IQisnormallydistributedwithameanof100andastandarddeviationof15.Supposeone individualisrandomlychosen.Let X = IQofanindividual. a. X ______________,_______ b. FindtheprobabilitythatthepersonhasanIQgreaterthan120.Includeasketchofthe graphandwriteaprobabilitystatement. c. Mensaisanorganizationwhosemembershavethetop2%ofallIQs.Findtheminimum IQneededtoqualifyfortheMensaorganization.Sketchthegraphandwritetheprobabilitystatement. d. Themiddle50%ofIQsfallbetweenwhattwovalues?Sketchthegraphandwritethe probabilitystatement. Exercise6.8.3 Solutiononp.240. ThepercentoffatcaloriesthatapersoninAmericaconsumeseachdayisnormallydistributed withameanofabout36andastandarddeviationof10.Supposethatoneindividualisrandomly chosen.Let X = percentoffatcalories. a. X ______________,_______ b. Findtheprobabilitythatthepercentoffatcaloriesapersonconsumesismorethan40. Graphthesituation.Shadeintheareatobedetermined. c. Findthemaximumnumberforthelowerquarterofpercentoffatcalories.Sketchthe graphandwritetheprobabilitystatement. Exercise6.8.4 Supposethatthedistanceofyballshittotheouteldinbaseballisnormallydistributedwith ameanof250feetandastandarddeviationof50feet. a. If X = distanceinfeetforayball,then X ______________,_______ b. Ifoneyballisrandomlychosenfromthisdistribution,whatistheprobabilitythatthis balltraveledfewerthan220feet?Sketchthegraph.ScalethehorizontalaxisX.Shade theregioncorrespondingtotheprobability.Findtheprobability. c. Findthe80thpercentileofthedistributionofyballs.Sketchthegraphandwritethe probabilitystatement. 8 Thiscontentisavailableonlineat.

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229 Exercise6.8.9 Solutiononp.240. ThuyDau,NgocBui,SamSu,andLanVoungconductedasurveyastohowlongcustomersat Luckyclaimedtowaitinthecheckoutlineuntiltheirturn.Let X = timeinline.Belowarethe orderedrealdatainminutes: 0.50 4.25 5 6 7.25 1.75 4.25 5.25 6 7.25 2 4.25 5.25 6.25 7.25 2.25 4.25 5.5 6.25 7.75 2.25 4.5 5.5 6.5 8 2.5 4.75 5.5 6.5 8.25 2.75 4.75 5.75 6.5 9.5 3.25 4.75 5.75 6.75 9.5 3.75 5 6 6.75 9.75 3.75 5 6 6.75 10.75 a. Calculatethesamplemeanandthesamplestandarddeviation. b. Constructahistogram.Startthe x )]TJ/F132 9.9626 Tf 10.131 0 Td [(axis at )]TJ/F58 9.9626 Tf 8.194 0 Td [(0 .375 andmakebarwidthsof2minutes. c. Drawasmoothcurvethroughthemidpointsofthetopsofthebars. d. Inwords,describetheshapeofyourhistogramandsmoothcurve. e. Letthesamplemeanapproximate m andthesamplestandarddeviationapproximate s Thedistributionof X canthenbeapproximatedby X f. Usethedistributioninetocalculatetheprobabilitythatapersonwillwaitfewerthan 6.1minutes. g. Determinethecumulativerelativefrequencyforwaitinglessthan6.1minutes. h. Whyaren'ttheanswerstofandgexactlythesame? i. Whyaretheanswerstofandgascloseastheyare? j. Ifonly10customersweresurveyedinsteadof50,doyouthinktheanswerstofandg wouldhavebeenclosertogetherorfartherapart?Explainyourconclusion. Exercise6.8.10 SupposethatRicardoandAnitaattenddifferentcolleges.Ricardo'sGPAisthesameastheaverageGPAathisschool.Anita'sGPAis0.70standarddeviationsaboveherschoolaverage.In completesentences,explainwhyeachofthefollowingstatementsmaybefalse. a. Ricardo'sactualGPAislowerthanAnita'sactualGPA. b. Ricardoisnotpassingsincehisz-scoreiszero. c. Anitaisinthe70thpercentileofstudentsathercollege. Exercise6.8.11 Solutiononp.241. BelowisthenumberofAIDScasesforSantaClaraCountybyyearofdiagnosis.

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230 CHAPTER6.THENORMALDISTRIBUTION Year #cases Year #cases Year #cases Year #cases 1983 10 1990 225 1997 170 2004 95 1984 26 1991 243 1998 137 2005 98 1985 60 1992 357 1999 137 2006 112 1986 76 1993 382 2000 121 2007 81 1987 134 1994 277 2001 119 1988 151 1995 249 2002 131 1989 175 1996 197 2003 116 a. CalculatethesamplemeanandthesamplestandarddeviationforthenumberofAIDS casesthedata. b. Constructahistogramofthedata. c. Drawasmoothcurvethroughthemidpointsofthetopsofthebarsofthehistogram. d. Inwords,describetheshapeofyourhistogramandsmoothcurve. e. Letthesamplemeanapproximate m andthesamplestandarddeviationapproximate s Thedistributionof X canthenbeapproximatedby X f. UsethedistributioninetocalculatetheprobabilitythatthenumberofAIDScasesisless than150. g. DeterminethecumulativerelativefrequencythatthenumberofAIDScasesislessthan 150.Hint:OrderthedataandcountthenumberofAIDScasesthatarelessthan150. DividebythetotalnumberofAIDScases. h. Whyaren'ttheanswerstofandgexactlythesame? 6.8.1TryTheseMultipleChoiceQuestions Thequestionsbelowrefertothefollowing: Thepatientrecoverytimefromaparticularsurgicalprocedure isnormallydistributedwithameanof5.3daysandastandarddeviationof2.1days. Exercise6.8.12 Solutiononp.241. Whatisthemedianrecoverytime? A. 2.7 B. 5.3 C. 7.4 D. 2.1 Exercise6.8.13 Solutiononp.241. Whatisthez-scoreforapatientwhotakes10daystorecover? A. 1.5 B. 0.2 C. 2.2 D. 7.3 Exercise6.8.14 Solutiononp.241. Whatistheprobabilityofspendingmorethan2daysinrecovery?

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231 A. 0.0580 B. 0.8447 C. 0.0553 D. 0.9420 Exercise6.8.15 Solutiononp.241. The90thpercentileforrecoverytimesis? A. 8.89 B. 7.07 C. 7.99 D. 4.32 Thequestionsbelowrefertothefollowing: Thelengthoftimetondaparkingspaceat9A.M.followsa normaldistributionwithameanof5minutesandastandarddeviationof2minutes. Exercise6.8.16 Solutiononp.241. Basedupontheaboveinformationandnumericallyjustied,wouldyoubesurprisedifittook lessthan1minutetondaparkingspace? A. Yes B. No C. Unabletodetermine Exercise6.8.17 Solutiononp.241. Findtheprobabilitythatittakesatleast8minutestondaparkingspace. A. 0.0001 B. 0.9270 C. 0.1862 D. 0.0668 Exercise6.8.18 Solutiononp.241. Seventypercentofthetime,ittakesmorethanhowmanyminutestondaparkingspace? A. 1.24 B. 2.41 C. 3.95 D. 6.05 Exercise6.8.19 Solutiononp.241. Ifthemeanissignicantlygreaterthanthestandarddeviation,whichofthefollowingstatements istrue? I. Thedatacannotfollowtheuniformdistribution. II. Thedatacannotfollowtheexponentialdistribution.. III. Thedatacannotfollowthenormaldistribution. A. Ionly B. IIonly C. IIIonly D. I,II,andIII

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232 CHAPTER6.THENORMALDISTRIBUTION 6.9Review 9 Thenexttwoquestionsreferto: X U 3,13 Exercise6.9.1 Solutiononp.241. Explainwhichofthefollowingarefalseandwhicharetrue. af x = 1 10 ,3 x 13 bThereisnomode. cThemedianislessthanthemean. dP X > 10 = P X 6 Exercise6.9.2 Solutiononp.241. Calculate: aMean bMedian c65thpercentile. Exercise6.9.3 Solutiononp.241. Whichofthefollowingistruefortheaboveboxplot? a25%ofthedataareatmost5. bThereisaboutthesameamountofdatafrom45asthereisfrom57. cTherearenodatavaluesof3. d50%ofthedataare4. Exercise6.9.4 Solutiononp.241. If P G j H = P G ,thenwhichofthefollowingiscorrect? AG and H aremutuallyexclusiveevents. BP G = P H CKnowingthat H hasoccurredwillaffectthechancethat G willhappen. DG and H areindependentevents. Exercise6.9.5 Solutiononp.242. If P J = 0 3, P K = 0 6,and J and K areindependentevents,thenexplainwhicharecorrect andwhichareincorrect. AP J and K = 0 BP J or K = 0 9 CP J or K = 0 .72 DP J 6 = P J j K 9 Thiscontentisavailableonlineat.

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233 Exercise6.9.6 Solutiononp.242. Onaverage,5studentsfromeachhighschoolclassgetfullscholarshipsto4-yearcolleges.Assume thatmosthighschoolclasseshaveabout500students. X =thenumberofstudentsfromahighschoolclassthatgetfullscholarshipsto4-yearschool. Whichofthefollowingisthedistributionof X ? A. P B. B,5 C. Exp/5 D. N,.01.99/500

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234 CHAPTER6.THENORMALDISTRIBUTION 6.10Lab1:NormalDistributionLapTimes 10 ClassTime: Names: 6.10.1StudentLearningOutcome: Thestudentwillcompareandcontrastempiricaldataandatheoreticaldistributiontodetermineif TerryVogel'slaptimestacontinuousdistribution. 6.10.2Directions: Roundtherelativefrequenciesandprobabilitiesto4decimalplaces.Carryallotherdecimalanswersto2 places. 6.10.3CollecttheData 1.UsethedatafromTerriVogel'sLogBookSection14.3.1:LapTimes.UseaStratiedSampling MethodbyLapRaces120andarandomnumbergeneratortopick6laptimesfromeachstratum. RecordthelaptimesbelowforLaps27. _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ 2.Constructahistogram.Make5-6intervals.Sketchthegraphusingarulerandpencil.Scaletheaxes. 10 Thiscontentisavailableonlineat.

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235 Figure6.4 3.Calculatethefollowing. a. x = b. s = 4.Drawasmoothcurvethroughthetopsofthebarsofthehistogram.Use12completesentencesto describethegeneralshapeofthecurve.Keepitsimple.Doesthegraphgostraightacross,doesit haveaV-shape,doesithaveahumpinthemiddleorateitherend,etc.? 6.10.4AnalyzetheDistribution Usingyoursamplemean,samplestandarddeviation,andhistogramtohelp,whatwastheapproximate theoreticaldistributionofthedata? X Howdoesthehistogramhelpyouarriveattheapproximatedistribution? 6.10.5DescribetheData UsetheDatafromthesectiontitled"CollecttheData"tocompletethefollowingstatements. TheIQRgoesfrom__________to__________. IQR=__________.IQR=Q3-Q1 The15thpercentileis: The85thpercentileis: Themedianis: Theempiricalprobabilitythatarandomlychosenlaptimeismorethan130seconds= Explainthemeaningofthe85thpercentileofthisdata.

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236 CHAPTER6.THENORMALDISTRIBUTION 6.10.6TheorecticalDistribution Usingthetheoreticaldistributionfromthesectiontitled"AnalysetheDistribution"completethefollowing statements: TheIQRgoesfrom__________to__________. IQR= The15thpercentileis: The85thpercentileis: Themedianis: Theprobabilitythatarandomlychosenlaptimeismorethan130seconds= Explainthemeaningthe85thpercentileofthisdistribution. 6.10.7DiscussionQuestions Dothedatafromthesectiontitled"CollecttheData"giveacloseapproximationtothetheoretical distibutioninthesectiontitled"AnalyzetheDistribution"?Incompletesentencesandcomparingthe resultinthesectionstitled"DescribetheData"and"TheoreticalDistribution",explainwhyorwhy not. 6.11Lab2:NormalDistributionPinkieLength 11 ClassTime: Names: 6.11.1StudentLearningOutcomes: Thestudentwillcompareempiricaldataandatheoreticaldistributiontodetermineifaneveryday experimenttsacontinuousdistribution. 6.11.2CollecttheData Measurethelengthofyourpinkiengerincm. 1.Randomlysurvey30adults.Roundtothenearest0.5cm. _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ _______ 11 Thiscontentisavailableonlineat.

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237 2.Constructahistogram.Make5-6intervals.Sketchthegraphusingarulerandpencil.Scaletheaxes. 3.CalculatetheFollowing a x = b s = 4.Drawasmoothcurvethroughthetopofthebarsofthehistogram.Use1-2completesentencesto describethegeneralshapeofthecurve.Keepisimple.Doesthegraphstraightacross,doesithavea V-shape,doesithaveahumpinthemiddleorateitherend,etc.? 6.11.3AnalyzetheDistribution Usingyoursamplemena,samplestandarddeviation,andhistogramtohelp,whatwastheapproximate theoreticaldistributionofthedatafromthesectiontitled"CollecttheData"? X Howdoesthehistogramhelpyouarriceattheapproximatedistribution? 6.11.4DescribetheData Usingthedatainthesectiontitled"CollecttheData"completethefollowingstatements.Hint:orderthe data R EMEMBER : IQR = Q 3 )]TJ/F132 9.9626 Tf 10.355 0 Td [(Q 1 IQR= 15thpercentileis: 85thpercentileis: Medianis: Whatistheempiricalprobabilitythatarandomlychosenpinkielengthismorethan6.5cm? Explainthemeaningthe85thpercentileofthisdata.

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238 CHAPTER6.THENORMALDISTRIBUTION 6.11.5TheoreticalDistribution UsingthetheoreticalDistributioninthesectiontitled"AnalyzetheDistribution" IQR= 15thpercentileis: 85thpercentileis: Medianis: Whatistheempiricalprobabilitythatarandomlychosenpinkielengthismorethan6.5cm? Explainthemeaningthe85thpercentileofthisdata. 6.11.6DiscussionQuestions Dothedatafromthesectionentitled"CollecttheData"givecloseapproximationtothetheoretical distributionin"AnalyzetheData"Incompletesentencesandcomparingtheresultsinthesections titled"DescribetheData"and"TheoreticalDistribution",explainwhyorwhynot. 6.11.7DotheExperiment: 1.Measurethelengthofyourpinkiengerincm. a. Randomlysurvey30adults.Recordthelengths.Roundtothenearest0.5cm. b. Constructahistogram.Make56intervals.Sketchthegraphusingarulerandpencil.Scale theaxes. i. x = ii. s = c. Drawasmoothcurvethroughthetopofthebarsofthehistogram.Use1-2completesentencestodescribethegeneralshapeofthecurve.Keepitsimple.Doesthegraphgo straightacross,doesithaveaV-shape,doesithaveahumpinthemiddleorateitherend, etc.? 2.Usingyoursamplemean,samplestandarddeviation,andhistogramtohelp,whatwastheapproximatetheoreticaldistributionofthedatafrom?

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239 a. X b. Howdoesthehistogramhelpyouarriveattheapproximatedistribution? 3.Usingthedatain,completethefollowingHint:orderthedata: R EMEMBER : IQR = Q 3 )]TJ/F132 9.9626 Tf 10.355 0 Td [(Q 1 a. TheIQRgoesfrom_______to_______. b. IQR= c. 15thpercentile= d. 85thpercentile= e. Median= f. Whatistheempiricalprobabilitythatarandomlychosenpinkielengthismorethan6.5cm? g. Explainthemeaningthe85thpercentileofthisdata. 4.Usingthetheoreticaldistributionin: a. TheIQRgoesfrom_______to_______. b. IQR= c. 15thpercentile= d. 85thpercentile= e. Median= f. Whatistheempiricalprobabilitythatarandomlychosenpinkielengthismorethan6.5cm? g. Explainthemeaningthe85thpercentileofthisdistribution. 5.Dothedatafromgiveacloseapproximationtothetheoreticaldistributionin?Incomplete sentencesandcomparingtheresultinand,explainwhyorwhynot.

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240 CHAPTER6.THENORMALDISTRIBUTION SolutionstoExercisesinChapter6 Example6.2,Problem1p.219 Thisz-scoretellsyouthat x = 10is 2.5 standarddeviationstothe right ofthemean 5 Example6.2,Problem2p.219 z = -4 .Thisz-scoretellsyouthat x = )]TJ/F58 9.9626 Tf 8.194 0 Td [(3is 4 standarddeviationstothe left ofthemean. SolutionstoPractice:TheNormalDistribution SolutiontoExercise6.7.3p.225 b. 3,0 .1979 SolutiontoExercise6.7.4p.225 b. 2 8,6,0 .7694 SolutiontoExercise6.7.5p.226 b. 0 .70 ,4 .78 years SolutionstoHomework SolutiontoExercise6.8.1p.227 a. N 66 ,2.5 b. 0.5404 c. No d. Between64.7and67.3inches SolutiontoExercise6.8.3p.227 a. N 36 10 b. 0.3446 c. 29.3 SolutiontoExercise6.8.5p.228 a. thetimeinhoursa4-year-oldinChinaspendsunsupervisedperday b. N 3,1 5 c. 0.0912 d. 0 e. 2.21hours SolutiontoExercise6.8.7p.228 a. Thedurationofacriminaltrial b. N 21 ,7 c. 0.3341 d. 22.77 SolutiontoExercise6.8.9p.229 a. Thesamplemeanis5.51andthesamplestandarddeviationis2.15 e. N 5 .51 ,2 .15

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241 f. 0.6081 g. 0.64 SolutiontoExercise6.8.11p.229 a. Thesamplemeanis155.16andthesamplestandarddeviationis92.1605. e. N 155 .16 ,92 .1605 f. 0.4315 g. 0.3408 SolutiontoExercise6.8.12p.230 B SolutiontoExercise6.8.13p.230 C SolutiontoExercise6.8.14p.230 D SolutiontoExercise6.8.15p.231 C SolutiontoExercise6.8.16p.231 C SolutiontoExercise6.8.17p.231 D SolutiontoExercise6.8.18p.231 C SolutiontoExercise6.8.19p.231 B SolutionstoReview SolutiontoExercise6.9.1p.232 aTrue bTrue cFalsethemedianandthemeanarethesameforthissymmetricdistribution dTrue SolutiontoExercise6.9.2p.232 a8 b8 cP X < k = 0 .65 = k )]TJ/F58 9.9626 Tf 10.131 0 Td [(3 1 10 k = 9 5 SolutiontoExercise6.9.3p.232 aFalse 3 4 ofthedataareatmost5 bTrueeachquartilehas25%ofthedata cFalsethatisunknown dFalse50%ofthedataare4orless SolutiontoExercise6.9.4p.232 D

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242 CHAPTER6.THENORMALDISTRIBUTION SolutiontoExercise6.9.5p.232 AFalse J and K areindependent,sotheyarenotmutuallyexclusivewhichwouldimplydependency BFalse CTruesince P J and K 6 = 0,then P J or K < 0 .09 DFalse P J and K 6 = 0areindependentwhichimplies P J = P J j K SolutiontoExercise6.9.6p.233 A

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Chapter7 TheCentralLimitTheorem 7.1TheCentralLimitTheorem 1 7.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: RecognizetheCentralLimitTheoremproblems. Classifycontinuouswordproblemsbytheirdistributions. ApplyandinterprettheCentralLimitTheoremforAverages. ApplyandinterprettheCentralLimitTheoremforSums. 7.1.2Introduction Whatdoesitmeantobeaverage?Whyarewesoconcernedwithaverages?Tworeasonsarethattheygive usamiddlegroundforcomparisonandtheyareeasytocalculate.Inthischapter,youwillstudyaverages andtheCentralLimitTheorem. TheCentralLimitTheorem CLTforshortisoneofthemostpowerfulandusefulideasinallofstatistics. Bothalternativesareconcernedwithdrawingnitesamplesofsize n fromapopulationwithaknown mean, m ,andaknownstandarddeviation, s .Therstalternativesaysthatifwecollectsamplesofsize n and n is"largeenough,"calculateeachsample'smean,andcreateahistogramofthosemeans,thenthe resultinghistogramwilltendtohaveanapproximatenormalbellshape.Thesecondalternativesaysthat ifweagaincollectsamplesofsizenthatare"largeenough,"calculatethesumofeachsampleandcreatea histogram,thentheresultinghistogramwillagaintendtohaveanormalbell-shape. Ineithercase,itdoesnotmatterwhatthedistributionoftheoriginalpopulationis,orwhetheryoueven needtoknowit.Theimportantfactisthatthesamplemeansaveragesandthesumstendtofollowthe normaldistribution. And,therestyouwilllearninthischapter. Thesizeofthesample, n ,dependsontheoriginalpopulationfromwhichthesamplesaredrawn.Ifthe originalpopulationisfarfromnormalthenmoreobservationsareneededforthesampleaveragesorthe samplesumstobenormal. Samplingisdonewithreplacement. Dothefollowingexampleinclass: Suppose8ofyouroll1fairdie10times,7ofyouroll2fairdice10times, 9ofyouroll5fairdice10times,and11ofyouroll10fairdice10times.The8,7,9,and11wererandomly chosen. 1 Thiscontentisavailableonlineat. 243

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244 CHAPTER7.THECENTRALLIMITTHEOREM Eachtimeapersonrollsmorethanonedie,he/shecalculatesthe average ofthefacesshowing.Forexample, onepersonmightroll5fairdiceandgeta2,2,3,4,6ononeroll. Theaverageis 2 + 2 + 3 + 4 + 6 5 = 3.4.The3.4isoneaveragewhen5fairdicearerolled.Thissameperson wouldrollthe5dice9moretimesandcalculate9moreaveragesforatotalof10averages. Yourinstructorwillpassoutthedicetoseveralpeopleasdescribedabove.Rollyourdice10times.For eachroll,recordthefacesandndtheaverage.Roundtothenearest0.5. Yourinstructorandpossiblyyouwillproduceonegraphitmightbeahistogramfor1die,onegraph for2dice,onegraphfor5dice,andonegraphfor10dice.Sincethe"average"whenyourollonedie,isjust thefaceonthedie,whatdistributiondothese"averages"appeartoberepresenting? Drawthegraphfortheaveragesusing2dice. Dotheaveragesshowanykindofpattern? Drawthegraphfortheaveragesusing5dice. Doyouseeanypatternemerging? Finally,drawthegraphfortheaveragesusing10dice. Doyouseeanypatterntothegraph?Whatcanyou concludeasyouincreasethenumberofdice? Asthenumberofdicerolledincreasesfrom1to2to5to10,thefollowingishappening: 1.Theaverageoftheaveragesremainsapproximatelythesame. 2.Thespreadoftheaveragesthestandarddeviationoftheaveragesgetssmaller. 3.Thegraphappearssteeperandthinner. YouhavejustdemonstratedtheCentralLimitTheoremCLT. TheCentralLimitTheoremtellsyouthatasyouincreasethenumberofdice, thesamplemeansaverages tendtowardanormaldistribution. 7.2TheCentralLimitTheoremforSampleMeansAverages 2 Suppose X isarandomvariablewithadistributionthatmaybeknownorunknownitcanbeanydistribution.Usingasubscriptthatmatchestherandomvariable,suppose: a. m X =themeanof X b. s X =thestandarddeviationof X Ifyoudrawrandomsamplesofsize n ,thenas n increases,therandomvariable X whichconsistsofsample means,tendstobe normallydistributed and X N m X s X p n TheCentralLimitTheorem forSampleMeansAveragessaysthatifyoukeepdrawinglargerandlarger sampleslikerolling1,2,5,and,nally,10diceand calculatingtheirmeans thesamplemeansaverages formtheirown normaldistribution .Thisdistributionhasthesamemeanastheoriginaldistributionanda variancethatequalstheoriginalvariancedividedby n ,thesamplesize. n isthenumberofvaluesthatare averagedtogethernotthenumberoftimestheexperimentisdone. Therandomvariable X hasadifferentz-scoreassociatedwithit. x isonesample z = x )]TJ/F134 9.9626 Tf 10.256 0 Td [(m X s X p n .1 2 Thiscontentisavailableonlineat.

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245 m X isboththeaverageof X andof X s X = s X p n = standarddeviationof X andiscalledthe standarderrorofthemean. Example7.1 Anunknowndistributionhasameanof90andastandarddeviationof15.Samplesofsize n =25 aredrawnrandomlyfromthepopulation. Problem1 Findtheprobabilitythatthe samplemean isbetween85and92. Solution Let X =onevaluefromtheoriginalunknownpopulation.Theprobabilityquestionasksyouto ndaprobabilityforthe samplemeanoraverage Let X = themeanoraverageofasampleofsize25.Since m X = 90, s X = 15,and n = 25; then X N 90, 15 p 25 Find P )]TJ/F58 9.9626 Tf 4.687 -8.074 Td [(85 < X < 92 Drawagraph. P )]TJ/F58 9.9626 Tf 4.687 -8.075 Td [(85 < X < 92 = 0.6997 Theprobabilitythatthesamplemeanisbetween85and92is0.6997. TI-83: normalcdf lowervalue,uppervalue,meanforaverages, stdev foraverages stdev =standarddeviation Theparameterlistisabbreviatedlower,upper, m s p n normalcdf 85,92,90, 15 p 25 = 0.6997 Problem2 Findtheaveragevaluethatis2standarddeviationsabovethethemeanoftheaverages. Solution Tondtheaveragevaluethatis2standarddeviationsabovethemeanoftheaverages,usethe formula value= m X + ofSTDEVs s X p n

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246 CHAPTER7.THECENTRALLIMITTHEOREM value=90 + 2 15 p 25 = 96 So,theaveragevaluethatis2standarddeviationsabovethemeanoftheaveragesis96. Example7.2 Thelengthoftime,inhours,ittakesan"over40"groupofpeopletoplayonesoccermatchis normallydistributedwitha meanof2hours anda standarddeviationof0.5hours .A sampleof size n =50 isdrawnrandomlyfromthepopulation. Problem1 Findtheprobabilitythatthe samplemean isbetween1.8hoursand2.3hours. Solution Let X =thetime,inhours,ittakestoplayonesoccermatch. Theprobabilityquestionasksyoutondaprobabilityforthe samplemeanoraveragetime,in hours ,ittakestoplayonesoccermatch. Let X =the average time,inhours,ittakestoplayonesoccermatch. Problem2 If m X = _________, s X = __________,and n = ___________,then X N______,______ bythe CentralLimitTheoremforAveragesofSampleMeans. Find P )]TJ/F58 9.9626 Tf 4.687 -8.075 Td [(1.8 < X < 2.3 .Drawagraph. P )]TJ/F58 9.9626 Tf 4.687 -8.075 Td [(1.8 < X < 2.3 = 0.9977 normalcdf 1.8,2.3,2, .5 p 50 = 0.9977 Theprobabilitythatthesamplemeanisbetween1.8hoursand2.3hoursis______. 7.3TheCentralLimitTheoremforSums 3 Suppose X isarandomvariablewithadistributionthatmaybe knownorunknown itcanbeanydistribution.Suppose: a. m X = themeanof X b. s X = thestandarddeviationof X Ifyoudrawrandomsamplesofsize n ,thenas n increases,therandomvariable S X whichconsistsofsums tendstobe normallydistributed and S X N )]TJ/F132 9.9626 Tf 4.812 -8.074 Td [(n m X p n s X TheCentralLimitTheoremforSums saysthatifyoukeepdrawinglargerandlargersamplesandtaking theirsums,thesumsformtheirownnormaldistribution. Thedistributionhasameanequaltotheoriginal meanmultipliedbythesamplesizeandastandarddeviationequaltotheoriginalstandarddeviation multipliedbythesquarerootofthesamplesize. Therandomvariable S X hasthefollowingz-scoreassociatedwithit: 3 Thiscontentisavailableonlineat.

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247 a. S X isonesum. b. z = S X )]TJ/F132 7.5716 Tf 6.322 0 Td [(n m X p n s X a. n m X = themeanof S X b. p n s X = standarddeviationof S X Example7.3 Anunknowndistributionhasameanof90andastandarddeviationof15.Asampleofsize80is drawnrandomlyfromthepopulation. Problem a. Findtheprobabilitythatthesumofthe80valuesorthetotalofthe80valuesismore than7500. b. Findthesumthatis1.5standarddeviationsbelowthemeanofthesums. Solution Let X =onevaluefromtheoriginalunknownpopulation.Theprobabilityquestionasksyouto ndaprobabilityfor thesumortotalof80values. S X =thesumortotalof80values.Since m X = 90, s X = 15,and s X = 80,then S X N 80 90, p 80 15 a. meanofthesums= n m X = 80 90 = 7200 b. standarddeviationofthesums= p n s X = p 80 15 c. sumof80values= S x = 7500 Find P S X > 7500 Drawagraph. P S X > 7500 = 0.0127 normalcdf lowervalue,uppervalue,meanofsums, stdev ofsums Theparameterlistisabbreviatedlower,upper, n m X p n s X normalcdf ,1E99,80 90, p 80 15 = 0.0127 Reminder: 1 E 99 = 10 99 .Pressthe EE keyforE.

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248 CHAPTER7.THECENTRALLIMITTHEOREM 7.4UsingtheCentralLimitTheorem 4 Itisimportantforyoutounderstandwhentousethe CLT .Ifyouarebeingaskedtondtheprobabilityof anaverageormean,usetheCLTformeansoraverages.Ifyouarebeingaskedtondtheprobabilityofa sumortotal,usetheCLTforsums.Thisalsoappliestopercentilesforaveragesandsums. N OTE :Ifyouarebeingaskedtondtheprobabilityofan individual value,do not usetheCLT. Usethedistributionofitsrandomvariable. 7.4.1LawofLargeNumbers TheLawofLargeNumberssaysthatifyoutakesamplesoflargerandlargersizefromanypopulation,then themean x ofthesamplegetscloserandcloserto m .FromtheCentralLimitTheorem,weknowthatas n getslargerandlarger,thesampleaveragesfollowanormaldistribution.Thelargerngets,thesmallerthe standarddeviationgets.Rememberthatthestandarddeviationfor X is s p n .Thismeansthatthesample mean x mustbeclosetothepopulationmean m .Wecansaythat m isthevaluethatthesampleaverages approachas n getslarger.TheCentralLimitTheoremillustratestheLawofLargeNumbers. Example7.4 Astudyinvolvingstressisdoneonacollegecampusamongthestudents. Thestressscoresfollow auniformdistribution withtheloweststressscoreequalto1andthehighestequalto5.Usinga sampleof75students,nd: a. Theprobabilitythatthe averagestressscore forthe75studentsislessthan2. b. The90thpercentileforthe averagestressscore forthe75students. c. Theprobabilitythatthe totalofthe75stressscores islessthan200. d. The90thpercentileforthe totalstressscore forthe75students. Let X =onestressscore. Problemsaandbaskyoutondaprobabilityorapercentileforan average or mean .Problemsc anddaskyoutondaprobabilityorapercentilefora totalorsum .Thesamplesize, n ,isequal to75. Sincetheindividualstressscoresfollowauniformdistribution, X U 1,5 where a = 1and b = 5SeethechapteronContinuousRandomVariablesSection5.1. m X = a + b 2 = 1 + 5 2 = 3 s X = q b )]TJ/F132 7.5716 Tf 6.435 0 Td [(a 2 12 = q 5 )]TJ/F58 7.5716 Tf 6.227 0 Td [(1 2 12 = 1.15 Forproblemsaandb,let X =theaveragestressscoreforthe75students.Then, X N 3, 1.15 p 75 where n = 75. Problem1 Find P )]TJETq1 0 0 1 130.238 153.8 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 130.512 145.516 Td [(X < 2 .Drawthegraph. Solution P )]TJETq1 0 0 1 107.424 124.908 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 107.698 116.624 Td [(X < 2 = 0 Theprobabilitythattheaveragestressscoreislessthan2isabout0. 4 Thiscontentisavailableonlineat.

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249 normalcdf 1,2,3, 1.15 p 75 = 0 R EMINDER :Thesmalleststressscoreis1.Therefore,thesmallestaveragefor75stressscoresis1. Problem2 Findthe90thpercentilefortheaverageof75stressscores.Drawagraph. Solution Let k =the90thprecentile. Find k where P )]TJETq1 0 0 1 164.858 411.681 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 165.132 403.397 Td [(X < k = 0.90. k = 3.2 The90thpercentilefortheaverageof75scoresisabout3.2.Thismeansthat90%ofalltheaverages of75stressscoresareatmost3.2and10%areatleast3.2. invNorm .90,3, 1.15 p 75 = 3.2 Forproblemscandd,let S X =thesumofthe75stressscores.Then, S X N h 75 3 p 75 1.15 i Problem3 Find P S X < 200 .Drawthegraph. Solution Themeanofthesumof75stressscoresis75 3 = 225

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250 CHAPTER7.THECENTRALLIMITTHEOREM Thestandarddeviationofthesumof75stressscoresis p 75 1.15 = 9.96 P S X < 200 = 0 Theprobabilitythatthetotalof75scoresislessthan200isabout0. normalcdf 75,200,75 3, p 75 1.15 = 0. R EMINDER :Thesmallesttotalof75stressscoresis75sincethesmallestsinglescoreis1. Problem4 Findthe90thpercentileforthetotalof75stressscores.Drawagraph. Solution Let k =the90thpercentile. Find k where P S X < k = 0.90. k = 237.8 The90thpercentileforthesumof75scoresisabout237.8.Thismeansthat90%ofallthesumsof 75scoresarenomorethan237.8and10%arenolessthan237.8. invNorm .90,75 3, p 75 1.15 = 237.8 Example7.5 Thedistributionofagesofstatisticsstudentsatacertaincollegehasan exponentialdistribution withameanageof22years.Eightystatisticsstudentsarerandomlyselected.Find

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251 a. Theprobabilitythatthe averageage ofthe80statisticsstudentsismorethan20. b. The95thpercentileforthe averageage ofthe80statisticsstudents. Let X =theageofonestatisticsstudent.Then X Exp 1 22 Chapter5. m = 22and s = 22. n = 80. Let X =theaverageageofthe80statisticsstudents.Then X N 22, 22 p 80 bytheCLTforSampleMeansorAverages Problem1 Find P )]TJETq1 0 0 1 130.238 564.157 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 130.512 555.873 Td [(X > 20 Drawthegraph. Solution P )]TJETq1 0 0 1 107.424 535.266 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 107.698 526.982 Td [(X > 20 = 0.7919 Theprobabilitythattheaveragestressscoreismorethan20is0.7919. normalcdf 20,1 E 99,22, 22 p 80 R EMINDER :1 E 99 = 10 99 and )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 E 99 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(10 99 .Pressthe EE keyforE. Problem2 Findthe95thpercentilefortheaverageof75stressscores.Drawagraph. Solution Let k =the95thpercentile. Find k where P )]TJETq1 0 0 1 164.858 212.076 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 165.132 203.792 Td [(X < k = 0.95 k = 26.0

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252 CHAPTER7.THECENTRALLIMITTHEOREM The95thpercentilefortheaverageageof80statisticsstudentsatacertaincommunitycollegeis about26.0.Thismeansthat95%oftheaverageagesofstatisticsstudentsareatmost26.0and10% areatleast26.0. invNorm .95,22, 22 p 80 = 26.0

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253 7.5SummaryofFormulas 5 Formula7.1: CentralLimitTheoremforSampleMeansAverages X N m X s X p n MeanforAverages )]TJETq1 0 0 1 300.582 649.579 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 300.856 641.295 Td [(X : m X Formula7.2: CentralLimitTheoremforSampleMeansAveragesZ-ScoreandStandardErrorof theMean z = x )]TJ/F134 7.5716 Tf 6.322 0 Td [(m X s X p n StandardErroroftheMeanStandardDeviationforAverages )]TJETq1 0 0 1 444.185 608.005 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 444.459 599.721 Td [(X : s X p n Formula7.3: CentralLimitTheoremforSums S X N n m X p n s X MeanforSums S X : n m X Formula7.4: CentralLimitTheoremforSumsZ-ScoreandStandardDeviationforSums z = S x )]TJ/F132 7.5716 Tf 6.322 0 Td [(n m X p n s X StandardDeviationforSums S X : p n s X 5 Thiscontentisavailableonlineat.

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254 CHAPTER7.THECENTRALLIMITTHEOREM 7.6Practice:TheCentralLimitTheorem 6 7.6.1StudentLearningOutcomes ThestudentwillexplorethepropertiesofdatathroughtheCentralLimitTheorem. 7.6.2Given Yoonieisapersonnelmanagerinalargecorporation.Eachmonthshemustreview16oftheemployees. Frompastexperience,shehasfoundthatthereviewstakeherapproximately4hourseachtodowitha populationstandarddeviationof1.2hours.Let X betherandomvariablerepresentingthetimeittakes hertocompleteonereview.Assume X isnormallydistributed.Let X betherandomvariablerepresenting theaveragetimetocompletethe16reviews.Let S X bethetotaltimeittakesYoonietocompleteallofthe month'sreviews. 7.6.3Distribution Completethedistributions. 1. X 2. X 3. S X 7.6.4GraphingProbability Foreachproblembelow: aSketchthegraph.Labelandscalethehorizontalaxis.Shadetheregioncorrespondingtothe probability. bCalculatethevalue. Exercise7.6.1 Solutiononp.274. Findtheprobabilitythat one reviewwilltakeYooniefrom3.5to4.25hours. a. b. P ________ < X < ________ = _______ Exercise7.6.2 Solutiononp.274. Findtheprobabilitythatthe average ofamonth'sreviewswilltakeYooniefrom3.5to4.25hrs. 6 Thiscontentisavailableonlineat.

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255 a. b. P = _______ Exercise7.6.3 Solutiononp.274. Findthe95thpercentileforthe average timetocompleteonemonth'sreviews. a. b. The95thPercentile= Exercise7.6.4 Solutiononp.274. Findtheprobabilitythatthe sum ofthemonth'sreviewstakesYooniefrom60to65hours. a. b. TheProbability= Exercise7.6.5 Solutiononp.274. Findthe95thpercentileforthe sum ofthemonth'sreviews.

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256 CHAPTER7.THECENTRALLIMITTHEOREM a. b. The95thpercentile= 7.6.5DiscussionQuestion Exercise7.6.6 WhatcausestheprobabilitiesinExercise7.6.1andExercise7.6.2todiffer?

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257 7.7Homework 7 Exercise7.7.1 Solutiononp.274. X N 60 ,9 .Supposethatyouformrandomsamplesof25fromthisdistribution.Let X bethe randomvariableofaverages.Let S X betherandomvariableofsums.For c-f ,sketchthegraph, shadetheregion,labelandscalethehorizontalaxisfor X ,andndtheprobability. a. Sketchthedistributionsof X and X onthesamegraph. b. X c. P )]TJETq1 0 0 1 128.784 583.351 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 129.057 575.067 Td [(X < 60 = d. Findthe30thpercentile. e. P )]TJ/F132 9.9626 Tf 4.688 -8.075 Td [(56 < X < 62 = f. P )]TJ/F132 9.9626 Tf 4.688 -8.075 Td [(18 < X < 58 = g. S X h. Findtheminimumvaluefortheupperquartile. i. P 1400 < S X < 1550 = Exercise7.7.2 Determinewhichofthefollowingaretrueandwhicharefalse.Then,incompletesentences, justifyyouranswers. a. Whenthesamplesizeislarge,themeanof X isapproximatelyequaltothemeanof X b. Whenthesamplesizeislarge, X isapproximatelynormallydistributed. c. Whenthesamplesizeislarge,thestandarddeviationof X isapproximatelythesameas thestandarddeviationof X Exercise7.7.3 Solutiononp.274. ThepercentoffatcaloriesthatapersoninAmericaconsumeseachdayisnormallydistributed withameanofabout36andastandarddeviationofabout10.Supposethat16individualsare randomlychosen. Let X = averagepercentoffatcalories. a. X~ ____________,______ b. Forthegroupof16,ndtheprobabilitythattheaveragepercentoffatcaloriesconsumed ismorethan5.Graphthesituationandshadeintheareatobedetermined. c. Findtherstquartilefortheaveragepercentoffatcalories. Exercise7.7.4 Previously,DeAnzastatisticsstudentsestimatedthattheamountofchangedaytimestatistics studentscarryisexponentiallydistributedwithameanof\$0.88.Supposethatwerandomlypick 25daytimestatisticsstudents. a. Inwords, X = b. X~ c. Inwords, X = d. X~ ____________,______ e. Findtheprobabilitythatanindividualhadbetween\$0.80and\$1.00.Graphthesituation andshadeintheareatobedetermined. f. Findtheprobabilitythattheaverageofthe25studentswasbetween\$0.80and\$1.00. Graphthesituationandshadeintheareatobedetermined. 7 Thiscontentisavailableonlineat.

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258 CHAPTER7.THECENTRALLIMITTHEOREM g. Explainthewhythereisadifferenceineandf. Exercise7.7.5 Solutiononp.274. Supposethatthedistanceofyballshittotheouteldinbaseballisnormallydistributedwith ameanof250feetandastandarddeviationof50feet.Werandomlysample49yballs. a. If X = averagedistanceinfeetfor49yballs,then X~ ______________,_______ b. Whatistheprobabilitythatthe49ballstraveledanaverageoflessthan240feet?Sketch thegraph.Scalethehorizontalaxisfor X .Shadetheregioncorrespondingtotheprobability.Findtheprobability. c. Findthe80thpercentileofthedistributionoftheaverageof49yballs. Exercise7.7.6 Supposethattheweightofopenboxesofcerealinahomewithchildrenisuniformlydistributed from2to6pounds.Werandomlysurvey64homeswithchildren. a. Inwords, X = b. X~ c. m X = d. s X = e. Inwords, S X = f. S X~ g. Findtheprobabilitythatthetotalweightofopenboxesislessthan250pounds. h. Findthe35thpercentileforthetotalweightofopenboxesofcereal. Exercise7.7.7 Solutiononp.274. Supposethatthedurationofaparticulartypeofcriminaltrialisknowntohaveameanof21days andastandarddeviationof7days.Werandomlysample9trials. a. Inwords, S X = b. S X~ c. Findtheprobabilitythatthetotallengthofthe9trialsisatleast225days. d. 90percentofthetotalof9ofthesetypesoftrialswilllastatleasthowlong? Exercise7.7.8 AccordingtotheInternalRevenueService,theaveragelengthoftimeforanindividualtocompleterecordkeep,learn,prepare,copy,assembleandsendIRSForm1040is10.53hourswithout anyattachedschedules.Thedistributionisunknown.Letusassumethatthestandarddeviation is2hours.Supposewerandomlysample36taxpayers. a. Inwords, X = b. Inwords, X = c. X~ d. Wouldyoubesurprisedifthe36taxpayersnishedtheirForm1040sinanaverageof morethan12hours?Explainwhyorwhynotincompletesentences. e. WouldyoubesurprisedifonetaxpayernishedhisForm1040inmorethan12hours?In acompletesentence,explainwhy. Exercise7.7.9 Solutiononp.275. Supposethatacategoryofworldclassrunnersareknowntorunamarathonmilesinan averageof145minuteswithastandarddeviationof14minutes.Consider49oftheraces. Let X = theaverageofthe49races.

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259 a. X~ b. Findtheprobabilitythattherunnerwillaveragebetween142and146minutesinthese49 marathons. c. Findthe80thpercentilefortheaverageofthese49marathons. d. Findthemedianoftheaveragerunningtimes. Exercise7.7.10 Theattentionspanofatwoyear-oldisexponentiallydistributedwithameanofabout8minutes. Supposewerandomlysurvey60twoyear-olds. a. Inwords, X = b. X~ c. Inwords, X = d. X~ e. Beforedoinganycalculations,whichdoyouthinkwillbehigher?Explainwhy. i. theprobabilitythatanindividualattentionspanislessthan10minutes;or ii. theprobabilitythattheaverageattentionspanforthe60childrenislessthan10 minutes?Why? f. Calculatetheprobabilitiesinparte. g. Explainwhythedistributionfor X isnotexponential. Exercise7.7.11 Solutiononp.275. Supposethatthelengthofresearchpapersisuniformlydistributedfrom10to25pages.We surveyaclassinwhich55researchpaperswereturnedintoaprofessor.Weareinterestedinthe averagelengthoftheresearchpapers. a. Inwords, X = b. X~ c. m X = d. s X = e. Inwords, X = f. X~ g. Inwords, S X = h. S X~ i. Withoutdoinganycalculations,doyouthinkthatit'slikelythattheprofessorwillneedto readatotalofmorethan1050pages?Why? j. Calculatetheprobabilitythattheprofessorwillneedtoreadatotalofmorethan1050 pages. k. Whyisitsounlikelythattheaveragelengthofthepaperswillbelessthan12pages? Exercise7.7.12 Thelengthofsongsinacollector'sCDcollectionisuniformlydistributedfrom2to3.5minutes. Supposewerandomlypick5CDsfromthecollection.Thereisatotalof43songsonthe5CDs. a. Inwords, X = b. X~ c. Inwords, X= d. X~ e. Findtherstquartilefortheaveragesonglength. f. TheIQRinterquartilerangefortheaveragesonglengthisfrom_______to_______.

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260 CHAPTER7.THECENTRALLIMITTHEOREM Exercise7.7.13 Solutiononp.275. Salariesforteachersinaparticularelementaryschooldistrictarenormallydistributedwitha meanof\$44,000andastandarddeviationof\$6500.Werandomlysurvey10teachersfromthat district. a. Inwords, X = b. Inwords, X = c. X~ d. Inwords, S X = e. S X~ f. Findtheprobabilitythattheteachersearnatotalofover\$400,000. g. Findthe90thpercentileforanindividualteacher'ssalary. h. Findthe90thpercentilefortheaverageteachers'salary. i. Ifwesurveyed70teachersinsteadof10,graphically,howwouldthatchangethedistributionfor X ? j. Ifeachofthe70teachersreceiveda\$3000raise,graphically,howwouldthatchangethe distributionfor X ? Exercise7.7.14 ThedistributionofincomeinsomeThirdWorldcountriesisconsideredwedgeshapedmany verypoorpeople,veryfewmiddleincomepeople,andfewtomanywealthypeople.Supposewe pickacountrywithawedgedistribution.Lettheaveragesalarybe\$2000peryearwithastandard deviationof\$8000.Werandomlysurvey1000residentsofthatcountry. a. Inwords, X = b. Inwords, X = c. X~ d. Howisitpossibleforthestandarddeviationtobegreaterthantheaverage? e. Whyisitmorelikelythattheaverageofthe1000residentswillbefrom\$2000to\$2100 thanfrom\$2100to\$2200? Exercise7.7.15 Solutiononp.275. TheaveragelengthofamaternitystayinaU.S.hospitalissaidtobe2.4dayswithastandarddeviationof0.9days.Werandomlysurvey80womenwhorecentlyborechildreninaU.S.hospital. a. Inwords, X = b. Inwords, X = c. X~ d. Inwords, S X = e. S X~ f. Isitlikelythatanindividualstayedmorethan5daysinthehospital?Whyorwhynot? g. Isitlikelythattheaveragestayforthe80womenwasmorethan5days?Whyorwhy not? h. Whichismorelikely: i. anindividualstayedmorethan5days;or ii. theaveragestayof80womenwasmorethan5days? i. Ifweweretosumupthewomen'sstays,isitlikelythat,collectivelytheyspentmorethan ayearinthehospital?Whyorwhynot? Exercise7.7.16 In1940theaveragesizeofaU.S.farmwas174acres.Let'ssaythatthestandarddeviationwas55 acres.Supposewerandomlysurvey38farmersfrom1940.Source:U.S.Dept.ofAgriculture

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261 a. Inwords, X = b. Inwords, X = c. X~ d. TheIQRfor X isfrom_______acresto_______acres. Exercise7.7.17 Solutiononp.275. Thestockclosingpricesof35U.S.semiconductormanufacturersaregivenbelow.Source: Wall StreetJournal 8.625;30.25;27.625;46.75;32.875;18.25;5;0.125;2.9375;6.875;28.25;24.25;21;1.5;30.25;71;43.5; 49.25;2.5625;31;16.5;9.5;18.5;18;9;10.5;16.625;1.25;18;12.875;7;2.875;2.875;60.25;29.25 a. Inwords, X = b.i. x = ii. s x = iii. n = c. Constructahistogramofthedistributionoftheaverages.Startat x = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0 .0005 .Makebar widthsof10. d. Inwords,describethedistributionofstockprices. e. Randomlyaverage5stockpricestogether.Usearandomnumbergenerator.Continue averaging5piecestogetheruntilyouhave10averages.Listthose10averages. f. Usethe10averagesfrometocalculate: i. x = ii. s x = g. Constructahistogramofthedistributionoftheaverages.Startat x = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0 .0005 .Makebar widthsof10. h. Doesthishistogramlooklikethegraphinc? i. In1-2completesentences,explainwhythegraphseitherlookthesameorlookdifferent? j. BaseduponthetheoryoftheCentralLimitTheorem, X~ Exercise7.7.18 UsetheInitialPublicOfferingdataSection14.3.2:StockPricesseeTableofContentstodothis problem. a. Inwords, X = b.i. m X = ii. s X = iii. n = c. Constructahistogramofthedistribution.Startat x = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0 .50 .Makebarwidthsof\$5. d. Inwords,describethedistributionofstockprices. e. Randomlyaverage5stockpricestogether.Usearandomnumbergenerator.Continue averaging5piecestogetheruntilyouhave15averages.Listthose15averages. f. Usethe15averagesfrometocalculatethefollowing: i. x = ii. s x = g. Constructahistogramofthedistributionoftheaverages.Startat x = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0 .50 .Makebar widthsof\$5. h. Doesthishistogramlooklikethegraphinc?Explainanydifferences. i. In1-2completesentences,explainwhythegraphseitherlookthesameorlookdifferent? j. BaseduponthetheoryoftheCentralLimitTheorem, X~

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262 CHAPTER7.THECENTRALLIMITTHEOREM 7.7.1Trythesemultiplechoicequestions. Thenexttwoquestionsrefertothefollowinginformation: Thetimetowaitforaparticularruralbus isdistributeduniformlyfrom0to75minutes.100ridersarerandomlysampledtolearnhowlongthey waited. Exercise7.7.19 Solutiononp.275. The90thpercentilesampleaveragewaittimeinminutesforasampleof100ridersis: A. 315.0 B. 40.3 C. 38.5 D. 65.2 Exercise7.7.20 Solutiononp.275. Wouldyoubesurprised,baseduponnumericalcalculations,ifthesampleaveragewaittimein minutesfor100riderswaslessthan30minutes? A. Yes B. No C. Thereisnotenoughinformation. Exercise7.7.21 Solutiononp.275. WhichofthefollowingisNOTTRUEaboutthedistributionforaverages? A. Themean,medianandmodeareequal B. Theareaunderthecurveisone C. Thecurvenevertouchesthex-axis D. Thecurveisskewedtotheright Thenexttwoquestionsrefertothefollowinginformation: Theaveragecostofunleadedgasolineinthe BayAreaoncefollowedanunknowndistributionwithameanof\$2.59andastandarddeviationof\$0.10. SixteengasstationsfromtheBayAreaarerandomlychosen.Weareinterestedintheaveragecostof gasolineforthe16gasstations. Exercise7.7.22 Solutiononp.275. Thedistributiontousefortheaveragecostofgasolineforthe16gasstationsis A. X N 2.59,0.10 B. X N 2.59, 0.10 p 16 C. X N 2.59, 0.10 16 D. X N 2.59, 16 0.10 Exercise7.7.23 Solutiononp.275. Whatistheprobabilitythattheaveragepricefor16gasstationsisover\$2.69? A. Almostzero B. 0.1587 C. 0.0943 D. Unknown

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263 7.8Review 8 Thenextthreequestionsrefertothefollowinginformation: Richard'sFurnitureCompanydeliversfurniturefrom10A.M.to2P.M.continuouslyanduniformly.Weareinterestedinhowlonginhourspastthe 10A.M.starttimethatindividualswaitfortheirdelivery. Exercise7.8.1 Solutiononp.276. X A. U 0,4 B. U 10 ,2 C. Exp 2 D. N 2,1 Exercise7.8.2 Solutiononp.276. Theaveragewaittimeis: A. 1hour B. 2hour C. 2.5hour D. 4hour Exercise7.8.3 Solutiononp.276. Supposethatitisnowpastnoononadeliveryday.Theprobabilitythatapersonmustwaitat least1 1 2 more hoursis: A. 1 4 B. 1 2 C. 3 4 D. 3 8 Exercise7.8.4 Solutiononp.276. Given: X~Exp 1 3 a. Find P X > 1 b. Calculatetheminimumvaluefortheupperquartile. c. Find P X = 1 3 Exercise7.8.5 Solutiononp.276. 40%offull-timestudentstook4yearstograduate 30%offull-timestudentstook5yearstograduate 20%offull-timestudentstook6yearstograduate 10%offull-timestudentstook7yearstograduate Theexpectedtimeforfull-timestudentstograduateis: A. 4years B. 4.5years C. 5years D. 5.5years 8 Thiscontentisavailableonlineat.

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264 CHAPTER7.THECENTRALLIMITTHEOREM Exercise7.8.6 Solutiononp.276. Whichofthefollowingdistributionsisdescribedbythefollowingexample? Manypeoplecanrunashortdistanceofunder2miles,butasthedistanceincreases,fewerpeople canrunthatfar. A. Binomial B. Uniform C. Exponential D. Normal Exercise7.8.7 Solutiononp.276. Thelengthoftimetobrushone'steethisgenerallythoughttobeexponentiallydistributedwith ameanof 3 4 minutes.Findtheprobabilitythatarandomlyselectedpersonbrusheshis/herteeth lessthan 3 4 minutes. A. 0.5 B. 3 4 C. 0.43 D. 0.63 Exercise7.8.8 Solutiononp.276. Whichdistributionaccuratelydescribesthefollowingsituation? Thechancethatateenageboyregularlygiveshismotherakissgoodnightandheshould!!is about20%.Fourteenteenageboysarerandomlysurveyed. X = thenumberofteenageboysthatregularlygivetheirmotherakissgoodnight A. B 14 ,0 .20 B. P 2 8 C. N 2 8,2 .24 D. Exp 1 0 .20

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265 7.9Lab1:CentralLimitTheoremPocketChange 9 ClassTime: Names: 7.9.1StudentLearningOutcomes: ThestudentwillexaminepropertiesoftheCentralLimitTheorem. N OTE :Thislabworksbestwhensamplingfromseveralclassesandcombiningdata. 7.9.2CollecttheData 1.Countthechangeinyourpocket.Donotincludebills. 2.Randomlysurvey30classmates.Recordthevaluesofthechange. __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 3.Constructahistogram.Make5-6intervals.Sketchthegraphusingarulerandpencil.Scaletheaxes. 9 Thiscontentisavailableonlineat.

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266 CHAPTER7.THECENTRALLIMITTHEOREM Figure7.1 4.Caluclatethefollowing: a. x = b. s = c. n = 1surveyingonepersonatatime 5.Drawasmoothcurvethroughthetopsofthebarsofthehistogram.Use12completesentencesto describethegeneralshapeofthecurve. 7.9.3CollectingAveragesofPairs Repeatsteps1-5ofthesectionabovetitled"CollecttheData"withoneexception.Insteadofrecording thechangeof30classmates,recordtheaveragechangeof30pairs. 1.Randomlysurvey30 pairs ofclassmates.Recordthevaluesoftheaverageoftheirchange. __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 2.Constructahistogram.Scaletheaxesusingthesamescalingyoudidforthesectiontitled"Collecting theData".Sketchthegraphusingarulerandapencil.

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267 Figure7.2 3.Calculatethefollowing: a. x = b. s = c. n = 2surveyingonepersonatatime 4.Drawasmoothcurvethroughtopsofthebarsofthehistogram.Use12completesentencesto describethegeneralshapeofthecurve. 7.9.4CollectingAveragesofGroupsofFive Repeatsteps15ofpartIwithoneexception.Insteadofrecordingthechangeof30classmates,record theaveragechangeof30groupsof5. 1.Randomlysurvey30 groupsof5 classmates.Recordthevaluesoftheaverageoftheir change. __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 2.Constructahistogram.Scaletheaxesusingthesamescalingyoudidforsectiontitled"Collectthe Data".Sketchthegraphusingarulerandapencil.

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268 CHAPTER7.THECENTRALLIMITTHEOREM Figure7.3 3.Calculatethefollowing: a. x = b. s = c. n = 5surveyingvepeopleatatime 4.Drawasmoothcurvethroughtopsofthebarsofthehistogram.Use12completesentencesto describethegeneralshapeofthecurve. 7.9.5DiscussionQuestions 1.As n changed,whydidtheshapeofthedistributionofthedatachange?Use12completesentences toexplainwhathappened. 2.Inthesectiontitled"CollecttheData",whatwastheapproximatedistributionofthedata? X 3.Inthesectiontitled"CollectingAveragesofGroupsofFive",whatwastheapproximatedistribution ofthedata? X 4.In12completesentences,explainanydifferencesinyouranswerstoprevioustwoquestions.

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269 7.10Lab2:CentralLimitTheoremCookieRecipes 10 ClassTime: Names: 7.10.1StudentLearningOutcomes: ThestudentwillexaminepropertiesoftheCentralLimitTheorem. 7.10.2Given: X =lengthoftimeindaysthatacookierecipelastedattheOlmsteadHomestead.Assumethateachof thedifferentrecipesmakesthesamequantityofcookies. Recipe# X Recipe# X Recipe# X Recipe# X 1 1 16 2 31 3 46 2 2 5 17 2 32 4 47 2 3 2 18 4 33 5 48 11 4 5 19 6 34 6 49 5 5 6 20 1 35 6 50 5 6 1 21 6 36 1 51 4 7 2 22 5 37 1 52 6 8 6 23 2 38 2 53 5 9 5 24 5 39 1 54 1 10 2 25 1 40 6 55 1 11 5 26 6 41 1 56 2 12 1 27 4 42 6 57 4 13 1 28 1 43 2 58 3 14 3 29 6 44 6 59 6 15 2 30 2 45 2 60 5 Calculatethefollowing: a. m x = b. s x = 7.10.3CollecttheData Usearandomnumbergeneratortorandomlyselect4samplesofsize n = 5fromthegivenpopulation. Recordyoursamplesbelow.Then,foreachsample,calculatethemeantothenearesttenth.Recordthemin thespacesprovided.Recordthesamplemeansfortherestoftheclass. 10 Thiscontentisavailableonlineat.

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270 CHAPTER7.THECENTRALLIMITTHEOREM 1.Completethetable: Sample1 Sample2 Sample3 Sample4 Samplemeansfromothergroups: Means: x = x = x = x = 2.Calculatethefollowing: a. x = b. s x = 3.Again,usearandomnumbergeneratortorandomlyselect4samplesfromthepopulation.Thistime, makethesamplesofsize n = 10 .Recordthesamplesbelow.Asbefore,foreachsample,calculatethe meantothenearesttenth.Recordtheminthespacesprovided.Recordthesamplemeansfortherest oftheclass. Sample1 Sample2 Sample3 Sample4 Samplemeansfromothergroups: Means: x = x = x = x = 4.Calculatethefollowing: a. x = b. s x = 5.Fortheoriginalpopulation,constructahistogram.Makeintervalswithbarwidth=1day.Sketchthe graphusingarulerandpencil.Scaletheaxes.

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271 Figure7.4 6.Drawasmoothcurvethroughthetopsofthebarsofthehistogram.Use12completesentencesto describethegeneralshapeofthecurve. 7.10.4RepeattheProcedureforN=5 1.Forthesampleof n = 5days averagedtogether,constructahistogramoftheaverages.Makeintervals with barwidths= 1 2 day .Sketchthegraphusingarulerandpencil.Scaletheaxes.

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272 CHAPTER7.THECENTRALLIMITTHEOREM Figure7.5 2.Drawasmoothcurvethroughthetopsofthebarsofthehistogram.Use12completesentencesto describethegeneralshapeofthecurve. 7.10.5RepeattheProcedureforN=10 1.Forthesampleof n = 10days averagedtogether,constructahistogramoftheaverages.Makeintervalswith barwidths= 1 2 day .Sketchthegraphusingarulerandpencil.Scaletheaxes.

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273 Figure7.6 2.Drawasmoothcurvethroughthetopsofthebarsofthehistogram.Use12completesentencesto describethegeneralshapeofthecurve. 7.10.6ComparetheData 1.Comparethethreehistogramsyouhavemade,theoneforthepopulationandthetwoforthesample means.Inthreetovesentences,describethesimilaritiesanddifferences. 2.StatethetheoreticalaccordingtotheCLTdistributionsforthesamplemeans. a. n = 5: X b. n = 10: X 3.Arethesamplemeansfor n=5 and n=10 closetothetheoreticalmean, m x ?Explainwhyorwhy not. 4.Whichofthetwodistributionsofsamplemeanshasthesmallerstandarddeviation?Why? 5.Asnchanged,whydidtheshapeofthedistributionofthedatachange?Use12completesentences toexplainwhathappened. N OTE : ThislabwasdesignedandcontributedbyCarolOlmstead.

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274 CHAPTER7.THECENTRALLIMITTHEOREM SolutionstoExercisesinChapter7 Example7.2,Problem2p.246 m X = 2 s X = 0.5 n = 50 ,and X N 2, 0.5 p 50 SolutionstoPractice:TheCentralLimitTheorem SolutiontoExercise7.6.1p.254 b. 3.5,4.25,0.2441 SolutiontoExercise7.6.2p.254 b. 0.7499 SolutiontoExercise7.6.3p.255 b. 4.49hours SolutiontoExercise7.6.4p.255 b. 0.3802 SolutiontoExercise7.6.5p.255 b71.90 SolutionstoHomework SolutiontoExercise7.7.1p.257 b. Xbar~N 60 9 p 25 c. 0.5000 d. 59.06 e. 0.8536 f. 0.1333 h. 1530.35 i. 0.8536 SolutiontoExercise7.7.3p.257 a. N 36 10 p 16 b. 1 c. 34.31 SolutiontoExercise7.7.5p.258 a. N 250 50 p 49 b. 0.0808 c. 256.01feet SolutiontoExercise7.7.7p.258 a. Thetotallengthoftimefor9criminaltrials b. N 189,21

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275 c. 0.0432 d. 162.09 SolutiontoExercise7.7.9p.258 a. N 145 14 p 49 b. 0.6247 c. 146.68 d. 145minutes SolutiontoExercise7.7.11p.259 b. U 10 25 c. 17.5 d. N p 225 12 f. N 17.5 0.5839 h. N 962.5 32.11 j. 0.0032 SolutiontoExercise7.7.13p.260 c. N 44 000 6500 p 10 e. N 440,000 p 10 6500 f. 0.9742 g. \$52,330 h. \$46,634 SolutiontoExercise7.7.15p.260 c. N 2 4, 0 9 p 80 e. N 192, 8.05 h. Individual SolutiontoExercise7.7.17p.261 b. \$20.71;\$17.31;35 d. Exponentialdistribution, X Exp 1/20.71 f. \$20.71;\$11.14 j. N 20.71 17.31 p 5 SolutiontoExercise7.7.19p.262 B SolutiontoExercise7.7.20p.262 A SolutiontoExercise7.7.21p.262 D SolutiontoExercise7.7.22p.262 B SolutiontoExercise7.7.23p.262 A

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276 CHAPTER7.THECENTRALLIMITTHEOREM SolutionstoReview SolutiontoExercise7.8.1p.263 A SolutiontoExercise7.8.2p.263 B SolutiontoExercise7.8.3p.263 A SolutiontoExercise7.8.4p.263 a. 0.7165 b. 4.16 c. 0 SolutiontoExercise7.8.5p.263 C SolutiontoExercise7.8.6p.264 C SolutiontoExercise7.8.7p.264 D SolutiontoExercise7.8.8p.264 A

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Chapter8 CondenceIntervals 8.1CondenceIntervals 1 8.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Calculateandinterpretcondenceintervalsforonepopulationaverageandonepopulationproportion. Interpretthestudent-tprobabilitydistributionasthesamplesizechanges. Discriminatebetweenproblemsapplyingthenormalandthestudent-tdistributions. 8.1.2Introduction Supposeyouaretryingtodeterminetheaveragerentofatwo-bedroomapartmentinyourtown.Youmight lookintheclassiedsectionofthenewspaper,writedownseveralrentslisted,andaveragethemtogether. Youwouldhaveobtainedapointestimateofthetruemean.Ifyouaretryingtodeterminethepercentof timesyoumakeabasketwhenshootingabasketball,youmightcountthenumberofshotsyoumakeand dividethatbythenumberofshotsyouattempted.Inthiscase,youwouldhaveobtainedapointestimate forthetrueproportion. Weusesampledatatomakegeneralizationsaboutanunknownpopulation.Thispartofstatisticsiscalled "inferentialstatistics." Thesampledatahelpustomakeestimatesofpopulationparameters.Werealize thatthepointestimateismostlikelynottheexactvalueofthe populationparameter ,butclosetoit.After calculatingpointestimates,weconstructcondenceintervalsinwhichwebelievetheparameterlies. Inthischapter,youwilllearntoconstructandinterpretcondenceintervals.Youwillalsolearnanew distribution,theStudent-t,andhowitisusedwiththeseintervals. Ifyouworkedinthemarketingdepartmentofanentertainmentcompany,youmightbeinterestedinthe averagenumberofcompactdiscsCD'saconsumerbuyspermonth.Ifso,youcouldconductasurvey andcalculatethesampleaverage, x ,andthesamplestandarddeviation, s .Youwoulduse x toestimate thepopulationmeanand s toestimatethepopulationstandarddeviation.Thesamplemean, x ,isthe pointestimate forthepopulationmean, m .Thesamplestandarddeviation, s ,isthepointestimateforthe populationstandarddeviation, s 1 Thiscontentisavailableonlineat. 277

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278 CHAPTER8.CONFIDENCEINTERVALS A condenceinterval isanothertypeofestimatebut,insteadofbeingjustonenumber,itisanintervalof numbers.SupposefortheCDexamplewedonotknowthepopulationmean m butwedoknowthatthe populationstandarddeviationis s = 1andoursamplesizeis100.ThenbytheCentralLimitTheorem,the standarddeviationforthesamplemeanis s p n = 1 p 100 = 0.1. The EmpiricalRule ,whichappliestobell-shapeddistributions,saysthatinapproximately95%ofthesamples,thesamplemean, x ,willbewithintwostandarddeviationsofthepopulationmean m .ForourCD example,twostandarddeviationsis 2 0.1 = 0.2.Thesamplemean x iswithin0.2unitsof m Because x iswithin0.2unitsof m ,whichisunknown,then m iswithin0.2unitsof x in95%ofthesamples. Thepopulationmean m iscontainedinanintervalwhoselowernumberiscalculatedbytakingthesample meanandsubtractingtwostandarddeviations 2 0.1 andwhoseuppernumberiscalculatedbytaking thesamplemeanandaddingtwostandarddeviations.Inotherwords, m isbetween x )]TJ/F58 9.9626 Tf 10.194 0 Td [(0.2and x + 0.2in 95%ofallthesamples. FortheCDexample,supposethatasampleproducedasamplemean x = 2.Thentheunknownpopulation mean m isbetween x )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.2 = 2 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.2 = 1.8and x + 0.2 = 2 + 0.2 = 2.2 Wesaythatweare 95%condent thattheunknownpopulationmeannumberofCDsisbetween1.8and 2.2. The95%condenceintervalis.8,2.2. The95%condenceintervalimpliestwopossibilities.Eithertheinterval.8,2.2containsthetruemean m oroursampleproducedan x thatisnotwithin0.2unitsofthetruemean m .Thesecondpossibilityhappens foronly5%ofallthesamples%-95%. Rememberthatacondenceintervaliscreatedforanunknownpopulationparameterlikethepopulation mean, m .Acondenceintervalhastheform pointestimate-marginoferror,pointestimate+marginoferror Themarginoferrordependsonthecondencelevelorpercentageofcondence. 8.1.3OptionalCollaborativeClassroomActivity Haveyourinstructorrecordthenumberofmealseachstudentinyourclasseatsoutinaweek.Assume thatthestandarddeviationisknowntobe3meals.Constructanapproximate95%condenceintervalfor thetrueaveragenumberofmealsstudentseatouteachweek. 1.Calculatethesamplemean. 2. s = 3and n = thenumberofstudentssurveyed. 3.Constructtheinterval x )]TJ/F58 9.9626 Tf 10.132 0 Td [(2 s p n x + 2 s p n Wesayweareapproximately95%condentthatthetrueaveragenumberofmealsthatstudentseatoutin aweekisbetween__________and___________.

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279 8.2CondenceInterval,SinglePopulationMean,PopulationStandard DeviationKnown,Normal 2 Toconstructacondenceintervalforasingleunknownpopulationmean m wherethepopulationstandard deviationisknown, weneed x asanestimatefor m andamarginoferror.Here,themarginoferror iscalledthe errorboundforapopulationmean abbreviated EBM .Themarginoferrordependson the condencelevel abbreviated CL .Thecondencelevelistheprobabilitythatthecondenceinterval producedcontainsthetruepopulationparameter.Mostoften,itisthechoiceofthepersonconstructingthe condenceintervaltochooseacondencelevelof90%orhigherbecausehewantstobereasonablycertain ofhisconclusions. Example8.1 Supposethesamplemeanis7andtheerrorboundforthemeanis2.5. Problem x = _______and EBM = _______. Thecondenceintervalis 7 )]TJ/F58 9.9626 Tf 10.131 0 Td [(2.5,7 + 2.5 IfthecondencelevelCLis95%,thenwesayweare95%condentthatthetruepopulation meanisbetween4.5and9.5. Acondenceintervalforapopulationmeanwithaknownstandarddeviationisbasedonthe factthatthesamplemeansfollowanapproximatelynormaldistribution.Supposewehaveconstructedthe90%condenceinterval,15where x = 10and EBM = 5.Togeta90%condence interval,wemustincludethecentral90%ofthesamplemeans.Ifweincludethecentral90%,we leaveoutatotalof10%or5%ineachtailofthenormaldistribution.Tocapturethecentral90% ofthesamplemeans,wemustgoout1.645standarddeviationsoneithersideofthecalculated samplemean.The1.645isthez-scorefromastandardnormaltablethathasareatotherightequal to0.05%areaintherighttail.Thegraphshowsthegeneralsituation. Tosummarize,resultingfromtheCentralLimitTheorem, X isnormallydistributed,thatis, X N m X s X p n Sincethepopulationstandarddeviation, s ,isknown,weuseanormalcurve. Thecondencelevel, CL ,is CL = 1 )]TJ/F134 9.9626 Tf 10.256 0 Td [(a .Eachofthetailscontainsanareaequalto a 2 2 Thiscontentisavailableonlineat.

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280 CHAPTER8.CONFIDENCEINTERVALS Thez-scorethathasareatotherightof a 2 is z a 2 Forexample,if a 2 = 0.025,thenareatotheright = 0.025andareatotheleft = 1 )]TJ/F58 9.9626 Tf 10.265 0 Td [(0.025 = 0.975 and z a 2 = z 0.025 = 1.96usingacalculator,computerortable.UsingtheTI83+or84calculator function, invNorm ,youcanverifythisresult. invNorm .975,0,1 = 1.96. Theerrorboundformulaforasinglepopulationmeanwhenthevarianceisknownis EBM = z a 2 s p n Thecondenceintervalhastheformat x )]TJ/F132 9.9626 Tf 10.405 0 Td [(EBM x + EBM Thegraphgivesapictureoftheentiresituation. CL + a 2 + a 2 = CL + a = 1. Example8.2 Problem1 Supposescoresonexamsinstatisticsarenormallydistributedwithanunknownpopulationmean andapopulationstandarddeviationof3points.Asampleof36scoresistakenandgivesasample meansampleaveragescoreof68.Finda90%condenceintervalforthetruepopulationmean ofstatisticsexamscores. Therstsolutionisstep-by-step. ThesecondsolutionusestheTI-83+andTI-84calculators. Solution Tondthecondenceinterval,youneedthesamplemean, x ,andtheEBM. a. x = 68 b. EBM = z a 2 s p n c. s = 3 d. n = 36 CL = 0.90so a = 1 )]TJ/F132 9.9626 Tf 10.216 0 Td [(CL = 1 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0.90 = 0.10 Since a 2 = 0.05,then z a 2 = z .05 = 1.645

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281 fromacalculator,computerorstandardnormaltable Therefore, EBM = 1.645 3 p 36 = 0.8225 Thisgives x )]TJ/F132 9.9626 Tf 10.405 0 Td [(EBM = 68 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.8225 = 67.18 and x + EBM = 68 + 0.8225 = 68.82 The90%condenceintervalis .18,68.82. Solution TheTI-83+andTI-84caculatorssimplifythiswholeprocedure.Press STAT andarrowoverto TESTS .Arrowdownto 7:ZInterval .Press ENTER .Arrowto Stats andpress ENTER .Arrowdown andenter3for s ,68for x ,36for n ,and.90for C-level .Arrowdownto Calculate andpress ENTER .Thecondenceintervalisto3decimalplaces.178,68.822. Wecanndtheerrorboundfromthecondenceinterval. Fromtheuppervalue,subtractthe samplemean or subtractthelowervaluefromtheuppervalueanddividebytwo.Theresultis theerrorboundforthemeanEBM. EBM = 68.822 )]TJ/F58 9.9626 Tf 10.132 0 Td [(68 = 0.822or EBM = 68.822 )]TJ/F58 7.5716 Tf 6.227 0 Td [(67.178 2 = 0.822 Wecaninterpretthecondenceintervalintwoways: 1.Weare90%condentthatthetruepopulationmeanforstatisticsexamscoresisbetween 67.178and68.822. 2.Ninetypercentofallcondenceintervalsconstructedinthiswaycontainthetrueaverage statisticsexamscore.Forexample,ifweconstructed100ofthesecondenceintervals,we wouldexpect90ofthemtocontainthetruepopulationmeanexamscore. Nowforthesameproblem,nda95%condenceintervalforthetruepopulationmeanofscores. Drawthegraph.Thesamplemean,standarddeviation,andsamplesizeare: Problem2 a. x = b. s = c. n = Thecondencelevelis CL = 0.95.Graph: Thecondenceintervalisusetechnology Problem3 x )]TJ/F132 9.9626 Tf 10.405 0 Td [(EBM x + EBM = _______,_______ .Theerrorbound EBM = _______. Solution x )]TJ/F132 9.9626 Tf 10.405 0 Td [(EBM x + EBM = .02,68.98 .Theerrorbound EBM = 0.98. Wecansaythatweare95%condentthatthetruepopulationmeanforstatisticsexamscoresis between67.02and68.98andthat95%ofallcondenceintervalsconstructedinthiswaycontain thetrueaveragestatisticsexamscore. Example8.3 Supposewechangethepreviousproblem. Problem1 Leaveeverythingthesameexceptthesamplesize. a. x = 68

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282 CHAPTER8.CONFIDENCEINTERVALS b. s = 3 c. z a 2 = 1.645 Solution Ifwe increase thesamplesize n to100,we decrease theerrorbound. EBM = z a 2 s p n = 1.645 3 p 100 = 0.4935 Solution Ifwe decrease thesamplesize n to25,we increase theerrorbound. EBM = z a 2 s p n = 1.645 3 p 25 = 0.987 Problem2 Leaveeverythingthesameexceptforthecondencelevel.Weincreasethecondencelevelfrom 0.90to0.95. a. x = 68 b. s = 3 c. z a 2 = 1.645 Solution a b Figure8.1 The90%condenceintervalis.18,68.82.The95%condenceintervalis.02,68.98.The 95%condenceintervaliswider. Ifyoulookatthegraphs,becausethearea0.95islargerthanthe area0.90,itmakessensethatthe95%condenceintervaliswider. 8.3CondenceInterval,SinglePopulationMean,StandardDeviation Unknown,Student-T 3 Inpractice,werarelyknowthepopulation standarddeviation .Inthepast,whenthesamplesizewaslarge, thisdidnotpresentaproblemtostatisticians.Theyusedthesamplestandarddeviation s asanestimate 3 Thiscontentisavailableonlineat.

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283 for s andproceededasbeforetocalculatea condenceinterval withcloseenoughresults.However, statisticiansranintoproblemswhenthesamplesizewassmall.Asmallsamplesizecausedinaccuracies inthecondenceinterval.WilliamS.GossettoftheGuinnessbreweryinDublin,Irelandranintothisvery problem.Hisexperimentswithhopsandbarleyproducedveryfewsamples.Justreplacing s with s didnot produceaccurateresultswhenhetriedtocalculateacondenceinterval.Herealizedthathecouldnotuse anormaldistributionforthecalculation.Thisproblemledhimto"discover"whatiscalledthe Student-t distribution .ThenamecomesfromthefactthatGossetwroteunderthepenname"Student." Upuntilthemid1990s,statisticiansusedthe normaldistribution approximationforlargesamplesizes andonlyusedtheStudent-tdistributionforsamplesizesofatmost30.Withthecommonuseofgraphing calculatorsandcomputers,thepracticeistousetheStudent-tdistributionwhenever s isusedasanestimate for s Ifyoudrawasimplerandomsampleofsize n fromapopulationthathasapproximatelyanormaldistributionwithmean m andunknownpopulationstandarddeviation s andcalculatethet-score t = x )]TJ/F134 7.5716 Tf 6.322 0 Td [(m s p n thenthet-scoresfollowa Student-tdistributionwith n )]TJ/F58 9.9626 Tf 10.358 0 Td [(1 degreesoffreedom .Thet-scorehasthesame interpretationasthe z-score .Itmeasureshowfar x isfromitsmean m .Foreachsamplesize n ,thereisa differentStudent-tdistribution. The degreesoffreedom n )]TJ/F58 9.9626 Tf 10.634 0 Td [(1,comefromthesamplestandarddeviation s .InChapter2,weused n deviations x )]TJETq1 0 0 1 141.972 445.861 cm[]0 d 0 J 0.398 w 0 0 m 5.499 0 l SQBT/F132 9.9626 Tf 142.266 439.649 Td [(xvalues tocalculate s .Becausethesumofthedeviationsis0,wecanndthelastdeviation onceweknowtheother n )]TJ/F58 9.9626 Tf 10.145 0 Td [(1deviations.Theother n )]TJ/F58 9.9626 Tf 10.145 0 Td [(1deviationscanchangeorvaryfreely. Wecallthe number n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 thedegreesoffreedomdf. ThefollowingaresomefactsabouttheStudent-tdistribution: 1.ThegraphfortheStudent-tdistributionissimilartothenormalcurve. 2.TheStudent-tdistributionhasmoreprobabilityinitstailsthanthenormalbecausethespreadis somewhatgreaterthanthenormal. 3.Theunderlyingpopulationofobservationsisnormalwithunknownpopulationmean m andunknownpopulationstandarddeviation s AStudent-ttablethereisoneinthebook-seetheTableofContentsgivest-scoresgiventhedegreesof freedomandtheright-tailedprobability.Thetableisverylimited. Calculatorsandcomputerscaneasily calculateanyStudent-tprobabilities. ThenotationfortheStudent-tdistributionisusingTastherandomvariable T t df where df = n )]TJ/F58 9.9626 Tf 10.132 0 Td [(1. Ifthepopulationstandarddeviationis notknown ,thenthe errorboundforapopulationmean formulais: EBM = t a 2 s p n t a 2 isthet-scorewithareatotherightequalto a 2 s =thesamplestandarddeviation Themechanicsforcalculatingtheerrorboundandthecondenceintervalarethesameaswhen s isknown. Example8.4 Supposeyoudoastudyofacupuncturetodeterminehoweffectiveitisinrelievingpain.You measuresensoryratesfor15subjectswiththeresultsgivenbelow.Usethesampledatatoconstructa95%condenceintervalforthemeansensoryrateforthepopulationassumednormal fromwhichyoutookthedata.

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284 CHAPTER8.CONFIDENCEINTERVALS 8.6;9.4;7.9;6.8;8.3;7.3;9.2;9.6;8.7;11.4;10.3;5.4;8.1;5.5;6.9 Note: Therstsolutionisstep-by-step. ThesecondsolutionusestheTI-83+andTI-84calculators. Solution Tondthecondenceinterval,youneedthesamplemean, x ,andtheEBM. x = 8.2267 s = 1.6722 n = 15 CL = 0.95so a = 1 )]TJ/F132 9.9626 Tf 10.131 0 Td [(CL = 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.95 = 0.05 EBM = t a 2 s p n a 2 = 0.025 t a 2 = t .025 = 2.14 Student-ttablewith df = 15 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 14 Therefore, EBM = 2.14 1.6722 p 15 = 0.924 Thisgives x )]TJ/F132 9.9626 Tf 10.131 0 Td [(EBM = 8.2267 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.9240 = 7.3 and x + EBM = 8.2267 + 0.9240 = 9.15 The95%condenceintervalis .30,9.15 Youare95%condentorsurethatthetruepopulationaveragesensoryrateisbetween7.30and 9.15. Solution TI-83+orTI-84:Usethefunction 8:TInterval in STATTESTS .Onceyouarein TESTS ,press 8:TInterval andarrowto Data .Press ENTER .Arrowdownandenterthelistnamewhereyou putthedatafor List ,enter1for Freq ,andenter.95for C-level .Arrowdownto Calculate and press ENTER .Thecondenceintervalis.3006,9.1527 8.4CondenceIntervalforaPopulationProportion 4 Duringanelectionyear,weseearticlesinthenewspaperthatstate condenceintervals intermsofproportionsorpercentages.Forexample,apollforaparticularcandidaterunningforpresidentmightshow thatthecandidatehas40%ofthevotewithin3percentagepoints.Often,electionpollsarecalculatedwith 95%condence.So,thepollsterswouldbe95%condentthatthetrueproportionofvoterswhofavored thecandidatewouldbebetween0.37and0.43 0.40 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.03,0.40 + 0.03 Investorsinthestockmarketareinterestedinthetrueproportionofstocksthatgoupanddowneachweek. BusinessesthatsellpersonalcomputersareinterestedintheproportionofhouseholdsintheUnitedStates thatownpersonalcomputers.Condenceintervalscanbecalculatedforthetrueproportionofstocksthat goupordowneachweekandforthetrueproportionofhouseholdsintheUnitedStatesthatownpersonal computers. 4 Thiscontentisavailableonlineat.

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285 Theproceduretondthecondenceinterval,thesamplesize,the errorbound, andthe condencelevel foraproportionissimilartothatforthepopulationmean.Theformulasaredifferent. Howdoyouknowyouaredealingwithaproportionproblem? First,theunderlying distributionisbinomial .Thereisnomentionofameanoraverage.If X isabinomialrandomvariable,then X B n p where n =thenumberoftrialsand p =theprobabilityofasuccess.Toformaproportion,take X ,therandomvariableforthenumberofsuccessesanddivideitby n ,thenumberoftrialsorthesamplesize.The randomvariable P 'read"Pprime"isthatproportion, P = X n Sometimestherandomvariableis P ,read"Phat". When n islarge,wecanusethe normaldistribution toapproximatethebinomial. X N )]TJ/F132 9.9626 Tf 4.812 -8.075 Td [(n p p n p q Ifwedivideallvaluesoftherandomvariableby n ,themeanby n ,andthestandarddeviationby n ,we getanormaldistributionofproportionswith P ',calledtheestimatedproportion,astherandomvariable. Recallthataproportion=thenumberofsuccessesdividedby n X n = P N n p n p n p q n Byalgebra, p n p q n = q p q n P 'followsanormaldistributionforproportions : P N p q p q n Thecondenceintervalhastheform p )]TJ/F132 9.9626 Tf 10.131 0 Td [(EBP p + EBP p = x n p '=the estimatedproportion ofsuccesses p 'isa pointestimate for p ,thetrueproportion x =the number ofsuccesses. n =thesizeofthesample Theerrorboundforaproportionis EBP = z a 2 q p q n q = 1 )]TJ/F132 9.9626 Tf 10.803 0 Td [(p Thisformulaisactuallyverysimilartotheerrorboundformulaforamean.Thedifferenceisthestandard deviation.Forameanwherethepopulationstandarddeviationisknown,thestandarddeviationis s p n Foraproportion,thestandarddeviationis q p q n However,intheerrorboundformula,thestandarddeviationis q p q n Intheerrorboundformula, p 'and q 'areestimatesof p and q .Theestimatedproportions p 'and q 'areused because p and q arenotknown. p 'and q 'arecalculatedfromthedata. p 'istheestimatedproportionof successes. q 'istheestimatedproportionoffailures.

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286 CHAPTER8.CONFIDENCEINTERVALS Whenastudygivesamarginoferrorof"+or-3percentagepoints",thisisdeterminedbeforethesurvey isdone.Since p 'and q 'areunknown, themostconservativechoiceis p = 0.5 and q = 0.5,becausethese valuesgivethelargeststandarddeviation,errorbound,andcondenceinterval. N OTE :Forthenormaldistributionofproportions,thez-scoreformulaisasfollows. If P N p q p q n thenthez-scoreformulais z = p )]TJ/F132 7.5716 Tf 6.738 0 Td [(p p p q n Example8.5 Supposethatasampleof500householdsinPhoenixwastakenlastMaytodeterminewhetherthe oldestchildhadgivenhis/hermotheraMother'sDaycard.Ofthe500households,421responded yes.Computea95%condenceintervalforthetrueproportionofallPhoenixhouseholdswhose oldestchildgavehis/hermotheraMother'sDaycard. Note: Therstsolutionisstep-by-step. ThesecondsolutionusestheTI-83+andTI-84calculators. Solution Let X =thenumberofoldestchildrenwhogavetheirmothersMother'sDaycardlastMay. X is binomial. X B 500, 421 500 Tocalculatethecondenceinterval,youmustnd p ', q ',and EBP n = 500 x =thenumberofsuccesses = 421 p = x n = 421 500 = 0.842 q = 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p = 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.842 = 0.158 Since CL = 0.95,then a = 1 )]TJ/F132 9.9626 Tf 10.131 0 Td [(CL = 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.95 = 0.05 a 2 = 0.025. Then z a 2 = z .025 = 1.96usingacalculator,computer,orstandardnormaltable. Rememberthattheareatotheright=0.025andtherefore,areatotheleftis0.975. Thez-scorethatcorrespondsto0.975is1.96. EBP = z a 2 q p q n = 1.96 r h .842 .158 500 i = 0.032 p )]TJ/F132 9.9626 Tf 10.131 0 Td [(EBP = 0.842 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0.032 = 0.81 p + EBP = 0.842 + 0.032 = 0.874 Thecondenceintervalforthetruebinomialpopulationproportionis p )]TJ/F132 9.9626 Tf 10.131 0 Td [(EBP p + EBP = 0.810,0.874 Weare95%condentthatbetween81%and87.4%oftheoldestchildreninhouseholdsinPhoenix gavetheirmothersaMother'sDaycardlastMay. Wecanalsosaythat95%ofthecondenceintervalsconstructedinthiswaycontainthetrue proportionofoldestchildreninPhoenixwhogavetheirmothersaMother'sDaycardlastMay.

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287 Solution TI-83+andTI-84:Press STAT andarrowoverto TESTS .Arrowdownto A:PropZint .Press ENTER Enter421for x ,500for n ,and.95for C-Level .Arrowdownto Calculate andpress ENTER .The condenceintervalis.81003,0.87397. Example8.6 Foraclassproject,apoliticalsciencestudentatalargeuniversitywantstodeterminethepercent ofstudentsthatareregisteredvoters.Hesurveys500studentsandndsthat300areregistered voters.Computea90%condenceintervalforthetruepercentofstudentsthatareregistered votersandinterpretthecondenceinterval. Solution x = 300and n = 500.UsingaTI-83+or84calculator,the90%condenceintervalforthetrue percentofstudentsthatareregisteredvotersis.564,0.636. Interpretation: Weare90%condentthatthetruepercentofstudentsthatareregisteredvotersisbetween 56.4%and63.6%. Ninetypercent%ofallcondenceintervalsconstructedinthiswaycontainthetrue percentofstudentsthatareregisteredvoters.

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288 CHAPTER8.CONFIDENCEINTERVALS 8.5SummaryofFormulas 5 Formula8.1: Generalformofacondenceinterval lowervalue uppervalue = pointestimate )]TJ/F132 9.9626 Tf 10.131 0 Td [(errorbound pointestimate + errorbound Formula8.2: Tondtheerrorboundwhenyouknowthecondenceinterval errorbound = uppervalue )]TJ/F132 9.9626 Tf 10.131 0 Td [(pointestimate OR errorbound = uppervalue )]TJ/F132 7.5716 Tf 6.228 0 Td [(lowervalue 2 Formula8.3: SinglePopulationMean,KnownStandardDeviation,NormalDistribution UsetheNormalDistributionforMeansSection7.2 EBM = z a 2 s p n Thecondenceintervalhastheformat x )]TJ/F132 9.9626 Tf 10.131 0 Td [(EBM x + EBM Formula8.4: SinglePopulationMean,UnknownStandardDeviation,Student-tDistribution UsetheStudent-tDistributionwithdegreesoffreedom df = n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1. EBM = t a 2 s p n Formula8.5: SinglePopulationProportion,NormalDistribution UsetheNormalDistributionforasinglepopulationproportion p = x n EBP = z a 2 q p q n p + q = 1 Thecondenceintervalhastheformat p )]TJ/F132 9.9626 Tf 10.131 0 Td [(EBP p + EBP Formula8.6: PointEstimates x isapointestimatefor m p 'isapointestimatefor r s isapointestimatefor s 5 Thiscontentisavailableonlineat.

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289 8.6Practice1:CondenceIntervalsforAverages,KnownPopulation StandardDeviation 6 8.6.1StudentLearningOutcomes ThestudentwillexplorethepropertiesofCondenceIntervalsforAveragesandgainknowledgeof populationstandarddeviation. 8.6.2Given TheaverageageforallFoothillCollegestudentsforFall2005was32.7.Thepopulationstandarddeviation hasbeenprettyconsistentat15.Twenty-veWinter2006studentswererandomlyselected.Theaverage ageforthesamplewas30.4.WeareinterestedinthetrueaverageageforWinter2006FoothillCollege students.http://research.fhda.edu/factbook/FHdemofs/Fact_sheet_fh_2005f.pdf 7 Let X = theageofaWinter2006FoothillCollegestudent 8.6.3CalculatingtheCondenceInterval Exercise8.6.1 Solutiononp.314. x = Exercise8.6.2 Solutiononp.314. n = Exercise8.6.3 Solutiononp.314. 15=insertsymbolhere Exercise8.6.4 Solutiononp.314. DenetheRandomVariable, X ,inwords. X = Exercise8.6.5 Solutiononp.314. Whatis x estimating? Exercise8.6.6 Solutiononp.314. Is s x known? Exercise8.6.7 Solutiononp.314. Asaresultofyouranswerto,statetheexactdistributiontousewhencalculatingtheCondenceInterval. 8.6.4ExplainingtheCondenceInterval Constructa95%CondenceIntervalforthetrueaverageageofWinter2006FoothillCollegestudents. Exercise8.6.8 Solutiononp.314. Howmuchareaisinbothtailscombined? a = ________ Exercise8.6.9 Solutiononp.314. Howmuchareaisineachtail? a 2 = ________ Exercise8.6.10 Solutiononp.314. Identifythefollowingspecications: 6 Thiscontentisavailableonlineat. 7 http://research.fhda.edu/factbook/FHdemofs/Fact_sheet_fh_2005f.pdf

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290 CHAPTER8.CONFIDENCEINTERVALS a. lowerlimit= b. upperlimit= c. errorbound= Exercise8.6.11 Solutiononp.314. The95%CondenceIntervalis:__________________ Exercise8.6.12 Fillintheblanksonthegraphwiththeareas,upperandlowerlimitsoftheCondenceInterval,and thesamplemean. Figure8.2 Exercise8.6.13 Inonecompletesentence,explainwhattheintervalmeans. 8.6.5DiscussionQuestions Exercise8.6.14 Usingthesamemean,standarddeviationandlevelofcondence,supposethat n were69instead of25.Wouldtheerrorboundbecomelargerorsmaller?Howdoyouknow? Exercise8.6.15 Usingthesamemean,standarddeviationandsamplesize,howwouldtheerrorboundchangeif thecondencelevelwerereducedto90%?Why?

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291 8.7Practice2:CondenceIntervalsforAverages,UnknownPopulation StandardDeviation 8 8.7.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofcondenceintervalsforaverages,aswellastheproperties ofanunknownpopulationstandarddeviation. 8.7.2Given Thefollowingrealdataaretheresultofarandomsurveyof39nationalagswithreplacementbetween picksfromvariouscountries.Weareinterestedinndingacondenceintervalforthetrueaveragenumber ofcolorsonanationalag.Let X = thenumberofcolorsonanationalag. X Freq. 1 1 2 7 3 18 4 7 5 6 8.7.3CalculatingtheCondenceInterval Exercise8.7.1 Solutiononp.314. Calculatethefollowing: a. x = b. s x = c. n = Exercise8.7.2 Solutiononp.315. DenetheRandomVariable, X ,inwords. X = __________________________ Exercise8.7.3 Solutiononp.315. Whatis x estimating? Exercise8.7.4 Solutiononp.315. Is s x known? Exercise8.7.5 Solutiononp.315. Asaresultofyouranswerto,statetheexactdistributiontousewhencalculatingtheCondenceInterval. 8 Thiscontentisavailableonlineat.

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292 CHAPTER8.CONFIDENCEINTERVALS 8.7.4CondenceIntervalfortheTrueAverageNumber Constructa95%CondenceIntervalforthetrueaveragenumberofcolorsonnationalags. Exercise8.7.6 Solutiononp.315. Howmuchareaisinbothtailscombined? a = Exercise8.7.7 Solutiononp.315. Howmuchareaisineachtail? a 2 = Exercise8.7.8 Solutiononp.315. Calculatethefollowing: a. lowerlimit= b. upperlimit= c. errorbound= Exercise8.7.9 Solutiononp.315. The95%CondenceIntervalis: Exercise8.7.10 Fillintheblanksonthegraphwiththeareas,upperandlowerlimitsoftheCondenceInterval, andthesamplemean. Figure8.3 Exercise8.7.11 Inonecompletesentence,explainwhattheintervalmeans. 8.7.5DiscussionQuestions Exercise8.7.12 Usingthesame x s x ,andlevelofcondence,supposethat n were69insteadof39.Wouldthe errorboundbecomelargerorsmaller?Howdoyouknow? Exercise8.7.13 Usingthesame x s x ,and n = 39 ,howwouldtheerrorboundchangeifthecondencelevelwere reducedto90%?Why?

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293 8.8Practice3:CondenceIntervalsforProportions 9 8.8.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofthecondenceintervalsforproportions. 8.8.2Given TheIceChaletoffersdozensofdifferentbeginningice-skatingclasses.Alloftheclassnamesareputintoa bucket.The5P.M.,Mondaynight,ages8-12,beginningice-skatingclasswaspicked.Inthatclasswere64 girlsand16boys.Supposethatweareinterestedinthetrueproportionofgirls,ages8-12,inallbeginning ice-skatingclassesattheIceChalet. 8.8.3EstimatedDistribution Exercise8.8.1 Whatisbeingcounted? Exercise8.8.2 Solutiononp.315. Inwords,denetheRandomVariable X X = Exercise8.8.3 Solutiononp.315. Calculatethefollowing: a. x = b. n = c. p = Exercise8.8.4 Solutiononp.315. Statetheestimateddistributionof X X Exercise8.8.5 Solutiononp.315. DeneanewRandomVariable P '.Whatis p 'estimating? Exercise8.8.6 Solutiononp.315. Inwords,denetheRandomVariable P '. P = Exercise8.8.7 Statetheestimateddistributionof P '. P 8.8.4ExplainingtheCondenceInterval Constructa92%CondenceIntervalforthetrueproportionofgirlsintheage8-12beginningice-skating classesattheIceChalet. Exercise8.8.8 Solutiononp.315. Howmuchareaisinbothtailscombined? a = Exercise8.8.9 Solutiononp.315. Howmuchareaisineachtail? a 2 = Exercise8.8.10 Solutiononp.315. Calculatethefollowing: a. lowerlimit= 9 Thiscontentisavailableonlineat.

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294 CHAPTER8.CONFIDENCEINTERVALS b. upperlimit= c. errorbound= Exercise8.8.11 Solutiononp.316. The92%CondenceIntervalis: Exercise8.8.12 Fillintheblanksonthegraphwiththeareas,upperandlowerlimitsoftheCondenceInterval,and thesampleproportion. Figure8.4 Exercise8.8.13 Inonecompletesentence,explainwhattheintervalmeans. 8.8.5DiscussionQuestions Exercise8.8.14 Usingthesame p 'andlevelofcondence,supposethatnwereincreasedto100.Wouldtheerror boundbecomelargerorsmaller?Howdoyouknow? Exercise8.8.15 Usingthesame p 'and n = 80 ,howwouldtheerrorboundchangeifthecondencelevelwere increasedto98%?Why? Exercise8.8.16 Ifyoudecreasedtheallowableerrorbound,whywouldtheminimumsamplesizeincreasekeepingthesamelevelofcondence?

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295 8.9Homework 10 N OTE :Ifyouareusingastudent-tdistributionforahomeworkproblembelow,youmayassume thattheunderlyingpopulationisnormallydistributed.Ingeneral,youmustrstprovethat assumption,though. Exercise8.9.1 Solutiononp.316. Amongvariousethnicgroups,thestandarddeviationofheightsisknowntobeapproximately3 inches.Wewishtoconstructa95%condenceintervalfortheaverageheightofmaleSwedes.48 maleSwedesaresurveyed.Thesamplemeanis71inches.Thesamplestandarddeviationis2.8 inches. a.i. x = ________ ii. s = ________ iii. s x = ________ iv. n = ________ v. n )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = ________ b. DenetheRandomVariables X and X ,inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa95%condenceintervalforthepopulationaverageheightofmaleSwedes. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. e. Whatwillhappentothelevelofcondenceobtainedif1000maleSwedesaresurveyed insteadof48?Why? Exercise8.9.2 InsixpackagesofTheFlintstonesRealFruitSnackstherewere5Bam-Bamsnackpieces.The totalnumberofsnackpiecesinthesixbagswas68.Wewishtocalculatea96%condenceinterval forthepopulationproportionofBam-Bamsnackpieces. a. DenetheRandomVariables X and P ',inwords. b. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice c. Calculate p '. d. Constructa96%condenceintervalforthepopulationproportionofBam-Bamsnack piecesperbag. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. e. Doyouthinkthatsixpackagesoffruitsnacksyieldenoughdatatogiveaccurateresults? Whyorwhynot? Exercise8.9.3 Solutiononp.316. Arandomsurveyofenrollmentat35communitycollegesacrosstheUnitedStatesyieldedthe followingguressource: MicrosoftBookshelf :6414;1550;2109;9350;21828;4300;5944;5722; 2825;2044;5481;5200;5853;2750;10012;6357;27000;9414;7681;3200;17500;9200;7380;18314; 6557;13713;17768;7493;2771;2861;1263;7285;28165;5080;11622.Assumetheunderlying populationisnormal. 10 Thiscontentisavailableonlineat.

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296 CHAPTER8.CONFIDENCEINTERVALS a.i. x = ii. s x = ________ iii. n = ________ iv. n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = ________ b. DenetheRandomVariables X and X ,inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa95%condenceintervalforthepopulationaverageenrollmentatcommunity collegesintheUnitedStates. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. e. Whatwillhappentotheerrorboundandcondenceintervalif500communitycolleges weresurveyed?Why? Exercise8.9.4 Fromastackof IEEESpectrum magazines,announcementsfor84upcomingengineeringconferenceswererandomlypicked.Theaveragelengthoftheconferenceswas3.94days,withastandard deviationof1.28days.Assumetheunderlyingpopulationisnormal. a. DenetheRandomVariables X and X ,inwords. b. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. c. Constructa95%condenceintervalforthepopulationaveragelengthofengineeringconferences. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. Exercise8.9.5 Solutiononp.316. Supposethatacommitteeisstudyingwhetherornotthereiswasteoftimeinourjudicialsystem. Itisinterestedintheaverageamountoftimeindividualswasteatthecourthousewaitingtobe calledforservice.Thecommitteerandomlysurveyed81people.Thesampleaveragewas8hours withasamplestandarddeviationof4hours. a.i. x = ________ ii. s x = ________ iii. n = ________ iv. n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = ________ b. DenetheRandomVariables X and X ,inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa95%condenceintervalforthepopulationaveragetimewasted. a. Statethecondenceinterval. b. Sketchthegraph. c. Calculatetheerrorbound. e. Explaininacompletesentencewhatthecondenceintervalmeans. Exercise8.9.6 Supposethatanaccountingrmdoesastudytodeterminethetimeneededtocompleteoneperson'staxforms.Itrandomlysurveys100people.Thesampleaverageis23.6hours.Thereisa knownstandarddeviationof7.0hours.Thepopulationdistributionisassumedtobenormal. a.i. x = ________

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297 ii. s = ________ iii. s x = ________ iv. n = ________ v. n )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = ________ b. DenetheRandomVariables X and X ,inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa90%condenceintervalforthepopulationaveragetimetocompletethetax forms. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. e. Ifthermwishedtoincreaseitslevelofcondenceandkeeptheerrorboundthesameby takinganothersurvey,whatchangesshoulditmake? f. Ifthermdidanothersurvey,kepttheerrorboundthesame,andonlysurveyed49people,whatwouldhappentothelevelofcondence?Why? g. Supposethatthermdecidedthatitneededtobeatleast96%condentofthepopulation averagelengthoftimetowithin1hour.Howwouldthenumberofpeopletherm surveyschange?Why? Exercise8.9.7 Solutiononp.316. Asampleof16smallbagsofthesamebrandofcandieswasselected.Assumethatthepopulation distributionofbagweightsisnormal.Theweightofeachbagwasthenrecorded.Themean weightwas2ounceswithastandarddeviationof0.12ounces.Thepopulationstandarddeviation isknowntobe0.1ounce. a.i. x = ________ ii. s = ________ iii. s x = ________ iv. n = ________ v. n )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = ________ b. DenetheRandomVariable X ,inwords. c. DenetheRandomVariable X ,inwords. d. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. e. Constructa90%condenceintervalforthepopulationaverageweightofthecandies. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. f. Constructa98%condenceintervalforthepopulationaverageweightofthecandies. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. g. Incompletesentences,explainwhythecondenceintervalinfislargerthanthecondenceintervaline. h. Incompletesentences,giveaninterpretationofwhattheintervalinfmeans. Exercise8.9.8 Apharmaceuticalcompanymakestranquilizers.Itisassumedthatthedistributionforthelength oftimetheylastisapproximatelynormal.Researchersinahospitalusedthedrugonarandom sampleof9patients.Theeffectiveperiodofthetranquilizerforeachpatientinhourswasas follows:2.7;2.8;3.0;2.3;2.3;2.2;2.8;2.1;and2.4.

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298 CHAPTER8.CONFIDENCEINTERVALS a.i. x = ________ ii. s x = ________ iii. n = ________ iv. n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = ________ b. DenetheRandomVariable X ,inwords. c. DenetheRandomVariable X ,inwords. d. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. e. Constructa95%condenceintervalforthepopulationaveragelengthoftime. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. f. Whatdoesitmeantobe%condentinthisproblem? Exercise8.9.9 Solutiononp.317. Supposethat14childrenweresurveyedtodeterminehowlongtheyhadtousetrainingwheels. Itwasrevealedthattheyusedthemanaverageof6monthswithasamplestandarddeviationof 3months.Assumethattheunderlyingpopulationdistributionisnormal. a.i. x = ________ ii. s x = ________ iii. n = ________ iv. n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = ________ b. DenetheRandomVariable X ,inwords. c. DenetheRandomVariable X ,inwords. d. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. e. Constructa99%condenceintervalforthepopulationaveragelengthoftimeusingtrainingwheels. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. f. Whywouldtheerrorboundchangeifthecondencelevelwasloweredto90%? Exercise8.9.10 Insurancecompaniesareinterestedinknowingthepopulationpercentofdriverswhoalways buckleupbeforeridinginacar. a. Whendesigningastudytodeterminethispopulationproportion,whatistheminimum numberyouwouldneedtosurveytobe95%condentthatthepopulationproportion isestimatedtowithin0.03? b. Ifitwaslaterdeterminedthatitwasimportanttobemorethan95%condentandanew surveywascommissioned,howwouldthataffecttheminimumnumberyouwould needtosurvey?Why? Exercise8.9.11 Solutiononp.317. Supposethattheinsurancecompaniesdiddoasurvey.Theyrandomlysurveyed400driversand foundthat320claimedtoalwaysbuckleup.Weareinterestedinthepopulationproportionof driverswhoclaimtoalwaysbuckleup. a.i. x = ________ ii. n = ________ iii. p = ________

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299 b. DenetheRandomVariables X and P ',inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa95%condenceintervalforthepopulationproportionthatclaimtoalways buckleup. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. e. Ifthissurveyweredonebytelephone,list3difcultiesthecompaniesmighthavein obtainingrandomresults. Exercise8.9.12 Unoccupiedseatsonightscauseairlinestoloserevenue.Supposealargeairlinewantstoestimateitsaveragenumberofunoccupiedseatsperightoverthepastyear.Toaccomplishthis,the recordsof225ightsarerandomlyselectedandthenumberofunoccupiedseatsisnotedforeach ofthesampledights.Thesamplemeanis11.6seatsandthesamplestandarddeviationis4.1 seats. a.i. x = ________ ii. s x = ________ iii. n = ________ iv. n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = ________ b. DenetheRandomVariables X and X ,inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa92%condenceintervalforthepopulationaveragenumberofunoccupied seatsperight. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. Exercise8.9.13 Solutiononp.317. Accordingtoarecentsurveyof1200people,61%feelthatthepresidentisdoinganacceptable job.Weareinterestedinthepopulationproportionofpeoplewhofeelthepresidentisdoingan acceptablejob. a. DenetheRandomVariables X and P ',inwords. b. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. c. Constructa90%condenceintervalforthepopulationproportionofpeoplewhofeelthe presidentisdoinganacceptablejob. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. Exercise8.9.14 Asurveyoftheaverageamountofcentsoffthatcouponsgivewasdonebyrandomlysurveying onecouponperpagefromthecouponsectionsofarecentSanJoseMercuryNews.Thefollowing datawerecollected:20;75;50;65;30;55;40;40;30;55;\$1.50;40;65;40.Assumethe underlyingdistributionisapproximatelynormal. a.i. x = ________ ii. s x = ________ iii. n = ________

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300 CHAPTER8.CONFIDENCEINTERVALS iv. n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = ________ b. DenetheRandomVariables X and X ,inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa95%condenceintervalforthepopulationaverageworthofcoupons. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. e. Ifmanyrandomsamplesweretakenofsize14,whatpercentofthecondentintervals constructedshouldcontainthepopulationaverageworthofcoupons?Explainwhy. Exercise8.9.15 Solutiononp.317. Anarticleregardinginterracialdatingandmarriagerecentlyappearedinthe WashingtonPost .Of the1709randomlyselectedadults,315identiedthemselvesasLatinos,323identiedthemselves asblacks,254identiedthemselvesasAsians,and779identiedthemselvesaswhites.Inthis survey,86%ofblackssaidthattheirfamilieswouldwelcomeawhitepersonintotheirfamilies. AmongAsians,77%wouldwelcomeawhitepersonintotheirfamilies,71%wouldwelcomea Latino,and66%wouldwelcomeablackperson. a. Weareinterestedinndingthe95%condenceintervalforthepercentofblackfamilies thatwouldwelcomeawhitepersonintotheirfamilies.DenetheRandomVariables X and P ',inwords. b. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. c. Constructa95%condenceinterval i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. Exercise8.9.16 Refertotheproblemabove. a. Constructthe95%condenceintervalsforthethreeAsianresponses. b. Eventhoughthethreepointestimatesaredifferent,doanyofthecondenceintervals overlap?Which? c. Foranyintervalsthatdooverlap,inwords,whatdoesthisimplyaboutthesignicanceof thedifferencesinthetrueproportions? d. Foranyintervalsthatdonotoverlap,inwords,whatdoesthisimplyaboutthesignicanceofthedifferencesinthetrueproportions? Exercise8.9.17 Solutiononp.317. Acampdirectorisinterestedintheaveragenumberofletterseachchildsendsduringhis/her campsession.Thepopulationstandarddeviationisknowntobe2.5.Asurveyof20campersis taken.Theaveragefromthesampleis7.9withasamplestandarddeviationof2.8. a.i. x = ________ ii. s = ________ iii. s x = ________ iv. n = ________ v. n )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = ________ b. DenetheRandomVariables X and X ,inwords. c. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. d. Constructa90%condenceintervalforthepopulationaveragenumberofletterscampers sendhome.

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301 i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. e. Whatwillhappentotheerrorboundandcondenceintervalif500campersaresurveyed? Why? Exercise8.9.18 StanfordUniversityconductedastudyofwhetherrunningishealthyformenandwomenover age50.Duringthersteightyearsofthestudy,1.5%ofthe451membersofthe50-PlusFitness Associationdied.Weareinterestedintheproportionofpeopleover50whorananddiedinthe sameeightyearperiod. a. DenetheRandomVariables X and P ',inwords. b. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. c. Constructa97%condenceintervalforthepopulationproportionofpeopleover50who rananddiedinthesameeightyearperiod. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. d. Explainwhata%condenceintervalmeansforthisstudy. Exercise8.9.19 Solutiononp.317. Inarecentsampleof84usedcarssalescosts,thesamplemeanwas\$6425withastandarddeviation of\$3156.Assumetheunderlyingdistributionisapproximatelynormal. a. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. b. DenetheRandomVariable X ,inwords. c. Constructa95%condenceintervalforthepopulationaveragecostofausedcar. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. d. Explainwhata%condenceintervalmeansforthisstudy. Exercise8.9.20 Atelephonepollof1000adultAmericanswasreportedinanissueof TimeMagazine .Oneofthe questionsaskedwasWhatisthemainproblemfacingthecountry?20%answeredcrime.We areinterestedinthepopulationproportionofadultAmericanswhofeelthatcrimeisthemain problem. a. DenetheRandomVariables X and P ',inwords. b. Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. c. Constructa95%condenceintervalforthepopulationproportionofadultAmericans whofeelthatcrimeisthemainproblem. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. d. Supposewewanttolowerthesamplingerror.Whatisonewaytoaccomplishthat? e. ThesamplingerrorgivenbyYankelovichPartners,Inc.whichconductedthepollis 3%.In1-3completesentences,explainwhatthe 3%represents.

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302 CHAPTER8.CONFIDENCEINTERVALS Exercise8.9.21 Solutiononp.318. Refertotheaboveproblem.Anotherquestioninthepollwas[Howmuchare]youworried aboutthequalityofeducationinourschools?63%respondedalot.Weareinterestedinthe populationproportionofadultAmericanswhoareworriedalotaboutthequalityofeducationin ourschools. 1.DenetheRandomVariables X and P ',inwords. 2.Whichdistributionshouldyouuseforthisproblem?Explainyourchoice. 3.Constructa95%condenceintervalforthepopulationproportionofadultAmericansworriedalotaboutthequalityofeducationinourschools. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. 4.ThesamplingerrorgivenbyYankelovichPartners,Inc.whichconductedthepollis 3%. In1-3completesentences,explainwhatthe 3%represents. Exercise8.9.22 Sixdifferentnationalbrandsofchocolatechipcookieswererandomlyselectedatthesupermarket. Thegramsoffatperservingareasfollows:8;8;10;7;9;9.Assumetheunderlyingdistributionis approximatelynormal. a. Calculatea90%condenceintervalforthepopulationaveragegramsoffatperservingof chocolatechipcookiessoldinsupermarkets. i. Statethecondenceinterval. ii. Sketchthegraph. iii. Calculatetheerrorbound. b. Ifyouwantedasmallererrorboundwhilekeepingthesamelevelofcondence,what shouldhavebeenchangedinthestudybeforeitwasdone? c. Gotothestoreandrecordthegramsoffatperservingofsixbrandsofchocolatechip cookies. d. Calculatetheaverage. e. Istheaveragewithintheintervalyoucalculatedinparta?Didyouexpectittobe?Why orwhynot? Exercise8.9.23 Acondenceintervalforaproportionisgiventobe0.22,0.34.Whydoesn'tthelowerlimitof thecondenceintervalmakepracticalsense?Howshoulditbechanged?Why? 8.9.1Trythesemultiplechoicequestions. Thenextthreeproblemsrefertothefollowing: AccordingaFieldPollconductedFebruary817,2005,79% ofCaliforniaadultsactualresultsare400outof506surveyedfeelthateducationandourschoolsisone ofthetopissuesfacingCalifornia.Wewishtoconstructa90%condenceintervalforthetrueproportion ofCaliforniaadultswhofeelthateducationandtheschoolsisoneofthetopissuesfacingCalifornia. Exercise8.9.24 Solutiononp.318. Apointestimateforthetruepopulationproportionis: A. 0.90 B. 1.27 C. 0.79

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303 D. 400 Exercise8.9.25 Solutiononp.318. A90%condenceintervalforthepopulationproportionis: A. .761,0.820 B. .125,0.188 C. .755,0.826 D. .130,0.183 Exercise8.9.26 Solutiononp.318. Theerrorboundisapproximately A. 1.581 B. 0.791 C. 0.059 D. 0.030 Thenexttwoproblemsrefertothefollowing: Aqualitycontrolspecialistforarestaurantchaintakesarandomsampleofsize12tochecktheamountof sodaservedinthe16oz.servingsize.Thesampleaverageis13.30withasamplestandarddeviationis 1.55.Assumetheunderlyingpopulationisnormallydistributed. Exercise8.9.27 Solutiononp.318. Findthe95%CondenceIntervalforthetruepopulationmeanfortheamountofsodaserved. A. .42,14.18 B. .32,14.29 C. .50,14.10 D. Impossibletodetermine Exercise8.9.28 Solutiononp.318. Whatistheerrorbound? A. 0.87 B. 1.98 C. 0.99 D. 1.74 Exercise8.9.29 Solutiononp.318. Whatismeantbytheterm%condentwhenconstructingacondenceintervalforamean? A. Ifwetookrepeatedsamples,approximately90%ofthesampleswouldproducethesame condenceinterval. B. Ifwetookrepeatedsamples,approximately90%ofthecondenceintervalscalculated fromthosesampleswouldcontainthesamplemean. C. Ifwetookrepeatedsamples,approximately90%ofthecondenceintervalscalculated fromthosesampleswouldcontainthetruevalueofthepopulationmean. D. Ifwetookrepeatedsamples,thesamplemeanwouldequalthepopulationmeaninapproximately90%ofthesamples.

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304 CHAPTER8.CONFIDENCEINTERVALS Thenexttwoproblemsrefertothefollowing: FivehundredandelevenhomesinacertainsouthernCaliforniacommunityarerandomlysurveyed todetermineiftheymeetminimalearthquakepreparednessrecommendations.Onehundredseventy-three ofthehomessurveyedmettheminimumrecommendationsforearthquakepreparednessand338did not. Exercise8.9.30 Solutiononp.318. FindtheCondenceIntervalatthe90%CondenceLevelforthetruepopulationproportionof southernCaliforniacommunityhomesmeetingatleasttheminimumrecommendationsforearthquakepreparedness. A. .2975,0.3796 B. .6270,6959 C. .3041,0.3730 D. .6204,0.7025 Exercise8.9.31 Solutiononp.318. Thepointestimateforthepopulationproportionofhomesthatdonotmeettheminimumrecommendationsforearthquakepreparednessis: A. 0.6614 B. 0.3386 C. 173 D. 338 8.10Review 11 Thenextthreeproblemsrefertothefollowingsituation: Supposethatasampleof15randomlychosen peoplewereputonaspecialweightlossdiet.Theamountofweightlost,inpounds,followsanunknown distributionwithmeanequalto12poundsandstandarddeviationequalto3pounds. Exercise8.10.1 Solutiononp.318. Tondtheprobabilitythattheaverageofthe15peoplelosenomorethan14pounds,therandom variableshouldbe: A. Thenumberofpeoplewholostweightonthespecialweightlossdiet B. Thenumberofpeoplewhowereonthediet C. Theaverageamountofweightlostby15peopleonthespecialweightlossdiet D. Thetotalamountofweightlostby15peopleonthespecialweightlossdiet Exercise8.10.2 Solutiononp.318. Findtheprobabilityaskedforinthepreviousproblem. Exercise8.10.3 Solutiononp.318. Findthe90thpercentilefortheaverageamountofweightlostby15people. Thenextthreequestionsrefertothefollowingsituation: Thetimeofoccurrenceoftherstaccidentduring rush-hourtrafcatamajorintersectionisuniformlydistributedbetweenthethreehourinterval4p.m.to 7p.m.Let X =theamountoftimehoursittakesfortherstaccidenttooccur. So,ifanaccidentoccursat4p.m.,theamountoftime,inhours,ittookfortheaccidenttooccuris _______. 11 Thiscontentisavailableonlineat.

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305 m = _______ s 2 = _______ Exercise8.10.4 Solutiononp.318. Whatistheprobabilitythatthetimeofoccurrenceiswithinthersthalf-hourorthelasthourof theperiodfrom4to7p.m.? A. Cannotbedeterminedfromtheinformationgiven B. 1 6 C. 1 2 D. 1 3 Exercise8.10.5 Solutiononp.318. The20thpercentileoccursafterhowmanyhours? A. 0.20 B. 0.60 C. 0.50 D. 1 Exercise8.10.6 Solutiononp.318. AssumeRamonhaskepttrackofthetimesfortherstaccidentstooccurfor40differentdays.Let C =thetotalcumulativetime.Then C followswhichdistribution? A. U 0,3 B. Exp 1 3 C. N 60 30 D. N 1 5,0 .01875 Exercise8.10.7 Solutiononp.318. Usingtheinformationinquestion#6,ndtheprobabilitythatthetotaltimeforallrstaccidents tooccurismorethan43hours. Thenexttwoquestionsrefertothefollowingsituation: Thelengthoftimeaparentmustwaitforhis childrentocleantheirroomsisuniformlydistributedinthetimeintervalfrom1to15days. Exercise8.10.8 Solutiononp.318. Howlongmustaparentexpecttowaitforhischildrentocleantheirrooms? A. 8days B. 3days C. 14days D. 6days Exercise8.10.9 Solutiononp.318. Whatistheprobabilitythataparentwillwaitmorethan6daysgiventhattheparenthasalready waitedmorethan3days? A. 0.5174 B. 0.0174 C. 0.7500 D. 0.2143

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306 CHAPTER8.CONFIDENCEINTERVALS Thenextveproblemsrefertothefollowingstudy: Twentypercentofthestudentsatalocalcommunity collegeliveinwithinvemilesofthecampus.Thirtypercentofthestudentsatthesamecommunitycollege receivesomekindofnancialaid.Ofthosewholivewithinvemilesofthecampus,75%receivesome kindofnancialaid. Exercise8.10.10 Solutiononp.319. Findtheprobabilitythatarandomlychosenstudentatthelocalcommunitycollegedoesnotlive withinvemilesofthecampus. A. 80% B. 20% C. 30% D. Cannotbedetermined Exercise8.10.11 Solutiononp.319. Findtheprobabilitythatarandomlychosenstudentatthelocalcommunitycollegeliveswithin vemilesofthecampusorreceivessomekindofnancialaid. A. 50% B. 35% C. 27.5% D. 75% Exercise8.10.12 Solutiononp.319. Basedupontheaboveinformation,arelivinginstudenthousingwithinvemilesofthecampus andreceivingsomekindofnancialaidmutuallyexclusive? A. Yes B. No C. Cannotbedetermined Exercise8.10.13 Solutiononp.319. Theinterestratechargedonthenancialaidis_______data. A. quantitativediscrete B. quantitativecontinuous C. qualitativediscrete D. qualitative Exercise8.10.14 Solutiononp.319. Whatfollowsisinformationaboutthestudentswhoreceivenancialaidatthelocalcommunity college. 1stquartile=\$250 2ndquartile=\$700 3rdquartile=\$1200 Theseamountsarefortheschoolyear.Ifasampleof200studentsistaken,howmanyare expectedtoreceive\$250ormore? A. 50 B. 250 C. 150 D. Cannotbedetermined

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307 Thenexttwoproblemsrefertothefollowinginformation: P A = 0 2, P B = 0 3, A and B are independentevents. Exercise8.10.15 Solutiononp.319. P AANDB = A. 0.5 B. 0.6 C. 0 D. 0.06 Exercise8.10.16 Solutiononp.319. P AORB = A. 0.56 B. 0.5 C. 0.44 D. 1 Exercise8.10.17 Solutiononp.319. If H and D aremutuallyexclusiveevents, P H = 0 .25 P D = 0 .15 ,then P H j D A. 1 B. 0 C. 0.40 D. 0.0375

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308 CHAPTER8.CONFIDENCEINTERVALS 8.11Lab1:CondenceIntervalHomeCosts 12 ClassTime: Names: 8.11.1StudentLearningOutcomes: Thestudentwillcalculatethe90%condenceintervalfortheaveragecostofahomeintheareain whichthisschoolislocated. Thestudentwillinterpretcondenceintervals. Thestudentwillexaminetheeffectsthatchangingconditionshasonthecondenceinterval. 8.11.2CollecttheData ChecktheRealEstatesectioninyourlocalnewspaper.Note:manypapersonlylistthemonedayper week.Also,wewillassumethathomescomeupforsalerandomly.Recordthesalespricesfor35randomly selectedhomesrecentlylistedinthecounty. 1.Completethetable: __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 8.11.3DescribetheData 1.Computethefollowing: a. x = b. s x = c. n = 2.DenetheRandomVariable X ,inwords. X = 3.Statetheestimateddistributiontouse.Usebothwordsandsymbols. 8.11.4FindtheCondenceInterval 1.Calculatethecondenceintervalandtheerrorbound. a. CondenceInterval: b. ErrorBound: 2.Howmuchareaisinbothtailscombined? a = 12 Thiscontentisavailableonlineat.

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309 3.Howmuchareaisineachtail? a 2 = 4.Fillintheblanksonthegraphwiththeareaineachsection.Then,llinthenumberlinewiththe upperandlowerlimitsofthecondenceintervalandthesamplemean. Figure8.5 5.Somestudentsthinkthata90%condenceintervalcontains90%ofthedata.Usethelistofdataon therstpageandcounthowmanyofthedatavaluesliewithinthecondenceinterval.Whatpercent isthis?Isthispercentcloseto90%?Explainwhythispercentshouldorshouldnotbecloseto90%. 8.11.5DescribetheCondenceInterval 1.Intwotothreecompletesentences,explainwhataCondenceIntervalmeansingeneral,asifyou weretalkingtosomeonewhohasnottakenstatistics. 2.Inonetotwocompletesentences,explainwhatthisCondenceIntervalmeansforthisparticular study. 8.11.6UsetheDatatoConstructCondenceIntervals 1.Usingtheaboveinformation,constructacondenceintervalforeachcondencelevel given. Condencelevel EBM/ErrorBound CondenceInterval 50% 80% 95% 99% 2.WhathappenstotheEBMasthecondencelevelincreases?Doesthewidthofthecondenceinterval increaseordecrease?Explainwhythishappens.

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310 CHAPTER8.CONFIDENCEINTERVALS 8.12Lab2:CondenceIntervalPlaceofBirth 13 ClassTime: Names: 8.12.1StudentLearningOutcomes: Thestudentwillcalculatethe90%condenceintervalforproportionofstudentsinthisschoolthat wereborninthisstate. Thestudentwillinterpretcondenceintervals. Thestudentwillexaminetheeffectsthatchangingconditionshasonthecondenceinterval. 8.12.2CollecttheData 1.Surveythestudentsinyourclass,askingthemiftheywereborninthisstate.Let X =thenumberthat wereborninthisstate. a. n = ____________ b. x = ____________ 2.DenetheRandomVariable P 'inwords. 3.Statetheestimateddistributiontouse. 8.12.3FindtheCondenceIntervalandErrorBound 1.Calculatethecondenceintervalandtheerrorbound. a. CondenceInterval: b. ErrorBound: 2.Howmuchareaisinbothtailscombined? a = 3.Howmuchareaisineachtail? a 2 = 4.Fillintheblanksonthegraphwiththeareaineachsection.Then,llinthenumberlinewiththe upperandlowerlimitsofthecondenceintervalandthesampleproportion. Figure8.6 13 Thiscontentisavailableonlineat.

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311 8.12.4DescribetheCondenceInterval 1.Intwotothreecompletesentences,explainwhataCondenceIntervalmeansingeneral,asifyou weretalkingtosomeonewhohasnottakenstatistics. 2.Inonetotwocompletesentences,explainwhatthisCondenceIntervalmeansforthisparticular study. 3.Usingtheaboveinformation,constructacondenceintervalforeachgivencondencelevel given. Condencelevel EBP/ErrorBound CondenceInterval 50% 80% 95% 99% 4.WhathappenstotheEBPasthecondencelevelincreases?Doesthewidthofthecondenceinterval increaseordecrease?Explainwhythishappens.

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312 CHAPTER8.CONFIDENCEINTERVALS 8.13Lab3:CondenceIntervalWomens'Heights 14 ClassTime: Names: 8.13.1StudentLearningOutcomes: Thestudentwillcalculatea90%condenceintervalusingthegivendata. Thestudentwillexaminetherelationshipbetweenthecondencelevelandthepercentofconstructed intervalsthatcontainthepopulationaverage. 8.13.2Given: 1. Heightsof100WomeninInches 59.4 71.6 69.3 65.0 62.9 66.5 61.7 55.2 67.5 67.2 63.8 62.9 63.0 63.9 68.7 65.5 61.9 69.6 58.7 63.4 61.8 60.6 69.8 60.0 64.9 66.1 66.8 60.6 65.6 63.8 61.3 59.2 64.1 59.3 64.9 62.4 63.5 60.9 63.3 66.3 61.5 64.3 62.9 60.6 63.8 58.8 64.9 65.7 62.5 70.9 62.9 63.1 62.2 58.7 64.7 66.0 60.5 64.7 65.4 60.2 65.0 64.1 61.1 65.3 64.6 59.2 61.4 62.0 63.5 61.4 65.5 62.3 65.5 64.7 58.8 66.1 64.9 66.9 57.9 69.8 58.5 63.4 69.2 65.9 62.2 60.0 58.1 62.5 62.4 59.1 66.4 61.2 60.4 58.7 66.7 67.5 63.2 56.6 67.7 62.5 Listedabovearetheheightsof100women.Usearandomnumbergeneratortorandomlyselect10 datavalues. 2.Calculatethesamplemeanandsamplestandarddeviation.Assumethatthepopulationstandard deviationisknowntobe3.3.Withthesevalues,constructa90%condenceintervalforyoursample of10values.Writethecondenceintervalyouobtainedintherstspaceofthetablebelow. 14 Thiscontentisavailableonlineat.

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313 3.Nowwriteyourcondenceintervalontheboard.Asothersintheclasswritetheircondenceintervalsontheboard,copythemintothetablebelow: 90%CondenceIntervals __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 8.13.3DiscussionQuestions 1.Theactualpopulationmeanforthe100heightsgivenaboveis m = 63. 4.Usingtheclasslistingof condenceintervals,counthowmanyofthemcontainthepopulationmean m ;i.e.,forhowmany intervalsdoesthevalueof m liebetweentheendpointsofthecondenceinterval? 2.Dividethisnumberbythetotalnumberofcondenceintervalsgeneratedbytheclasstodetermine thepercentofcondenceintervalsthatcontainthemean m .Writethispercentbelow. 3.Isthepercentofcondenceintervalsthatcontainthepopulationmean m closeto90%? 4.Supposewehadgenerated100condenceintervals.Whatdoyouthinkwouldhappentothepercent ofcondenceintervalsthatcontainedthepopulationmean? 5.Whenweconstructa90%condenceinterval,wesaythatweare 90%condentthatthetruepopulationmeanlieswithinthecondenceinterval. Usingcompletesentences,explainwhatwemeanby thisphrase. 6.Somestudentsthinkthata90%condenceintervalcontains90%ofthedata.Usethelistofdatagiven ontherstpageandcounthowmanyofthedatavaluesliewithinthecondenceintervalthatyou generatedonthatpage.Howmanyofthe100datavaluesliewithinyourcondenceinterval?What percentisthis?Isthispercentcloseto90%? 7.Explainwhyitdoesnotmakesensetocountdatavaluesthatlieinacondenceinterval.Thinkabout therandomvariablethatisbeingusedintheproblem. 8.Supposeyouobtainedtheheightsof10womenandcalculatedacondenceintervalfromthisinformation.Withoutknowingthepopulationmean m ,wouldyouhaveanywayofknowing forcertain if yourintervalactuallycontainedthevalueof m ?Explain. N OTE : ThislabwasdesignedandcontributedbyDianeMathios.

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314 CHAPTER8.CONFIDENCEINTERVALS SolutionstoExercisesinChapter8 Example8.1p.279 x = 7and EBM = 2.5. Example8.2,Problem2p.281 a. x = 68 b. s = 3 c. n = 36 SolutionstoPractice1:CondenceIntervalsforAverages,KnownPopulationStandardDeviation SolutiontoExercise8.6.1p.289 30.4 SolutiontoExercise8.6.2p.289 25 SolutiontoExercise8.6.3p.289 s SolutiontoExercise8.6.4p.289 theageofWinter2006Foothillstudents SolutiontoExercise8.6.5p.289 m SolutiontoExercise8.6.6p.289 yes SolutiontoExercise8.6.7p.289 Normal SolutiontoExercise8.6.8p.289 0.05 SolutiontoExercise8.6.9p.289 0.025 SolutiontoExercise8.6.10p.289 a. 24.52 b. 36.28 c. 5.88 SolutiontoExercise8.6.11p.290 24 52 36.28 SolutionstoPractice2:CondenceIntervalsforAverages,UnknownPopulationStandardDeviation SolutiontoExercise8.7.1p.291 a. 3.26 b. 1.02

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315 c. 39 SolutiontoExercise8.7.2p.291 theaveragenumberofcolorsof39ags SolutiontoExercise8.7.3p.291 m SolutiontoExercise8.7.4p.291 No SolutiontoExercise8.7.5p.291 t 38 SolutiontoExercise8.7.6p.292 0.05 SolutiontoExercise8.7.7p.292 0.025 SolutiontoExercise8.7.8p.292 a. 2.93 b. 3.59 c. 0.33 SolutiontoExercise8.7.9p.292 2.93;3.59 SolutionstoPractice3:CondenceIntervalsforProportions SolutiontoExercise8.8.2p.293 Thenumberofgirls,age8-12,inthebeginningiceskatingclass SolutiontoExercise8.8.3p.293 a. 64 b. 80 c. 0.08 SolutiontoExercise8.8.4p.293 B 80,0.80 SolutiontoExercise8.8.5p.293 p SolutiontoExercise8.8.6p.293 Theproportionofgirls,age8-12,inthebeginningiceskatingclass. SolutiontoExercise8.8.8p.293 0.80 SolutiontoExercise8.8.9p.293 0.04 SolutiontoExercise8.8.10p.293 a. 0.72 b. 0.88 c. 0.08

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316 CHAPTER8.CONFIDENCEINTERVALS SolutiontoExercise8.8.11p.294 0.72;0.88 SolutionstoHomework SolutiontoExercise8.9.1p.295 a.i. 71 ii. 3 iii. 2.8 iv. 48 v. 47 c. 71 3 p 48 d.i. CI:.15,71.85 iii. EB=0.85 SolutiontoExercise8.9.3p.295 a.i. 8629 ii. 6944 iii. 35 iv. 34 c. t 34 d.i. CI:,11,014 iii. EB=2385 e. Itwillbecomesmaller SolutiontoExercise8.9.5p.296 a.i. 8 ii. 4 iii. 81 iv. 80 c. t 80 d.i. CI:.12,8.88 iii. EB=0.88 SolutiontoExercise8.9.7p.297 a.i. 2 ii. 0.1 iii. 0.12 iv. 16 v. 15 b. theweightof1smallbagofcandies c. theaverageweightof16smallbagsofcandies d. 2, 0 1 p 16 e.i. CI:.96,2.04 iii. EB=0.04 f.i. CI:.94,2.06 iii. EB=0.06

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317 SolutiontoExercise8.9.9p.298 a.i. 6 ii. 3 iii. 14 iv. 13 b. thetimeforachildtoremovehistrainingwheels c. theaveragetimefor14childrentoremovetheirtrainingwheels. d. t 13 e.i. CI:.58,8.42 iii. EB=2.42 SolutiontoExercise8.9.11p.298 a.i. 320 ii. 400 iii. 0.80 c. N 0 .80 p 0 .61 0 .39 1200 d.i. CI:.76,0.84 iii. EB=0.02 SolutiontoExercise8.9.13p.299 b. N 0 .61 p 0 .61 0 .39 1200 c.i. CI:.59,0.63 iii. EB=0.02 SolutiontoExercise8.9.15p.300 b. N 0 .86 p 0 .86 0 .14 323 c.i. CI:.8229,0.8984 iii. EB=0.038 SolutiontoExercise8.9.17p.300 a.i. 7.9 ii. 2.5 iii. 2.8 iv. 20 v. 19 c. N 7 9, 2 5 p 20 d.i. CI:.98,8.82 iii. EB:0.92 SolutiontoExercise8.9.19p.301 a. t 83 b. averagecostof84usedcars c.i. CI:.10,7109.90 iii. EB=684.90

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318 CHAPTER8.CONFIDENCEINTERVALS SolutiontoExercise8.9.21p.302 b. N 0 .63 p 0 .63 0 .37 1000 c.i. CI:.60,0.66 iii. EB=0.03 SolutiontoExercise8.9.24p.302 C SolutiontoExercise8.9.25p.303 A SolutiontoExercise8.9.26p.303 D SolutiontoExercise8.9.27p.303 B SolutiontoExercise8.9.28p.303 C SolutiontoExercise8.9.29p.303 C SolutiontoExercise8.9.30p.304 C SolutiontoExercise8.9.31p.304 A SolutionstoReview SolutiontoExercise8.10.1p.304 C SolutiontoExercise8.10.2p.304 0.9951 SolutiontoExercise8.10.3p.304 12.99 SolutiontoExercise8.10.4p.305 C SolutiontoExercise8.10.5p.305 B SolutiontoExercise8.10.6p.305 C SolutiontoExercise8.10.7p.305 0.9990 SolutiontoExercise8.10.8p.305 A SolutiontoExercise8.10.9p.305 C

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319 SolutiontoExercise8.10.10p.306 A SolutiontoExercise8.10.11p.306 B SolutiontoExercise8.10.12p.306 B SolutiontoExercise8.10.13p.306 B SolutiontoExercise8.10.14p.306 C. 150 SolutiontoExercise8.10.15p.307 D SolutiontoExercise8.10.16p.307 C SolutiontoExercise8.10.17p.307 B

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320 CHAPTER8.CONFIDENCEINTERVALS

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323 "donotreject H o "ifthesampleinformationfavorsthenullhypothesis,meaningthatthereisnotenough informationtorejectthenull. MathematicalSymbolsUsedin H o and H a : H o H a equal = notequal 6 = or greaterthan > or lessthan < greaterthanorequalto lessthan < lessthanorequalto morethan > N OTE : H o alwayshasasymbolwithanequalinit. H a neverhasasymbolwithanequalinit. Thechoiceofsymboldependsonthewordingofthehypothesistest. 9.2.1OptionalCollaborativeClassroomActivity Bringtoclassanewspaper,somenewsmagazines,andsomeInternetarticles.Ingroups,ndarticlesfrom whichyourgroupcanwriteanullandalternatehypotheses.Discussyourhypotheseswiththerestofthe class. 9.3OutcomesandtheTypeIandTypeIIErrors 3 Whenyouperformahypothesistest,therearefouroutcomes.Theoutcomesaresummarizedinthefollowingtable: Action True False Donotreject H o CorrectOutcome TypeIIerror Reject H o TypeIError CorrectOutcome H o =thenullhypothesis Thefouroutcomesinthetableare: Thedecisionisto notreject H o when,infact, H o istruecorrectdecision. Thedecisionisto reject H o when,infact, H o istrue incorrectdecisionknownasa TypeIerror Thedecisionisto notreject H o when,infact, H o isfalse incorrectdecisionknownasa TypeIIerror Thedecisionisto reject H o when,infact, H o isfalse correctdecision whoseprobabilityiscalledthe PoweroftheTest Eachoftheerrorsoccurswithaparticularprobability.TheGreekletters a and b representtheprobabilities. a =probabilityofaTypeIerror= PTypeIerror =probabilityofrejectingthenullhypothesiswhenthe nullhypothesisistrue. b =probabilityofaTypeIIerror= PTypeIIerror =probabilityofnotrejectingthenullhypothesiswhen thenullhypothesisisfalse. a and b shouldbeassmallaspossiblebecausetheyareprobabilitiesoferrors.Theyarerarely0. 3 Thiscontentisavailableonlineat.

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324 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION ThePoweroftheTestis1 )]TJ/F134 9.9626 Tf 10.504 0 Td [(b .Ideally,wewantahighpowerthatisascloseto1aspossible. ThefollowingareexamplesofTypeIandTypeIIerrors. Example9.5 Supposethenullhypothesis, H o ,is:Frank'srockclimbingequipmentissafe. TypeIerror :Frankconcludesthathisrockclimbingequipmentmaynotbesafewhen,infact,it reallyissafe. TypeIIerror :Frankconcludesthathisrockclimbingequipmentissafewhen,infact, itisnotsafe. a =probability thatFrankthinkshisrockclimbingequipmentmaynotbesafewhen,infact,it reallyis. b =probability thatFrankthinkshisrockclimbingequipmentissafewhen,infact,itis not. Noticethat,inthiscase,theerrorwiththegreaterconsequenceistheTypeIIerror.IfFrankthinks hisrockclimbingequipmentissafe,hewillgoaheadanduseit. Example9.6 Supposethenullhypothesis, H o ,is:Thevictimofanautomobileaccidentisalivewhenhearrives attheemergencyroomofahospital. TypeIerror :Theemergencycrewconcludesthatthevictimisdeadwhen,infact,thevictimis alive. TypeIIerror :Theemergencycrewconcludesthatthevictimisalivewhen,infact,thevictim isdead. a =probability thattheemergencycrewthinksthevictimisdeadwhen,infact,heisreallyalive = PTypeIerror b =probability thattheemergencycrewthinksthevictimisalivewhen,infact, heisdead= PTypeIIerror TheerrorwiththegreaterconsequenceistheTypeIerror.Iftheemergencycrewthinksthevictim isdead,theywillnottreathim. 9.4DistributionNeededforHypothesisTesting 4 Earlierinthecourse,wediscussedsamplingdistributions. Particulardistributionsareassociatedwith hypothesistesting. Performtestsofapopulationmeanusinga normaldistribution ora student-tdistribution. Remember,useastudent-tdistributionwhenthepopulation standarddeviation isunknownand thepopulationfromwhichthesampleistakenisnormal.Performtestsofapopulationproportionusing anormaldistributionusually n islarge. Ifyouaretestinga singlepopulationmean ,thedistributionforthetestisfor averages : X N m X s X p n or t df SeeChapters7and8 Thepopulationparameteris m .Theestimatedvaluepointestimatefor m is x ,thesamplemean. Ifyouaretestinga singlepopulationproportion ,thedistributionforthetestisforproportionsorpercentages: P N p q p q n SeeChapter8 4 Thiscontentisavailableonlineat.

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325 Thepopulationparameteris p .Theestimatedvaluepointestimatefor p is p '. p = x n where x isthe numberofsuccessesand n isthesamplesize. 9.5Assumption 5 Whenyouperforma hypothesistest ofasinglepopulationmean m usinga Student-tdistribution often calledat-test,therearefundamentalassumptionsthatneedtobemetinorderforthetesttoworkproperly. Yourdatashouldbea simplerandomsample thatcomesfromapopulationthatisapproximately normally distributed .Youusethesample standarddeviation toapproximatethepopulationstandarddeviation. Notethatifthesamplesizeislargerthan30,at-testwillworkevenifthepopulationisnotapproximately normallydistributed. Whenyouperforma hypothesistestofasinglepopulationmean m usinganormaldistributionoftencalled az-test,youtakeasimplerandomsamplefromthepopulation.Thepopulationyouaretestingisnormally distributedoryoursamplesizeislargerthan30orboth.Youknowthevalueofthepopulationstandard deviation. Whenyouperforma hypothesistestofasinglepopulationproportion p ,youtakeasimplerandomsample fromthepopulation.Youmustmeettheconditionsfora binomialdistribution whicharethereareacertain number n ofindependenttrials,eachtrialhasthesameprobabilityofasuccess p ,andtheoutcomesofany trialaresuccessorfailure.Theshapeofthebinomialdistributionneedstobesimilartotheshapeofthe normaldistribution.Toensurethis,thequantities np and nq mustbothbegreaterthanve np > 5and nq > 5.Thenthebinomialdistributionofsampleestimatedproportioncanbeapproximatedbythe normaldistributionwith m = np and s = p npq .Rememberthat q = 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p 9.6RareEvents 6 Supposeyoumakeanassumptionaboutapropertyofthepopulationthisassumptionisthe nullhypothesis .Thenyougathersampledatarandomly.Ifthesamplehaspropertiesthatwouldbevery unlikely tooccuriftheassumptionistrue,thenyouwouldconcludethatyourassumptionaboutthepopulationis probablyincorrect.Rememberthatyourassumptionisjustan assumption -itisnotafactanditmayor maynotbetrue.Butyoursampledataisrealanditisshowingyouafactthatseemstocontradictyour assumption. Forexample,DidiandAliareatabirthdaypartyofaverywealthyfriend.Theyhurrytoberstinlineto grabaprizefromatallbasketthattheycannotseeinsidebecausetheywillbeblindfolded.Thereare200 plasticbubblesinthebasketandDidiandAlihavebeentoldthatthereisonlyonewitha\$100bill.Didi istherstpersontoreachintothebasketandpulloutabubble.Herbubblecontainsthe\$100bill.The probabilityofthishappeningis 1 200 = 0.005.Becausethisissounlikely,Aliishopingthatwhatthetwo ofthemweretoldiswrongandtherearemore\$100billsinthebasket.A"rareevent"hasoccurredDidi gettingthe\$100billsoAlidoubtstheassumptionaboutonlyone\$100billbeinginthebasket. 9.7UsingtheSampletoSupportOneoftheHypotheses 7 Usethesampledatatocalculatetheactualsignicanceofthetestorthe p-value .Thep-valueisthe probabilitythatanoutcomeofthedataforexample,thesamplemeanwillhappenpurelybychance whenthenullhypothesisistrue 5 Thiscontentisavailableonlineat. 6 Thiscontentisavailableonlineat. 7 Thiscontentisavailableonlineat.

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326 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Alargep-valuecalculatedfromthedataindicatesthattheoutcomeofthedataishappeningpurelyby chance.Thedatasupportsthe nullhypothesis sowedonotrejectit.Thesmallerthep-value,themore unlikelytheoutcome,andthestrongertheevidenceisagainstthenullhypothesis.Wewouldrejectthenull hypothesisiftheevidenceisstronglyagainstthenullhypothesis. Thep-valueissometimescalledthe computed a becauseitiscalculatedfromthedata.Youcanthinkofit astheprobabilityofincorrectlyrejectingthenullhypothesiswhenthenullhypothesisisactuallytrue. Drawagraphthatshowsthep-value.Thehypothesistestiseasiertoperformifyouuseagraphbecause youseetheproblemmoreclearly. Example9.7:toillustratethep-value Supposeabakerclaimsthathisbreadheightismorethan15cm,ontheaverage.Severalofhis customersdonotbelievehim.Topersuadehiscustomersthatheisright,thebakerdecidesto doahypothesistest.Hebakes10loavesofbread.Theaverageheightofthesampleloavesis17 cm.Thebakerknowsfrombakinghundredsofloavesofbreadthatthe standarddeviation forthe heightis0.5cm. Thenullhypothesiscouldbe H o : m 15Thealternatehypothesisis H a : m > 15 Thewords "ismorethan" translatesasa" > "so" m > 15"goesintothealternatehypothesis.The nullhypothesismustcontradictthealternatehypothesis. Since s isknown s = 0.5cm.,thedistributionforthetestisnormalwithmean m = 15and standarddeviation s p n = 0.5 p 10 = 0.16. Supposethenullhypothesisistruetheaverageheightoftheloavesisnomorethan15cm.Then istheaverageheightcmcalculatedfromthesampleunexpectedlylarge?Thehypothesistest worksbyaskingthequestionhow unlikely thesampleaveragewouldbeifthenullhypothesis weretrue.Thegraphshowshowfaroutthesampleaverageisonthenormalcurve.Howfar outthesampleaverageisonthenormalcurveismeasuredbythep-value.Thep-valueisthe probabilitythat,ifweweretotakeothersamples,anyothersampleaveragewouldfallatleastas faroutas17cm. Thep-value,then,istheprobabilitythatasampleaverageisthesameorgreaterthan17cm. whenthepopulationmeanis,infact,15cm. Wecancalculatethisprobabilityusingthenormal distributionforaveragesfromChapter7. p-value = P )]TJETq1 0 0 1 148.674 127.381 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 148.948 119.097 Td [(X > 17 whichisapproximately0. Ap-valueofapproximately0tellsusthatitishighlyunlikelythataloafofbreadrisesnomore than15cm,ontheaverage.Thatis,almost0%ofallloavesofbreadwouldbeatleastashighas17 cm. purelybyCHANCE .Becausetheoutcomeof17cm.isso unlikelymeaningitishappening

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327 NOTbychancebutisarareevent ,weconcludethattheevidenceisstronglyagainstthenull hypothesistheaverageheightisatmost15cm..Thereissufcientevidencethatthetrueaverage heightforthepopulationofthebaker'sloavesofbreadisgreaterthan15cm. 9.8DecisionandConclusion 8 Asystematicwaytomakeadecisionofwhethertorejectornotrejectthe nullhypothesis istocompare the p-value anda preconceived a alsocalleda"signicancelevel" .Apreconceived a istheprobabilityof a TypeIerror rejectingthenullhypothesiswhenthenullhypothesisistrue.Itmayormaynotbegiven toyouatthebeginningoftheproblem. Whenyoumakea decision torejectornotreject H o ,doasfollows: If a > p-value ,reject H o .Theresultsofthesampledataaresignicant.Thereissufcientevidenceto concludethat H o isanincorrectbeliefandthatthe alternativehypothesis H a ,maybecorrect. If a p-value ,donotreject H o .Theresultsofthesampledataarenotsignicant.Thereisnotsufcient evidencetoconcludethatthealternativehypothesis, H a ,maybecorrect. Whenyou"donotreject H o ",itdoesnotmeanthatyoushouldbelievethat H o istrue.Itsimply meansthatthesampledatahas failed toprovidesufcientevidencetocastseriousdoubtaboutthe truthfulnessof H o Conclusion: Afteryoumakeyourdecision,writeathoughtful conclusion aboutthehypothesesintermsof thegivenproblem. 9.9AdditionalInformation 9 Ina hypothesistest problem,youmayseewordssuchas"thelevelofsignicanceis1%."The"1%"is thepreconceived a Thestatisticiansettingupthehypothesistestselectsthevalueof a touse before collectingthesample data. Ifnolevelofsignicanceisgiven,wegenerallycanuse a = 0.05 Whenyoucalculatethe p-value anddrawthepicture,thep-valueisinthelefttail,therighttail,or splitevenlybetweenthetwotails.Forthisreason,wecallthehypothesistestleft,right,ortwotailed. The alternatehypothesis H a ,tellsyouifthetestisleft,right,ortwo-tailed.Itisthe key toconducting theappropriatetest. H a never hasasymbolthatcontainsanequalsign. Thefollowingexamplesillustratealeft,right,andtwo-tailedtest. Example9.8 H o : m = 5 H a : m < 5 Testofasinglepopulationmean. H a tellsyouthetestisleft-tailed.Thepictureofthep-valueisas follows: 8 Thiscontentisavailableonlineat. 9 Thiscontentisavailableonlineat.

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328 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Example9.9 H o : p 0.2 H a : p > 0.2 Thisisatestofasinglepopulationproportion. H a tellsyouthetestis right-tailed .Thepictureof thep-valueisasfollows: Example9.10 H o : m = 50 H a : m 6 = 50 Thisisatestofasinglepopulationmean. H a tellsyouthetestis two-tailed .Thepictureofthe p-valueisasfollows. 9.10SummaryoftheHypothesisTest 10 The hypothesistest itselfhasanestablishedprocess.Thiscanbesummarizedasfollows: 10 Thiscontentisavailableonlineat.

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329 1.Determine H o and H a .Remember,theyarecontradictory. 2.Determinetherandomvariable. 3.Determinethedistributionforthetest. 4.Drawagraph,calculatetheteststatistic,andusetheteststatistictocalculatethe p-value .Az-score andat-scoreareexamplesofteststatistics. 5.Comparethepreconceived a withthep-value,makeadecisionrejectorcannotreject H o ,andwrite aclearconclusionusingEnglishsentences. Noticethatinperformingthehypothesistest,youuse a andnot b b isneededtohelpdeterminethe samplesizeofthedatathatisusedincalculatingthep-value.Rememberthatthequantity1 )]TJ/F134 9.9626 Tf 10.636 0 Td [(b iscalled the PoweroftheTest .Ahighpowerisdesirable.Ifthepoweristoolow,statisticianstypicallyincreasethe samplesizewhilekeeping a thesame.Ifthepowerislow,thenullhypothesismightnotberejectedwhen itshouldbe. 9.11Examples 11 Example9.11 Jeffrey,asaneight-yearold, establishedanaveragetimeof16.43seconds forswimmingthe25yardfreestyle,witha standarddeviationof0.8seconds .Hisdad,Frank,thoughtthatJeffreycould swimthe25-yardfreestylefasterbyusinggoggles.FrankboughtJeffreyanewpairofexpensive gogglesandtimedJeffreyfor 1525-yardfreestyleswims .Forthe15swims, Jeffrey'saverage timewas16seconds.FrankthoughtthatthegoggleshelpedJeffreytoswimfasterthanthe16.43 seconds. Conductahypothesistestusingapreconceived a = 0.05. Solution SettinguptheHypothesisTest: Sincetheproblemisaboutameanaverage,thisisa testofasinglepopulationmean H o : m = 16.43 H a : m < 16.43 ForJeffreytoswimfaster,histimewillbelessthan16.43seconds.The" < "tellsyouthisislefttailed. Calculatingthedistributionneeded: Randomvariable: X =theaveragetimetoswimthe25-yardfreestyle. Distributionforthetest: X isnormalpopulation standarddeviation isknown: s = 0.8 X N m s X p n Therefore, X N 16.43, 0.8 p 15 m = 16.43comesfrom H 0 andnotthedata. s = 0.8,and n = 15. Calculatethep-valueusingthenormaldistributionforamean: p-value = P )]TJ/F132 9.9626 Tf -0.725 -8.882 Td [(X < 16 = 0.0187wherethesamplemeanintheproblemisgivens16. p-value = 0.0187Thisiscalledthe actuallevelofsignicance .Thep-valueistheareatotheleft ofthesamplemean,16. 11 Thiscontentisavailableonlineat.

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330 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Graph: Figure9.1 m = 16.43comesfrom H o .Ourassumptionis m = 16.43. Interpretationofthep-value:If H o istrue ,thereisa0.0187probability.87%thatJeffrey'smean oraveragetimetoswimthe25-yardfreestyleis16secondsorless.Becausea1.87%chanceis small,themeantimeof16secondsorlessisnothappeningrandomly.Itisarareevent. Compare a andthep-value: a = 0.05 p-value = 0.0187 a > p-value Makeadecision: Since a > p-value ,reject H o Thismeansthatyoureject m = 16.43.Inotherwords,youdonotthinkJeffreyswimsthe25-yard freestylein16.43secondsbutfasterwiththenewgoggles. Conclusion: Atthe5%signicancelevel,weconcludethatJeffreyswimsfasterusingthenew goggles.ThesampledatashowthereissufcientevidencethatJeffrey'smeantimetoswimthe 25-yardfreestyleislessthan16.43seconds. Thep-valuecaneasilybecalculatedusingtheTI-83+andtheTI-84calculators: Press STAT andarrowoverto TESTS .Press 1:Z-Test .Arrowoverto Stats andpress ENTER .Arrow downandenter16.43for m 0 nullhypothesis,.8for s ,16forthesamplemean,and15for n .Arrow downto m :alternatehypothesisandarrowoverto < m 0 .Press ENTER .Arrowdownto Calculate andpress ENTER .Thecalculatornotonlycalculatesthep-value p = 0.0187butitalsocalculates theteststatisticz-scoreforthesamplemean. m < 16.43isthealternatehypothesis.Dothisset ofinstructionsagainexceptarrowto Draw insteadof Calculate .Press ENTER .Ashadedgraph appearswith z = )]TJ/F58 9.9626 Tf 8.194 0 Td [(2.08teststatisticand p = 0.0187p-value.Makesurewhenyouuse Draw thatnootherequationsarehighlightedin Y = andtheplotsareturnedoff. WhenthecalculatordoesaZ-Test,the Z-Test functionndsthep-valuebydoinganormalprobabilitycalculationusingthe CentralLimitTheorem Chapter7Section7.1: P )]TJETq1 0 0 1 104.933 110.326 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 105.207 102.042 Td [(X < 16 = 2ndDISTRnormcdf )]TJ/F58 9.9626 Tf 8.195 0 Td [(10 99,16,16.43,0.8/ p 15 TheTypeIandTypeIIerrorsforthisproblemareasfollows:

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332 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Calculatingthedistributionneeded: Randomvariable: X =theaverageweightliftedbythefootballplayers. Distributionforthetest: Itisnormalbecause s isknown. X N 275, 55 p 30 x = 286.2poundsfromthedata n = 30 s = 55pounds Alwaysuse s ifyouknowit. Weassume m = 275poundsunlessourdatashows usotherwise. Calculatethep-valueusingthenormaldistributionforamean: p-value = P X > 286.2 = 0.1323wherethesamplemeaniscalculatedas286.2poundsfromthe data. Interpretationofthep-value: If H o istrue,thenthereisa0.1323probability.23%thatthe footballplayerscanliftameanoraverageweightof286.2poundsormore.Becausea13.23% chanceislargeenough,ameanweightliftof286.2poundsormoreishappeningrandomlyandis notarareevent. Figure9.3 Compare a andthep-value: a = 0.025 p-value = 0.1323 a < p-value Makeadecision: Since a < p-value ,donotreject H o Conclusion: Atthe2.5%levelofsignicance,fromthesampledata,thereisnotsufcientevidence toconcludethatthetruemeanweightliftedismorethan275pounds. Thep-valuecaneasilybecalculatedusingtheTI-83+andtheTI-84calculators: Putthedataandfrequenciesintolists.Press STAT andarrowoverto TESTS .Press 1:Z-Test .Arrow overto Data andpress ENTER .Arrowdownandenter275for m 0 ,55for s ,thenameofthelistwhere youputthedata,andthenameofthelistwhereyouputthefrequencies.Arrowdownto m :and arrowoverto > m 0 .Press ENTER .Arrowdownto Calculate andpress ENTER .Thecalculatornot onlycalculatesthep-value p = 0.1331,alittledifferentfromtheabovecalculation-initwe usedthesamplemeanroundedtoonedecimalplaceinsteadofthedatabutitalsocalculatesthe

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333 teststatisticz-scoreforthesamplemean,thesamplemean,andthesamplestandarddeviation. m > 275isthealternatehypothesis.Dothissetofinstructionsagainexceptarrowto Draw instead of Calculate .Press ENTER .Ashadedgraphappearswith z = 1.112teststatisticand p = 0.1331 p-value.Makesurewhenyouuse Draw thatnootherequationsarehighlightedin Y = andthe plotsareturnedoff. Example9.13 Statisticsstudentsbelievethattheaveragescoreontherststatisticstestis65.Astatisticsinstructorthinkstheaveragescoreishigherthan65.Hesamplestenstatisticsstudentsandobtains thescores65;65;70;67;66;63;63;68;72;71.Heperformsahypothesistestusinga5%levelof signicance.Thedataarefromanormaldistribution. Solution SettinguptheHypothesisTest: A5%levelofsignicancemeansthat a = 0.05.Thisisatestofa singlepopulationmean H o : m = 65 H a : m > 65 Sincetheinstructorthinkstheaveragescoreishigher,usea" > ".The" > "meansthetestis right-tailed. Calculatingthedistributionneeded: Randomvariable: X =averagescoreontherststatisticstest. Distributionforthetest: Ifyoureadtheproblemcarefully,youwillnoticethatthereis nopopulationstandarddeviationgiven .Youareonlygiven n = 10sampledatavalues.Noticealsothatthe datacomefromanormaldistribution.Thismeansthatthedistributionforthetestisastudent-t. Use t df .Therefore,thedistributionforthetestis t 9 where n = 10and df = 10 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = 9. Calculatethep-valueusingtheStudent-tdistribution: p-value = P X > 67 = 0.0396wherethesamplemeanandsamplestandarddeviationarecalculatedas67poundsand3.1972fromthedata. Interpretationofthep-value: Ifthenullhypothesisistrue,thenthereisa0.0396probability.96% thatthesamplemeanis67poundsormore. Figure9.4

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334 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Compare a andthep-value: Since a = .05and p-value = 0.0396.Therefore, a > p-value Makeadecision: Since a > p-value ,reject H o Thismeansyoureject m = 65.Inotherwords,youbelievetheaveragetestscoreismorethan65. Conclusion: Ata5%levelofsignicance,thesampledatashowsufcientevidencethatthemean averagetestscoreismorethan65,justasthemathinstructorthinks. Thep-valuecaneasilybecalculatedusingtheTI-83+andtheTI-84calculators: Putthedataintoalist.Press STAT andarrowoverto TESTS .Press 2:T-Test .Arrowoverto Data andpress ENTER .Arrowdownandenter65for m 0 ,thenameofthelistwhereyouputthe data,and1for Freq: .Arrowdownto m :andarrowoverto > m 0 .Press ENTER .Arrowdown to Calculate andpress ENTER .Thecalculatornotonlycalculatesthep-value p = 0.0396butit alsocalculatestheteststatistict-scoreforthesamplemean,thesamplemean,andthesample standarddeviation. m > 65isthealternatehypothesis.Dothissetofinstructionsagainexcept arrowto Draw insteadof Calculate .Press ENTER .Ashadedgraphappearswith t = 1.9781test statisticand p = 0.0396p-value.Makesurewhenyouuse Draw thatnootherequationsare highlightedin Y = andtheplotsareturnedoff. Example9.14 Joonbelievesthat50%ofrst-timebridesintheUnitedStatesareyoungerthantheirgrooms. Sheperformsahypothesistesttodetermineifthepercentageis thesameordifferentfrom50% Joonsamples 100rst-timebrides and 53 replythattheyareyoungerthantheirgrooms.Forthe hypothesistest,sheusesa1%levelofsignicance. Solution SettinguptheHypothesisTest: The1%levelofsignicancemeansthat a = 0.01.Thisisa testofasinglepopulationproportion H o : p = 0.50 H a : p 6 = 0.50 Thewords "isthesameordifferentfrom" tellyouthisisatwo-tailedtest. Calculatingthedistributionneeded: Randomvariable: P '=thepercentofofrst-timebrideswhoareyoungerthantheirgrooms. Distribution Distributionforthetest: Theproblemcontainsnomentionofanaverage.Theinformationisgiven intermsofpercentages.Usethedistributionfor P ',theestimatedproportion. P N p q p q n Therefore, P N 0.5, q 5 5 100 where p = 0.50, q = 1 )]TJ/F132 9.9626 Tf 11.044 0 Td [(p = 0.50,and n = 100. Calculatethep-valueusingthenormaldistributionforproportions: p-value = P P < 0.47or P > 0.53 = 0.5485 where x = 53, p = x n = 53 100 = 0.53

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335 Interpretationofthep-value: Ifthenullhypothesisistrue,thereis0.5485probability.85%that thesampleestimatedproportion p 'is0.53ormoreOR0.47orlessseethegraphbelow. Figure9.5 m = p = 0.50comesfrom H o ,thenullhypothesis. p = 0.53.Sincethecurveissymmetricalandthetestistwo-tailed,the p 'forthelefttailisequalto 0.50 )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.03 = 0.47where m = p = 0.50..03isthedifferencebetween0.53and0.50. Compare a andthep-value: Since a = 0.01and p-value = 0.5485.Therefore, a < p-value Makeadecision: Since a < p-value ,youcannotreject H o Conclusion: Atthe1%levelofsignicance,thesampledatadonotshowsufcientevidencethat thepercentageofrst-timebrideswhoareyoungerthantheirgroomsisdifferentfrom50%. Thep-valuecaneasilybecalculatedusingtheTI-83+andtheTI-84calculators: Press STAT andarrowoverto TESTS .Press 5:1-PropZTest .Enter.5for p 0 and100for n .Arrow downto Prop andarrowto notequals p 0 .Press ENTER .Arrowdownto Calculate andpress ENTER .Thecalculatorcalculatesthep-value p = 0.5485andtheteststatisticz-score. Propnot equals .5isthealternatehypothesis.Dothissetofinstructionsagainexceptarrowto Draw instead of Calculate .Press ENTER .Ashadedgraphappearswith z = 0.6teststatisticand p = 0.5485 p-value.Makesurewhenyouuse Draw thatnootherequationsarehighlightedin Y = andthe plotsareturnedoff. TheTypeIandTypeIIerrorsareasfollows: TheTypeIerroristoconcludethattheproportionofrst-timebridesthatareyoungerthantheir groomsis50%when,infact,theproportionisactuallydifferentfrom50%.Rejectthenullhypothesiswhenthenullhtpothesisistrue. TheTypeIIerroristoconcludethattheproportionofrst-timebridesthatareyoungerthantheir groomsisdifferentfrom50%when,infact,theproportionisactually50%.Donotrejectthenull hypothesiswhenthenullhypothesisisfalse. Example9.15 Problem1 Supposetheproportionofhouseholdsthathavethreetelephonenumbersis30%.Thetelephone companyhasreasontobelievethattheproportionofhouseholdsislessthan30%.Beforethey

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336 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION startabigadvertisingcampaign,theyconductahypothesistest.Theirmarketingpeoplesurvey 150householdswiththeresultthat43ofthehouseholdshavethreetelephonenumbers. Solution SettinguptheHypothesisTest: H o : p = 0.30 H a : p < 0.30 Calculatingthedistributionneeded: The randomvariable is P '=proportionofhouseholdsthathavethreetelephonenumbers. The distribution forthehypothesistestis P N 0.30, q 0.30 0.70 150 Problem2 Calculatethep-valueusingthenormaldistributionforproportions: Calculatethep-valueusingthenormaldistributionforproportions: Thevaluethathelpsdeterminethep-valueis p '.Calculate p 'below. p = x n where x =thenumberofsuccesses Problem3 Whatisa success forthisproblem? Problem4 x = 43, n = 150,and p = ________usingtheformulaabove. Drawthegraphforthisproblem.Drawthehorizontalaxis.Labelandshadeappropriately. Compare a andthep-value: Fillintheblanks. Problem5 p-value=_________ Problem6 Makeadecision._____________Reject/Donotreject H 0 because____________. Writeaconclusion. ThenextexampleisapoemwrittenbyastatisticsstudentnamedNicoleHart.Thesolutiontotheproblem followsthepoem.Noticethatthehypothesistestisforasinglepopulationproportion.Thismeansthatthe nullandalternatehypothesesusetheparameter p .Thedistributionforthetestisnormal.Theestimated proportion p 'istheproportionofeaskilledtothetotaleasfoundonFido.Thisissampleinformation. Theproblemgivesapreconceived a = 0.01,forcomparison,anda95%condenceintervalcomputation. Thepoemiscleverandhumorous,sopleaseenjoyit! N OTE :Noticethesolutionsheetthathasthesolution.LookintheTableofContentsforthetopic "SolutionSheets."Usecopiesoftheappropriatesolutionsheetforhomeworkproblems.

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337 Example9.16 Mydoghassomanyfleas, Theydonotcomeoffwithease. Asforshampoo,Ihavetriedmanytypes EvenonecalledBubbleHype, Whichonlykilled25%ofthefleas, UnfortunatelyIwasnotpleased. I'veusedallkindsofsoap, UntilIhadgiveuphope UntilonedayIsaw Anadthatputmeinawe. Ashampoousedfordogs CalledGOODENOUGHtoCleanaHog Guaranteedtokillmorefleas. IgaveFidoabath Andafterdoingthemath Hisnumberoffleas Starteddroppingby3's! Withhisoldshampoo Icounted42. Attheendofhisbath, Iredidthemath Andthenewshampoohadkilled17fleas. SonoIwaspleased. Nowitistimeforyoutohavesomefun Withthelevelofsignificancebeing.01, Youmusthelpmefigureout Usethenewshampooorgowithout? Solution SettinguptheHypothesisTest: H o : p = 0.25 H a : p > 0.25 Calculatingthedistributionneeded: Inwords,CLEARLYstatewhatyourrandomvariable X or P 'represents. P 0 =Theproportionofeasthatarekilledbythenewshampoo Statethedistributiontouseforthetest. Normal: N 0.25, q 0.25 1 )]TJ/F58 7.5716 Tf 6.228 0 Td [(0.25 42 TestStatistic: t = 2.3163

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338 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Calculatethep-valueusingthenormaldistributionforproportions: p-value = 0.0103 In12completesentences,explainwhatthep-valuemeansforthisproblem. Ifthenullhypothesisistruetheproportionis0.25,thenthereisa0.0103probabilitythatthe sampleestimatedproportionis0.4048 17 42 ormore. Usethepreviousinformationtosketchapictureofthissituation.CLEARLY,labelandscalethe horizontalaxisandshadetheregionscorrespondingtothep-value. Figure9.6 Compare a andthep-value: Indicatethecorrectdecisionrejectordonotrejectthenullhypothesis,thereasonforit,and writeanappropriateconclusion,usingCOMPLETESENTENCES. alpha decision reasonfordecision 0.01 Donotreject H o a < p-value Conclusion: Atthe1%levelofsignicance,thesampledatadonotshowsufcientevidencethat thepercentageofeathatarekilledbythenewshampooismorethan25%. Constructa95%CondenceIntervalforthetruemeanorproportion.Includeasketchofthe graphofthesituation.LabelthepointestimateandthelowerandupperboundsoftheCondence Interval.

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339 Figure9.7 CondenceInterval: 0.26,0.55 Weare95%condentthatthetruepopulationproportion p of easthatarekilledbythenewshampooisbetween26%and55%. N OTE :Thisisaweaktestsincethep-valueisveryclosetoalpha.Inreality,onewouldprobably domoretests.

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340 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION 9.12SummaryofFormulas 12 H o and H a arecontradictory. If H o has: equal = greaterthanorequalto lessthanorequalto then H a has: notequal 6 = or greaterthan > or lessthan < lessthan < greaterthan > If a p-value,thendonotreject H o If a > p-value,thenreject H o a ispreconceived.Itsvalueissetbeforethehypothesisteststarts.Thep-valueiscalculatedfromthedata. a =probabilityofaTypeIerror=PTypeIerror=probabilityofrejectingthenullhypothesiswhenthe nullhypothesisistrue. b =probabilityofaTypeIIerror=PTypeIIerror=probabilityofrejectingthenullhypothesiswhenthe nullhypothesisistrue. Ifthereisnogivenpreconceived a ,thenuse a = 0.05. TypesofHypothesisTests Singlepopulationmean, known populationvarianceorstandarddeviation: Normaltest Singlepopulationmean, unknown populationvarianceorstandarddeviation: Student-ttest Singlepopulationproportion: Normaltest 12 Thiscontentisavailableonlineat.

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341 9.13Practice1:SingleMean,KnownPopulationStandardDeviation 13 9.13.1StudentLearningOutcomes Thestudentwillexplorehypothesistestingwithsinglemeanandknownpopulationstandarddeviationdata. 9.13.2Given Supposethatarecentarticlestatedthattheaveragetimespentinjailbyarsttimeconvictedburglaris 2.5years.Astudywasthendonetoseeiftheaveragetimehasincreasedinthenewcentury.Arandom sampleof26rsttimeconvictedburglarsinarecentyearwaspicked.Theaveragelengthoftimeinjail fromthesurveywas3yearswithastandarddeviationof1.8years.Supposethatitissomehowknownthat thepopulationstandarddeviationis1.5.Conductahypothesistesttodetermineiftheaveragelengthof jailtimehasincreased. 9.13.3HypothesisTesting:SingleAverage Exercise9.13.1 Solutiononp.366. Isthisatestofaveragesorproportions? Exercise9.13.2 Solutiononp.366. Statethenullandalternativehypotheses. a. H o : b. H a : Exercise9.13.3 Solutiononp.366. Isthisaright-tailed,left-tailed,ortwo-tailedtest?Howdoyouknow? Exercise9.13.4 Solutiononp.366. WhatsymbolrepresentstheRandomVariableforthistest? Exercise9.13.5 Solutiononp.366. Inwords,denetheRandomVariableforthistest. Exercise9.13.6 Solutiononp.366. Isthepopulationstandarddeviationknownand,ifso,whatisit? Exercise9.13.7 Solutiononp.366. Calculatethefollowing: a. x = b. s = c. s x = d. n = Exercise9.13.8 Solutiononp.366. Sinceboth s and s x aregiven,whichshouldbeused?In1-2completesentences,explainwhy. Exercise9.13.9 Solutiononp.366. Statethedistributiontouseforthehypothesistest. Exercise9.13.10 Sketchagraphofthesituation.Labelthehorizontalaxis.Markthehypothesizedmeanandthe samplemean x .Shadetheareacorrespondingtothep-value. 13 Thiscontentisavailableonlineat.

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342 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Exercise9.13.11 Solutiononp.366. Findthep-value. Exercise9.13.12 Solutiononp.366. Atapre-conceived a = 0 .05 ,whatisyour: a. Decision: b. Reasonforthedecision: c. Conclusionwriteoutinacompletesentence: 9.13.4DiscussionQuestions Exercise9.13.13 Doesitappearthattheaveragejailtimespentforrsttimeconvictedburglarshasincreased? Whyorwhynot?

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343 9.14Practice2:SingleMean,UnknownPopulationStandardDeviation 14 9.14.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofhypothesistestingwithasinglemeanandunknownpopulationstandarddeviation. 9.14.2Given Arandomsurveyof75deathrowinmatesrevealedthattheaveragelengthoftimeondeathrowis17.4 yearswithastandarddeviationof6.3years.Conductahypothesistesttodetermineifthepopulation averagetimeondeathrowcouldlikelybe15years. 9.14.3HypothesisTesting:SingleAverage Exercise9.14.1 Solutiononp.367. Isthisatestofaveragesorproportions? Exercise9.14.2 Solutiononp.367. Statethenullandalternativehypotheses. a. H o : b. H a : Exercise9.14.3 Solutiononp.367. Isthisaright-tailed,left-tailed,ortwo-tailedtest?Howdoyouknow? Exercise9.14.4 Solutiononp.367. WhatsymbolrepresentstheRandomVariableforthistest? Exercise9.14.5 Solutiononp.367. Inwords,denetheRandomVariableforthistest. Exercise9.14.6 Solutiononp.367. Isthepopulationstandarddeviationknownand,ifso,whatisit? Exercise9.14.7 Solutiononp.367. Calculatethefollowing: a. x = b. 6 3 = c. n = Exercise9.14.8 Solutiononp.367. Whichtestshouldbeused?In1-2completesentences,explainwhy. Exercise9.14.9 Solutiononp.367. Statethedistributiontouseforthehypothesistest. Exercise9.14.10 Sketchagraphofthesituation.Labelthehorizontalaxis.Markthehypothesizedmeanandthe samplemean, x .Shadetheareacorrespondingtothep-value. 14 Thiscontentisavailableonlineat.

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344 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Figure9.8 Exercise9.14.11 Solutiononp.367. Findthep-value. Exercise9.14.12 Solutiononp.367. Atapre-conceived a = 0 .05 ,whatisyour: a. Decision: b. Reasonforthedecision: c. Conclusionwriteoutinacompletesentence: 9.14.4DiscussionQuestion Doesitappearthattheaveragetimeondeathrowcouldbe15years?Whyorwhynot?

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345 9.15Practice3:SingleProportion 15 9.15.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofhypothesistestingwithasingleproportion. 9.15.2Given TheNationalInstituteofMentalHealthpublishedanarticlestatingthatinanyone-yearperiod,approximately9.5percentofAmericanadultssufferfromdepressionoradepressiveillness. http://www.nimh.nih.gov/publicat/depression.cfmSupposethatinasurveyof100peopleinacertain town,sevenofthemsufferedfromdepressionoradepressiveillness.Conductahypothesistesttodetermineifthetrueproportionofpeopleinthattownsufferingfromdepressionoradepressiveillnessislower thanthepercentinthegeneraladultAmericanpopulation. 9.15.3HypothesisTesting:SingleProportion Exercise9.15.1 Solutiononp.367. Isthisatestofaveragesorproportions? Exercise9.15.2 Solutiononp.367. Statethenullandalternativehypotheses. a. H o : b. H a : Exercise9.15.3 Solutiononp.367. Isthisaright-tailed,left-tailed,ortwo-tailedtest?Howdoyouknow? Exercise9.15.4 Solutiononp.367. WhatsymbolrepresentstheRandomVariableforthistest? Exercise9.15.5 Solutiononp.367. Inwords,denetheRandomVariableforthistest. Exercise9.15.6 Solutiononp.368. Calculatethefollowing: ax = bn = cp-hat = Exercise9.15.7 Solutiononp.368. Calculate s x .Makesuretoshowhowyousetuptheformula. Exercise9.15.8 Solutiononp.368. Statethedistributiontouseforthehypothesistest. Exercise9.15.9 Sketchagraphofthesituation.Labelthehorizontalaxis.Markthehypothesizedmeanandthe sampleproportion,p-hat.Shadetheareacorrespondingtothep-value. 15 Thiscontentisavailableonlineat.

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346 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Exercise9.15.10 Solutiononp.368. Findthep-value Exercise9.15.11 Solutiononp.368. Atapre-conceived a = 0 .05 ,whatisyour: a. Decision: b. Reasonforthedecision: c. Conclusionwriteoutinacompletesentence: 9.15.4DiscusionQuestion Exercise9.15.12 Doesitappearthattheproportionofpeopleinthattownwithdepressionoradepressiveillness islowerthangeneraladultAmericanpopulation?Whyorwhynot?

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347 9.16Homework 16 Exercise9.16.1 Solutiononp.368. Someofthestatementsbelowrefertothenullhypothesis,sometothealternatehypothesis. Statethenullhypothesis, H o ,andthealternativehypothesis, H a ,intermsoftheappropriateparameter m or p a. Americansworkanaverageof34yearsbeforeretiring. b. Atmost60%ofAmericansvoteinpresidentialelections. c. TheaveragestartingsalaryforSanJoseStateUniversitygraduatesisatleast\$100,000per year. d. 29%ofhighschoolseniorsgetdrunkeachmonth. e. Fewerthan5%ofadultsridethebustoworkinLosAngeles. f. Theaveragenumberofcarsapersonownsinherlifetimeisnotmorethan10. g. AbouthalfofAmericansprefertoliveawayfromcities,giventhechoice. h. Europeanshaveanaveragepaidvacationeachyearofsixweeks. i. Thechanceofdevelopingbreastcancerisunder11%forwomen. j. Privateuniversitiescost,onaverage,morethan\$20,000peryearfortuition. Exercise9.16.2 Solutiononp.368. Fora-jabove,statetheTypeIandTypeIIerrorsincompletesentences. Exercise9.16.3 Fora-jabove,incompletesentences: a. StateaconsequenceofcommittingaTypeIerror. b. StateaconsequenceofcommittingaTypeIIerror. D IRECTIONS :Foreachofthewordproblems,useasolutionsheettodothehypothesistest. N OTE :Ifyouareusingastudent-tdistributionforahomeworkproblembelow,youmayassume thattheunderlyingpopulationisnormallydistributed.Ingeneral,youmustrstprovethat assumption,though. Exercise9.16.4 Aparticularbrandoftiresclaimsthatitsdeluxetireaveragesatleast50,000milesbeforeitneeds tobereplaced.Frompaststudiesofthistire,thestandarddeviationisknowntobe8000.Asurvey ofownersofthattiredesignisconducted.Fromthe28tiressurveyed,theaveragelifespanwas 46,500mileswithastandarddeviationof9800miles.Dothedatasupporttheclaimatthe5% level? Exercise9.16.5 Solutiononp.368. Fromgenerationtogeneration,theaverageagewhensmokersrststarttosmokevaries.However,thestandarddeviationofthatageremainsconstantofaround2.1years.Asurveyof40 smokersofthisgenerationwasdonetoseeiftheaveragestartingageisatleast19.Thesample averagewas18.1withasamplestandarddeviationof1.3.Dothedatasupporttheclaimatthe5% level? Exercise9.16.6 Thecostofadailynewspapervariesfromcitytocity.However,thevariationamongprices remainssteadywithastandarddeviationof6.Astudywasdonetotesttheclaimthatthe 16 Thiscontentisavailableonlineat.

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348 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION averagecostofadailynewspaperis35.Twelvecostsyieldanaveragecostof30withastandard deviationof4.Dothedatasupporttheclaimatthe1%level? Exercise9.16.7 Solutiononp.368. Anarticleinthe SanJoseMercuryNews statedthatstudentsintheCaliforniastateuniversity systemtakeanaverageof4.5yearstonishtheirundergraduatedegrees.Supposeyoubelieve thattheaveragetimeislonger.Youconductasurveyof49studentsandobtainasamplemeanof 5.1withasamplestandarddeviationof1.2.Dothedatasupportyourclaimatthe1%level? Exercise9.16.8 Theaveragenumberofsickdaysanemployeetakesperyearisbelievedtobeabout10.Members ofapersonneldepartmentdonotbelievethisgure.Theyrandomlysurvey8employees.The numberofsickdaystheytookforthepastyearareasfollows:12;4;15;3;11;8;6;8.Let x = thenumberofsickdaystheytookforthepastyear.Shouldthepersonnelteambelievethatthe averagenumberisabout10? Exercise9.16.9 Solutiononp.369. In1955, LifeMagazine reportedthatthe25year-oldmotherofthreeworked[onaverage]an80 hourweek.Recently,manygroupshavebeenstudyingwhetherornotthewomen'smovement has,infact,resultedinanincreaseintheaverageworkweekforwomencombiningemployment andat-homework.Supposeastudywasdonetodetermineiftheaverageworkweekhasincreased.81womenweresurveyedwiththefollowingresults.Thesampleaveragewas83;the samplestandarddeviationwas10.Doesitappearthattheaverageworkweekhasincreasedfor womenatthe5%level? Exercise9.16.10 Yourstatisticsinstructorclaimsthat60percentofthestudentswhotakeherElementaryStatistics classgothroughlifefeelingmoreenriched.Forsomereasonthatshecan'tquitegureout,most peopledon'tbelieveher.Youdecidetocheckthisoutonyourown.Yourandomlysurvey64of herpastElementaryStatisticsstudentsandndthat34feelmoreenrichedasaresultofherclass. Now,whatdoyouthink? Exercise9.16.11 Solutiononp.369. ANissanMotorCorporationadvertisementread,Theaverageman'sI.Q.is107.Theaverage browntrout'sI.Q.is4.Sowhycan'tmancatchbrowntrout?Supposeyoubelievethattheaverage browntrout'sI.Q.isgreaterthan4.Youcatch12browntrout.Ashpsychologistdeterminesthe I.Q.sasfollows:5;4;7;3;6;4;5;3;6;3;8;5.Conductahypothesistestofyourbelief. Exercise9.16.12 Refertothepreviousproblem.Conductahypothesistesttoseeifyourdecisionandconclusion wouldchangeifyourbeliefwerethattheaveragebrowntrout'sI.Q.is not 4. Exercise9.16.13 Solutiononp.369. Accordingtoanarticlein Newsweek ,thenaturalratioofgirlstoboysis100:105.InChina,the birthratiois100:114.7%girls.Supposeyoudon'tbelievethereportedguresofthepercent ofgirlsborninChina.Youconductastudy.Inthisstudy,youcountthenumberofgirlsandboys bornin150randomlychosenrecentbirths.Thereare60girlsand90boysbornofthe150.Based onyourstudy,doyoubelievethatthepercentofgirlsborninChinais46.7? Exercise9.16.14 Apolldonefor Newsweek foundthat13%ofAmericanshaveseenorsensedthepresenceofan angel.Acontingentdoubtsthatthepercentisreallythathigh.Itconductsitsownsurvey.Out of76Americanssurveyed,only2hadseenorsensedthepresenceofanangel.Asaresultofthe contingent'ssurvey,wouldyouagreewiththe Newsweek poll?Incompletesentences,alsogive threereasonswhythetwopollsmightgivedifferentresults.

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349 Exercise9.16.15 Solutiononp.369. Theaverageworkweekforengineersinastart-upcompanyisbelievedtobeabout60hours.A newlyhiredengineerhopesthatit'sshorter.Sheasks10engineeringfriendsinstart-upsforthe lengthsoftheiraverageworkweeks.Basedontheresultsthatfollow,shouldshecountonthe averageworkweektobeshorterthan60hours? Datalengthofaverageworkweek:70;45;55;60;65;55;55;60;50;55. Exercise9.16.16 UsetheLaptimedataforLap4seeTableofContentstotesttheclaimthatTerrinishesLap 4onaverageinlessthan129seconds.Usealltwentyracesgiven. Exercise9.16.17 UsetheInitialPublicOfferingdataseeTableofContentstotesttheclaimthattheaverage offerpricewas\$18pershare.Donotuseallthedata.Useyourrandomnumbergeneratorto randomlysurvey15prices. N OTE :Thefollowingquestionswerewrittenbypaststudents.Theyareexcellentproblems! Exercise9.16.18 18."AsianFamilyReunion"byChauNguyen Everytwoyearsitcomesaround Weallgettogetherfromdifferenttowns. Inmyhonestopinion It'snotatypicalfamilyreunion Notforty,orfifty,orsixty, Buthowaboutseventycompanions! Thekidswouldplay,scream,andshout Oneminutethey'rehappy,anotherthey'llpout. Theteenagerswouldlook,stare,andcompare Fromhowtheylooktowhattheywear. Themenwouldchatabouttheirbusiness Thattheymakemore,butneverless. Moneyisalwaystheirsubject Andthere'salwaystalkofmorenewprojects. Thewomengettiredfromallofthechats Theyheadtothekitchentosetoutthemats. Somewouldsitandsomewouldstand Eatingandtalkingwithplatesintheirhands. Thencomethegamesandthesongs Andsuddenly,everyonegetsalong! Withallthatlaughter,it'ssadtosay Thatitalwaysendsinthesameoldway. Theyhugandkissandsay"good-bye" Andthentheyallbegintocry! Isaythat60percentshedtheirtears Butmymomcounted35peoplethisyear. Shesaidthatboysandmenwillalwayshavetheirpride, Sowewon'teverseethemcry. Imyselfdon'tthinkshe'scorrect, Socouldyoupleasetrythisproblemtoseeifyouobject?

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350 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Exercise9.16.19 Solutiononp.369. "TheProblemwithAngels"byCyndyDowling Althoughthisproblemiswhollymine, Thecatalystcamefromthemagazine,Time. OnthemagazinecoverIdidfind Therealmofangelsticklingmymind. Inside,69%Ifoundtobe Inangels,Americansdobelieve. Then,itwastimetorisetothetask, Ninety-fivehighschoolandcollegestudentsIdidask. Viewingallasonegroup, Randomsamplingtogetthescoop. So,Iaskedeachtobetrue, "Doyoubelieveinangels?"Tellme,do! Hypothesizingatthestart, Totallybelievinginmyheart Thattheproportionwhosaidyes Wouldbeequalonthistest. Loandbehold,seventy-threedidarrive, Outofthesampleofninety-five. Nowyourjobhasjustbegun, Solvethisproblemandhavesomefun. Exercise9.16.20 "BlowingBubbles"bySondraPrull Studyingstatsjustmademetense, Ihadtofindsomesanedefense. Somelightandliftingsimpleplay Tofloatmymathanxietyaway. Blowingbubblesliftsmehigh Takesmytroublestothesky. POIK!They'regone,withallmystress Bubbletherapyisthebest. ThelabelsaideachtimeIblew Theaveragenumberofbubbleswouldbeatleast22. IblewandblewandthisIfound From64blows,theyallareround! Butthenumberofbubblesin64blows Variedwidely,thisIknow. 20perblowbecamethemean Theydeviatedby6,andnot16. Fromcountingbubbles,Isuredidrelax ButnowIgivetoyouyourtask. Was22areasonableguess? Findtheanswerandpassthistest!

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351 Exercise9.16.21 Solutiononp.369. 21."DalmatianDarnation"byKathySparling AgreedydogbreedernamedSpreckles Bredpuppieswithnumerousfreckles TheDalmatianshesought Possessedspotuponspot Themorespots,hethought,themoreshekels. Hiscompetitorsdidnotagree Thatfreckleswouldincreasethefee. Theysaid,``Spotsarequitenice Buttheydon'taffectprice; Oneshouldbreedforimprovedpedigree.'' Thebreedersdecidedtoprove Thisstrategywasawrongmove. Breedingonlyforspots Wouldwreakhavoc,theythought. Histheorytheywanttodisprove. TheyproposedacontesttoSpreckles Comparingdogpricestofreckles. Inrecordstheylookedup Onehundredonepups: Dalmatiansthatfetchedthemostshekels. TheyaskedMr.Sprecklestoname Anaveragespotcounthe'dclaim Tobringinbigbucks. SaidSpreckles,``Well,shucks, It'sforonehundredonethatIaim.'' Saidanamateurstatistician Whowantedtohelpwiththismission. ``Twenty-oneforthesample Standarddeviation'sample: Theyexaminedonehundredandone Dalmatiansthatfetchedagoodsum. Theycountedeachspot, Mark,freckleanddot Andtalliedupeveryone. Insteadofonehundredonespots Theyaveragedninetysixdots CantheymuzzleSpreckles' Obsessionwithfreckles Basedonallthedogdatathey'vegot?

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352 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Exercise9.16.22 "MacaroniandCheese,please!!"byNeddaMisherghiandRachelleHall AsapoorstarvingstudentIdon'thavemuchmoneytospendforeventhebarenecessities.So myfavoriteandmainstaplefoodismacaroniandcheese.It'shighintasteandlowincostand nutritionalvalue. Oneday,asIsatdowntodeterminethemeaningoflife,Igotaseriouscravingforthis,oh,so important,foodofmylife.SoIwentdownthestreettoGreatwaytogetaboxofmacaroniand cheese,butitwasSOexpensive!\$2.02!!!Canyoubelieveit?Itmademestopandthink.Theworld ischangingfast.Ihadthoughtthattheaveragecostofaboxthenormalsize,notsomesupergigantic-family-value-packwasatmost\$1,butnowIwasn'tsosure.However,Iwasdetermined tondout.Iwentto53oftheclosestgrocerystoresandsurveyedthepricesofmacaroniand cheese.HerearethedataIwroteinmynotebook: PriceperboxofMacandCheese: 5stores@\$2.02 15stores@\$0.25 3stores@\$1.29 6stores@\$0.35 4stores@\$2.27 7stores@\$1.50 5stores@\$1.89 8stores@0.75. IcouldseethatthecostsvariedbutIhadtositdowntogureoutwhetherornotIwasright.If itdoesturnoutthatthismouth-wateringdishisatmost\$1,thenI'llthrowabigcheesypartyin ournextstatisticslab,withenoughmacaroniandcheeseforjustme.Afterall,asapoorstarving studentIcan'tbeexpectedtofeedourclassofanimals! Exercise9.16.23 Solutiononp.369. "WilliamShakespeare:TheTragedyofHamlet,PrinceofDenmark"byJacquelineGhodsi THECHARACTERSinorderofappearance: HAMLET,PrinceofDenmarkandstudentofStatistics POLONIUS,Hamlet'stutor HOROTIO,friendtoHamletandfellowstudent Scene:Thegreatlibraryofthecastle,inwhichHamletdoeshislessons ActI Thedayisfair,butthefaceofHamletisclouded.Hepacesthelargeroom.Histutor,Polonius,is reprimandingHamletregardingthelatter'srecentexperience.Horatioisseatedatthelargetable atrightstage. POLONIUS:MyLord,howcans'tthouadmitthatthouhastseenaghost!Itisbutagmentof yourimagination! HAMLET:Ibegtodiffer;Iknowofacertaintythatve-and-seventyinonehundredofus,condemnedtothewhipsandscornsoftimeasweare,havegazeduponaspiritofhealth,orgoblin damn'd,betheirintentswickedorcharitable. POLONIUSIfthoudoestinsistuponthywretchedvisionthenletmeinvestyourtime;betrue tothyworkandspeaktomethroughthereasonofthenullandalternatehypotheses.Heturns

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353 toHoratio.DidnotHamlethimselfsay,Whatpieceofworkisman,hownobleinreason,how inniteinfaculties?Thenletnotthisfoolishnesspersist.Go,Horatio,makeasurveyofthree-andsixtyanddiscoverwhatthetrueproportionbe.Formypart,Iwillneversuccumbtothisfantasy, butdeemmantobedevoidofallreasonshouldthyproposalofatleastve-and-seventyinone hundredholdtrue. HORATIOtoHamlet:Whatshouldwedo,myLord? HAMLET:Gotothypurpose,Horatio. HORATIO:Towhatend,myLord? HAMLET:Thatyoumustteachme.Butletmeconjureyoubytherightsofourfellowship,bythe consonanceofouryouth,buttheobligationofourever-preservedlove,beevenanddirectwith me,whetherIamrightorno. Horatioexits,followedbyPolonius,leavingHamlettoponderalone. ActII Thenextday,Hamletawaitsanxiouslythepresenceofhisfriend,Horatio.Poloniusentersand placessomebooksuponthetablejustamomentbeforeHoratioenters. POLONIUS:So,Horatio,whatisitthoudidstrevealthroughthydeliberations? HORATIO:Inarandomsurvey,forwhichpurposethouthyselfsentmeforth,Ididdiscoverthat one-and-fortybelieveferventlythatthespiritsofthedeadwalkwithus.BeforemyGod,Imight notthisbelieve,withoutthesensibleandtrueavouchofmineowneyes. POLONIUS:Givethineownthoughtsnotongue,Horatio.PoloniusturnstoHamlet.Butlook to'tIchargeyou,myLord.ComeHoratio,letusgotogether,forthisisnotourtest.Horatioand Poloniusleavetogether. HAMLET:Toreject,ornotreject,thatisthequestion:whether`tisnoblerinthemindtosufferthe slingsandarrowsofoutrageousstatistics,ortotakearmsagainstaseaofdata,and,byopposing, endthem.Hamletresignedlyattendstohistask. Curtainfalls Exercise9.16.24 "Untitled"byStephenChen I'veoftenwonderedhowsoftwareisreleasedandsoldtothepublic.Ironically,Iworkforacompanythatsellsproductswithknownproblems.Unfortunately,mostoftheproblemsaredifcult tocreate,whichmakesthemdifculttox.IusuallyusethetestprogramX,whichteststheproduct,totrytocreateaspecicproblem.Whenthetestprogramisruntomakeanerroroccur,the likelihoodofgeneratinganerroris1%. So,armedwiththisknowledge,IwroteanewtestprogramYthatwillgeneratethesameerrorthat testprogramXcreates,butmoreoften.Tondoutifmytestprogramisbetterthantheoriginal, sothatIcanconvincethemanagementthatI'mright,Iranmytestprogramtondouthowoften Icangeneratethesameerror.WhenIranmytestprogram50times,Igeneratedtheerrortwice. Whilethismaynotseemmuchbetter,IthinkthatIcanconvincethemanagementtousemytest programinsteadoftheoriginaltestprogram.AmIright?

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354 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Exercise9.16.25 Solutiononp.369. JapaneseGirls'Names byKumiFuruichi ItusedtobeverytypicalforJapanesegirls'namestoendwithko.Thetrendmighthave startedaroundmygrandmothers'generationanditspeakmighthavebeenaroundmymother's generation.KomeanschildinChinesecharacter.Parentswouldnametheirdaughterswith koattachingtootherChinesecharacterswhichhavemeaningsthattheywanttheirdaughters tobecome,suchasSachikoahappychild,Yoshikoagoodchild,Yasukoahealthychild,and soon. However,InoticedrecentlythatonlytwooutofnineofmyJapanesegirlfriendsatthisschoolhave nameswhichendwithko.Moreandmore,parentsseemtohavebecomecreative,modernized, and,sometimes,westernizedinnamingtheirchildren. Ihaveafeelingthat,while70percentormoreofmymother'sgenerationwouldhavenameswith koattheend,theproportionhasdroppedamongmypeers.IwrotedownallmyJapanese friends',ex-classmates',co-workers,andacquaintances'namesthatIcouldremember.Beloware thenames.Somearerepeats.Testtoseeiftheproportionhasdroppedforthisgeneration. Ai,Akemi,Akiko,Ayumi,Chiaki,Chie,Eiko,Eri,Eriko,Fumiko,Harumi,Hitomi,Hiroko,Hiroko,Hidemi,Hisako,Hinako,Izumi,Izumi,Junko,Junko,Kana,Kanako,Kanayo,Kayo,Kayoko, Kazumi,Keiko,Keiko,Kei,Kumi,Kumiko,Kyoko,Kyoko,Madoka,Maho,Mai,Maiko,Maki, Miki,Miki,Mikiko,Mina,Minako,Miyako,Momoko,Nana,Naoko,Naoko,Naoko,Noriko, Rieko,Rika,Rika,Rumiko,Rei,Reiko,Reiko,Sachiko,Sachiko,Sachiyo,Saki,Sayaka,Sayoko, Sayuri,Seiko,Shiho,Shizuka,Sumiko,Takako,Takako,Tomoe,Tomoe,Tomoko,Touko,Yasuko, Yasuko,Yasuyo,Yoko,Yoko,Yoko,Yoshiko,Yoshiko,Yoshiko,Yuka,Yuki,Yuki,Yukiko,Yuko, Yuko. Exercise9.16.26 Phillip'sWishbySuzanneOsorio Mynephewlikestoplay Chasingthegirlsmakeshisday. Heaskedhismother Ifitisokay Togethisearpierced. Shesaid,``Noway!'' Topokeaholethroughyourear, IsnotwhatIwantforyou,dear. Hearguedhispointquitewell, Saysevenmymachopal,Mel, Hasgottenthisdone. It'salljustforfun. C'monplease,mom,please,whatthehell. AgainPhillipcomplainedtohismother, Sayinghalfhisfriendsincludingtheirbrothers Arepiercingtheirears Andtheyhavenofears Hewantstobeliketheothers. Shesaid,``Ithinkit'smuchless. Wemustdoahypothesistest. Andifyouareright,

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355 Iwon'tputupafight. But,ifnot,thenmycasewillrest.'' Weproceededtocallfiftyguys Toseewhosepredictionwouldfly. Nineteenofthefifty Saidpiercingwasnifty Andearringsthey'doccasionallybuy. Thenthere'stheotherthirty-one, Whosaidthey'dneverhavethisdone. Sonowthispoem'sfinished. Willhishopesbediminished, Orwillmynephewhavehisfun? Exercise9.16.27 Solutiononp.370. TheCravenbyMarkSalangsang Onceuponamorningdreary InstatsclassIwasweakandweary. Ponderingoverlastnight'shomework Whoseanswerswerenowontheboard ThisIdidandnothingmore. WhileInoddednearlynapping Suddenly,therecameatapping. Assomeonegentlyrapping, RappingmyheadasIsnore. Quoththeteacher,``Sleepnomore.'' ``Ineveryclassyoufallasleep,'' Theteachersaid,hisvoicewasdeep. ``SoatallyI'vebeguntokeep Ofeveryclassyounapandsnore. Thepercentagebeingforty-four.'' ``MydearteacherImustconfess, WhilesleepingiswhatIdobest. Thepercentage,Ithink,mustbeless, Apercentagelessthanforty-four.'' ThisIsaidandnothingmore. ``We'llsee,''hesaidandwalkedaway, Andfiftyclassesfromthatday HecountedtillthemonthofMay TheclassesinwhichInappedandsnored. Thenumberhefoundwastwenty-four. Atasignificancelevelof0.05, PleasetellmeamIstillalive? Ordidmygradejusttakeadive Plungingdownbeneaththefloor? UpontheeIherebyimplore.

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356 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Exercise9.16.28 ToastmastersInternationalcitesaFebruary2001reportbyGallopPollthat40%ofAmericansfear publicspeaking.Astudentbelievesthatlessthan40%ofstudentsatherschoolfearpublicspeaking.Sherandomlysurveys361schoolmatesandndsthat135reporttheyfearpublicspeaking. Conductahypothesistesttodetermineifthepercentatherschoolislessthan40%. Source: http://toastmasters.org/artisan/detail.asp?CategoryID=1&SubCategoryID=10&ArticleID=429&Page=1 17 Exercise9.16.29 Solutiononp.370. In2004,68%ofonlinecoursestaughtatcommunitycollegesnationwideweretaughtbyfull-time faculty.Totestif68%alsorepresentsCalifornia'spercentforfull-timefacultyteachingtheonline classes,LongBeachCityCollegeLBCC,CA,wasrandomlyselectedforcomparison.In2004, 34ofthe44onlinecoursesLBCCofferedweretaughtbyfull-timefaculty.Conductahypothesis testtodetermineif68%representsCA.NOTE:Foratruetest,usemoreCAcommunitycolleges. Sources: GrowingbyDegrees byAllenandSeaman;AmitSchitai,DirectorofInstructionalTechnologyandDistanceLearning,LBCC. N OTE :Foratruetest,usemoreCAcommunitycolleges. Exercise9.16.30 Accordingtoanarticlein TheNewYorkTimes /12/2004,19.3%ofNewYorkCityadults smokedin2003.Supposethatasurveyisconductedtodeterminethisyear'srate.Twelveout of70randomlychosenN.Y.Cityresidentsreplythattheysmoke.Conductahypothesistestto determineistherateisstill19.3%. Exercise9.16.31 Solutiononp.370. TheaverageageofDeAnzaCollegestudentsinWinter2006termwas26.6 yearsold.Aninstructorthinkstheaverageageforonlinestudentsisolderthan 26.6.Sherandomlysurveys56onlinestudentsandndsthatthesampleaverageis29.4withastandarddeviationof2.1.Conductahypothesistest. Source: http://research.fhda.edu/factbook/DAdemofs/Fact_sheet_da_2006w.pdf 18 Exercise9.16.32 In2004,registerednursesearnedanaverageannualsalaryof\$52,330.Asurveywasconducted of41Californianursedtodetermineiftheannualsalaryishigherthan\$52,330forCalifornia nurses.Thesampleaveragewas\$61,121withasamplestandarddeviationof\$7,489.Conducta hypothesistest. Source:http://stats.bls.gov/oco/ocos083.htm#earnings 19 Exercise9.16.33 Solutiononp.370. LaLecheLeagueInternationalreportsthattheaverageageofweaningachildfrombreastfeeding isage4to5worldwide.InAmerica,mostnursingmothersweantheirchildrenmuchearlier. Supposearandomsurveyisconductedof21U.S.motherswhorecentlyweanedtheirchildren. Theaverageweaningagewas9months/4yearwithastandarddeviationof4months.Conduct ahypothesistesttodetermineistheaverageweaningageintheU.S.islessthan4yearsold. Source:http://www.lalecheleague.org/Law/BAFeb01.html 20 17 http://toastmasters.org/artisan/detail.asp?CategoryID=1&SubCategoryID=10&ArticleID=429&Page=1 18 http://research.fhda.edu/factbook/DAdemofs/Fact_sheet_da_2006w.pdf 19 http://stats.bls.gov/oco/ocos083.htm#earnings 20 http://www.lalecheleague.org/Law/BAFeb01.html

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357 9.16.1Trythesemultiplechoicequestions. Exercise9.16.34 Solutiononp.370. Whenanewdrugiscreated,thepharmaceuticalcompanymustsubjectittotestingbeforereceivingthenecessarypermissionfromtheFoodandDrugAdministrationFDAtomarketthedrug. Supposethenullhypothesisisthedrugisunsafe.WhatistheTypeIIError? A. Toclaimthedrugissafewhenin,fact,itisunsafe B. Toclaimthedrugisunsafewhen,infact,itissafe. C. Toclaimthedrugissafewhen,infact,itissafe. D. Toclaimthedrugisunsafewhen,infact,itisunsafe Thenexttwoquestionsrefertothefollowinginformation: Overthepastfewdecades,public healthofcialshaveexaminedthelinkbetweenweightconcernsandteengirlssmoking.Researcherssurveyedagroupof273randomlyselectedteengirlslivinginMassachusettsbetween 12and15yearsold.Afterfouryearsthegirlsweresurveyedagain.Sixty-threesaidthey smokedtostaythin.Istheregoodevidencethatmorethanthirtypercentoftheteengirlssmoke tostaythin? Exercise9.16.35 Solutiononp.370. Thealternatehypothesisis A. p < 0 .30 B. p 0 .30 C. p 0 .30 D. p > 0 .30 Exercise9.16.36 Solutiononp.370. Afterconductingthetest,yourdecisionandconclusionare A. Reject H o :Morethan30%ofteengirlssmoketostaythin. B. Donotreject H o :Lessthan30%ofteengirlssmoketostaythin. C. Donotreject H o :Atmost30%ofteengirlssmoketostaythin. D. Reject H o :Lessthan30%ofteengirlssmoketostaythin. Thenextthreequestionsrefertothefollowinginformation: Astatisticsinstructorbelievesthatfewerthan 20%ofEvergreenValleyCollegeEVCstudentsattendedtheopeningnightmidnightshowingofthelatest HarryPottermovie.Shesurveys84ofherstudentsandndsthat11ofattendedthemidnightshowing. Exercise9.16.37 Solutiononp.370. Anappropriatealternativehypothesisis A. p = 0 .20 B. p > 0 .20 C. p < 0 .20 D. p 0 .20 Exercise9.16.38 Solutiononp.370. Ata1%levelofsignicance,anappropriateconclusionis: A. ThepercentofEVCstudentswhoattendedthemidnightshowingofHarryPotterisat least20%. B. ThepercentofEVCstudentswhoattendedthemidnightshowingofHarryPotterismore than20%.

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358 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION C. ThepercentofEVCstudentswhoattendedthemidnightshowingofHarryPotterisless than20%. D. Thereisnotenoughinformationtomakeadecision. Exercise9.16.39 Solutiononp.370. TheTypeIerrorisbelievingthatthepercentofEVCstudentswhoattendedis: A. atleast20%,wheninfact,itislessthan20%. B. 20%,wheninfact,itis20%. C. lessthan20%,wheninfact,itisatleast20%. D. lessthan20%,wheninfact,itislessthan20%. Thenexttwoquestionsrefertothefollowinginformation: ItisbelievedthatLakeTahoeCommunityCollegeLTCCIntermediateAlgebrastudentsgetlessthan7 hoursofsleeppernight,onaverage.Asurveyof22LTCCIntermediateAlgebrastudentsgeneratedan averageof7.24hourswithastandarddeviationof1.93hours.Atalevelofsignicanceof5%,doLTCC IntermediateAlgebrastudentsgetlessthan7hoursofsleeppernight,onaverage? Exercise9.16.40 Solutiononp.370. Thedistributiontobeusedforthistestis X A. N 7 .24 1 .93 p 22 B. N 7 .24 ,1 .93 C. t 22 D. t 21 Exercise9.16.41 Solutiononp.370. TheTypeIIerrorisIbelievethattheaveragenumberofhoursofsleepLTCCstudentsgetper night A. islessthan7hourswhen,infact,itisatleast7hours. B. islessthan7hourswhen,infact,itislessthan7hours. C. isatleast7hourswhen,infact,itisatleast7hours. D. isatleast7hourswhen,infact,itislessthan7hours. Thenextthreequestionsrefertothefollowinginformation: Anorganizationin1995reportedthatteenagers spentanaverageof4.5hoursperweekonthetelephone.Theorganizationthinksthat,in2007,theaverage ishigher.Fifteenrandomlychosenteenagerswereaskedhowmanyhoursperweektheyspendonthe telephone.Thesamplemeanwas4.75hourswithasamplestandarddeviationof2.0. Exercise9.16.42 Solutiononp.370. Thenullandalternatehypothesesare: A. H o : x = 4 5, H a : x > 4 5 B. H o : m 4 5 H a : m < 4 5 C. H o : m = 4 .75H a : m > 4 .75 D. H o : m = 4 5 H a : m > 4 5 Exercise9.16.43 Solutiononp.370. Atasignicancelevelof a = 0 .05 ,thecorrectconclusionis: A. Theaveragein2007ishigherthanitwasin1995. B. Theaveragein1995ishigherthanin2007.

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359 C. Theaverageisstillaboutthesameasitwasin1995. D. Thetestisinconclusive. Exercise9.16.44 Solutiononp.370. TheTypeIerroris: A. Toconcludetheaveragehoursperweekin2007ishigherthanin1995,wheninfact,itis higher. B. Toconcludetheaveragehoursperweekin2007ishigherthanin1995,wheninfact,itis thesame. C. Toconcludetheaveragehoursperweekin2007isthesameasin1995,wheninfact,itis higher. D. Toconcludetheaveragehoursperweekin2007isnohigherthanin1995,wheninfact, itisnothigher. 9.17Review 21 Exercise9.17.1 Solutiononp.371. 1.RebeccaandMattare14yearoldtwins.Matt'sheightis2standarddeviationsbelowthemean for14yearoldboys'height.Rebecca'sheightis0.10standarddeviationsabovethemeanfor14 yearoldgirls'height.Interpretthis. A. Mattis2.1inchesshorterthanRebecca B. Rebeccaisverytallcomparedtoother14yearoldgirls. C. RebeccaistallerthanMatt. D. Mattisshorterthantheaverage14yearoldboy. 2.ConstructahistogramoftheIPOdataseeTableofContents.Use5intervals. Thenextsixquestionsrefertothefollowinginformation: Ninetyhomeownerswereaskedthenumberof estimatestheyobtainedbeforehavingtheirhomesfumigated. X =thenumberofestimates. X Rel.Freq. CumulativeRel.Freq. 1 0.3 2 0.2 4 0.4 5 0.1 3.Calculatethefrequencies. 4.Completethecumulativerelativefrequencycolumn.Whatpercentoftheestimatesfellatorbelow4? Exercise9.17.2 Solutiononp.371. 5.Calculatethesamplemeanaandsamplestandarddeviationb. Exercise9.17.3 Solutiononp.371. 6.Calculatethemedian,M,therstquartile,Q1,thethirdquartile,Q3. 21 Thiscontentisavailableonlineat.

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360 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION Exercise9.17.4 Solutiononp.371. 7.Themiddle50%ofthedataarebetween_____and_____. 8.Constructaboxplotofthedata. Thenextthreequestionsrefertothefollowingtable: Seventy5thand6thgraderswereaskedtheirfavorite dinner. Pizza Hamburgers Spaghetti Friedshrimp 5thgrader 15 6 9 0 6thgrader 15 7 10 8 Exercise9.17.5 Solutiononp.371. 9.Findtheprobabilitythatonerandomlychosenchildisinthe6thgradeandprefersfriedshrimp. A. 32 70 B. 8 32 C. 8 8 D. 8 70 Exercise9.17.6 Solutiononp.371. 10.Findtheprobabilitythatachilddoesnotpreferpizza. A. 30 70 B. 30 40 C. 40 70 D. 1 Exercise9.17.7 Solutiononp.371. 11.Findtheprobabilityachildisinthe5thgradegiventhatthechildprefersspaghetti. A. 9 19 B. 9 70 C. 9 30 D. 19 70 Exercise9.17.8 Solutiononp.371. 12.Asampleofconvenienceisarandomsample. A. true B. false Exercise9.17.9 Solutiononp.371. 13.Astatisticisanumberthatisapropertyofthepopulation. A. true B. false Exercise9.17.10 Solutiononp.371. 14.Youshouldalwaysthrowoutanydatathatareoutliers.

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361 A. true B. false Exercise9.17.11 Solutiononp.371. 15.LeebakespiesforalittlerestaurantinFelton.Shegenerallybakes20piesinaday,onthe average. a. DenetheRandomVariable X b. Statethedistributionfor X c. FindtheprobabilitythatLeebakesmorethan25piesinanygivenday. Exercise9.17.12 Solutiononp.371. 16.SixdifferentbrandsofItaliansaladdressingwererandomlyselectedatasupermarket.The gramsoffatperservingare7,7,9,6,8,5.Assumethattheunderlyingdistributionisnormal. Calculatea95%condenceintervalforthepopulationaveragegramsoffatperservingofItalian saladdressingsoldinsupermarkets. Exercise9.17.13 Solutiononp.371. 17.Given:uniform,exponential,normaldistributions.Matcheachtoastatementbelow. a. mean=median 6 = mode b. mean > median > mode c. mean=median=mode

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362 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION 9.18Lab:HypothesisTestingofaSingleMeanandSingleProportion 22 ClassTime: Names: 9.18.1StudentLearningOutcomes: Thestudentwillselecttheappropriatedistributionstouseineachcase. Thestudentwillconducthypothesistestsandinterprettheresults. 9.18.2TelevisionSurvey Inarecentsurvey,itwasstatedthatAmericanswatchtelevisiononaveragefourhoursperday.Assume that s = 2.Usingyourclassasthesample,conductahypothesistesttodetermineiftheaveragefor studentsatyourschoolislower. 1. H o : 2. H a : 3.Inwords,denetherandomvariable.__________= 4.Thedistributiontouseforthetestis: 5.Calculatetheteststatisticusingyourdata. 22 Thiscontentisavailableonlineat.

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363 6.Drawagraphandlabelitappropriately.Shadetheactuallevelofsignicance. a. Graph: Figure9.9 b. Calculatethep-value: 7.Doyouordoyounotrejectthenullhypothesis?Why? 8.Writeaclearconclusionusingacompletesentence. 9.18.3LanguageSurvey Accordingtothe2000Census,about39.5%ofCaliforniansand17.9%ofallAmericansspeakalanguage otherthanEnglishathome.Usingyourclassasthesample,conductahypothesistesttodetermineifthe percentofthestudentsatyourschoolthatspeakalanguageotherthanEnglishathomeisdifferentfrom 39.5%. 1. H o : 2. H a : 3.Inwords,denetherandomvariable.__________= 4.Thedistributiontouseforthetestis: 5.Calculatetheteststatisticusingyourdata.

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364 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION 6.Drawagraphandlabelitappropriately.Shadetheactuallevelofsignicance. a. Graph: Figure9.10 b. Calculatethep-value: 7.Doyouordoyounotrejectthenullhypothesis?Why? 8.Writeaclearconclusionusingacompletesentence. 9.18.4JeansSurvey Supposethatyoungadultsownanaverageof3pairsofjeans.Survey8peoplefromyourclasstodetermine iftheaverageishigherthan3. 1. H o : 2. H a : 3.Inwords,denetherandomvariable.__________= 4.Thedistributiontouseforthetestis: 5.Calculatetheteststatisticusingyourdata.

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365 6.Drawagraphandlabelitappropriately.Shadetheactuallevelofsignicance. a. Graph: Figure9.11 b. Calculatethep-value: 7.Doyouordoyounotrejectthenullhypothesis?Why? 8.Writeaclearconclusionusingacompletesentence.

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366 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION SolutionstoExercisesinChapter9 Example9.15,Problem2p.336 p'= 43 150 Example9.15,Problem3p.336 Asuccessishavingthreetelephonenumbersinyourhousehold Example9.15,Problem4p.336 p = 43 150 Example9.15,Problem5p.336 p-value=0.3594 Example9.15,Problem6p.336 Assumingthat a =0.5, a < p-value .Donotreject H 0 becausethereisnotsufcientevidencethatthe proportionsofhouseholdsthathavethreetelephonesis30% SolutionstoPractice1:SingleMean,KnownPopulationStandardDeviation SolutiontoExercise9.13.1p.341 Averages SolutiontoExercise9.13.2p.341 aH o : m = 2 5or, H o : m 2 5 bH a : m > 2 5 SolutiontoExercise9.13.3p.341 right-tailed SolutiontoExercise9.13.4p.341 X SolutiontoExercise9.13.5p.341 Theaveragetimespentinjailfor26rsttimeconvictedburglars SolutiontoExercise9.13.6p.341 Yes,1.5 SolutiontoExercise9.13.7p.341 a. 3 b. 1.5 c. 1.8 d. 26 SolutiontoExercise9.13.8p.341 s SolutiontoExercise9.13.9p.341 X~N 2 5 1.5 p 26 SolutiontoExercise9.13.11p.342 0.0446 SolutiontoExercise9.13.12p.342 a. Rejectthenullhypothesis

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367 SolutionstoPractice2:SingleMean,UnknownPopulationStandardDeviation SolutiontoExercise9.14.1p.343 averages SolutiontoExercise9.14.2p.343 a. H o : m = 15 b. H a : m 6 = 15 SolutiontoExercise9.14.3p.343 two-tailed SolutiontoExercise9.14.4p.343 X SolutiontoExercise9.14.5p.343 theaveragetimespentondeathrow SolutiontoExercise9.14.6p.343 No SolutiontoExercise9.14.7p.343 a. 17.4 b. s c. 75 SolutiontoExercise9.14.8p.343 t )]TJ/F58 9.9626 Tf 8.194 0 Td [(test SolutiontoExercise9.14.9p.343 t 74 SolutiontoExercise9.14.11p.344 0.0015 SolutiontoExercise9.14.12p.344 a. Rejectthenullhypothesis SolutionstoPractice3:SingleProportion SolutiontoExercise9.15.1p.345 Proportions SolutiontoExercise9.15.2p.345 a. H o : p = 0 .095 b. H a : P < 0 .095 SolutiontoExercise9.15.3p.345 left-tailed SolutiontoExercise9.15.4p.345 P-hat SolutiontoExercise9.15.5p.345 theproportionofpeopleinthattownsufferingfromdepress.oradepr.illness

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368 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION SolutiontoExercise9.15.6p.345 a. 7 b. 100 c. 0.07 SolutiontoExercise9.15.7p.345 2.93 SolutiontoExercise9.15.8p.345 Normal SolutiontoExercise9.15.10p.346 0.1969 SolutiontoExercise9.15.11p.346 a. Donotrejectthenullhypothesis SolutionstoHomework SolutiontoExercise9.16.1p.347 a. H o : m = 34 ; H a : m 6 = 34 c. H o : m 100 000 ; H a : m < 100 000 d. H o : p = 0 .29 ; H a : p 6 = 0 .29 g. H o : p = 0 .50 ; H a : p 6 = 0 .50 i. H o : p 0 .11 ; H a : p < 0 .11 SolutiontoExercise9.16.2p.347 a. TypeIerror:Webelievetheaverageisnot34years,whenitreallyis34years.TypeIIerror:We believetheaverageis34years,whenitisnotreally34years. c. TypeIerror:Webelievetheaverageislessthan\$100,000,whenitreallyisatleast\$100,000.TypeII error:Webelievetheaverageisatleast\$100,000,whenitisreallylessthan\$100,000. d. TypeIerror:Webelievethattheproportionofh.s.seniorswhogetdrunkeachmonthisnot29%, whenitreallyis29%.TypeIIerror:Webelievethat29%ofh.s.seniorsgetdrunkeachmonth, whentheproportionisreallynot29%. i. TypeIerror:Webelievetheproportionislessthan11%,whenitisreallyatleast11%.TypeIIerror: WEbelievetheproportionisatleast11%,whenitreallyislessthan11%. SolutiontoExercise9.16.5p.347 e. z = )]TJ/F58 9.9626 Tf 8.194 0 Td [(2 .71 f. 0.0034 h. Decision:Rejectnull;Conclusion: m < 19 i. 17.449 18.757 SolutiontoExercise9.16.7p.348 e. 3.5 f. 0.0005 h. Decision:Rejectnull;Conclusion: m > 4 5 i. 4 .7553 ,5 .4447

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369 SolutiontoExercise9.16.9p.348 e. 2.7 f. 0.0042 h. Decision:RejectNull i. 80.789 85.211 SolutiontoExercise9.16.11p.348 d. t 11 e. 1.96 f. 0.0380 h. Decision:Rejectnullwhen a = 0 .05 ;donotrejectnullwhen a = 0 .01 i. 3 .8865 ,5 .9468 SolutiontoExercise9.16.13p.348 e. -1.64 f. 0.1000 h. Decision:Donotrejectnull i. 0 .3216 ,0 .4784 SolutiontoExercise9.16.15p.349 d. t 9 e. -1.33 f. 0.1086 h. Decision:Donotrejectnull i. 51.886 62.114 SolutiontoExercise9.16.19p.350 e. 1.65 f. 0.0984 h. Decision:Donotrejectnull i. 0 .6836 ,0 .8533 SolutiontoExercise9.16.21p.351 e. -2.39 f. 0.0093 h. Decision:Rejectnull i. 91.854 100.15 SolutiontoExercise9.16.23p.352 e. -1.82 f. 0.0345 h. Decision:Donotrejectnull i. 0 .5331 ,0 .7685 SolutiontoExercise9.16.25p.354 e. z = )]TJ/F58 9.9626 Tf 8.194 0 Td [(2 .99 f. 0.0014 h. Decision:Rejectnull;Conclusion: p < .70 i. 0 .4529 ,0 .6582

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370 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION SolutiontoExercise9.16.27p.355 e. 0.57 f. 0.7156 h. Decision:Donotrejectnull i. 0 .3415 ,0 .6185 SolutiontoExercise9.16.29p.356 e. 1.32 f. 0.1873 h. Decision:Donotrejectnull i. 0 .65 ,0 .90 SolutiontoExercise9.16.31p.356 e. 9.98 f. 0.0000 h. Decision:Rejectnull i. 28. 8, 30. 0 SolutiontoExercise9.16.33p.356 e. -44.7 f. 0.0000 h. Decision:Rejectnull i. 0 .60 ,0 .90 -inyears SolutiontoExercise9.16.34p.357 B SolutiontoExercise9.16.35p.357 D SolutiontoExercise9.16.36p.357 C SolutiontoExercise9.16.37p.357 C SolutiontoExercise9.16.38p.357 A SolutiontoExercise9.16.39p.358 C SolutiontoExercise9.16.40p.358 D SolutiontoExercise9.16.41p.358 D SolutiontoExercise9.16.42p.358 D SolutiontoExercise9.16.43p.358 C SolutiontoExercise9.16.44p.359 B

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371 SolutionstoReview SolutiontoExercise9.17.1p.359 D SolutiontoExercise9.17.2p.359 a. 2.8 b. 1.48 SolutiontoExercise9.17.3p.359 M = 3; Q 1 = 1; Q 3 = 4 SolutiontoExercise9.17.4p.360 1and4 SolutiontoExercise9.17.5p.360 D SolutiontoExercise9.17.6p.360 C SolutiontoExercise9.17.7p.360 A SolutiontoExercise9.17.8p.360 B SolutiontoExercise9.17.9p.360 B SolutiontoExercise9.17.10p.360 B SolutiontoExercise9.17.11p.361 b. P 20 c. 0.1122 SolutiontoExercise9.17.12p.361 CI: 5 .52 ,8 .48 SolutiontoExercise9.17.13p.361 a. uniform b. exponential c. normal

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372 CHAPTER9.HYPOTHESISTESTING:SINGLEMEANANDSINGLE PROPORTION

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Chapter10 HypothesisTesting:TwoMeans,Paired Data,TwoProportions 10.1HypothesisTesting:TwoPopulationMeansandTwoPopulation Proportions 1 10.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Classifyhypothesistestsbytype. Conductandinterprethypothesistestsfortwopopulationmeans,populationstandarddeviations known. Conductandinterprethypothesistestsfortwopopulationmeans,populationstandarddeviations unknown. Conductandinterprethypothesistestsfortwopopulationproportions. Conductandinterprethypothesistestsformatchedorpairedsamples. 10.1.2Introduction Studiesoftencomparetwogroups.Forexample,researchersareinterestedintheeffectaspirinhasin preventingheartattacks.Overthelastfewyears,newspapersandmagazineshavereportedaboutvarious aspirinstudiesinvolvingtwogroups.Typically,onegroupisgivenaspirinandtheothergroupisgivena placebo.Then,theheartattackrateisstudiedoverseveralyears. Thereareothersituationsthatdealwiththecomparisonoftwogroups.Forexample,studiescomparevariousdietandexerciseprograms.Politicianscomparetheproportionofindividualsfromdifferentincome bracketswhomightvoteforthem.StudentswhoareinterestedinwhetherSATorGREpreparatorycourses reallyhelpraisetheirscores. Inthepreviouschapter,youlearnedtoconducthypothesistestsonsinglemeansandsingleproportions. Youwillexpanduponthatinthischapter.Youwillcomparetwoaveragesortwoproportionstoeachother. Theprocedureisstillthesame,justexpanded. 1 Thiscontentisavailableonlineat. 373

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374 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Tocomparetwoaveragesortwoproportions,youworkwithtwogroups.Thegroupsareclassiedas independent and matchedpairs Independentgroups meanthatthetwosamplestakenareindependent, thatis,samplevaluesselectedfromonepopulationarenotrelatedinanywaytosamplevaluesselected fromtheotherpopulation. Matchedpairs refertomatchedorpairedsamples.Theparametertestedusingmatchedpairsisthepopulationmean.Theparameterstestedusingindependentgroupsareeither populationmeansorpopulationproportions. N OTE :Thischapterreliesoneitheracalculatororacomputertocalculatethedegreesoffreedom, theteststatistics,andp-values.TI-83+andTI-84instructionsareincludedaswellasthethetest statisticformulas.Becauseoftechnology,wedonotneedtoseparatetwopopulationmeans, independentgroups,populationvariancesunknownintolargeandsmallsamplesizes.Thesmall samplecasedependsontheassumptionthattheunknownpopulationvariancesareequal.Itis notnecessarytomakethatassumption. Thischapterdealswiththefollowinghypothesistests: Independentgroupssamplesareindependent 1.Twopopulationmeans. 2.Twopopulationproportions. Matchedorpairedsamples 1.Samplesizesareoftensmall. 2.Twomeasurementsaredrawnfromthesamepairofindividualsorobjects. 3.Twosamplesarecombinedtoformone sampleofdifferences 10.2ComparingTwoIndependentPopulationMeanswithUnknown PopulationStandardDeviations 2 1.Thetwoindependentsamplesaresimplerandomsamplesfromtwodistinctpopulations. 2.Bothpopulationsarenormallydistributedwiththepopulationmeansandstandarddeviationsunknown. Thecomparisonoftwopopulationmeansisverycommon.Adifferencebetweenthetwosamplesdepends onboththemeansandthestandarddeviations.Verydifferentmeanscanoccurbychanceifthereisgreat variationamongtheindividualsamples.Inordertoaccountforthevariation,wetakethedifferenceof thesamplemeans, X 1 X 2 ,anddividebythestandarderrorshownbelowinordertostandardizethe difference.Theresultisat-scoreteststatisticshownbelow. Becausewedonotknowthepopulationstandarddeviations,weestimatethemusingthetwosample standarddeviationsfromourindependentsamples.Forthehypothesistest,wecalculatetheestimated standarddeviation,or standarderror ,of thedifferenceinsamplemeans X 1 X 2 Thestandarderroris: s S 1 2 n 1 + S 2 2 n 2 .1 Theteststatistict-scoreiscalculatedasfollows: 2 Thiscontentisavailableonlineat.

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375 T-score x 1 )]TJETq1 0 0 1 291.266 677.497 cm[]0 d 0 J 0.398 w 0 0 m 9.684 0 l SQBT/F132 9.9626 Tf 291.56 671.285 Td [(x 2 )]TJ/F142 10.3811 Tf 10.256 -0.104 Td [( m 1 )]TJ/F134 9.9626 Tf 10.256 0 Td [(m 2 r S 1 2 n 1 + S 2 2 n 2 .2 where: s 1 and s 2 ,thesamplestandarddeviations,areestimatesof s 1 and s 2 ,respectively. s 1 and s 2 aretheunknownpopulationstandarddeviations. x 1 and x 2 arethesamplemeans. m 1 and m 2 arethepopulationmeans. The degreesoffreedomdf isasomewhatcomplicatedcalculation.However,acomputerorcalculatorcalculatesiteasily.Thedfsarenotalwaysawholenumber.Theteststatisticcalculatedaboveisapproximated bytheStudent-tdistributionwithdfsasfollows: Degreesoffreedom df = s 1 2 n 1 + s 2 2 n 2 2 1 n 1 )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 s 1 2 n 1 2 + 1 n 2 )]TJ/F58 7.5716 Tf 6.227 0 Td [(1 s 2 2 n 2 2 .3 Whenbothsamplesizes n 1 and n 2 areveorlarger,theStudent-tapproximationisverygood.Noticethat thesamplevariances s 1 2 and s 2 2 arenotpooled.Ifthequestioncomesup,donotpoolthevariances. N OTE :Itisnotnecessarytocomputethisbyhand.Acalculatororcomputereasilycomputesit. Example10.1:Independentgroups Theaverageamountoftimeboysandgirlsages7through11spendplayingsportseachdayis believedtobethesame.Anexperimentisdone,dataiscollected,resultinginthetablebelow: SampleSize AverageNumberof HoursPlayingSports PerDay SampleStandard Deviation Girls 9 2hours p 0.75 Boys 16 3.2hours 1.00 Problem Isthereadifferenceintheaverageamountoftimeboysandgirlsages7through11playsports eachday?Testatthe5%levelofsignicance. Solution Thepopulationstandarddeviationsarenotknown. Let g bethesubscriptforgirlsand b bethe subscriptforboys.Then, m g isthepopulationmeanforgirlsand m b isthepopulationmeanfor boys.Thisisatestoftwo independentgroups ,twopopulation means Randomvariable : X g )]TJETq1 0 0 1 200.745 148.034 cm[]0 d 0 J 0.398 w 0 0 m 11.685 0 l SQBT/F132 9.9626 Tf 201.019 139.75 Td [(X b =differenceintheaverageamountoftimegirlsandboysplaysports eachday. H o : m g = m b )]TJ/F134 9.9626 Tf 4.812 -8.075 Td [(m g )]TJ/F134 9.9626 Tf 10.255 0 Td [(m b = 0 H a : m g 6 = m b )]TJ/F134 9.9626 Tf 4.812 -8.075 Td [(m g )]TJ/F134 9.9626 Tf 10.255 0 Td [(m b 6 = 0

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376 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Thewords "thesame" tellyou H o hasan"=".Sincetherearenootherwordstoindicate H a ,then assume "isdifferent." Thisisatwo-tailedtest. Distributionforthetest: Use t df where df iscalculatedusingthe df formulaforindependent groups,twopopulationmeans.Usingacalculator, df isapproximately18.8462. Donotpoolthe variances. Calculatethep-valueusingaStudent-tdistribution: p-value=0.0054 Graph: Figure10.1 s g = p 0.75 s b = 1 So, x g )]TJETq1 0 0 1 129.809 325.483 cm[]0 d 0 J 0.398 w 0 0 m 9.593 0 l SQBT/F132 9.9626 Tf 130.103 319.272 Td [(x b = 2 )]TJ/F58 9.9626 Tf 10.131 0 Td [(3.2 = )]TJ/F58 9.9626 Tf 8.194 0 Td [(1.2 Halfthep-valueisbelow-1.2andhalfisabove1.2. Makeadecision: Since a > p-value,reject H o Thismeansyoureject m g = m b .Themeansaredifferent. Conclusion: Atthe5%levelofsignicance,thesampledatashowthereissufcientevidenceto concludethattheaveragenumberofhoursthatgirlsandboysaged7through11playsportsper dayisdifferent. N OTE :TI-83+andTI-84:Press STAT .Arrowoverto TESTS andpress 4:2-SampTTest .Arrowover toStatsandpress ENTER .Arrowdownandenter 2 fortherstsamplemean, .75 forSx1, 9 for n1, 3.2 forthesecondsamplemean, 1 forSx2,and 16 forn2.Arrowdownto m 1:andarrowto doesnotequal m 2.Press ENTER .ArrowdowntoPooled:andNo.Press ENTER .Arrowdownto Calculate andpress ENTER .Thep-valueisp=0.0054,thedfsareapproximately18.8462,andthe teststatisticis-3.14.DotheprocedureagainbutinsteadofCalculatedoDraw. Example10.2 Astudyisdonebyacommunitygroupintwoneighboringcollegestodeterminewhichonegraduatesstudentswithmoremathclasses.CollegeAsamples11graduates.Theiraverageis4math

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377 classeswithastandarddeviationof1.5mathclasses.CollegeBsamples9graduates.Theiraverageis3.5mathclasseswithastandarddeviationof1mathclass.Thecommunitygroupbelieves thatastudentwhograduatesfromcollegeA hastakenmoremathclasses, ontheaverage.Testat a1%signicancelevel.Answerthefollowingquestions. Problem1 Isthisatestoftwomeansortwoproportions? Problem2 Arethepopulationsstandarddeviationsknownorunknown? Problem3 Whichdistributiondoyouusetoperformthetest? Problem4 Whatistherandomvariable? Problem5 Whatarethenullandalternatehypothesis? Problem6 Isthistestright,left,ortwotailed? Problem7 Whatisthep-value? Problem8 Doyourejectornotrejectthenullhypothesis? Conclusion: Atthe1%levelofsignicance,fromthesampledata,thereisnotsufcientevidencetoconclude thatastudentwhograduatesfromcollegeAhastakenmoremathclasses,ontheaverage,thana studentwhograduatesfromcollegeB. 10.3ComparingTwoIndependentPopulationMeanswithKnownPopulationStandardDeviations 3 Eventhoughthissituationisnotlikelyknowingthepopulationstandarddeviationsisnotlikelybecause usuallyyouhavetwosetsofdata,thefollowingexampleillustrateshypothesistestingforindependent means,knownpopulationstandarddeviations.ThedistributionisNormalandisforthedifferenceof samplemeans, X 1 )]TJETq1 0 0 1 163.674 239.846 cm[]0 d 0 J 0.398 w 0 0 m 11.776 0 l SQBT/F132 9.9626 Tf 163.948 231.562 Td [(X 2 .Thenormaldistributionhasthefollowingformat: Normaldistribution X 1 )]TJETq1 0 0 1 239.062 203.378 cm[]0 d 0 J 0.398 w 0 0 m 11.776 0 l SQBT/F132 9.9626 Tf 239.336 195.094 Td [(X 2 N 2 4 u 1 )]TJ/F132 9.9626 Tf 10.255 0 Td [(u 2 s s 1 2 n 1 + s 2 2 n 2 3 5 .4 Thestandarddeviationis: s s 1 2 n 1 + s 2 2 n 2 .5 3 Thiscontentisavailableonlineat.

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378 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Theteststatisticz-scoreis: z = x 1 )]TJETq1 0 0 1 300.529 676.511 cm[]0 d 0 J 0.398 w 0 0 m 9.684 0 l SQBT/F132 9.9626 Tf 300.823 670.299 Td [(x 2 )]TJ/F142 10.3811 Tf 10.256 -0.105 Td [( m 1 )]TJ/F134 9.9626 Tf 10.256 0 Td [(m 2 r s 1 2 n 1 + s 2 2 n 2 .6 Example10.3 independentgroups,populationstandarddeviationsknown: Themeanlastingtimeof2competingoorwaxesistobecompared. Twentyoors arerandomlyassigned totesteachwax .The followingtableistheresult. Wax SampleMeanNumberofMonthsFloorWaxLast PopulationStandardDeviation 1 3 0.33 2 2.9 0.36 Problem Doesthedataindicatethat wax1ismoreeffectivethanwax2 ?Testata5%levelofsignicance. Solution Thisisatestoftwoindependentgroups,twopopulationmeans,populationstandarddeviations known. RandomVariable : X 1 )]TJETq1 0 0 1 203.792 438.583 cm[]0 d 0 J 0.398 w 0 0 m 11.776 0 l SQBT/F132 9.9626 Tf 204.066 430.299 Td [(X 2 = differenceintheaveragenumberofmonthsthecompetingoor waxeslast. H o : m 1 m 2 H a : m 1 > m 2 Thewords "ismoreeffective" saysthat wax1lastslongerthanwax2 ,ontheaverage."Longer"is a > symbolandgoesinto H a .Therefore,thisisaright-tailedtest. Distributionforthetest: Thepopulationstandarddeviationsareknownsothedistributionis normal.Usingtheformulaabove,thedistributionis: X 1 )]TJETq1 0 0 1 115.894 291.819 cm[]0 d 0 J 0.398 w 0 0 m 11.776 0 l SQBT/F132 9.9626 Tf 116.168 283.535 Td [(X 2 N 0, q 0.33 2 20 + 0.36 2 20 Since m 1 m 2 then m 1 )]TJ/F134 9.9626 Tf 10.256 0 Td [(m 2 0andthemeanforthenormaldistributionis0. Calculatethep-valueusingthenormaldistribution: p-value=0.1799

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379 Graph: Figure10.2 x 1 )]TJETq1 0 0 1 113.802 468.466 cm[]0 d 0 J 0.398 w 0 0 m 9.684 0 l SQBT/F132 9.9626 Tf 114.096 462.254 Td [(x 2 = 3 )]TJ/F58 9.9626 Tf 10.131 0 Td [(2.9 = 0.1 Compare a andthep-value: a = 0.05andp-value=0.1799.Therefore, a < p-value. Makeadecision: Since a < p-value,donotreject H o Conclusion: Atthe5%levelofsignicance,fromthesampledata,thereisnotsufcientevidence toconcludethatwax1lastslongerwax1ismoreeffectivethanwax2. N OTE :TI-83+andTI-84:Press STAT .Arrowoverto TESTS andpress 3:2-SampZTest .Arrowover to Stats andpress ENTER .Arrowdownandenter .33 forsigma1, .36 forsigma2, 3 fortherst samplemean, 20 forn1, 2.9 forthesecondsamplemean,and 20 forn2.Arrowdownto m 1:and arrowto > m 2.Press ENTER .Arrowdownto Calculate andpress ENTER .Thep-valueisp=0.1799 andtheteststatisticis0.9157.Dotheprocedureagainbutinsteadof Calculate do Draw 10.4ComparingTwoIndependentPopulationProportions 4 1.Thetwoindependentsamplesaresimplerandomsamplesthatareindependent. 2.Thenumberofsuccessesisatleastveandthenumberoffailuresisatleastveforeachofthe samples. Comparingtwoproportions,likecomparingtwomeans,iscommon.Iftwoestimatedproportionsaredifferent,itmaybeduetoadifferenceinthepopulationsoritmaybeduetochance.Ahypothesistestcanhelp determineifadifferenceintheestimatedproportions P A )]TJ/F132 9.9626 Tf 10.455 0 Td [(P B reectsadifferenceinthepopulations. Thedifferenceoftwoproportionsfollowsanapproximatenormaldistribution.Generally,thenullhypothesisstatesthatthetwoproportionsarethesame.Thatis, H o : p A = p B .Toconductthetest,weuseapooled proportion, p c 4 Thiscontentisavailableonlineat.

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380 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Thepooledproportioniscalculatedasfollows: p c = X A + X B n A + n B .7 Thedistributionforthedifferencesis: P A )]TJ/F132 9.9626 Tf 10.455 0 Td [(P B N 0, s p c 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p c 1 n A + 1 n B # .8 Theteststatisticz-scoreis: z = p A )]TJ/F132 9.9626 Tf 10.804 0 Td [(p B )]TJ/F142 10.3811 Tf 10.255 -0.105 Td [( p A )]TJ/F132 9.9626 Tf 10.804 0 Td [(p B r p c 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p c 1 n A + 1 n B .9 Example10.4:Twopopulationproportions Twotypesofmedicationforhivesarebeingtestedtodetermineifthereisa differenceinthepercentageofadultpatientreactions.Twenty outofarandom sampleof200 adultsgivenmedication Astillhadhives30minutesaftertakingthemedication. Twelve outofanother randomsampleof 200adults givenmedicationBstillhadhives30minutesaftertakingthemedication.Testata1% levelofsignicance. 10.4.1Determiningthesolution Thisisatestof2populationproportions. Problem Howdoyouknow? Let A and B bethesubscriptsformedicationAandmedicationB.Then p A and p B arethedesired populationproportions. RandomVariable: P A )]TJ/F132 9.9626 Tf 10.588 0 Td [(P B = differenceinthepercentagesofadultpatientswhodidnotreactafter30minutesto medicationAandmedicationB. H o : p A = p B p A )]TJ/F132 9.9626 Tf 10.804 0 Td [(p B = 0 H a : p A 6 = p B p A )]TJ/F132 9.9626 Tf 10.804 0 Td [(p B 6 = 0 Thewords "isadifference" tellyouthetestistwo-tailed. Distributionforthetest: Sincethisisatestoftwobinomialpopulationproportions,thedistributionisnormal: p c = X A + X B n A + n B = 20 + 12 200 + 200 = 0.081 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p c = 0.92 Therefore, P A )]TJ/F132 9.9626 Tf 10.455 0 Td [(P B N 0, r 0.08 0.92 1 200 + 1 200 P A )]TJ/F132 9.9626 Tf 10.455 0 Td [(P B followsanapproximatenormaldistribution. Calculatethep-valueusingthenormaldistribution: p-value=0.1404. EstimatedproportionforgroupA: p A = X A n A = 20 200 = 0.1

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381 EstimatedproportionforgroupB: p B = X B n B = 12 200 = 0.06 Graph: Figure10.3 P A )]TJ/F132 9.9626 Tf 10.455 0 Td [(P B = 0.1 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0.06 = 0.04. Halfthep-valueisbelow-0.04andhalfisabove0.04. Compare a andthep-value: a = 0.01andthe p-value = 0.1404. a < p-value. Makeadecision:Since a < p-value ,youcannotreject H o Conclusion: Ata1%levelofsignicance,fromthesampledata,thereisnotsufcientevidenceto concludethatthereisadifferenceinthepercentagesofadultpatientswhodidnotreactafter30 minutestomedicationAandmedicationB. TI-83+andTI-84:Press STAT .Arrowoverto TESTS andpress 6:2-PropZTest .Arrowdownand enter 20 for x 1, 200 for n 1, 12 for x 2,and 200 for n 2.Arrowdownto p1 :andarrowto doesnot equalp2 .Press ENTER .Arrowdownto Calculate andpress ENTER .Thep-valueis p = 0.1404and theteststatisticis1.47.Dotheprocedureagainbutinsteadof Calculate do Draw 10.5MatchedorPairedSamples 5 1.Simplerandomsamplingisused. 2.Differencesarecalculatedfromthematchedorpairedsamples. 3.Thematchedpairshavedifferencesthateithercomefromapopulationthatisnormalorthenumber ofdifferencesisgreaterthan30orboth. Inahypothesistestformatchedorpairedsamples,subjectsarematchedinpairsanddifferencesarecalculated.Thedifferencesarethedata.Thepopulationmeanforthedifferences, m d ,isthentestedusing aStudent-ttestforasinglepopulationmeanwith n )]TJ/F58 9.9626 Tf 10.612 0 Td [(1degreesoffreedomwhere n isthenumberof differences. 5 Thiscontentisavailableonlineat.

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382 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Theteststatistict-scoreis: t = x d )]TJ/F134 9.9626 Tf 10.256 0 Td [(m d s d p n .10 Example10.5:Matchedorpairedsamples Astudywasconductedtoinvestigatetheeffectivenessofhypnotisminreducingpain.Results forrandomlyselectedsubjectsareshowninthetable.The"before"valueismatchedtoan"after" value. Subject: A B C D E F G H Before 6.6 6.5 9.0 10.3 11.3 8.1 6.3 11.6 After 6.8 2.5 7.4 8.5 8.1 6.1 3.4 2.0 Problem Arethesensorymeasurements,onaverage,lowerafterhypnotism?Testata5%signicancelevel. Solution Corresponding"before"and"after"valuesformmatchedpairs. AfterData BeforeData Difference 6.8 6.6 0.2 2.4 6.5 -4.1 7.4 9 -1.6 8.5 10.3 -1.8 8.1 11.3 -3.2 6.1 8.1 -2 3.4 6.3 -2.9 2 11.6 -9.6 Thedata forthetest arethedifferences:{0.2,-4.1,-1.6,-1.8,-3.2,-2,-2.9,-9.6} Thesamplemeanandsamplestandarddeviationofthedifferencesare: x d = )]TJ/F58 9.9626 Tf 8.194 0 Td [(3.13and s d = 2.91Verifythesevalues. Let m d bethepopulationmeanforthedifferences.Weusethesubscript d todenote"differences." RandomVariable: X d =theaveragedifferenceofthesensorymeasurements H o : m d 0.11 Thereisnoimprovement. m d isthepopulationmeanofthedifferences. H a : m d < 0.12 Thereisimprovement.Thescoreshouldbelowerafterhypnotismsothedifferenceoughttobe negativetoindicateimprovement. Distributionforthetest: Thedistributionisastudent-twith df = n )]TJ/F58 9.9626 Tf 10.463 0 Td [(1 = 8 )]TJ/F58 9.9626 Tf 10.463 0 Td [(1 = 7.Use t 7 Noticethatthetestisforasinglepopulationmean.

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383 Calculatethep-valueusingtheStudent-tdistribution: p-value = 0.0095 Graph: Figure10.4 X d istherandomvariableforthedifferences. Thesamplemeanandsamplestandarddeviationofthedifferencesare: x d = )]TJ/F58 9.9626 Tf 8.194 0 Td [(3.13 s d = 2.91 Compare a andthep-value: a = 0.05and p-value = 0.0095. a > p-value Makeadecision: Since a > p-value ,reject H o Thismeansthat m d < 0andthereisimprovement. Conclusion: Ata5%levelofsignicance,fromthesampledata,thereissufcientevidencetoconcludethatthesensorymeasurements,onaverage,arelowerafterhypnotism.Hypnotismappears tobeeffectiveinreducingpain. N OTE :FortheTI-83+andTI-84calculators,youcaneithercalculatethedifferencesaheadoftime after-before andputthedifferencesintoalistoryoucanputthe after dataintoarstlistand the before dataintoasecondlist.Thengotoathirdlistandarrowuptothename.Enter1stlist name-2ndlistname.Thecalculatorwilldothesubtractionandyouwillhavethedifferencesin thethirdlist. N OTE :TI-83+andTI-84:Useyourlistofdifferencesasthedata.Press STAT andarrowoverto TESTS .Press 2:T-Test .Arrowoverto Data andpress ENTER .Arrowdownandenter 0 for m 0 ,the nameofthelistwhereyouputthedata,and 1 forFreq:.Arrowdownto m :andarrowoverto < m 0 .Press ENTER .Arrowdownto Calculate andpress ENTER .Thep-valueis0.0094andthetest statisticis-3.04.Dotheseinstructionsagainexceptarrowto Draw insteadof Calculate .Press ENTER .

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384 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Example10.6 Acollegefootballcoachwasinterestedinwhetherthecollege'sstrengthdevelopmentclassincreasedhisplayers'maximumliftinpoundsonthebenchpressexercise.Heasked4ofhis playerstoparticipateinastudy.Theamountofweighttheycouldeachliftwasrecordedbefore theytookthestrengthdevelopmentclass.Aftercompletingtheclass,theamountofweightthey couldeachliftwasagainmeasured.Thedataareasfollows: Weightinpounds Player1 Player2 Player3 Player4 Amountofweightedliftedpriortotheclass 205 241 338 368 Amountofweightliftedaftertheclass 295 252 330 360 Thecoachwantstoknowifthestrengthdevelopmentclassmakeshisplayersstronger,onaverage. Problem Recordthe differences data.Calculatethedifferencesbysubtractingtheamountofweightlifted priortotheclassfromtheweightliftedaftercompletingtheclass.Thedataforthedifferencesare: {90,11,-8,-8} Usingthedifferencesdata,calculatethesamplemeanandthesamplestandarddeviation. x d = 21.3 s d = 46.7 Usingthedifferencedata,thisbecomesatestofasingle__________llintheblank. Denetherandomvariable: X d = averagedifferenceinthemaximumliftperplayer. Thedistributionforthehypothesistestis t 3 H o : m d 0 H a : m d > 0 Graph: Figure10.5 Calculatethep-value: Thep-valueis0.2150 Decision: Ifthelevelofsignicanceis5%,thedecisionistonotrejectthenullhypothesisbecause a < p-value .

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385 Whatistheconclusion? Example10.7 SeveneighthgradersatKennedyMiddleSchoolmeasuredhowfartheycouldpushtheshot-put withtheirdominantwritinghandandtheirweakernon-writinghand.Theythoughtthatthey couldpushequaldistanceswitheitherhand.Thefollowingdatawascollected. Distance infeet using Student1 Student2 Student3 Student4 Student5 Student6 Student7 Dominant Hand 30 26 34 17 19 26 20 Weaker Hand 28 14 27 18 17 26 16 Problem Conductahypothesistest todeterminewhetherthedifferencesindistancesbetweenthechildren's dominantversusweakerhandsissignicant. HINT :useat-testonthedifferencedata. C HECK :Theteststatisticis2.18andthep-valueis0.0716. Whatisyourconclusion?

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386 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS 10.6SummaryofTypesofHypothesisTests 6 Rule10.1: TwoPopulationMeans 1.Populationsareindependentandpopulationstandarddeviationsareunknown. 2.Populationsareindependentandpopulationstandarddeviationsareknownnotlikely. Rule10.2: MatchedorPairedSamples Rule10.3: TwoPopulationProportions 1.Populationsareindependent. DECISIONANDCONCLUSION Thefollowinghypothesestestsfortwoindependentpopulationmeanshavethesamedecisionandconclusion. H o : m 1 m 2 H a : m 1 < m 2 H o : m 2 m 1 H a : m 2 > m 1 Thefollowinghypothesestestsfortwopopulationproportionshavethesamedecisionandconclusion. H o : p 1 p 2 H a : p 1 < p 2 H o : p 2 p 1 H a : p 2 > p 1 6 Thiscontentisavailableonlineat.

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387 10.7Practice1:HypothesisTestingforTwoProportions 7 10.7.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofhypothesistestingwithtwoproportions. 10.7.2Given Inthe2000Census,2.4percentoftheU.S.populationreportedbeingtwoormoreraces.However,the percentvariestremendouslyfromstatetostate.http://www.census.gov/prod/2001pubs/c2kbr01-6.pdf Supposethattworandomsurveysareconducted.Intherstrandomsurvey,outof1000NorthDakotans, only9peoplereportedbeingoftwoormoreraces.Inthesecondrandomsurvey,outof500Nevadans, 17peoplereportedbeingoftwoormoreraces.Conductahypothesistesttodetermineifthepopulation percentsarethesameforthetwostatesorifthepercentforNevadaisstatisticallyhigherthanforNorth Dakota. 10.7.3HypothesisTesting:TwoAverages Exercise10.7.1 Solutiononp.409. Isthisatestofaveragesorproportions? Exercise10.7.2 Solutiononp.409. Statethenullandalternativehypotheses. a. H 0 : b. H a : Exercise10.7.3 Solutiononp.409. Isthisaright-tailed,left-tailed,ortwo-tailedtest?Howdoyouknow? Exercise10.7.4 WhatistheRandomVariableofinterestforthistest? Exercise10.7.5 Inwords,denetheRandomVariableforthistest. Exercise10.7.6 Solutiononp.410. WhichdistributionNormalorstudent-twouldyouuseforthishypothesistest? Exercise10.7.7 Explainwhyyouchosethedistributionyoudidfortheabovequestion. Exercise10.7.8 Solutiononp.410. Calculatetheteststatistic. Exercise10.7.9 Sketchagraphofthesituation.Labelthehorizontalaxis.Markthehypothesizeddifferenceand thesampledifference.Shadetheareacorrespondingtothe p )]TJ/F58 9.9626 Tf 8.195 0 Td [(value. 7 Thiscontentisavailableonlineat.

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388 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Figure10.6 Exercise10.7.10 Solutiononp.410. Findthe p )]TJ/F58 9.9626 Tf 8.195 0 Td [(value: Exercise10.7.11 Solutiononp.410. Atapre-conceived a = 0 .05 ,whatisyour: a. Decision: b. Reasonforthedecision: c. Conclusionwriteoutinacompletesentence: 10.7.4DiscussionQuestion Exercise10.7.12 DoesitappearthattheproportionofNevadanswhoaretwoormoreracesishigherthanthe proportionofNorthDakotans?Whyorwhynot?

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389 10.8Practice2:HypothesisTestingforTwoAverages 8 10.8.1StudentLearningOutcome Thestudentwillexplorethepropertiesofhypothesistestingwithtwoaverages. 10.8.2Given TheU.S.CenterforDiseaseControlreportsthattheaveragelifeexpectancyforwhitesbornin1900was 47.6yearsandfornonwhitesitwas33.0years.http://www.cdc.gov/nchs/data/dvs/nvsr53_06t12.pdf Supposethatyourandomlysurveydeathrecordsforpeoplebornin1900inacertaincounty.Ofthe124 whites,theaveragelifespanwas45.3yearswithastandarddeviationof12.7years.Ofthe82nonwhites, theaveragelifespanwas34.1yearswithastandarddeviationof15.6years.Conductahypothesistestto seeiftheaveragelifespansinthecountywerethesameforwhitesandnonwhites. 10.8.3HypothesisTesting:TwoAverages Exercise10.8.1 Solutiononp.410. Isthisatestofaveragesorproportions? Exercise10.8.2 Solutiononp.410. Statethenullandalternativehypotheses. a. H 0 : b. H a : Exercise10.8.3 Solutiononp.410. Isthisaright-tailed,left-tailed,ortwo-tailedtest?Howdoyouknow? Exercise10.8.4 Solutiononp.410. WhatistheRandomVariableofinterestforthistest? Exercise10.8.5 Solutiononp.410. Inwords,denetheRandomVariableforthistest. Exercise10.8.6 WhichdistributionNormalorstudent-twouldyouuseforthishypothesistest? Exercise10.8.7 Explainwhyyouchosethedistributionyoudidfortheabovequestion. Exercise10.8.8 Solutiononp.410. Calculatetheteststatistic. Exercise10.8.9 Sketchagraphofthesituation.Labelthehorizontalaxis.Markthehypothesizeddifferenceand thesampledifference.Shadetheareacorrespondingtothe p )]TJ/F58 9.9626 Tf 8.195 0 Td [(value. 8 Thiscontentisavailableonlineat.

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390 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Figure10.7 Exercise10.8.10 Solutiononp.410. Findthe p )]TJ/F58 9.9626 Tf 8.195 0 Td [(value: Exercise10.8.11 Solutiononp.410. Atapre-conceived a = 0 .05 ,whatisyour: a. Decision: b. Reasonforthedecision: c. Conclusionwriteoutinacompletesentence: 10.8.4DiscussionQuestion Exercise10.8.12 Doesitappearthattheaveragesarethesame?Whyorwhynot?

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391 10.9Homework 9 ForquestionsExercise10.9.1-Exercise10.9.10,indicatewhichofthefollowingchoicesbestidentiesthe hypothesistest. A. Independentgroupmeans,populationstandarddeviationsand/orvariancesknown B. Independentgroupmeans,populationstandarddeviationsand/orvariancesunknown C. Matchedorpairedsamples D. Singlemean E. 2proportions F. Singleproportion Exercise10.9.1 Solutiononp.410. Apowderdietistestedon49peopleandaliquiddietistestedon36differentpeople.Thepopulationstandarddeviationsare2poundsand3pounds,respectively.Ofinterestiswhetherthe liquiddietyieldsahigheraverageweightlossthanthepowderdiet. Exercise10.9.2 Twochocolatebarsaretaste-testedonconsumers.Ofinterestiswhetheralargerpercentageof consumerswillpreferonebarovertheother. Exercise10.9.3 Solutiononp.410. TheaveragenumberofEnglishcoursestakeninatwoyeartimeperiodbymaleandfemalecollegestudentsisbelievedtobeaboutthesame.Anexperimentisconductedanddataarecollected from9malesand16females. Exercise10.9.4 Afootballleaguereportedthattheaveragenumberoftouchdownspergamewas5.Astudyis donetodetermineiftheaveragenumberoftouchdownshasdecreased. Exercise10.9.5 Solutiononp.410. AstudyisdonetodetermineifstudentsintheCaliforniastateuniversitysystemtakelongerto graduatethanstudentsenrolledinprivateuniversities.100studentsfromboththeCaliforniastate universitysystemandprivateuniversitiesaresurveyed.Fromyearsofresearch,itisknownthat thepopulationstandarddeviationsare1.5811yearsand1year,respectively. Exercise10.9.6 AccordingtoaYWCARapeCrisisCenternewsletter,75%ofrapevictimsknowtheirattackers.A studyisdonetoverifythis. Exercise10.9.7 Solutiononp.410. Accordingtoarecentstudy,U.S.companieshaveanaveragematernity-leaveofsixweeks. Exercise10.9.8 Arecentdrugsurveyshowedanincreaseinuseofdrugsandalcoholamonglocalhighschool studentsascomparedtothenationalpercent.Supposethatasurveyof100localyouthsand100 nationalyouthsisconductedtoseeifthepercentageofdrugandalcoholuseishigherlocallythan nationally. Exercise10.9.9 Solutiononp.410. AnewSATstudycourseistestedon12individuals.Pre-courseandpost-coursescoresare recorded.OfinterestistheaverageincreaseinSATscores. Exercise10.9.10 UniversityofMichiganresearchersreportedinthe JournaloftheNationalCancerInstitute that quittingsmokingisespeciallybenecialforthoseunderage49.InthisAmericanCancerSociety 9 Thiscontentisavailableonlineat.

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392 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS study,theriskprobabilityofdyingoflungcancerwasaboutthesameasforthosewhohadnever smoked. 10.9.1Foreachproblembelow,llinahypothesistestsolutionsheet. N OTE :Ifyouareusingastudent-tdistributionforahomeworkproblembelow,includingfor paireddata,youmayassumethattheunderlyingpopulationisnormallydistributed.Ingeneral, youmustrstprovethatassumption,though. Exercise10.9.11 Solutiononp.411. Apowderdietistestedon49peopleandaliquiddietistestedon36differentpeople.Ofinterest iswhethertheliquiddietyieldsahigheraverageweightlossthanthepowderdiet.Thepowder dietgrouphadanaverageweightlossof42poundswithastandarddeviationof12pounds.The liquiddietgrouphadanaverageweightlossof45poundswithastandarddeviationof14pounds. Exercise10.9.12 TheaveragenumberofEnglishcoursestakeninatwoyeartimeperiodbymaleandfemalecollegestudentsisbelievedtobeaboutthesame.Anexperimentisconductedanddataarecollected from29malesand16females.Themalestookanaverageof3Englishcourseswithastandard deviationof0.8.Thefemalestookanaverageof4Englishcourseswithastandarddeviationof 1.0.Aretheaveragesstatisticallythesame? Exercise10.9.13 Solutiononp.411. AstudyisdonetodetermineifstudentsintheCaliforniastateuniversitysystemtakelongerto graduatethanstudentsenrolledinprivateuniversities.100studentsfromboththeCaliforniastate universitysystemandprivateuniversitiesaresurveyed.Supposethatfromyearsofresearch,it isknownthatthepopulationstandarddeviationsare1.5811yearsand1year,respectively.The followingdataarecollected.TheCaliforniastateuniversitysystemstudentstookonaverage4.5 yearswithastandarddeviationof0.8.Theprivateuniversitystudentstookonaverage4.1years withastandarddeviationof0.3. Exercise10.9.14 AnewSATstudycourseistestedon12individuals.Pre-courseandpost-coursescoresare recorded.OfinterestistheaverageincreaseinSATscores.Thefollowingdataiscollected: Pre-coursescore Post-coursescore 1200 1300 960 920 1010 1100 840 880 1100 1070 1250 1320 860 860 1330 1370 790 770 990 1040 1110 1200 740 850

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393 Exercise10.9.15 Solutiononp.411. Arecentdrugsurveyshowedanincreaseinuseofdrugsandalcoholamonglocalhighschool seniorsascomparedtothenationalpercent.Supposethatasurveyof100localseniorsand100 nationalseniorsisconductedtoseeifthepercentageofdrugandalcoholuseishigherlocallythan nationally.Locally,65seniorsreportedusingdrugsoralcoholwithinthepastmonth,while60 nationalseniorsreportedusingthem. Exercise10.9.16 Astudentatafour-yearcollegeclaimsthataverageenrollmentatfouryearcollegesishigherthan attwoyearcollegesintheUnitedStates.Twosurveysareconducted.Ofthe35twoyearcolleges surveyed,theaverageenrollmentwas5068withastandarddeviationof4777.Ofthe35four-year collegessurveyed,theaverageenrollmentwas5466withastandarddeviationof8191.Source: MicrosoftBookshelf Exercise10.9.17 Solutiononp.411. AstudywasconductedbytheU.S.Armytoseeifapplyingantiperspiranttosoldiers'feetfora fewdaysbeforeamajorhikewouldhelpcutdownonthenumberofblisterssoldiershadontheir feet.Intheexperiment,forthreenightsbeforetheywentona13-milehike,agroupof328West Pointcadetsputanalcohol-basedantiperspirantontheirfeet.Acontrolgroupof339soldiers putonasimilar,butinactive,preparationontheirfeet.Onthedayofthehike,thetemperature reached83 F.Attheendofthehike,21%ofthesoldierswhohadusedtheantiperspirantand48% ofthecontrolgrouphaddevelopedfootblisters.Conductahypothesistesttoseeifthepercent ofsoldiersusingtheantiperspirantwassignicantlylowerthanthecontrolgroup.Source:U.S. Armystudyreportedin JournaloftheAmericanAcademyofDermatologists Exercise10.9.18 Weareinterestedinwhetherthepercentsoffemalesuicidevictimsforages15to24arethesame forthewhiteandtheblackracesintheUnitedStates.Werandomlypickoneyear,1992,tocompare theraces.ThenumberofsuicidesestimatedintheUnitedStatesin1992forwhitefemalesis4930. 580wereaged15to24.Theestimateforblackfemalesis330.40wereaged15to24.Wewilllet femalesuicidevictimsbeourpopulation.Source :theNationalCenterforHealthStatistics,U.S. Dept.ofHealthandHumanServices Exercise10.9.19 Solutiononp.411. AtRachel's11thbirthdayparty,8girlsweretimedtoseehowlonginsecondstheycouldhold theirbreathinarelaxedposition.Afteratwo-minuterest,theytimedthemselveswhilejumping. Thegirlsthoughtthatthejumpingwouldnotaffecttheirtimes,onaverage.Testtheirhypothesis. Relaxedtimeseconds Jumpingtimeseconds 26 21 47 40 30 28 22 21 23 25 45 43 37 35 29 32 Exercise10.9.20 ElizabethMjelde,anarthistoryprofessor,wasinterestedinwhetherthevaluefromtheGolden Ratioformula, larger + smallerdimension largerdimension wasthesameintheWhitneyExhibitforworksfrom1900

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394 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS 1919asforworksfrom19201942.37earlyworksweresampled.Theyaveraged1.74with astandarddeviationof0.11.65ofthelaterworksweresampled.Theyaveraged1.746witha standarddeviationof0.1064.DoyouthinkthatthereisasignicantdifferenceintheGolden Ratiocalculation?Source: datafromWhitneyExhibitonloantoSanJoseMuseumofArt Exercise10.9.21 Solutiononp.411. Oneofthequestionsinastudyofmaritalsatisfactionofdualcareercoupleswastoratethestatement,I'mpleasedwiththewaywedividetheresponsibilitiesforchildcare.Theratingswent from1stronglyagreeto5stronglydisagree.Belowaretenofthepairedresponsesforhusbands andwives.Conductahypothesistesttoseeiftheaveragedifferenceinthehusband'sversusthe wife'ssatisfactionlevelisnegativemeaningthat,withinthepartnership,thehusbandishappier thanthewife. Wife'sscore 2 2 3 3 4 2 1 1 2 4 Husband'sscore 2 2 1 3 2 1 1 1 2 4 Exercise10.9.22 Tenindividualswentonalowfatdietfor12weekstolowertheircholesterol.Evaluatethedata below.Doyouthinkthattheircholesterollevelsweresignicantlylowered? Startingcholesterollevel Endingcholesterollevel 140 140 220 230 110 120 240 220 200 190 180 150 190 200 360 300 280 300 260 240 Exercise10.9.23 Solutiononp.411. Averageentrylevelsalariesforcollegegraduateswithmechanicalengineeringdegreesand electricalengineeringdegreesarebelievedtobeapproximatelythesame.Source: http:// www.graduatingengineer.com 10 .Arecruitingofcethinksthattheaveragemechanicalengineeringsalaryisactuallylowerthantheaverageelectricalengineeringsalary.Therecruiting ofcerandomlysurveys50entrylevelmechanicalengineersand60entrylevelelectricalengineers.Theiraveragesalarieswere\$46,100and\$46,700,respectively.Theirstandarddeviations were\$3450and\$4210,respectively.Conductahypothesistesttodetermineifyouagreethatthe averageentrylevelmechanicalengineeringsalaryislowerthantheaverageentrylevelelectrical engineeringsalary. Exercise10.9.24 Arecentyearwasrandomlypickedfrom1985tothepresent.Inthatyear,therewere2051Hispanic studentsatCabrilloCollegeoutofatotalof12,328students.AtLakeTahoeCollege,therewere 10 http://www.graduatingengineer.com/

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395 321Hispanicstudentsoutofatotalof2441students.Ingeneral,doyouthinkthatthepercent ofHispanicstudentsatthetwocollegesisbasicallythesameordifferent?Source: Chancellor's Ofce,CaliforniaCommunityColleges,November1994 Exercise10.9.25 Solutiononp.411. Eightrunnerswereconvincedthattheaveragedifferenceintheirindividualtimesforrunningone mileversusracewalkingonemilewasatmost2minutes.Belowaretheirtimes.Doyouagree thattheaveragedifferenceisatmost2minutes? Runningtimeminutes Racewalkingtimeminutes 5.1 7.3 5.6 9.2 6.2 10.4 4.8 6.9 7.1 8.9 4.2 9.5 6.1 9.4 4.4 7.9 Exercise10.9.26 Marketingcompanieshavecollecteddataimplyingthatteenagegirlsusemoreringtonesontheir cellularphonesthanteenageboysdo.Inoneparticularstudyof40randomlychosenteenagegirls andboysofeachwithcellularphones,theaveragenumberofringtonesforthegirlswas3.2 withastandarddeviationof1.5.Theaveragefortheboyswas1.7withastandarddeviationof 0.8.Conductahypothesistesttodetermineiftheaveragesareapproximatelythesameorifthe girls'averageishigherthantheboys'average. Exercise10.9.27 Solutiononp.411. Whileherhusbandspent2hourspickingoutnewspeakers,astatisticiandecidedtodetermine whetherthepercentofmenwhoenjoyshoppingforelectronicequipmentishigherthanthepercentofwomenwhoenjoyshoppingforelectronicequipment.ThepopulationwasSaturdayafternoonshoppers.Outof67men,24saidtheyenjoyedtheactivity.8ofthe24womensurveyed claimedtoenjoytheactivity.Interprettheresultsofthesurvey. Exercise10.9.28 Weareinterestedinwhetherchildren'seducationalcomputersoftwarecostsless,onaverage,than children'sentertainmentsoftware.36educationalsoftwaretitleswererandomlypickedfroma catalog.Theaveragecostwas\$31.14withastandarddeviationof\$4.69.35entertainmentsoftware titleswererandomlypickedfromthesamecatalog.Theaveragecostwas\$33.86withastandard deviationof\$10.87.Decidewhetherchildren'seducationalsoftwarecostsless,onaverage,than children'sentertainmentsoftware.Source: EducationalResources ,Decembercatalog Exercise10.9.29 Solutiononp.412. Parentsofteenageboysoftencomplainthatautoinsurancecostsmore,onaverage,forteenage boysthanforteenagegirls.Agroupofconcernedparentsexaminesarandomsampleofinsurance bills.Theaverageannualcostfor36teenageboyswas\$679.For23teenagegirls,itwas\$559.From pastyears,itisknownthatthepopulationstandarddeviationforeachgroupis\$180.Determine whetherornotyoubelievethattheaveragecostforautoinsuranceforteenageboysisgreaterthan thatforteenagegirls.

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396 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Exercise10.9.30 Agroupoftransferboundstudentswonderediftheywillspendthesameaverageamountontexts andsupplieseachyearattheirfour-yearuniversityastheyhaveattheircommunitycollege.They conductedarandomsurveyof54studentsattheircommunitycollegeand66studentsattheir localfour-yearuniversity.Thesamplemeanswere\$947and\$1011,respectively.Thepopulation standarddeviationsareknowntobe\$254and\$87,respectively.Conductahypothesistestto determineiftheaveragesarestatisticallythesame. Exercise10.9.31 Solutiononp.412. JoanNguyenrecentlyclaimedthattheproportionofcollegeagemaleswithatleastonepierced earisashighastheproportionofcollegeagefemales.Sheconductedasurveyinherclasses.Out of107males,20hadatleastonepiercedear.Outof92females,47hadatleastonepiercedear.Do youbelievethattheproportionofmaleshasreachedtheproportionoffemales? Exercise10.9.32 Somemanufacturersclaimthatnon-hybridsedancarshavealoweraveragemilespergallon mpgthanhybridones.Supposethatconsumerstest21hybridsedansandgetanaverage31 mpgwithastandarddeviationof7mpg.Thirty-onenon-hybridsedansaverage22mpgwitha standarddeviationof4mpg.Supposethatthepopulationstandarddeviationsareknowntobe6 and3,respectively.Conductahypothesistesttothemanufacturersclaim. QuestionsExercise10.9.33Exercise10.9.37refertotheTerriVogel'sdatasetseeTableofContents. Exercise10.9.33 Solutiononp.412. UsingthedatafromLap1only,conductahypothesistesttodetermineiftheaveragetimefor completingalapinracesisthesameasitisinpractices. Exercise10.9.34 RepeatthetestinExercise10.9.33,butuseLap5datathistime. Exercise10.9.35 Solutiononp.412. RepeatthetestinExercise10.9.33,butthistimecombinethedatafromLaps1and5. Exercise10.9.36 In23completesentences,explainindetailhowyoumightuseTerriVogel'sdatatoanswerthe followingquestion.DoesTerriVogeldrivefasterinracesthanshedoesinpractices? Exercise10.9.37 Solutiononp.412. IstheproportionofracelapsTerricompletesslowerthan130secondslessthantheproportionof practicelapsshecompletesslowerthan135seconds? Exercise10.9.38 "ToBreakfastorNottoBreakfast?"byRichardAyore IntheAmericansociety,birthdaysareoneofthosedaysthateveryonelooksforwardto.Peopleof differentagesandpeergroupsgathertomarkthe18th,20th,...birthdays.Duringthistime,one looksbacktoseewhatheorshehadachievedforthepastyear,andalsofocusesaheadformore tocome. If,byanychance,Iaminvitedtooneoftheseparties,myexperienceisalwaysdifferent.Instead ofdancingaroundwithmyfriendswhilethemusicisbooming,Igetcarriedawaybymemories ofmyfamilybackhomeinKenya.IrememberthegoodtimesIhadwithmybrothersandsister whilewedidourdailyroutine. Everymorning,Irememberwewenttotheshambagardentoweedourcrops.Irememberone dayarguingwithmybrotherastowhyhealwaysremainedbehindjusttojoinusanhourlater.In hisdefense,hesaidthathepreferredwaitingforbreakfastbeforehecametoweed.Hesaid,This iswhyIalwaysworkmorehoursthanyouguys!

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397 Andso,toprovehiswrongorright,wedecidedtogiveitatry.Onedaywewenttoworkasusual withoutbreakfast,andrecordedthetimewecouldworkbeforegettingtiredandstopping.On thenextday,weallatebreakfastbeforegoingtowork.Werecordedhowlongweworkedagain beforegettingtiredandstopping.Ofinterestwasouraverageincreaseinworktime.Thoughnot sure,mybrotherinsistedthatitismorethantwohours.Usingthedatabelow,solveourproblem. Workhourswithbreakfast Workhourswithoutbreakfast 8 6 7 5 9 5 5 4 9 7 8 7 10 7 7 5 6 6 9 5 10.9.2Trythesemultiplechoicequestions. ForquestionsExercise10.9.39Exercise10.9.40,usethefollowinginformation. AnewAIDSpreventiondrugswastriedonagroupof224HIVpositivepatients.Forty-vepatients developedAIDSafterfouryears.Inacontrolgroupof224HIVpositivepatients,68developedAIDSafter fouryears.Wewanttotestwhetherthemethodoftreatmentreducestheproportionofpatientsthatdevelop AIDSafterfouryearsoriftheproportionsofthetreatedgroupandtheuntreatedgroupstaythesame. Letthesubscript t =treatedpatientand ut =untreatedpatient. Exercise10.9.39 Solutiononp.412. Theappropriatehypothesesare: A. H o : p t < p ut and H a : p t p ut B. H o : p t p ut and H a : p t > p ut C. H o : p t = p ut and H a : p t 6 = p ut D. H o : p t = p ut and H a : p t < p ut Exercise10.9.40 Solutiononp.412. Ifthe p -valueis0.0062whatistheconclusionuse a = 5? A. Themethodhasnoeffect. B. ThemethodreducestheproportionofHIVpositivepatientsthatdevelopAIDSafterfour years. C. ThemethodincreasestheproportionofHIVpositivepatientsthatdevelopAIDSafter fouryears. D. Thetestdoesnotdeterminewhetherthemethodhelpsordoesnothelp.

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398 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Exercise10.9.41 Solutiononp.412. LesleyE.Taninvestigatedtherelationshipbetweenleft-handednessandright-handednessand motorcompetenceinpreschoolchildren.Randomsamplesof41left-handersand41right-handers weregivenseveraltestsofmotorskillstodetermineifthereisevidenceofadifferencebetweenthe childrenbasedonthisexperiment.Theexperimentproducedthemeansandstandarddeviations shownbelow.Determinetheappropriatetestandbestdistributiontouseforthattest. Left-handed Right-handed Samplesize 41 41 Samplemean 97.5 98.1 Samplestandarddeviation 17.5 19.2 A. Twoindependentmeans,normaldistribution B. Twoindependentmeans,student-tdistribution C. Matchedorpairedsamples,student-tdistribution D. Twopopulationproportions,normaldistribution ForquestionsExercise10.9.42Exercise10.9.43,usethefollowinginformation. Anexperimentisconductedtoshowthatbloodpressurecanbeconsciouslyreducedinpeopletrainedina biofeedbackexerciseprogram.Sixsubjectswererandomlyselectedandthebloodpressuremeasurementswererecordedbeforeandafterthetraining.Thedifferencebetweenbloodpressureswascalculated after )]TJ/F132 9.9626 Tf 10.131 0 Td [(before producingthefollowingresults: x d = )]TJ/F132 9.9626 Tf 8.194 0 Td [(10. 2 s d = 8 4.Usingthedata,testthehypothesis thatthebloodpressurehasdecreasedafterthetraining, Exercise10.9.42 Solutiononp.412. Thedistributionforthetestis A. t 5 B. t 6 C. N )]TJ/F132 9.9626 Tf 8.194 0 Td [(10. 2,8 4 D. N )]TJ/F132 9.9626 Tf 8.195 0 Td [(10. 2, 8 4 p 6 Exercise10.9.43 Solutiononp.412. If a = 0 .05 ,the p -valueandtheconclusionare A. 0.0014;thebloodpressuredecreasedafterthetraining B. 0.0014;thebloodpressureincreasedafterthetraining C. 0.0155;thebloodpressuredecreasedafterthetraining D. 0.0155;thebloodpressureincreasedafterthetraining ForquestionsExercise10.9.44Exercise10.9.45,usethefollowinginformation. TheEasternandWesternMajorLeagueSoccerconferenceshaveanewReserveDivisionthatallowsnew playerstodeveloptheirskills.AsofMay25,2005,theReserveDivisionteamsscoredthefollowingnumber ofgoalsfor2005.

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399 Western Eastern LosAngeles9 D.C.United9 FCDallas3 Chicago8 ChivasUSA4 Columbus7 RealSaltLake3 NewEngland6 Colorado4 MetroStars5 SanJose4 KansasCity3 ConductahypothesistesttodetermineiftheWesternReserveDivisionteamsscore,onaverage,fewergoals thantheEasternReserveDivisionteams.Subscripts: 1 WesternReserveDivision W ; 2 EasternReserve Division E Exercise10.9.44 Solutiononp.412. The exact distributionforthehypothesistestis: A. Thenormaldistribution. B. Thestudent-tdistribution. C. Theuniformdistribution. D. Theexponentialdistribution. Exercise10.9.45 Solutiononp.412. Ifthelevelofsignicanceis0.05,theconclusionis: A. The W Divisionteamsscore,onaverage,fewergoalsthanthe E teams. B. The W Divisionteamsscore,onaverage,moregoalsthanthe E teams. C. The W teamsscore,onaverage,aboutthesamenumberofgoalsasthe E teamsscore. D. Unabletodetermine. QuestionsExercise10.9.46Exercise10.9.48refertothefollowing. AresearcherisinterestedindeterminingifacertaindrugvaccinepreventsWestNiledisease.Thevaccine withthedrugisadministeredto36peopleandanother36peoplearegivenavaccinethatdoesnotcontain thedrug.Ofthegroupthatgetsthevaccinewiththedrug,onegetsWestNiledisease.Ofthegroupthat getsthevaccinewithoutthedrug,three3getWestNiledisease.Conductahypothesistesttodetermine iftheproportionofpeoplethatgetthevaccinewithoutthedrugandgetWestNilediseaseismorethanthe proportionofpeoplethatgetthevaccinewiththedrugandgetWestNiledisease. Drugsubscript:groupwhogetthevaccinewiththedrug. NoDrugsubscript:groupwhogetthevaccinewithoutthedrug Exercise10.9.46 Solutiononp.412. Thisisatestof: A. atestoftwoproportions B. atestoftwoindependentmeans C. atestofasinglemean D. atestofmatchedpairs. Exercise10.9.47 Solutiononp.412. Anappropriatenullhypothesisis:

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400 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS A. p NoDrug p Drug B. p NoDrug p Drug C. m NoDrug m Drug D. p NoDrug > p Drug Exercise10.9.48 Solutiononp.413. The p -valueis0.1517.Ata1%levelofsignicance,theappropriateconclusionis A. theproportionofpeoplethatgetthevaccinewithoutthedrugandgetWestNiledisease islessthantheproportionofpeoplethatgetthevaccinewiththedrugandgetWest Niledisease. B. theproportionofpeoplethatgetthevaccinewithoutthedrugandgetWestNiledisease ismorethantheproportionofpeoplethatgetthevaccinewiththedrugandgetWest Niledisease. C. theproportionofpeoplethatgetthevaccinewithoutthedrugandgetWestNiledisease ismorethanorequaltotheproportionofpeoplethatgetthevaccinewiththedrugand getWestNiledisease. D. theproportionofpeoplethatgetthevaccinewithoutthedrugandgetWestNiledisease isnomorethantheproportionofpeoplethatgetthevaccinewiththedrugandget WestNiledisease. QuestionsExercise10.9.49andExercise10.9.50refertothefollowing: Agolfinstructorisinterestedindeterminingifhernewtechniqueforimprovingplayers'golfscoresis effective.Shetakesfournewstudents.Sherecordstheir18-holesscoresbeforelearningthetechnique andthenafterhavingtakenherclass.Sheconductsahypothesistest.Thedataareasfollows. Player1 Player2 Player3 Player4 Averagescorebeforeclass 83 78 93 87 Averagescoreafterclass 80 80 86 86 Exercise10.9.49 Solutiononp.413. Thisisatestof: A. atestoftwoindependentmeans B. atestoftwoproportions C. atestofasingleproportion D. atestofmatchedpairs. Exercise10.9.50 Solutiononp.413. Thecorrectdecisionis: A. Reject H o B. Donotreject H o C. Thetestisinconclusive QuestionsExercise10.9.51andExercise10.9.52refertothefollowing: Supposeastatisticsinstructorbelievesthatthereisnosignicantdifferencebetweentheaverageclass scoresofhertwoclassesonExam2.Theaverageandstandarddeviationforher8:30classof35students

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401 were75.86and16.91.Theaverageandstandarddeviationforher11:30classof37studentswere75.41and 19.73.:30subscriptreferstothe8:30class.:30subscriptreferstothe11:30class. Exercise10.9.51 Solutiononp.413. Anappropriatealternatehypothesisforthehypothesistestis: A. m 8: 30 > m 11 : 30 B. m 8: 30 < m 11 : 30 C. m 8: 30 = m 11 : 30 D. m 8: 30 6 = m 11 : 30 Exercise10.9.52 Solutiononp.413. Aconcludingstatementis: A. The11:30classaverageisbetterthanthe8:30classaverage. B. The8:30classaverageisbetterthanthe11:30classaverage. C. Thereisnosignicantdifferencebetweentheaveragesofthetwoclasses. D. Thereisasignicantdifferencebetweentheaveragesofthetwoclasses. 10.10Review 11 Thenextthreequestionsrefertothefollowinginformation: InasurveyatKirkwoodSkiResortthefollowinginformationwasrecorded: SportParticipationbyAge 010 11-20 21-40 40+ Ski 10 12 30 8 Snowboard 6 17 12 5 Supposethatonepersonfromoftheabovewasrandomlyselected. Exercise10.10.1 Solutiononp.413. Findtheprobabilitythatthepersonwasaskierorwasage1120. Exercise10.10.2 Solutiononp.413. Findtheprobabilitythatthepersonwasasnowboardergivenhe/shewasage2140. Exercise10.10.3 Solutiononp.413. Explainwhichofthefollowingaretrueandwhicharefalse. a. SportandAgeareindependentevents. b. Skiandage1120aremutuallyexclusiveevents. c. P Skiandage21 )]TJ/F132 9.9626 Tf 10.132 0 Td [(40 < P Ski j age21 )]TJ/F132 9.9626 Tf 10.132 0 Td [(40 d. P Snowboardorage 0 )]TJ/F132 9.9626 Tf 10.132 0 Td [(10 < P Snowboard j age 0 )]TJ/F132 9.9626 Tf 10.131 0 Td [(10 11 Thiscontentisavailableonlineat.

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402 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS Exercise10.10.4 Solutiononp.413. Theaveragelengthoftimeapersonwithabrokenlegwearsacastisapproximately6weeks. Thestandarddeviationisabout3weeks.Thirtypeoplewhohadrecentlyhealedfrombroken legswereinterviewed.Statethedistributionthatmostaccuratelyreectstotaltimetohealforthe thirtypeople. Exercise10.10.5 Solutiononp.413. Thedistributionfor X isUniform.Whatcanwesayforcertainaboutthedistributionfor X when n = 1? A. Thedistributionfor X isstillUniformwiththesamemeanandstandarddev.asthe distributionfor X B. Thedistributionfor X isNormalwiththedifferentmeanandadifferentstandarddeviationasthedistributionfor X C. Thedistributionfor X isNormalwiththesamemeanbutalargerstandarddeviationthan thedistributionfor X D. Thedistributionfor X isNormalwiththesamemeanbutasmallerstandarddeviation thanthedistributionfor X Exercise10.10.6 Solutiononp.413. Thedistributionfor X isuniform.Whatcanwesayforcertainaboutthedistributionfor X when n = 50? A. Thedistributionfor X isstilluniformwiththesamemeanandstandarddeviationas thedistributionfor X B. Thedistributionfor X isNormalwiththesamemeanbutalargerstandarddeviation asthedistributionfor X C. Thedistributionfor X isNormalwithalargermeanandalargerstandarddeviation thanthedistributionfor X D. Thedistributionfor X isNormalwiththesamemeanbutasmallerstandarddeviation thanthedistributionfor X Thenextthreequestionsrefertothefollowinginformation: Agroupofstudentsmeasuredthelengthsofallthecarrotsinave-poundbagofbabycarrots.They calculatedtheaveragelengthofbabycarrotstobe2.0incheswithastandarddeviationof0.25inches. Supposewerandomlysurvey16ve-poundbagsofbabycarrots. Exercise10.10.7 Solutiononp.413. Statetheapproximatedistributionfor X ,thedistributionfortheaveragelengthsofbabycarrots in16ve-poundbags. X~ Exercise10.10.8 Explainwhywecannotndtheprobabilitythatoneindividualrandomlychosencarrotisgreater than2.25inches. Exercise10.10.9 Solutiononp.413. Findtheprobabilitythat X isbetween2and2.25inches. Thenextthreequestionsrefertothefollowinginformation: Atthebeginningoftheterm,theamountoftimeastudentwaitsinlineatthecampusstoreisnormally distributedwithameanof5minutesandastandarddeviationof2minutes. Exercise10.10.10 Solutiononp.413. Findthe90thpercentileofwaitingtimeinminutes.

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403 Exercise10.10.11 Solutiononp.413. Findthemedianwaitingtimeforonestudent. Exercise10.10.12 Solutiononp.413. Findtheprobabilitythattheaveragewaitingtimefor40studentsisatleast4.5minutes.

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404 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS 10.11Lab:HypothesisTestingforTwoMeansandTwoProportions 12 ClassTime: Names: 10.11.1StudentLearningOutcomes: Thestudentwillselecttheappropriatedistributionstouseineachcase. Thestudentwillconducthypothesistestsandinterprettheresults. 10.11.2Supplies: Thebusinesssectionfromtwoconsecutivedays'newspapers 3smallpackagesofM&Ms 5smallpackagesofReesesPieces 10.11.3IncreasingStocksSurvey Lookatyesterday'snewspaperbusinesssection.Conductahypothesistesttodetermineiftheproportion ofNewYorkStockExchangeNYSEstocksthatincreasedisgreaterthantheproportionofNASDAQstocks thatincreased.Asrandomlyaspossible,choose40NYSEstocksand32NASDAQstocksandcompletethe followingstatements. 1. H o 2. H a 3.Inwords,denetheRandomVariable.____________= 4.Thedistributiontouseforthetestis: 5.Calculatetheteststatisticusingyourdata. 12 Thiscontentisavailableonlineat.

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405 6.Drawagraphandlabelitappropriately.Shadetheactuallevelofsignicance. a. Graph: Figure10.8 b. Calculatethep-value: 7.Doyourejectornotrejectthenullhypothesis?Why? 8.Writeaclearconclusionusingacompletesentence. 10.11.4DecreasingStocksSurvey Randomlypick8stocksfromthenewspaper.Usingtwoconsecutivedays'businesssections,testwhether thestockswentdown,onaverage,forthesecondday. 1. H o 2. H a 3.Inwords,denetheRandomVariable.____________= 4.Thedistributiontouseforthetestis: 5.Calculatetheteststatisticusingyourdata.

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406 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS 6.Drawagraphandlabelitappropriately.Shadetheactuallevelofsignicance. a. Graph: Figure10.9 b. Calculatethep-value: 7.Doyourejectornotrejectthenullhypothesis?Why? 8.Writeaclearconclusionusingacompletesentence. 10.11.5CandySurvey BuythreesmallpackagesofM&Msand5smallpackagesofReesesPiecessamenetweightastheM&Ms. Testwhetherornottheaveragenumberofcandypiecesperpackageisthesameforthetwobrands. 1. H o : 2. H a : 3.Inwords,denetherandomvariable.__________= 4.Whatdistributionshouldbeusedforthistest? 5.Calculatetheteststatisticusingyourdata.

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407 6.Drawagraphandlabelitappropriately.Shadetheactuallevelofsignicance. a. Graph: Figure10.10 b. Calculatethep-value: 7.Doyourejectornotrejectthenullhypothesis?Why? 8.Writeaclearconclusionusingacompletesentence. 9.Explainhowyourresultsmightdifferif10peoplepooledtheirrawdatatogetherandthetestwere redone. 10.Wouldthisnewtestortheoriginalonebemoreaccurate?Explainyouranswerincompletesentences. 10.11.6ShoeSurvey Testwhetherwomenhave,onaverage,morepairsofshoesthanmen.Includeallformsofsneakers,shoes, sandals,andboots.Useyourclassasthesample. 1. H o 2. H a 3.Inwords,denetheRandomVariable.____________= 4.Thedistributiontouseforthetestis: 5.Calculatetheteststatisticusingyourdata.

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408 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS 6.Drawagraphandlabelitappropriately.Shadetheactuallevelofsignicance. a. Graph: Figure10.11 b. Calculatethep-value: 7.Doyourejectornotrejectthenullhypothesis?Why? 8.Writeaclearconclusionusingacompletesentence.

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409 SolutionstoExercisesinChapter10 Example10.2,Problem1p.377 twomeans Example10.2,Problem2p.377 unknown Example10.2,Problem3p.377 Student-t Example10.2,Problem4p.377 X A )]TJETq1 0 0 1 100.708 554.677 cm[]0 d 0 J 0.398 w 0 0 m 12.957 0 l SQBT/F132 9.9626 Tf 100.982 546.393 Td [(X B Example10.2,Problem5p.377 H o : m A m B H a : m A > m B Example10.2,Problem6p.377 right Example10.2,Problem7p.377 0.1928 Example10.2,Problem8p.377 Donotreject. Example10.4p.380 Theproblemasksforadifferenceinpercentages. Example10.6p.384 means;Ata5%levelofsignicance,fromthesampledata,thereisnotsufcientevidencetoconcludethat thestrengthdevelopmentclasshelpedtomaketheplayersstronger,onaverage. Example10.7p.385 H 0 : m d equals0; H a : m d doesnotequal0;Donotrejectthenull;Ata5%signicancelevel,fromthesample data,thereisnotsufcientevidencetoconcludethatthedifferencesindistancesbetweenthechildren's dominantversusweakerhandsissignicantthereisnotsufcientevidencetoshowthatthechildren couldpushtheshot-putfurtherwiththeirdominanthand.Alphaandthep-valueareclosesothetestis notstrong. SolutionstoPractice1:HypothesisTestingforTwoProportions SolutiontoExercise10.7.1p.387 Proportions SolutiontoExercise10.7.2p.387 a. H 0 : PN = PND a. H a : PN > PND SolutiontoExercise10.7.3p.387 right-tailed

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410 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS SolutiontoExercise10.7.6p.387 Normal SolutiontoExercise10.7.8p.387 3.50 SolutiontoExercise10.7.10p.388 0.0002 SolutiontoExercise10.7.11p.388 a. Rejectthenullhypothesis SolutionstoPractice2:HypothesisTestingforTwoAverages SolutiontoExercise10.8.1p.389 Averages SolutiontoExercise10.8.2p.389 a. H 0 : m W = m NW b. H a : m W 6 = m NW SolutiontoExercise10.8.3p.389 two-tailed SolutiontoExercise10.8.4p.389 X W )]TJETq1 0 0 1 102.526 404.155 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 102.8 395.871 Td [(X NW SolutiontoExercise10.8.5p.389 student-t SolutiontoExercise10.8.8p.389 5.42 SolutiontoExercise10.8.10p.390 0.0000 SolutiontoExercise10.8.11p.390 a. Rejectthenullhypothesis SolutionstoHomework SolutiontoExercise10.9.1p.391 A SolutiontoExercise10.9.3p.391 B SolutiontoExercise10.9.5p.391 A SolutiontoExercise10.9.7p.391 D SolutiontoExercise10.9.9p.391 C

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411 SolutiontoExercise10.9.11p.392 d. t 68.44 e. -1.04 f. 0.1519 h. Dec:donotrejectnull SolutiontoExercise10.9.13p.392 StandardNormal e. z = 2 .14 f. 0.0163 h. Decision:Rejectnullwhen a = 0 .05 ;Donotrejectnullwhen a = 0 .01 SolutiontoExercise10.9.15p.393 e. 0.73 f. 0.2326 h. Decision:Donotrejectnull SolutiontoExercise10.9.17p.393 e. -7.33 f. 0 h. Decision:Rejectnull SolutiontoExercise10.9.19p.393 d. t 7 e. -1.51 f. 0.1755 h. Decision:Donotrejectnull SolutiontoExercise10.9.21p.394 d. t 9 e. t = )]TJ/F58 9.9626 Tf 8.194 0 Td [(1 .86 f. 0.0479 h. Decision:Rejectnull,butrunanothertest SolutiontoExercise10.9.23p.394 d. t 108 e. t = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0 .82 f. 0.2066 h. Decision:Donotrejectnull SolutiontoExercise10.9.25p.395 d. t 7 e. t = 2 .9850 f. 0.0103 h. Decision:Rejectnull;Theaveragedifferenceismorethan2minutes. SolutiontoExercise10.9.27p.395 e. 0.22 f. 0.4133

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412 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS h. Decision:Donotrejectnull SolutiontoExercise10.9.29p.395 e. z = 2 .50 f. 0.0063 h. Decision:Rejectnull SolutiontoExercise10.9.31p.396 e. -4.82 f. 0 h. Decision:Rejectnull SolutiontoExercise10.9.33p.396 d. t 20.32 e. -4.70 f. 0.0001 h. Decision:Rejectnull SolutiontoExercise10.9.35p.396 d. t 40.94 e. -5.08 f. 0 h. Decision:Rejectnull SolutiontoExercise10.9.37p.396 e. -0.95 f. 0.1705 h. Decision:Donotrejectnull SolutiontoExercise10.9.39p.397 D SolutiontoExercise10.9.40p.397 B SolutiontoExercise10.9.41p.398 B SolutiontoExercise10.9.42p.398 A SolutiontoExercise10.9.43p.398 C SolutiontoExercise10.9.44p.399 B SolutiontoExercise10.9.45p.399 C SolutiontoExercise10.9.46p.399 A SolutiontoExercise10.9.47p.399 A

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413 SolutiontoExercise10.9.48p.400 D SolutiontoExercise10.9.49p.400 D SolutiontoExercise10.9.50p.400 B SolutiontoExercise10.9.51p.401 D SolutiontoExercise10.9.52p.401 C SolutionstoReview SolutiontoExercise10.10.1p.401 77 100 SolutiontoExercise10.10.2p.401 12 42 SolutiontoExercise10.10.3p.401 a. False b. False c. True d. False SolutiontoExercise10.10.4p.402 N 180 16.43 SolutiontoExercise10.10.5p.402 A SolutiontoExercise10.10.6p.402 C SolutiontoExercise10.10.7p.402 N 2 .25 p 16 SolutiontoExercise10.10.9p.402 0.5000 SolutiontoExercise10.10.10p.402 7.6 SolutiontoExercise10.10.11p.403 5 SolutiontoExercise10.10.12p.403 0.9431

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414 CHAPTER10.HYPOTHESISTESTING:TWOMEANS,PAIREDDATA,TWO PROPORTIONS

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Chapter11 TheChi-SquareDistribution 11.1TheChi-SquareDistribution 1 11.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Interpretthechi-squareprobabilitydistributionasthesamplesizechanges. Conductandinterpretchi-squaregoodness-of-thypothesistests. Conductandinterpretchi-squaretestofindependencehypothesis.tests. Conductandinterpretchi-squaresinglevariancehypothesistestsoptional. 11.1.2Introduction Haveyoueverwonderediflotterynumberswereevenlydistributedorifsomenumbersoccurredwitha greaterfrequency?Howaboutifthetypesofmoviespeoplepreferredweredifferentacrossdifferentage groups?Whataboutifacoffeemachinewasdispensingapproximatelythesameamountofcoffeeeach time?Youcouldanswerthesequestionsbyconductingahypothesistest. Youwillnowstudyanewdistribution,onethatisusedtodeterminetheanswerstotheaboveexamples. ThisdistributioniscalledtheChi-squaredistribution. Inthischapter,youwilllearnthethreemajorapplicationsoftheChi-squaredistribution: Thegoodness-of-ttest,whichdeterminesifdatatsaparticulardistribution,suchaswiththelottery example Thetestofindependence,whichdeterminesifeventsareindependent,suchaswiththemovieexample Thetestofasinglevariance,whichtestsvariability,suchaswiththecoffeeexample N OTE :Chi-squarecalculationsdependoncalculatorsorcomputersformostofthecalculations. TI-83+andTI-84calculatorinstructionsareinthethechapter. 1 Thiscontentisavailableonlineat. 415

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416 CHAPTER11.THECHI-SQUAREDISTRIBUTION 11.1.3OptionalCollaborativeClassroomActivity LookinthesportssectionofanewspaperorontheInternetforsomesportsdatabaseballaverages,basketballscores,golftournamentscores,footballodds,swimmingtimes,etc..Plotahistogramandaboxplot usingyourdata.Seeifyoucandetermineaprobabilitydistributionthatyourdatats.Haveadiscussion withtheclassaboutyourchoice. 11.2Notation 2 Thenotationforthechi-squaredistributionis: c 2 c 2 df where df = degreesoffreedomdependonhowchi-squareisbeingused.Ifyouwanttopracticecalculatingchi-squareprobabilitiesthenuse df = n )]TJ/F58 9.9626 Tf 10.203 0 Td [(1.Thedegreesoffreedomforthethreemajorusesareeach calculateddifferently. Forthe c 2 distribution,thepopulationmeanis m = df andthepopulationstandarddeviationis s = p 2 df Therandomvariableisshownas c 2 butmaybeanyuppercaseletter. Therandomvariableforachi-squaredistributionwith k degreesoffreedomisthesumof k independent, squarednormalvariables. c 2 = Z 1 2 + Z 2 2 + ... + Z k 2 11.3FactsAbouttheChi-SquareDistribution 3 1.Thecurveisnonsymmetricalandskewedtotheright. 2.Thereisadifferentchi-squarecurveforeach df 2 Thiscontentisavailableonlineat. 3 Thiscontentisavailableonlineat.

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417 a b Figure11.1 3.Theteststatisticforanytestisalwaysgreaterthanorequaltozero. 4.When df > 90,thechi-squarecurveapproximatesthenormal.For X c 2 1000 themean, m = df = 1000 andthestandarddeviation, s = p 2 1000 = 44.7.Therefore, X N 1000,44.7 ,approximately. 5.Themean, m ,islocatedjusttotherightofthepeak. Figure11.2 11.4Goodness-of-FitTest 4 Inthistypeofhypothesistest,youdeterminewhetherthedata "t" aparticulardistributionornot.Forexample,youmaysuspectyourunknowndatatabinomialdistribution.Youuseachi-squaretestmeaning thedistributionforthehypothesistestischi-squaretodetermineifthereisatornot. Thenullandthe alternatehypothesesforthistestmaybewritteninsentencesormaybestatedasequationsorinequalities. 4 Thiscontentisavailableonlineat.

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418 CHAPTER11.THECHI-SQUAREDISTRIBUTION Theteststatisticforagoodness-of-ttestis: S n O )]TJ/F132 9.9626 Tf 10.405 0 Td [(E 2 E .1 where: O =observedvaluesdata E =expectedvaluesfromtheory n =thenumberofdifferentdatacellsorcategories Theobservedvaluesarethedatavaluesandtheexpectedvaluesarethevaluesyouwouldexpecttogetif thenullhypothesisweretrue. Thereare n termsoftheform O )]TJ/F132 7.5716 Tf 6.436 0 Td [(E 2 E Thedegreesoffreedomare df=numberofcolumns-1numberofrows-1 Thegoodness-of-ttestisalmostalwaysrighttailed. Iftheobservedvaluesandthecorrespondingexpected valuesarenotclosetoeachother,thentheteststatisticcangetverylargeandwillbewayoutintheright tailofthechi-squarecurve. Example11.1 Absenteeismofcollegestudentsfrommathclassesisamajorconcerntomathinstructorsbecause missingclassappearstoincreasethedroprate.Threestatisticsinstructorswonderedwhether theabsenteeratewasthe same foreverydayoftheschoolweek.Theytookasampleofabsent studentsfromthreeoftheirstatisticsclassesduringoneweekoftheterm.Theresultsofthesurvey appearinthetable. Monday Tuesday Wednesday Thursday Friday #ofstudentsabsent 28 22 18 20 32 Determinethenullandalternatehypothesesneededtorunagoodness-of-ttest. Sincetheinstructorswonderwhethertheabsenteerateisthesameforeveryschoolday,wecould sayinthenullhypothesisthatthedata "t" auniformdistribution. H o : Therateatwhichcollegestudentsareabsentfromtheirstatisticsclasstsauniformdistribution. Thealternatehypothesisistheoppositeofthenullhypothesis. H a : Therateatwhichcollegestudentsareabsentfromtheirstatisticsclassdoesnottauniform distribution. Problem1 Howmanystudentsdoyou expect tobeabsentonanygivenschoolday? Solution Thetotalnumberofstudentsinthesampleis120. Ifthenullhypothesisweretrue, youwould divide120by5toget24absencesexpectedperday. Theexpectednumberisbasedonatruenull hypothesis. Problem2 Whatarethedegreesoffreedom df ? Solution Thereare5daysoftheweekor5"cells"orcategories.

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419 df = no cells )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 5 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 4 Example11.2 Employersparticularlywanttoknowwhichdaysoftheweekemployeesareabsentinave dayworkweek.Mostemployerswouldliketobelievethatemployeesareabsentequallyduring theweek.Thatis,theaveragenumberoftimesanemployeeisabsentisthesameonMonday, Tuesday,Wednesday,Thursday,orFriday.Supposeasampleof20absentdayswastakenandthe daysabsentweredistributedasfollows: DayoftheWeekAbsent Monday Tuesday Wednesday Thursday Friday NumberofAbsences 5 4 2 3 6 Problem Forthepopulationofemployees,dotheabsentdaysoccurwithequalfrequenciesduringave dayworkweek?Testata5%signicancelevel. Solution Thenullandalternatehypothesesare: H o :Theabsentdaysoccurwithequalfrequencies,thatis,theytauniformdistribution. H a :Theabsentdaysoccurwithunequalfrequencies,thatis,theydonottauniformdistribution. Iftheabsentdaysoccurwithequalfrequencies,then,outof20absentdays,therewouldbe4 absencesonMonday,4onTuesday,4onWednesday,4onThursday,and4onFriday.These numbersarethe expected E values.Thevaluesinthetablearethe observed O valuesordata. Thistime,calculatethe c 2 teststatisticbyhand.Makeachartwiththefollowingheadings: Expected E values Observed O values O )]TJ/F132 9.9626 Tf 10.406 0 Td [(E O )]TJ/F132 9.9626 Tf 10.406 0 Td [(E 2 O )]TJ/F132 7.5716 Tf 6.436 0 Td [(E 2 E Nowaddsumthelastcolumn.Verifythatthesumis2.5.Thisisthe c 2 teststatistic. Tondthep-value,calculate P )]TJ/F134 9.9626 Tf 4.812 -8.075 Td [(c 2 > 2.5 .Thistestisright-tailed. The dfs arethe numberofcells )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 4. Next,completeagraphliketheonebelowwiththeproperlabelingandshading.Youshould shadetherighttail.Itwillbea"large"righttailforthisexamplebecausethep-valueis"large."

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420 CHAPTER11.THECHI-SQUAREDISTRIBUTION Useacomputerorcalculatortondthep-value.Youshouldget p-value = 0.6446. Thedecisionistonotrejectthenullhypothesis. Conclusion: Ata5%levelofsignicance,fromthesampledata,thereisnotsufcientevidenceto concludethattheabsentdaysdonotoccurwithequalfrequencies. TI-83+andTI-84: Press 2ndDISTR .Arrowdownto c 2 cdf .Press ENTER .Enter .5,1E99,4 Roundedto4places,youshouldsee0.6446whichisthep-value. N OTE :TI-83+andsomeTI-84calculatorsdonothaveaspecialprogramfortheteststatisticforthe goodness-of-ttest.ThenextexampleExample11-3hasthecalculatorinstructions.Thenewer TI-84calculatorshavein STATTESTS thetest Chi2GOF .Torunthetest,puttheobservedvalues thedataintoarstlistandtheexpectedvaluesthevaluesyouexpectifthenullhypothesisis trueintoasecondlist.Press STATTESTS and Chi2GOF .EnterthelistnamesfortheObservedlist andtheExpectedlist.Enterwhateverelseisaskedandpress calculate or draw .Makesureyou clearanylistsbeforeyoustart.Seebelow. N OTE : ToClearListsinthecalculators: Gointo STATEDIT andarrowuptothelistnameareaof theparticularlist.Press CLEAR andthenarrowdown.Thelistwillbecleared.Or,youcanpress STAT andpress4for ClrList .Enterthelistnameandpress ENTER Example11.3 OnestudyindicatesthatthenumberoftelevisionsthatAmericanfamilieshaveisdistributedthis isthe given distributionfortheAmericanpopulationasfollows: NumberofTelevisions Percent 0 10 1 16 2 55 3 11 over3 8 Thetablecontainsexpected E percents.

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421 Arandomsampleof600familiesinthefarwesternUnitedStatesresultedinthefollowingdata: NumberofTelevisions Frequency 0 66 1 119 2 340 3 60 over3 15 Total=600 Thetablecontainsobserved O frequencyvalues. Problem Atthe1%signicancelevel,doesitappearthatthedistribution"numberoftelevisions"offar westernUnitedStatesfamiliesisdifferentfromthedistributionfortheAmericanpopulationasa whole? Solution ThisproblemasksyoutotestwhetherthefarwesternUnitedStatesfamiliesdistributiontsthe distributionoftheAmericanfamilies.Thistestisalwaysright-tailed. Thersttablecontainsexpectedpercentages.Togetexpected E frequencies,multiplythepercentageby600.Theexpectedfrequenciesare: NumberofTelevisions Percent ExpectedFrequency 0 10 0.10 600 = 60 1 16 0.16 600 = 96 2 55 0.55 600 = 330 3 11 0.11 600 = 66 over3 8 0.08 600 = 48 Therefore,theexpectedfrequenciesare60,96,330,66,and48.IntheTIcalculators,youcanletthe calculatordothemath.Forexample,insteadof60,enter.10*600. H o :The"numberoftelevisions"distributionoffarwesternUnitedStatesfamiliesisthesameas the"numberoftelevisions"distributionoftheAmericanpopulation. H a :The"numberoftelevisions"distributionoffarwesternUnitedStatesfamiliesisdifferentfrom the"numberoftelevisions"distributionoftheAmericanpopulation. Distributionforthetest: c 2 4 where df = thenumberofcells )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = 5 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 4. N OTE : df 6 = 600 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 Calculatetheteststatistic: c 2 = 29.65

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422 CHAPTER11.THECHI-SQUAREDISTRIBUTION Graph: Probabilitystatement: p-value = P )]TJ/F134 9.9626 Tf 4.812 -8.075 Td [(c 2 > 29.65 = 0.000006. Compare a andthep-value: a = 0.01 p-value = 0.000006 So, a > p-value Makeadecision: Since a > p-value ,reject H o Thismeansyourejectthebeliefthatthedistributionforthefarwesternstatesisthesameasthat oftheAmericanpopulationasawhole. Conclusion: Atthe1%signicancelevel,fromthedata,thereissufcientevidencetoconclude thatthe"numberoftelevisions"distributionforthefarwesternUnitedStatesisdifferentfromthe "numberoftelevisions"distributionfortheAmericanpopulationasawhole. N OTE :TI-83+andsomeTI-84calculators:Press STAT and ENTER .Makesuretoclearlists L1 L2 ,and L3 iftheyhavedatainthemseethenoteattheendofExample11-2.Into L1 ,put theobservedfrequencies 66 119 349 60 15 .Into L2 ,puttheexpectedfrequencies .10*600, .16*600 .55*600 .11*600 .08*600 .Arrowovertolist L3 anduptothenamearea "L3" .Enter L1-L2/L2 and ENTER .Press 2ndQUIT .Press 2ndLIST andarrowoverto MATH .Press 5 .You shouldsee "sum"EnterL3 .Roundedto2decimalplaces,youshouldsee 29.65 .Press 2nd DISTR .Press 7 orArrowdownto 7: c 2cdf andpress ENTER .Enter .65,1E99,4 .Roundedto4 places,youshouldsee 5.77E-6=.000006 roundedto6decimalplaceswhichisthep-value. Example11.4 Supposeyouiptwocoins100times.Theresultsare20HH,27HT,30TH,and23TT.Arethe coinsfair?Testata5%signicancelevel. Solution Thisproblemcanbesetupasagoodness-of-tproblem.Thesamplespaceforippingtwofair coinsis{HH,HT,TH,TT}.Outof100ips,youwouldexpect25HH,25HT,25TH,and25TT. Thisistheexpecteddistribution.Thequestion,"Arethecoinsfair?"isthesameassaying,"Does thedistributionofthecoinsHH,27HT,30TH,23TTttheexpecteddistribution?" RandomVariable: Let X =thenumberofheadsinoneipofthetwocoins. X takesonthevalue 0,1,2.Thereare0,1,or2headsintheipof2coins.Therefore,the numberofcellsis3 .Since X =thenumberofheads,theobservedfrequenciesare20for2heads,57for1head,and23for

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423 0headsorbothtails.Theexpectedfrequenciesare25for2heads,50for1head,and25for0 headsorbothtails.Thistestisright-tailed. H o :Thecoinsarefair. H a :Thecoinsarenotfair. Distributionforthetest: c 2 2 where df = 3 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 2. Calculatetheteststatistic: c 2 = 2.14 Graph: Probabilitystatement: p-value = P )]TJ/F134 9.9626 Tf 4.812 -8.075 Td [(c 2 > 2.14 = 0.3430 Compare a andthep-value: a = 0.05 p-value = 0.3430 So, a < p-value Makeadecision: Since a < p-value ,donotreject H o Conclusion: Thecoinsarefair. N OTE :TI-83+andsomeTI-84calculators:Press STAT and ENTER .Makesureyouclearlists L1 L2 ,and L3 iftheyhavedatainthem.Into L1 ,puttheobservedfrequencies 20 57 23 .Into L2 ,put theexpectedfrequencies 25 50 25 .Arrowovertolist L3 anduptothenamearea "L3" .Enter L1-L2/L2 and ENTER .Press 2ndQUIT .Press 2ndLIST andarrowoverto MATH .Press 5 .You shouldsee "sum" EnterL3 .Roundedto2decimalplaces,youshouldsee 2.14 .Press 2ndDISTR Arrowdownto 7: c 2cdf orpress 7 .Press ENTER .Enter 2.14,1E99,2 .Roundedto4places,you shouldsee .3430 whichisthep-value. N OTE :ForthenewerTI-84calculators,check STATTESTS toseeifyouhave Chi2GOF .Ifyoudo, seethecalculatorinstructionsaNOTEbeforeExample11-3

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424 CHAPTER11.THECHI-SQUAREDISTRIBUTION 11.5TestofIndependence 5 Testsofindependenceinvolveusinga contingencytable ofobserveddatavalues.YourstsawacontingencytablewhenyoustudiedprobabilityintheProbabilityTopicsSection3.1chapter. Theteststatisticforatestofindependenceissimilartothatofagoodness-of-ttest: S i j O )]TJ/F132 9.9626 Tf 10.406 0 Td [(E 2 E .2 where: O =observedvalues E =expectedvalues i =thenumberofrowsinthetable j =thenumberofcolumnsinthetable Thereare i j termsoftheform O )]TJ/F132 7.5716 Tf 6.436 0 Td [(E 2 E Atestofindependencedetermineswhethertwofactorsareindependentornot. Yourstencounteredthe termindependenceinChapter3.Asareview,considerthefollowingexample. Example11.5 Suppose A =aspeedingviolationinthelastyearand B =acarphoneuser.If A and B areindependentthen P AANDB = P A P B AANDB istheeventthatadriverreceivedaspeeding violationlastyearandisalsoacarphoneuser.Suppose,inastudyofdriverswhoreceivedspeedingviolationsinthelastyearandwhousecarphones,that755peopleweresurveyed.Outofthe 755,70hadaspeedingviolationand685didnot;305werecarphoneusersand450werenot. Let y =expectednumberofcarphoneuserswhoreceivedspeedingviolations. If A and B areindependent,then P AANDB = P A P B .Bysubstitution, y 755 = 70 755 305 755 Solvefor y : y = 70 305 755 = 28.3 About28peoplefromthesampleareexpectedtobecarphoneusersandtoreceivespeeding violations. Inatestofindependence,westatethenullandalternatehypothesesinwords.Sincethecontingencytableconsistsof twofactors ,thenullhypothesisstatesthatthefactorsare independent andthealternatehypothesisstatesthattheyare notindependentdependent .Ifwedoatestof independenceusingtheexampleabove,thenthenullhypothesisis: H o :Beingacarphoneuserandreceivingaspeedingviolationareindependentevents. Ifthenullhypothesisweretrue,wewouldexpectabout28peopletobecarphoneusersandto receiveaspeedingviolation. Thetestofindependenceisalwaysright-tailed becauseofthecalculationoftheteststatistic.Ifthe expectedandobservedvaluesarenotclosetogether,thentheteststatisticisverylargeandway outintherighttailofthechi-squarecurve,likegoodness-of-t. 5 Thiscontentisavailableonlineat.

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425 Thedegreesoffreedomforthetestofindependenceare: df=numberofcolumns-1numberofrows-1 Thefollowingformulacalculatesthe expectednumber E : E = rowtotalcolumntotal totalnumbersurveyed Example11.6 Inavolunteergroup,adults21andoldervolunteerfromonetoninehourseachweektospend timewithadisabledseniorcitizen.Theprogramrecruitsamongcommunitycollegestudents, four-yearcollegestudents,andnonstudents.Thefollowingtableisa sample oftheadultvolunteersandthenumberofhourstheyvolunteerperweek. NumberofHoursWorkedPerWeekbyVolunteerTypeObserved TypeofVolunteer 1-3Hours 4-6Hours 7-9Hours CommunityCollegeStudents 111 96 48 Four-YearCollegeStudents 96 133 61 Nonstudents 91 150 53 Thetablecontains Problem Arethenumberofhoursvolunteered independent ofthetypeofvolunteer? Solution The observedtable andthequestionattheendoftheproblem,"Arethenumberofhoursvolunteeredindependentofthetypeofvolunteer?"tellyouthisisatestofindependence.Thetwo factorsare numberofhoursvolunteered and typeofvolunteer .Thistestisalwaysright-tailed. H o :Thenumberofhoursvolunteeredis independent ofthetypeofvolunteer. H a :Thenumberofhoursvolunteeredis dependent onthetypeofvolunteer. Theexpectedtableis: NumberofHoursWorkedPerWeekbyVolunteerTypeExpected TypeofVolunteer 1-3Hours 4-6Hours 7-9Hours CommunityCollegeStudents 90.57 115.19 49.24 Four-YearCollegeStudents 103.00 131.00 56.00 Nonstudents 104.42 132.81 56.77 Thetablecontains expected E valuesdata. Forexample,thecalculationfortheexpectedfrequencyforthetopleftcellis E = rowtotalcolumntotal totalnumbersurveyed = 255 298 839 = 90.57 Calculatetheteststatistic: c 2 = 12.99calculatororcomputer Distributionforthetest: c 2 4

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426 CHAPTER11.THECHI-SQUAREDISTRIBUTION df = 3columns )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 3rows )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 2 2 = 4 Graph: Probabilitystatement: p-value = P )]TJ/F134 9.9626 Tf 4.812 -8.074 Td [(c 2 > 12.99 = 0.0113 Compare a andthe p-value : Sinceno a isgiven,assume a = 0.05. p-value = 0.0113. a > p-value Makeadecision: Since a > p-value ,reject H o .Thismeansthatthefactorsarenotindependent. Conclusion: Ata5%levelofsignicance,fromthedata,thereissufcientevidencetoconclude thatthenumberofhoursvolunteeredandthetypeofvolunteeraredependentononeanother. Fortheaboveexample,iftherehadbeenanothertypeofvolunteer,teenagers,whatwouldthe degreesoffreedombe? N OTE :Calculatorinstructionsfollow. TI-83+andTI-84calculator:Pressthe MATRX keyandarrowoverto EDIT .Press 1:[A] .Press 3 ENTER3ENTER .EnterthetablevaluesbyrowfromExample11-6.Press ENTER aftereach.Press 2ndQUIT .Press STAT andarrowoverto TESTS .Arrowdownto C: c 2-TEST .Press ENTER .You shouldsee Observed:[A]andExpected:[B] .Arrowdownto Calculate .Press ENTER .Thetest statisticis12.9909andthe p-value = 0.0113.Dotheprocedureasecondtimebutarrowdownto Draw insteadof calculate Example11.7 DeAnzaCollegeisinterestedintherelationshipbetweenanxietylevelandtheneedtosucceed inschool.Arandomsampleof400studentstookatestthatmeasuredanxietylevelandneedto succeedinschool.Thetableshowstheresults.DeAnzaCollegewantstoknowifanxietylevel andneedtosucceedinschoolareindependentevents.

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427 NeedtoSucceedinSchoolvs.AnxietyLevel Needto Succeedin School HighAnxiety Med-high Anxiety Medium Anxiety Med-low Anxiety LowAnxiety RowTotal HighNeed 35 42 53 15 10 155 Medium Need 18 48 63 33 31 193 LowNeed 4 5 11 15 17 52 ColumnTotal 57 95 127 63 58 400 Problem1 Howmanyhighanxietylevelstudentsareexpectedtohaveahighneedtosucceedinschool? Solution Thecolumntotalforahighanxietylevelis57.Therowtotalforhighneedtosucceedinschoolis 155.Thesamplesizeortotalsurveyedis400. E = rowtotalcolumntotal totalsurveyed = 155 57 400 = 22.09 Theexpectednumberofstudentswhohaveahighanxietylevelandahighneedtosucceedin schoolisabout22. Problem2 Howmanystudentsdoyouexpecttohavealowneedtosucceedinschoolandamed-lowlevel ofanxiety? Solution Thecolumntotalforamed-lowanxietylevelis63.Therowtotalforalowneedtosucceedin schoolis52.Thesamplesizeortotalsurveyedis400. Problem3 a. E = rowtotalcolumntotal totalsurveyed = b. Theexpectednumberofstudentswhohaveamed-lowanxietylevelandalowneedto succeedinschoolisabout: 11.6TestofaSingleVarianceOptional 6 Atestofasinglevarianceassumesthattheunderlyingdistributionis normal .Thenullandalternate hypothesesarestatedintermsofthe populationvariance orpopulationstandarddeviation.Thetest statisticis: n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 s 2 s 2 .3 where: 6 Thiscontentisavailableonlineat.

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428 CHAPTER11.THECHI-SQUAREDISTRIBUTION n =thetotalnumberofdata s 2 =samplevariance s 2 =populationvariance Youmaythinkof s astherandomvariableinthistest.Thedegreesoffreedomare df = n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1. Atestofasinglevariancemayberight-tailed,left-tailed,ortwo-tailed. Thefollowingexamplewillshowyouhowtosetupthenullandalternatehypotheses.Thenulland alternatehypothesescontainstatementsaboutthepopulationvariance. Example11.8 Mathinstructorsarenotonlyinterestedinhowtheirstudentsdoonexams,onaverage,but howtheexamscoresvary.Tomanyinstructors,thevarianceorstandarddeviationmaybemore importantthantheaverage. Supposeamathinstructorbelievesthatthestandarddeviationforhisnalexamis5points.One ofhisbeststudentsthinksotherwise.Thestudentclaimsthatthestandarddeviationismorethan 5points.Ifthestudentweretoconductahypothesistest,whatwouldthenullandalternate hypothesesbe? Solution Eventhoughwearegiventhepopulationstandarddeviation,wecansetthetestupusingthe populationvarianceasfollows. H o : s 2 = 5 2 H a : s 2 > 5 2 Example11.9 Withindividuallinesatitsvariouswindows,apostofcendsthatthestandarddeviation fornormallydistributedwaitingtimesforcustomersonFridayafternoonis7.2minutes.The postofceexperimentswithasinglemainwaitinglineandndsthatforarandomsampleof25 customers,thewaitingtimesforcustomershaveastandarddeviationof3.5minutes. Withasignicancelevelof5%,testtheclaimthat asinglelinecauseslowervariationamong waitingtimesshorterwaitingtimesforcustomers Solution Sincetheclaimisthatasinglelinecauseslowervariation,thisisatestofasinglevariance.The parameteristhepopulationvariance, s 2 ,orthepopulationstandarddeviation, s RandomVariable: Thesamplestandarddeviation, s ,istherandomvariable.Let s =standard deviationforthewaitingtimes. H o : s 2 = 7.2 2 H a : s 2 < 7.2 2 Theword "lower" tellsyouthisisaleft-tailedtest. Distributionforthetest: c 2 24 ,where: n =thenumberofcustomerssampled df = n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 25 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = 24

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429 Calculatetheteststatistic: c 2 = n )]TJ/F58 7.5716 Tf 6.227 0 Td [(1 s 2 s 2 = 25 )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 3.5 2 7.2 2 = 5.67 where n = 25, s = 3.5,and s = 7.2. Graph: Probabilitystatement: p-value = P )]TJ/F134 9.9626 Tf 4.812 -8.074 Td [(c 2 < 5.67 = 0.000042 Compare a andthep-value: a = 0.05 p-value = 0.000042 a > p-value Makeadecision: Since a > p-value ,reject H o Thismeansthatyoureject s 2 = 7.2 2 .Inotherwords,youdonotthinkthevariationinwaiting timesis7.2minutes,butlower. Conclusion: Ata5%levelofsignicance,fromthedata,thereissufcientevidencetoconclude thatasinglelinecausesalowervariationamongthewaitingtimes or withasingleline,thecustomerwaitingtimesvarylessthan7.2minutes. TI-83+andTI-84calculators :In 2ndDISTR ,use 7: c 2cdf .Thesyntaxis lower,upper,df for theparameterlist.ForExample11-9, c 2cdf-1E99,5.67,24 .The p-value = 0.000042.

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430 CHAPTER11.THECHI-SQUAREDISTRIBUTION 11.7SummaryofFormulas 7 Formula11.1: TheChi-squareProbabilityDistribution m = df and s = p 2 df Formula11.2: Goodness-of-FitHypothesisTest Usegoodness-of-ttotestwhetheradatasettsaparticularprobabilitydistribution. Thedegreesoffreedomare numberofcellsorcategories-1 Theteststatisticis S n O )]TJ/F132 7.5716 Tf 6.436 0 Td [(E 2 E ,where O =observedvaluesdata, E =expectedvaluesfrom theory,and n =thenumberofdifferentdatacellsorcategories. Thetestisright-tailed. Formula11.3: TestofIndependence Usethetestofindependencetotestwhethertwofactorsareindependentornot. Thedegreesoffreedomareequalto numberofcolumns-1numberofrows-1 Theteststatisticis S i j O )]TJ/F132 7.5716 Tf 6.436 0 Td [(E 2 E where O =observedvalues, E =expectedvalues, i =thenumber ofrowsinthetable,and j =thenumberofcolumnsinthetable. Thetestisright-tailed. Ifthenullhypothesisistrue,theexpectednumber E = rowtotalcolumntotal totalsurveyed Formula11.4: TestofaSingleVariance Usethetesttodeterminevariation. Thedegreesoffreedomarethenumberofsamples-1. Theteststatisticis n )]TJ/F58 7.5716 Tf 6.227 0 Td [(1 s 2 s 2 ,where n =thetotalnumberofdata, s 2 =samplevariance,and s 2 =populationvariance. Thetestmaybeleft,right,ortwo-tailed. 7 Thiscontentisavailableonlineat.

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431 11.8Practice1:Goodness-of-FitTest 8 11.8.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofgoodness-of-ttestdata. 11.8.2Given Thefollowingdataarereal.ThecumulativenumberofAIDScasesreportedforSantaClaraCountythrough December31,2003,isbrokendownbyethnicityasfollows: Ethnicity NumberofCases White 2032 Hispanic 897 African-American 372 Asian,PacicIslander 168 NativeAmerican 20 Total=3489 ThepercentageofeachethnicgroupinSantaClaraCountyisasfollows: Ethnicity Percentageoftotalcountypopulation Numberexpectedroundto2 decimalplaces White 47.79% 1667.39 Hispanic 24.15% African-American 3.55% Asian,PacicIslander 24.21% NativeAmerican 0.29% Total=100% 11.8.3ExpectedResults IftheethnicityofAIDSvictimsfollowedtheethnicityofthetotalcountypopulation,llintheexpected numberofcasesperethnicgroup. 11.8.4Goodness-of-FitTest Performagoodness-of-ttesttodeterminewhetherthemake-upofAIDScasesfollowstheethnicityofthe generalpopulationofSantaClaraCounty. Exercise11.8.1 H o : 8 Thiscontentisavailableonlineat.

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432 CHAPTER11.THECHI-SQUAREDISTRIBUTION Exercise11.8.2 H a : Exercise11.8.3 Isthisaright-tailed,left-tailed,ortwo-tailedtest? Exercise11.8.4 Solutiononp.455. degreesoffreedom= Exercise11.8.5 Solutiononp.455. Chi 2 teststatistic= Exercise11.8.6 Solutiononp.455. p-value= Exercise11.8.7 Graphthesituation.Labelandscalethehorizontalaxis.Markthemeanandteststatistic.Shade intheregioncorrespondingtothep-value. Let a = 0.05 Decision: ReasonfortheDecision: Conclusionwriteoutincompletesentences: 11.8.5DiscussionQuestion Exercise11.8.8 DoesitappearthatthepatternofAIDScasesinSantaClaraCountycorrespondstothedistributionofethnicgroupsinthiscounty?Whyorwhynot?

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433 11.9Practice2:ContingencyTables 9 11.9.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofcontingencytables. Conductahypothesistesttodetermineifsmokinglevelandethnicityareindependent. 11.9.2CollecttheData CopythedataprovidedinProbabilityTopicsPractice2:CalculatingProbabilitiesintothetablebelow. SmokingLevelsbyEthnicityObserved Smoking LevelPer Day African American Native Hawaiian Latino Japanese Americans White TOTALS 1-10 11-20 21-30 31+ TOTALS 11.9.3Hypothesis Statethehypotheses. H o : H a : 11.9.4ExpectedValues Enterexpectedvaluesintheabovebelow.Roundtotwodecimalplaces. 11.9.5AnalyzetheData Calculatethefollowingvalues: Exercise11.9.1 Solutiononp.455. Degreesoffreedom= Exercise11.9.2 Solutiononp.455. Chi 2 teststatistic= Exercise11.9.3 Solutiononp.455. p-value= Exercise11.9.4 Solutiononp.455. Isthisaright-tailed,left-tailed,ortwo-tailedtest?Explainwhy. 9 Thiscontentisavailableonlineat.

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434 CHAPTER11.THECHI-SQUAREDISTRIBUTION 11.9.6GraphtheData Exercise11.9.5 Graphthesituation.Labelandscalethehorizontalaxis.Markthemeanandteststatistic.Shade intheregioncorrespondingtothep-value. 11.9.7Conclusions Statethedecisionandconclusioninacompletesentenceforthefollowingpreconceivedlevelsof a Exercise11.9.6 Solutiononp.455. a = 0 .05 a. Decision: b. Reasonforthedecision: c. Conclusionwriteoutinacompletesentence: Exercise11.9.7 a = 0.01 a. Decision: b. Reasonforthedecision: c. Conclusionwriteoutinacompletesentence:

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435 11.10Practice3:TestofaSingleVariance 10 11.10.1StudentLearningOutcomes Thestudentwillexplorethepropertiesofdatawithatestofasinglevariance. 11.10.2Given Supposeanairlineclaimsthatitsightsareconsistentlyontimewithanaveragedelayofatmost15minutes.Itclaimsthattheaveragedelayissoconsistentthatthevarianceisnomorethan150minutes.Doubtingtheconsistencypartoftheclaim,adisgruntledtravelercalculatesthedelaysforhisnext25ights.The averagedelayforthose25ightsis22minuteswithastandarddeviationof15minutes. 11.10.3SampleVariance Exercise11.10.1 Isthetravelerdisputingtheclaimabouttheaverageoraboutthevariance? Exercise11.10.2 Solutiononp.455. Asamplestandarddeviationof15minutesisthesameasasamplevarianceof__________minutes. Exercise11.10.3 Isthisaright-tailed,left-tailed,ortwo-tailedtest? 11.10.4HypothesisTest Performahypothesistestontheconsistencypartoftheclaim. Exercise11.10.4 H o : Exercise11.10.5 H a : Exercise11.10.6 Solutiononp.455. Degreesoffreedom= Exercise11.10.7 Solutiononp.455. Chi 2 teststatistic= Exercise11.10.8 Solutiononp.455. p-value= Exercise11.10.9 Graphthesituation.Labelandscalethehorizontalaxis.Markthemeanandteststatistic.Shade thep-value. 10 Thiscontentisavailableonlineat.

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436 CHAPTER11.THECHI-SQUAREDISTRIBUTION Exercise11.10.10 Let a = 0 .05 Decision: Conclusionwriteoutinacompletesentence: 11.10.5DiscussionQuestions Exercise11.10.11 Howdidyouknowtotestthevarianceinsteadofthemean? Exercise11.10.12 Ifanadditionaltestweredoneontheclaimoftheaveragedelay,whichdistributionwouldyou use? Exercise11.10.13 Ifanadditionaltestwasdoneontheclaimoftheaveragedelay,but45ightsweresurveyed, whichdistributionwouldyouuse?

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437 11.11Homework 11 Exercise11.11.1 a. Explainwhythegoodnessofttestandthetestforindependencearegenerallyright tailedtests. b. Ifyoudidaleft-tailedtest,whatwouldyoubetesting? 11.11.1WordProblems Foreachwordproblem,useasolutionsheettosolvethehypothesistestproblem.Roundexpectedfrequencytotwodecimalplaces. Exercise11.11.2 A6-sideddieisrolled120times.Fillintheexpectedfrequencycolumn.Then,conductahypothesistesttodetermineifthedieisfair.Thedatabelowaretheresultofthe120rolls. FaceValue Frequency ExpectedFrequency 1 15 2 29 3 16 4 15 5 30 6 15 Exercise11.11.3 Solutiononp.456. ThemaritalstatusdistributionoftheU.S.malepopulation,age15andolder,isasshownbelow. Source:U.S.CensusBureau,CurrentPopulationReports MaritalStatus Percent ExpectedFrequency nevermarried 31.3 married 56.1 widowed 2.5 divorced/separated 10.1 Supposethatarandomsampleof400U.S.youngadultmales,1824yearsold,yieldedthe followingfrequencydistribution.WeareinterestedinwhetherthisagegroupofmaleststhedistributionoftheU.S.adultpopulation.Calculatethefrequencyonewouldexpectwhensurveying 400people.Fillintheabovetable,roundingtotwodecimalplaces. 11 Thiscontentisavailableonlineat.

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438 CHAPTER11.THECHI-SQUAREDISTRIBUTION MaritalStatus Frequency nevermarried 140 married 238 widowed 2 divorced/separated 20 Thenexttwoquestionsrefertothefollowinginformation: TherealdatabelowarefromtheCalifornia ReinvestmentCommitteeandtheCaliforniaEconomicCensus.Thedataconcernthepercentofloansmade bytheSmallBusinessAdministrationforSantaClaraCountyinrecentyears. Source:SanJoseMercury News EthnicGroup PercentofLoans PercentofPopulation PercentofBusinessesOwned Asian 22.48 16.79 12.17 Black 1.15 3.51 1.61 Latino 6.19 21.00 6.51 White 66.97 58.09 79.70 Exercise11.11.4 Performagoodness-of-ttesttodeterminewhetherthepercentofbusinessesownedinSanta ClaraCountytsthepercentofthepopulation,basedonethnicity. Exercise11.11.5 Solutiononp.456. Performagoodness-of-ttesttodeterminewhetherthepercentofloanststhepercentofthe businessesownedinSantaClaraCounty,basedonethnicity. Exercise11.11.6 TheCityofSouthLakeTahoehasanAsianpopulationof1419people,outofatotalpopulation of23,609 Source:U.S.CensusBureau,Census2000 .Conductagoodnessofttesttodetermine iftheself-reportedsub-groupsofAsiansareevenlydistributed. Race Frequency ExpectedFrequency AsianIndian 131 Chinese 118 Filipino 1045 Japanese 80 Korean 12 Vietnamese 9 Other 24 Exercise11.11.7 Solutiononp.456. LongBeachisacityinLosAngelesCountyL.A.C.ThepopulationofLongBeachis461,522;the populationofL.A.C.is9,519,338 Source:U.S.CensusBureau,Census2000 .Conductagoodness ofttesttodetermineiftheracialdemographicsofLongBeachtthatofL.A.C.

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439 Race Percent,L.A.C. Expected#,L.B. Actual#,L.B. AmericanIndianandAlaskaNative 0.8 3692 3,881 Asian 11.9 55,591 BlackorAfricanAmerican 9.8 68,618 NativeHawaiianandOtherPacicIslander 0.3 5,605 White,includingHispanic/Latino 48.7 208,410 Other 23.5 95,107 Twoormoreraces 5.0 24,310 Exercise11.11.8 UCLAconductedasurveyofmorethan263,000collegefreshmenfrom385collegesinfall2005. Theresultsofstudentexpectedmajorsbygenderwerereportedin TheChronicleofHigherEducation/2/06. Conductagoodnessofttesttodetermineifthemaledistributiontsthefemale distribution. Major Women Men Arts&Humanities 14.0% 11.4% BiologicalSciences 8.4% 6.7% Business 13.1% 22.7% Education 13.0% 5.8% Engineering 2.6% 15.6% PhysicalSciences 2.6% 3.6% Professional 18.9% 9.3% SocialSciences 13.0% 7.6% Technical 0.4% 1.8% Other 5.8% 8.2% Undecided 8.0% 6.6% Exercise11.11.9 Solutiononp.456. ArecentdebateaboutwhereintheUnitedStatesskiersbelievetheskiingisbestpromptedthe followingsurvey.Testtoseeifthebestskiareaisindependentoftheleveloftheskier. U.S.SkiArea Beginner Intermediate Advanced Tahoe 20 30 40 Utah 10 30 60 Colorado 10 40 50 Exercise11.11.10 Carmanufacturersareinterestedinwhetherthereisarelationshipbetweenthesizeofcaran individualdrivesandthenumberofpeopleinthedriver'sfamilythatis,whethercarsizeand familysizeareindependent.Totestthis,supposethat800carownerswererandomlysurveyed withthefollowingresults.Conductatestforindependence.

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440 CHAPTER11.THECHI-SQUAREDISTRIBUTION FamilySize Sub&Compact Mid-size Full-size Van&Truck 1 20 35 40 35 2 20 50 70 80 3-4 20 50 100 90 5+ 20 30 70 70 Exercise11.11.11 Solutiononp.456. Collegestudentsmaybeinterestedinwhetherornottheirmajorshaveanyeffectonstarting salariesaftergraduation.Supposethat300recentgraduatesweresurveyedastotheirmajors incollegeandtheirstartingsalariesaftergraduation.Belowarethedata.Conductatestfor independence. Major < \$30,000 \$30,000-\$39,999 \$40,000+ English 5 20 5 Engineering 10 30 60 Nursing 10 15 15 Business 10 20 30 Psychology 20 30 20 Exercise11.11.12 Sometravelagentsclaimthathoneymoonhotspotsvaryaccordingtoageofthebrideandgroom. Supposethat280EastCoastrecentbrideswereinterviewedastowheretheyspenttheirhoneymoons.Theinformationisgivenbelow.Conductatestforindependence. Location 20-29 30-39 40-49 50andover NiagaraFalls 15 25 25 20 Poconos 15 25 25 10 Europe 10 25 15 5 VirginIslands 20 25 15 5 Exercise11.11.13 Solutiononp.456. Amanagerofasportsclubkeepsinformationconcerningthemainsportinwhichmembers participateandtheirages.Totestwhetherthereisarelationshipbetweentheageofamember andhisorherchoiceofsport,643membersofthesportsclubarerandomlyselected.Conducta testforindependence. Sport 18-25 26-30 31-40 41andover racquetball 42 58 30 46 tennis 58 76 38 65 swimming 72 60 65 33

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441 Exercise11.11.14 AmajorfoodmanufacturerisconcernedthatthesalesforitsskinnyFrenchfrieshavebeendecreasing.Asapartofafeasibilitystudy,thecompanyconductsresearchintothetypesoffriessold acrossthecountrytodetermineifthetypeoffriessoldisindependentoftheareaofthecountry. Theresultsofthestudyarebelow.Conductatestforindependence. TypeofFries Northeast South Central West skinnyfries 70 50 20 25 curlyfries 100 60 15 30 steakfries 20 40 10 10 Exercise11.11.15 Solutiononp.456. AccordingtoDanLenard,anindependentinsuranceagentintheBuffalo,N.Y.area,thefollowing isabreakdownoftheamountoflifeinsurancepurchasedbymalesinthefollowingagegroups. Heisinterestedinwhethertheageofthemaleandtheamountoflifeinsurancepurchasedare independentevents.Conductatestforindependence. AgeofMales None \$50,000-\$100,000 \$100,001-\$150,000 \$150,001-\$200,000 \$200,000+ 20-29 40 15 40 0 5 30-39 35 5 20 20 10 40-49 20 0 30 0 30 50+ 40 30 15 15 10 Exercise11.11.16 Supposethat600thirtyyearoldsweresurveyedtodeterminewhetherornotthereisarelationshipbetweenthelevelofeducationanindividualhasandsalary.Conductatestforindependence. AnnualSalary Notahighschool grad. Highschoolgraduate Collegegraduate Mastersordoctorate < \$30,000 15 25 10 5 \$30,000-\$40,000 20 40 70 30 \$40,000-\$50,000 10 20 40 55 \$50,000-\$60,000 5 10 20 60 \$60,000+ 0 5 10 150 Exercise11.11.17 Solutiononp.456. Aplantmanagerisconcernedherequipmentmayneedrecalibrating.Itseemsthattheactual weightofthe15oz.cerealboxesitllshasbeenuctuating.Thestandarddeviationshouldbe atmost 1 2 oz.Inordertodetermineifthemachineneedstoberecalibrated,84randomlyselected boxesofcerealfromthenextday'sproductionwereweighed.Thestandarddeviationofthe84 boxeswas0.54.Doesthemachineneedtoberecalibrated? Exercise11.11.18 Consumersmaybeinterestedinwhetherthecostofaparticularcalculatorvariesfromstoreto store.Basedonsurveying43stores,whichyieldedasamplemeanof\$84andasamplestandard deviationof\$12,testtheclaimthatthestandarddeviationisgreaterthan\$15.

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442 CHAPTER11.THECHI-SQUAREDISTRIBUTION Exercise11.11.19 Solutiononp.457. Isabella,anaccomplished BaytoBreakers runner,claimsthatthestandarddeviationforhertime torunthe7mileraceisatmost3minutes.Totestherclaim,Rupinderlooksup5ofherrace times.Theyare55minutes,61minutes,58minutes,63minutes,and57minutes. Exercise11.11.20 Airlinecompaniesareinterestedintheconsistencyofthenumberofbabiesoneachight,sothat theyhaveadequatesafetyequipment.Theyarealsointerestedinthevariationofthenumberof babies.Supposethatanairlineexecutivebelievestheaveragenumberofbabiesonightsis6with avarianceof9atmost.Theairlineconductsasurvey.Theresultsofthe18ightssurveyedgive asampleaverageof6.4withasamplestandarddeviationof3.9.Conductahypothesistestofthe airlineexecutive'sbelief. Exercise11.11.21 Solutiononp.457. Accordingtothe U.S.BureauoftheCensus,UnitedNations, in1994thenumberofbirthsper womaninChinawas1.8.Thisfertilityratehasbeenattributedtothelawpassedin1979restricting birthstooneperwoman.Supposethatagroupofstudentsstudiedwhetherornotthestandard deviationofbirthsperwomanwasgreaterthan0.75.Theyasked50womenacrossChinathe numberofbirthstheyhad.Belowaretheresults.Doesthestudents'surveyindicatethatthe standarddeviationisgreaterthan0.75? #ofbirths Frequency 0 5 1 30 2 10 3 5 Exercise11.11.22 Accordingtoanavidaquariest,theaveragenumberofshina20gallontankis10,witha standarddeviationof2.Hisfriend,alsoanaquariest,doesnotbelievethatthestandarddeviation is2.Shecountsthenumberofshin15other20gallontanks.Basedontheresultsthatfollow,do youthinkthatthestandarddeviationisdifferentfrom2?Data:11;10;9;10;10;11;11;10;12;9;7; 9;11;10;11 Exercise11.11.23 Solutiononp.457. Themanagerof"Frenchies"isconcernedthatpatronsarenotconsistentlyreceivingthesame amountofFrenchfrieswitheachorder.Thechefclaimsthatthestandarddeviationfora10 ounceorderoffriesisatmost1.5oz.,butthemanagerthinksthatitmaybehigher.Herandomly weighs49ordersoffries,whichyields:meanof11oz.,standarddeviationof2oz. 11.11.2Trythesetrue/falsequestions. Exercise11.11.24 Solutiononp.457. Asthedegreesoffreedomincrease,thegraphofthechi-squaredistributionlooksmoreandmore symmetrical. Exercise11.11.25 Solutiononp.457. Thestandarddeviationofthechi-squaredistributionistwicethemean. Exercise11.11.26 Solutiononp.457. Themeanandthemedianofthechi-squaredistributionarethesameif df = 24 .

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443 Exercise11.11.27 Solutiononp.457. InaGoodness-of-Fittest,theexpectedvaluesarethevalueswewouldexpectifthenullhypothesisweretrue. Exercise11.11.28 Solutiononp.457. Ingeneral,iftheobservedvaluesandexpectedvaluesofaGoodness-of-Fittestarenotclose together,thentheteststatisticcangetverylargeandonagraphwillbewayoutintherighttail. Exercise11.11.29 Solutiononp.457. ThedegreesoffreedomforaTestforIndependenceareequaltothesamplesizeminus1. Exercise11.11.30 Solutiononp.457. UseaGoodness-of-Fittesttodetermineifhighschoolprincipalsbelievethatstudentsareabsent equallyduringtheweekornot. Exercise11.11.31 Solutiononp.457. TheTestforIndependenceusestablesofobservedandexpecteddatavalues. Exercise11.11.32 Solutiononp.457. Thetesttousewhendeterminingifthecollegeoruniversityastudentchoosestoattendisrelated tohis/hersocioeconomicstatusisaTestforIndependence. Exercise11.11.33 Solutiononp.457. ThetesttousetodetermineifacoinisfairisaGoodness-of-Fittest. Exercise11.11.34 Solutiononp.458. InaTestofIndependence,theexpectednumberisequaltotherowtotalmultipliedbythecolumn totaldividedbythetotalsurveyed. Exercise11.11.35 Solutiononp.458. InaGoodness-ofFittest,ifthep-valueis0.0113,ingeneral,donotrejectthenullhypothesis. Exercise11.11.36 Solutiononp.458. ForaChi-Squaredistributionwithdegreesoffreedomof17,theprobabilitythatavalueisgreater than20is0.7258. Exercise11.11.37 Solutiononp.458. If df = 2,thechi-squaredistributionhasashapethatremindsusoftheexponential. 11.12Review 12 Thenexttwoquestionsrefertothefollowingrealstudy: ArecentsurveyofU.S.teenagepregnancywasansweredby720girls,age12-19.6%ofthegirlssurveyed saidtheyhavebeenpregnant. ParadeMagazine WeareinterestedinthetrueproportionofU.S.girls,age 12-19,whohavebeenpregnant. Exercise11.12.1 Solutiononp.458. Findthe95%condenceintervalforthetrueproportionofU.S.girls,age12-19,whohavebeen pregnant. Exercise11.12.2 Solutiononp.458. Thereportalsostatedthattheresultsofthesurveyareaccuratetowithin 3.7%atthe95% condencelevel.Supposethatanewstudyistobedone.Itisdesiredtobeaccuratetowithin2% ofthe95%condencelevel.Whatwillhappentotheminimumnumberthatshouldbesurveyed? Exercise11.12.3 Given: X Exp 1 3 .Sketchthegraphthatdepicts: P X > 1 12 Thiscontentisavailableonlineat.

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444 CHAPTER11.THECHI-SQUAREDISTRIBUTION Thenextfourquestionsrefertothefollowinginformation: Supposethatthetimethatownerskeeptheircarspurchasednewisnormallydistributedwithamean of7yearsandastandarddeviationof2years.Weareinterestedinhowlonganindividualkeepshiscar purchasednew.Ourpopulationispeoplewhobuytheircarsnew. Exercise11.12.4 Solutiononp.458. 60%ofindividualskeeptheircars atmost howmanyyears? Exercise11.12.5 Solutiononp.458. Supposethatwerandomlysurveyoneperson.Findtheprobabilitythatpersonkeepshis/hercar lessthan 2.5years. Exercise11.12.6 Solutiononp.458. Ifwearetopickindividuals10atatime,ndthedistributionforthe average carlengthownership. Exercise11.12.7 Solutiononp.458. Ifwearetopick10individuals,ndtheprobabilitythatthe sum oftheirownershiptimeismore than55years. Exercise11.12.8 Solutiononp.458. Forwhichdistributionisthemediannotequaltothemean? A. Uniform B. Exponential C. Normal D. Student-t Exercise11.12.9 Solutiononp.458. Comparethestandardnormaldistributiontothestudent-tdistribution,centeredat0.Explain whichofthefollowingaretrueandwhicharefalse. a. Asthenumbersurveyedincreases,theareatotheleftof-1forthestudent-tdistribution approachestheareaforthestandardnormaldistribution. b. Asthenumbersurveyedincreases,theareatotheleftof-1forthestandardnormaldistributionapproachestheareaforthestudent-tdistribution. c. Asthedegreesoffreedomdecrease,thegraphofthestudent-tdistributionlooksmorelike thegraphofthestandardnormaldistribution. d. Ifthenumbersurveyedislessthan30,thenormaldistributionshouldneverbeused. Thenextvequestionsrefertothefollowinginformation: Weareinterestedinthecheckingaccountbalanceofatwenty-year-oldcollegestudent.Werandomly survey16twenty-year-oldcollegestudents.Weobtainasamplemeanof\$640andasamplestandard deviationof\$150.Let X =checkingaccountbalanceofanindividualtwentyyearoldcollegestudent. Exercise11.12.10 Explainwhywecannotdeterminethedistributionof X Exercise11.12.11 Solutiononp.458. Ifyouweretocreateacondenceintervalorperformahypothesistestforthepopulationaverage checkingaccountbalanceof20-yearoldcollegestudents,whatdistributionwouldyouuse? Exercise11.12.12 Solutiononp.458. Findthe95%condenceintervalforthetrueaveragecheckingaccountbalanceofatwenty-yearoldcollegestudent.

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446 CHAPTER11.THECHI-SQUAREDISTRIBUTION Figure11.3 Exercise11.12.20 Solutiononp.459. Findthenumberofresponsesthatwere". Exercise11.12.21 Solutiononp.459. Findthemean,standarddeviation,allfourquartilesandIQR. Exercise11.12.22 Solutiononp.459. BaseduponresearchatDeAnzaCollege,itisbelievedthatabout19%ofthestudentpopulation speaksalanguageotherthanEnglishathome. Supposethatastudywasdonethisyeartoseeifthatpercenthasdecreased.Ninety-eightstudents wererandomlysurveyedwiththefollowingresults.Fourteensaidthattheyspeakalanguage otherthanEnglishathome. a. Stateanappropriate null hypothesis. b. Stateanappropriate alternate hypothesis. c. DenetheRandomVariable, P '. d. Calculatetheteststatistic. e. Calculatethep-value. f. Atthe5%levelofdecision,whatisyourdecisionaboutthenullhypothesis? g. WhatistheTypeIerror? h. WhatistheTypeIIerror? Exercise11.12.23 Assumethatyouareanemergencyparamediccalledintorescuevictimsofanaccident.You needtohelpapatientwhoisbleedingprofusely.Thepatientisalsoconsideredtobeahighrisk forcontractingAIDS.Assumethatthenullhypothesisisthatthepatientdoes not havetheHIV virus.WhatisaTypeIerror? Exercise11.12.24 Solutiononp.459. ItisoftensaidthatCaliforniansaremorecasualthantherestofAmericans.Supposethata surveywasdonetoseeiftheproportionofCalifornianprofessionalsthatwearjeanstoworkis

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447 greaterthantheproportionofnon-Californianprofessionals.Fiftyofeachwassurveyedwiththe followingresults.10Californianswearjeanstoworkand4non-Californianswearjeanstowork. C =Californianprofessional NC =non-Californianprofessional a. Stateappropriate null and alternate hypotheses. b. DenetheRandomVariable. c. Calculatetheteststatisticandp-value. d. Atthe5%levelofdecision,doyouacceptorrejectthenullhypothesis? e. WhatistheTypeIerror? f. WhatistheTypeIIerror? Thenexttwoquestionsrefertothefollowinginformation: AgroupofStatisticsstudentshavedevelopedatechniquethattheyfeelwilllowertheiranxietylevelon statisticsexams.Theymeasuredtheiranxietylevelatthestartofthequarterandagainattheendofthe quarter.Recordedisthepaireddatainthatorder:,900;,1050;,700;,1100;, 900;,900. Exercise11.12.25 Solutiononp.459. Thisisatestofpickthebestanswer: A. largesamples,independentmeans B. smallsamples,independentmeans C. dependentmeans Exercise11.12.26 Solutiononp.459. Statethedistributiontouseforthetest.

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448 CHAPTER11.THECHI-SQUAREDISTRIBUTION 11.13Lab1:Chi-SquareGoodness-of-Fit 13 ClassTime: Names: 11.13.1StudentLearningOutcome: Thestudentwillevaluatedatacollectedtodetermineiftheyteithertheuniformorexponential distributions. 11.13.2CollecttheData Gotoyourlocalsupermarket.Ask30peopleastheyleaveforthetotalamountontheirgroceryreceipts. Or,ask3cashiersforthelast10amounts.Besuretoincludetheexpresslane,ifitisopen. 1.Recordthevalues. __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 2.Constructahistogramofthedata.Make5-6intervals.Sketchthegraphusingarulerandpencil. Scaletheaxes. 13 Thiscontentisavailableonlineat.

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449 Figure11.4 3.Calculatethefollowing: a. x = b. s = c. s 2 = 11.13.3UniformDistribution Testtoseeifgroceryreceiptsfollowtheuniformdistribution. 1.Usingyourlowestandhighestvalues, X U _______,_______ 2.Dividethedistributionaboveintofths. 3.Calculatethefollowing: a. Lowestvalue= b. 20thpercentile= c. 40thpercentile= d. 60thpercentile= e. 80thpercentile= f. Highestvalue= 4.Foreachfth,counttheobservednumberofreceiptsandrecordit.Thendeterminetheexpected numberofreceiptsandrecordthat.

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450 CHAPTER11.THECHI-SQUAREDISTRIBUTION Fifth Observed Expected 1st 2nd 3rd 4th 5th 5. H o : 6. H a : 7.Whatdistributionshouldyouuseforahypothesistest? 8.Whydidyouchoosethisdistribution? 9.Calculatetheteststatistic. 10.Findthep-value. 11.Sketchagraphofthesituation.Labelandscalethex-axis.Shadetheareacorrespondingtothepvalue. Figure11.5 12.Stateyourdecision. 13.Stateyourconclusioninacompletesentence.

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451 11.13.4ExponentialDistribution Testtoseeifgroceryreceiptsfollowtheexponentialdistributionwithdecayparameter 1 x 1.Using 1 x asthedecayparameter, X Exp _______ 2.Calculatethefollowing: a. Lowestvalue= b. Firstquartile= c. 37thpercentile= d. Median= e. 63rdpercentile= f. 3rdquartile= g. Highestvalue= 3.Foreachcell,counttheobservednumberofreceiptsandrecordit.Thendeterminetheexpected numberofreceiptsandrecordthat. Cell Observed Expected 1st 2nd 3rd 4th 5th 6th 4. H o 5. H a 6.Whatdistributionshouldyouuseforahypothesistest? 7.Whydidyouchoosethisdistribution? 8.Calculatetheteststatistic. 9.Findthep-value. 10.Sketchagraphofthesituation.Labelandscalethex-axis.Shadetheareacorrespondingtothepvalue.

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452 CHAPTER11.THECHI-SQUAREDISTRIBUTION Figure11.6 11.Stateyourdecision. 12.Stateyourconclusioninacompletesentence. 11.13.5DiscussionQuestions 1.Didyourdatateitherdistribution?Ifso,which? 2.Ingeneral,doyouthinkit'slikelythatdatacouldtmorethanonedistribution?Incompletesentences,explainwhyorwhynot.

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453 11.14Lab2:Chi-SquareTestforIndependence 14 ClassTime: Names: 11.14.1StudentLearningOutcome: Thestudentwillevaluateifthereisasignicantrelationshipbetweenfavoritetypeofsnackand gender. 11.14.2CollecttheData 1.Usingyourclassasasample,completethefollowingchart. Favoritetypeofsnack sweetscandy&bakedgoods icecream chips&pretzels fruits&vegetables Total male female Total 2.Lookingattheabovechart,doesitappeartoyouthatthereisdependencebetweengenderandfavoritetypeofsnackfood?Whyorwhynot? 11.14.3DeterminetheClassication Conductahypothesistesttodetermineifthefactorsareindependent 1. H o : 2. H a : 3.Whatdistributionshouldyouuseforahypothesistest? 4.Whydidyouchoosethisdistribution? 5.Calculatetheteststatistic. 6.Findthep-value. 7.Sketchagraphofthesituation.Labelandscalethex-axis.Shadetheareacorrespondingtothepvalue. 14 Thiscontentisavailableonlineat.

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454 CHAPTER11.THECHI-SQUAREDISTRIBUTION Figure11.7 8.Stateyourdecision. 9.Stateyourconclusioninacompletesentence. 11.14.4DiscussionQuestions 1.IstheconclusionofyourstudythesameasordifferentfromyouranswertoI2above? 2.Whydoyouthinkthatoccurred?

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455 SolutionstoExercisesinChapter11 Example11.7,Problem3p.427 a. E = rowtotalcolumntotal totalsurveyed = 8.19 b. 8 SolutionstoPractice1:Goodness-of-FitTest SolutiontoExercise11.8.4p.432 degreesoffreedom=4 SolutiontoExercise11.8.5p.432 951.69 SolutiontoExercise11.8.6p.432 0 SolutionstoPractice2:ContingencyTables SolutiontoExercise11.9.1p.433 12 SolutiontoExercise11.9.2p.433 10301.8 SolutiontoExercise11.9.3p.433 0 SolutiontoExercise11.9.4p.433 right SolutiontoExercise11.9.6p.434 a. Rejectthenullhypothesis SolutionstoPractice3:TestofaSingleVariance SolutiontoExercise11.10.2p.435 225 SolutiontoExercise11.10.6p.435 24 SolutiontoExercise11.10.7p.435 36 SolutiontoExercise11.10.8p.435 0.0549

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456 CHAPTER11.THECHI-SQUAREDISTRIBUTION SolutionstoHomework SolutiontoExercise11.11.3p.437 a. Thedatatsthedistribution b. Thedatadoesnottthedistribution c. 3 e. 19.27 f. 0.0002 h. Decision:RejectNull;Conclusion:Datadoesnottthedistribution. SolutiontoExercise11.11.5p.438 c. 3 e. 10.91 f. 0.0122 g. Decision:Rejectnullwhen a = 0 .05 ;Conclusion:Percentofloansdoesnottthedistribution. Decision:Donotrejectnullwhen a = 0 .01 ;ConclusionPercentofloanststhedistribution. SolutiontoExercise11.11.7p.438 c. 6 e. 27,876 f. 0 h. Decision:Rejectnull;Conclusion:L.B.doesnottL.A.C. SolutiontoExercise11.11.9p.439 c. 4 e. 10.53 f. 0.0324 h. Decision:Rejectnull;Conclusion:Bestskiareaandlevelofskierarenotindependent. SolutiontoExercise11.11.11p.440 c. 8 e. 33.55 f. 0 h. Decision:Rejectnull;Conclusion:Majorandstartingsalaryarenotindependentevents. SolutiontoExercise11.11.13p.440 c. 6 e. 25.21 f. 0.0003 h. Decision:Rejectnull SolutiontoExercise11.11.15p.441 c. 12 e. 125.74 f. 0 h. Decision:Rejectnull SolutiontoExercise11.11.17p.441 c. 83 d. 96.81

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457 e. 0.1426 g. Decision:Donotrejectnull;Conclusion:Thestandarddeviationisatmost0.5oz. h. Itdoesnotneedtobecalibrated SolutiontoExercise11.11.19p.442 c. 4 d. 4.52 e. 0.3402 g. Decision:Donotrejectnull. h. No SolutiontoExercise11.11.21p.442 c. 49 d. 54.37 e. 0.2774 g. Decision:Donotrejectnull;Conclusion:Thestandarddeviationisatmost0.75. h. No SolutiontoExercise11.11.23p.442 a. s 2 1 5 2 c. 48 d. 85.33 e. 0.0007 g. Decision:Rejectnull. h. Yes SolutiontoExercise11.11.24p.442 True SolutiontoExercise11.11.25p.442 False SolutiontoExercise11.11.26p.442 False SolutiontoExercise11.11.27p.443 True SolutiontoExercise11.11.28p.443 True SolutiontoExercise11.11.29p.443 False SolutiontoExercise11.11.30p.443 True SolutiontoExercise11.11.31p.443 True SolutiontoExercise11.11.32p.443 True SolutiontoExercise11.11.33p.443 True

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458 CHAPTER11.THECHI-SQUAREDISTRIBUTION SolutiontoExercise11.11.34p.443 True SolutiontoExercise11.11.35p.443 False SolutiontoExercise11.11.36p.443 False SolutiontoExercise11.11.37p.443 True SolutionstoReview SolutiontoExercise11.12.1p.443 0 .0424 ,0 .0770 SolutiontoExercise11.12.2p.443 2401 SolutiontoExercise11.12.4p.444 7.5 SolutiontoExercise11.12.5p.444 0.0122 SolutiontoExercise11.12.6p.444 N 7,0 .63 SolutiontoExercise11.12.7p.444 0.9911 SolutiontoExercise11.12.8p.444 B SolutiontoExercise11.12.9p.444 a. True b. False c. False d. False SolutiontoExercise11.12.11p.444 student-twith df = 15 SolutiontoExercise11.12.12p.444 560.07 719.93 SolutiontoExercise11.12.13p.445 quantitative-continuous SolutiontoExercise11.12.14p.445 quantitative-discrete SolutiontoExercise11.12.15p.445 b. P 4 c. 0.0183

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459 SolutiontoExercise11.12.16p.445 greaterthan SolutiontoExercise11.12.17p.445 No; P X = 8 = 0 .0348 SolutiontoExercise11.12.18p.445 Youwilllose\$5 SolutiontoExercise11.12.19p.445 Becca SolutiontoExercise11.12.20p.446 14 SolutiontoExercise11.12.21p.446 Mean=3.2 Quartiles=1.85,2,3,and5 IQR=3 SolutiontoExercise11.12.22p.446 d. z = )]TJ/F58 9.9626 Tf 8.194 0 Td [(1 .19 e. 0.1171 f. Donotrejectthenull SolutiontoExercise11.12.24p.446 c. z = 1 .73 ; p = 0 .0419 d. Rejectthenull SolutiontoExercise11.12.25p.447 C SolutiontoExercise11.12.26p.447 t 5

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460 CHAPTER11.THECHI-SQUAREDISTRIBUTION

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Chapter12 LinearRegressionandCorrelation 12.1LinearRegressionandCorrelation 1 12.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Discussbasicideasoflinearregressionandcorrelation. Createandinterpretalineofbestt. Calculateandinterpretthecorrelationcoefcient. Calculateandinterpretoutliers. 12.1.2Introduction Professionalsoftenwanttoknowhowtwoormorevariablesarerelated.Forexample,istherearelationship betweenthegradeonthesecondmathexamastudenttakesandthegradeonthenalexam?Ifthereisa relationship,whatisitandhowstrongistherelationship? Inanotherexample,yourincomemaybedeterminedbyyoureducation,yourprofession,youryearsof experience,andyourability.Theamountyoupayarepairpersonforlaborisoftendeterminedbyaninitial amountplusanhourlyfee.Theseareallexamplesinwhichregressioncanbeused. Thetypeofdatadescribedintheexamplesis bivariate data-"bi"fortwovariables.Inreality,statisticians use multivariate data,meaningmanyvariables. Inthischapter,youwillbestudyingthesimplestformofregression,"linearregression"withoneindependentvariable x .Thisinvolvesdatathattsalineintwodimensions.Youwillalsostudycorrelationwhich measureshowstrongtherelationshipis. 12.2LinearEquations 2 Linearregressionfortwovariablesisbasedonalinearequationwithoneindependentvariable.Ithasthe form: y = a + bx .1 1 Thiscontentisavailableonlineat. 2 Thiscontentisavailableonlineat. 461

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462 CHAPTER12.LINEARREGRESSIONANDCORRELATION where a and b areconstantnumbers. x istheindependentvariable,and y isthedependentvariable. Typically,youchooseavaluetosubstitute fortheindependentvariableandthensolveforthedependentvariable. Example12.1 Thefollowingexamplesarelinearequations. y = 3 + 2x .2 y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.01 + 1.2x .3 Thegraphofalinearequationoftheform y = a + bx isa straightline .Anylinethatisnotverticalcanbe describedbythisequation. Example12.2 Figure12.1: Graphoftheequation y = )]TJ/F58 8.9664 Tf 7.374 0 Td [(1 + 2x Linearequationsofthisformoccurinapplicationsoflifesciences,socialsciences,psychology,business, economics,physicalsciences,mathematics,andotherareas. Example12.3 Aaron'sWordProcessingServiceAWPSdoeswordprocessing.Itsrateis\$32perhourplusa \$31.50one-timecharge.Thetotalcosttoacustomerdependsonthenumberofhoursittakesto dothewordprocessingjob. Problem Findtheequationthatexpressesthe totalcost intermsofthe numberofhours requiredtonish thewordprocessingjob. Solution Let x =thenumberofhoursittakestogetthejobdone. Let y =thetotalcosttothecustomer. The\$31.50isaxedcost.Ifittakes x hourstocompletethejob,then 32 x isthecostofthe wordprocessingonly.Thetotalcostis: y = 31.50 + 32x

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463 12.3SlopeandY-InterceptofaLinearEquation 3 Forthelinearequation y = a + bx b =slopeand a =y-intercept. Fromalgebrarecallthattheslopeisanumberthatdescribesthesteepnessofalineandthey-interceptis theycoordinateofthepoint 0, a wherethelinecrossesthey-axis. a b c Figure12.2: Threepossiblegraphsof y = a + bx .aIf b > 0,thelineslopesupwardtotheright.bIf b = 0,thelineishorizontal.cIf b < 0,thelineslopesdownwardtotheright. Example12.4 Svetlanatutorstomakeextramoneyforcollege.Foreachtutoringsession,shechargesaone timefeeof\$25plus\$15perhouroftutoring.Alinearequationthatexpressesthetotalamountof moneySvetlanaearnsforeachsessionshetutorsis y = 25 + 15x Problem Whataretheindependentanddependentvariables?Whatisthey-interceptandwhatisthe slope?Interpretthemusingcompletesentences. Solution TheindependentvariablexisthenumberofhoursSvetlanatutorseachsession.Thedependent variableyistheamount,indollars,Svetlanaearnsforeachsession. They-interceptis25a=25.Atthestartofthetutoringsession,Svetlanachargesaone-timefee of\$25thisiswhenx=0.Theslopeis15b=15.Foreachsession,Svetlanaearns\$15foreach hourshetutors. 12.4ScatterPlots 4 Beforewetakeupthediscussionoflinearregressionandcorrelation,weneedtoexamineawaytodisplay therelationbetweentwovariables x and y .Themostcommonandeasiestwayisa scatterplot .The followingexampleillustratesascatterplot. 3 Thiscontentisavailableonlineat. 4 Thiscontentisavailableonlineat.

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464 CHAPTER12.LINEARREGRESSIONANDCORRELATION Example12.5 Fromanarticleinthe WallStreetJournal :InEuropeandAsia,m-commerceisbecomingmore popular.M-commerceusershavespecialmobilephonesthatworklikeelectronicwalletsaswellas providephoneandInternetservices.Userscandoeverythingfrompayingforparkingtobuying aTVsetorsodafromamachinetobankingtocheckingsportsscoresontheInternet.Inthenext fewyears,willtherebearelationshipbetweentheyearandthenumberofm-commerceusers? Constructascatterplot.Let x =theyearandlet y =thenumberofm-commerceusers,inmillions. x year y #ofusers 2000 0.5 2002 20.0 2003 33.0 2004 47.0 a b Figure12.3: aTableshowingthenumberofm-commerceusersinmillionsbyyear.bScatterplot showingthenumberofm-commerceusersinmillionsbyyear. Ascatterplotshowsthe direction and strength ofarelationshipbetweenthevariables.Acleardirection happenswhenthereiseither: Highvaluesofonevariableoccurringwithhighvaluesoftheothervariableorlowvaluesofone variableoccurringwithlowvaluesoftheothervariable. Highvaluesofonevariableoccurringwithlowvaluesoftheothervariable. Youcandeterminethestrengthoftherelationshipbylookingatthescatterplotandseeinghowclosethe pointsaretoaline,apowerfunction,anexponentialfunction,ortosomeothertypeoffunction. Whenyoulookatascatterplot,youwanttonoticethe overallpattern andany deviations fromthepattern. Thefollowingscatterplotexamplesillustratetheseconcepts.

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465 aPositiveLinearPatternStrong bLinearPatternw/OneDeviation Figure12.4 aNegativeLinearPatternStrong bNegativeLinearPatternWeak Figure12.5 aExponentialGrowthPattern bNoPattern Figure12.6 Inthischapter,weareinterestedinscatterplotsthatshowalinearpattern.Linearpatternsarequitecommon.Thelinearrelationshipisstrongifthepointsareclosetoastraightline.Ifwethinkthatthepoints showalinearrelationship,wewouldliketodrawalineonthescatterplot.Thislinecanbecalculated throughaprocesscalled linearregression .However,weonlycalculatearegressionlineifoneofthevariableshelpstoexplainorpredicttheothervariable.If x istheindependentvariableand y thedependent variable,thenwecanusearegressionlinetopredict y foragivenvalueof x .

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466 CHAPTER12.LINEARREGRESSIONANDCORRELATION 12.5TheRegressionEquation 5 Datararelytsastraightlineexactly.Usually,youmustbesatisedwithroughpredictions.Typically,you haveasetofdatawhosescatterplotappearsto "t" astraightline.Thisiscalleda LineofBestFitorLeast SquaresLine 12.5.1OptionalCollaborativeClassroomActivity Ifyouknowaperson'spinkysmallestngerlength,doyouthinkyoucouldpredictthatperson'sheight? Collectdatafromyourclasspinkyngerlength,ininches.Theindependentvariable, x ,ispinkynger lengthandthedependentvariable, y ,isheight. Foreachsetofdata,plotthepointsongraphpaper.Makeyourgraphbigenoughand usearuler .Then "byeye"drawalinethatappearsto"t"thedata.Foryourline,picktwoconvenientpointsandusethem tondtheslopeoftheline.Findthey-interceptofthelinebyextendingyourlinessotheycrossthey-axis. Usingtheslopesandthey-intercepts,writeyourequationof"bestt".Doyouthinkeveryonewillhave thesameequation?Whyorwhynot? Usingyourequation,whatisthepredictedheightforapinkylengthof2.5inches? Example12.6 Arandomsampleof11statisticsstudentsproducedthefollowingdatawhere x isthethirdexam score,outof80,and y isthenalexamscore,outof200.Canyoupredictthenalexamscoreofa randomstudentifyouknowthethirdexamscore? 5 Thiscontentisavailableonlineat.

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467 xthirdexamscore ynalexamscore 65 175 67 133 71 185 71 163 66 126 75 198 67 153 70 163 71 159 69 151 69 159 a b Figure12.7: aTableshowingthescoresonthenalexambasedonscoresfromthethirdexam.bScatter plotshowingthescoresonthenalexambasedonscoresfromthethirdexam. Thethirdexamscore, x ,istheindependentvariableandthenalexamscore, y ,isthedependent variable.Wewillplotaregressionlinethatbest"ts"thedata.Ifeachofyouweretotaline"by eye",youwoulddrawdifferentlines.Wecanusewhatiscalleda least-squaresregressionline to obtainthebesttline. Considerthediagramshown.Eachpointofdataisofthetheform x y andeachpointofthelineofbest tusingleast-squareslinearregressionhastheform x y The y isread "yhat" andisthe estimatedvalueof y .Itisthevalueof y obtainedusingtheregressionline. Itisnotgenerallyequalto y fromdata.

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468 CHAPTER12.LINEARREGRESSIONANDCORRELATION Figure12.8 Theterm j y 0 )]TJ/F58 9.9626 Tf 12.16 0.144 Td [( y 0 j = e 0 iscalledthe "error"orresidual .Itisnotanerrorinthesenseofamistake,but measurestheverticaldistancebetweentheactualvalueof y andtheestimatedvalueof y e =theGreekletter epsilon Foreachdatapoint,youcancalculate, j y i )]TJ/F58 9.9626 Tf 11.835 0.144 Td [( y i j = e i for i = 1,2,3,...,11 Each e isaverticaldistance. Fortheexampleaboutthethirdexamscoresandthenalexamscoresforthe11statisticsstudents,there are11datapoints.Therefore,thereare11 e values.Ifyousquareeach e andadd,youget e 1 2 + e 2 2 + ... + e 11 2 = 11 S i=1 e 2 Thisiscalledthe SumofSquaredErrorsSSE Usingcalculus,youcanmakethe SSE aminimum.Whenyoumakethe SSE aminimum,youhavedeterminedthepointsthatareonthelineofbestt.Itturnsoutthatthelineofbestthastheequation: y = a + bx .4 where a = y )]TJ/F132 9.9626 Tf 10.255 0 Td [(b x and b = S x )]TJETq1 0 0 1 218.598 220.439 cm[]0 d 0 J 0.303 w 0 0 m 4.179 0 l SQBT/F132 7.5716 Tf 218.822 215.718 Td [(x y )]TJETq1 0 0 1 241.949 220.439 cm[]0 d 0 J 0.303 w 0 0 m 3.975 0 l SQBT/F132 7.5716 Tf 242.044 215.718 Td [(y S x )]TJETq1 0 0 1 230.171 211.309 cm[]0 d 0 J 0.303 w 0 0 m 4.179 0 l SQBT/F132 7.5716 Tf 230.395 206.588 Td [(x 2 x and y aretheaveragesofthe x valuesandthe y values,respectively.Thebesttlinealwayspasses throughthepoint x y Theslope b canbewrittenas b = r s y s x where s y =thestandarddeviationofthe y valuesand s x =the standarddeviationofthe x values. r isthecorrelationcoefcientwhichisdiscussedinthenextsection. N OTE :Manycalculatorsoranylinearregressionandcorrelationcomputerprogramcancalculate thebesttline.Thecalculationstendtobetediousifdonebyhand. Inthetechnologysection, thereareinstructionsforcalculatingthebesttline.

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469 Thegraphofthelineofbesttforthethirdexam/nalexamexampleisshownbelow: Figure12.9 Remember,thebesttlineiscalledthe leastsquaresregressionline itissometimesreferredtoasthe LSL whichisanacronymforleastsquaresline.Thebesttlineforthethirdexam/nalexamexamplehasthe equation: y = )]TJ/F58 9.9626 Tf 8.195 0 Td [(173.51 + 4.83x .5 Theideabehindndingthebesttlineisbasedontheassumptionthatthedataareactuallyscatteredabout astraightline.Remember,itisalwaysimportanttoplotascatterdiagramrstwhichmanycalculatorsand computerprogramscandotoseeifitisworthcalculatingthelineofbestt. Theslopeofthelineis4.83b=4.83.Wecaninterprettheslopeasfollows:Asthethirdexamscore increasesbyonepoint,thenalexamscoreincreasesby4.83points. N OTE :Ifthescatterplotindicatesthatthereisalinearrelationshipbetweenthevariables,thenit isreasonabletouseabesttlinetomakepredictionsfor y given x withinthedomainofx-values inthesampledata, butnotnecessarilyforx-valuesoutsidethatdomain. 12.6TheCorrelationCoefcient 6 Besideslookingatthescatterplotandseeingthatalineseemsreasonable,howcanyoutellifthelineisa goodpredictor?Usethecorrelationcoefcientasanotherindicatorbesidesthescatterplotofthestrength oftherelationshipbetween x and y .Thecorrelationcoefcient, r ,isdenedas: r = n S x y )]TJ/F142 7.8896 Tf 6.322 -0.079 Td [( S x S y q [ n S x 2 )]TJ/F142 7.8896 Tf 6.322 -0.079 Td [( S x 2 ] [ n S y 2 )]TJ/F142 7.8896 Tf 6.323 -0.079 Td [( S y 2 ] where: 6 Thiscontentisavailableonlineat.

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470 CHAPTER12.LINEARREGRESSIONANDCORRELATION )]TJ/F58 9.9626 Tf 21.105 0 Td [(1 r 1 n =thenumberofdatapoints Ifyoususpectalinearrelationshipbetween x and y ,then r canmeasurehowstrongitis. If r = 1,thereisperfectpositivecorrelation.If r = )]TJ/F58 9.9626 Tf 8.194 0 Td [(1,thereisperfectnegativecorrelation.Inboththese cases,theoriginaldatapointslieonastraightline.Ofcourse,intherealworld,thiswillnotgenerally happen. Theformulafor r looksformidable.However,manycalculatorsandanyregressionandcorrelationcomputerprogramcancalculate r .Thesignof r isthesameastheslope, b ,ofthebesttline. 12.7FactsAbouttheCorrelationCoefcientforLinearRegression 7 Apositive r meansthatwhen x increases, y increasesandwhen x decreases, y decreases positive correlation Anegative r meansthatwhen x increases, y decreasesandwhen x decreases, y increases negative correlation An r ofzeromeansthereisabsolutelynolinearrelationshipbetween x and y nocorrelation Highcorrelationdoesnotsuggestthat x causes y or y causes x .Wesay "correlationdoesnotimply causation." Forexample,everypersonwholearnedmathinthe17thcenturyisdead.However, learningmathdoesnotnecessarilycausedeath! aPositiveCorrelation bNegativeCorrelation cZeroCorrelation Figure12.10: aAscatterplotshowingdatawithapositivecorrelation.bAscatterplotshowingdata withanegativecorrelation.cAscatterplotshowingdatawithzerocorrelation. 7 Thiscontentisavailableonlineat.

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471 The95%CriticalValuesoftheSampleCorrelationCoefcientTableSection12.10attheendofthischapter beforetheSummarySection12.11 maybeusedtogiveyouagoodideaofwhetherthecomputedvalue of r issignicantornot .Compare r totheappropriatecriticalvalueinthetable.If r issignicant,thenyou maywanttousethelineforprediction. Example12.7 Supposeyoucomputed r = 0.801using n = 10datapoints. df = n )]TJ/F58 9.9626 Tf 10.554 0 Td [(2 = 10 )]TJ/F58 9.9626 Tf 10.554 0 Td [(2 = 8.The criticalvaluesassociatedwith df = 8are-0.632and+0.632.If r < negativecriticalvalue or r > positivecriticalvalue ,then r issignicant.Since r = 0.801and0.801 > 0.632, r issignicantandthe linemaybeusedforprediction.Ifyouviewthisexampleonanumberline,itwillhelpyou. Figure12.11: r isnotsignicantbetween-0.632and+0.632. r = 0.801 > + 0.632.Therefore, r issignicant. Example12.8 Supposeyoucomputed r = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.624with14datapoints. df = 14 )]TJ/F58 9.9626 Tf 10.175 0 Td [(2 = 12.Thecriticalvaluesare -0.532and0.532.Since )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.624 < )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.532, r issignicantandthelinemaybeusedforprediction Figure12.12: r = )]TJ/F58 8.9664 Tf 7.374 0 Td [(0.624 < )]TJ/F58 8.9664 Tf 7.374 0 Td [(0.532.Therefore, r issignicant. Example12.9 Supposeyoucomputed r = 0.776and n = 6. df = 6 )]TJ/F58 9.9626 Tf 10.531 0 Td [(2 = 4.Thecriticalvaluesare-0.811 and0.811.Since )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.811 < 0.776 < 0.811, r isnotsignicantandthelineshouldnotbeusedfor prediction. Figure12.13: )]TJ/F58 8.9664 Tf 7.375 0 Td [(0.811 < r = 0.776 < 0.811.Therefore, r isnotsignicant. N OTE :If r is-1or r is+1,thenallthedatapointslieexactlyonastraightline.Ifthelineis signicant,then withintherangeofthex-values, thelinecanbeusedtopredicta y value.Asan illustration,considerthethirdexam/nalexamexample.Thelineofbesttis: y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(173.51 + 4.83x with r = 0.6631 Canthelinebeusedforprediction? Givenathirdexamscore x value,canwesuccessfullypredictthe nalexamscorepredicted y value. Test r = 0.6631withitsappropriatecriticalvalue.

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472 CHAPTER12.LINEARREGRESSIONANDCORRELATION Usingthetablewith df = 11 )]TJ/F58 9.9626 Tf 10.366 0 Td [(2 = 9,thecriticalvaluesare-0.602and+0.602.Since0.6631 > 0.602, r is signicant. Because r issignicantandthescatterplotshowsareasonablelineartrend,thelinecanbeused topredictnalexamscores. Example12.10 Supposeyoucomputedthefollowingcorrelationcoefcients.Usingthetableattheendofthe chapter,determineif r issignicantandthelineofbesttassociatedwitheach r canbeusedto predicta y value.Ifithelps,drawanumberline. r = )]TJ/F58 9.9626 Tf 8.195 0 Td [(0.567andthesamplesize, n ,is19.The df = n )]TJ/F58 9.9626 Tf 10.318 0 Td [(2 = 17.Thecriticalvalueis-0.456. )]TJ/F58 9.9626 Tf 8.195 0 Td [(0.567 < )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.456so r issignicant. r = 0.708andthesamplesize, n ,is9.The df = n )]TJ/F58 9.9626 Tf 10.582 0 Td [(2 = 7.Thecriticalvalueis0.666. 0.708 > 0.666so r issignicant. r = 0.134andthesamplesize, n ,is14.The df = 14 )]TJ/F58 9.9626 Tf 9.957 0 Td [(2 = 12.Thecriticalvalueis0.532.0.134 isbetween-0.532and0.532so r isnotsignicant. r = 0andthesamplesize, n ,is5.Nomatterwhatthedfsare, r = 0isbetweenthetwo criticalvaluesso r isnotsignicant. 12.8Prediction 8 Theexamscores x -values rangefrom65to75.Supposeyouwanttoknowthenalexamscoreofstatistics studentswhoreceived73onthethirdexam. Since73isbetweenthe x -values65and75 ,substitute x = 73 intotheequation.Then: y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(173.51 + 4.83 73 = 179.08.7 Wepredictthatastatisticsstudentwhoreceivesa73onthethirdexamwillreceive179.08onthenal exam. Remember,donotusetheregressionequationtopredictvaluesoutsidethedomainof x Example12.11 Recallthethirdexam/nalexamexample. Problem1 Whatwouldyoupredictthenalexamscoretobeforastudentwhoscoreda66onthethird exam? Solution 145.27 Problem2 Whatwouldyoupredictthenalexamscoretobeforastudentwhoscoreda78onthethird exam? 8 Thiscontentisavailableonlineat.

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473 12.9Outliers 9 Insomedatasets,therearevalues points called outliers Outliersarepointsthatarefarfromtheleast squaresline. Theyhavelarge"errors."Outliersneedtobeexaminedclosely.Sometimes,forsomereason oranother,theyshouldnotbeincludedintheanalysisofthedata.Itispossiblethatanoutlierisaresultof erroneousdata.Othertimes,anoutliermayholdvaluableinformationaboutthepopulationunderstudy. Thekeyistocarefullyexaminewhatcausesadatapointtobeanoutlier. Example12.12 Inthethirdexam/nalexamexample,youcandetermineifthereisanoutlierornot.Ifthereis one,asanexercise,deleteitandttheremainingdatatoanewline.Forthisexample,thenewline oughttottheremainingdatabetter.Thismeansthe SSE shouldbesmallerandthecorrelation coefcientoughttobecloserto1or-1. Solution Computersandmanycalculatorscandetermineoutliersfromthedata.However,asanexercise, wewillgothroughthestepsthatareneededtocalculateanoutlier.Inthetablebelow,therst twocolumnsarethethirdexamandthenalexamdata.Thethirdcolumnshowsthey-hatvalues calculatedfromthelineofbestt. x y y 65 175 140 67 133 150 71 185 169 71 163 169 66 126 145 75 198 189 67 153 150 70 163 164 71 159 169 69 151 160 69 159 160 A Residual isthe Actualyvalue )]TJ/F132 9.9626 Tf 10.131 0 Td [(predictedyvalue = y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y Calculatetheabsolutevalueofeachresidual. Calculateeach j y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y j : 9 Thiscontentisavailableonlineat.

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474 CHAPTER12.LINEARREGRESSIONANDCORRELATION x y y j y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y j 65 175 140 j 175 )]TJ/F58 9.9626 Tf 10.131 0 Td [(140 j = 35 67 133 150 j 133 )]TJ/F58 9.9626 Tf 10.131 0 Td [(150 j = 17 71 185 169 j 185 )]TJ/F58 9.9626 Tf 10.131 0 Td [(169 j = 16 71 163 169 j 163 )]TJ/F58 9.9626 Tf 10.131 0 Td [(169 j = 6 66 126 145 j 126 )]TJ/F58 9.9626 Tf 10.131 0 Td [(145 j = 19 75 198 189 j 198 )]TJ/F58 9.9626 Tf 10.131 0 Td [(189 j = 9 67 153 150 j 153 )]TJ/F58 9.9626 Tf 10.131 0 Td [(150 j = 3 70 163 164 j 163 )]TJ/F58 9.9626 Tf 10.131 0 Td [(164 j = 1 71 159 169 j 159 )]TJ/F58 9.9626 Tf 10.131 0 Td [(169 j = 10 69 151 160 j 151 )]TJ/F58 9.9626 Tf 10.131 0 Td [(160 j = 9 69 159 160 j 159 )]TJ/F58 9.9626 Tf 10.131 0 Td [(160 j = 1 Squareeach j y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y j : 35 2 ;17 2 ;16 2 ;6 2 ;19 2 ;9 2 ;3 2 ;1 2 ;10 2 ;9 2 ;1 2 Then,addsumallthe j y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y j squaredterms: 11 S i=1 j y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y j 2 = 11 S i=1 e 2 Recallthat j y i )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y i j = e i = 35 2 + 17 2 + 16 2 + 6 2 + 19 2 + 9 2 + 3 2 + 1 2 + 10 2 + 9 2 + 1 2 = 2440 = SSE Next,calculate s ,thestandarddeviationofallthe j y )]TJ/F58 9.9626 Tf 11.962 0.145 Td [( y j = e valueswhere n =thetotalnumber ofdatapoints. Calculatethestandarddeviationof35;17;16;6;19;9;3;1;10;9;1. s = q SSE n )]TJ/F58 7.5716 Tf 6.228 0 Td [(2 Forthethirdexam/nalexamproblem, s = q 2440 11 )]TJ/F58 7.5716 Tf 6.228 0 Td [(2 = 16.47 Next,multiply s by1.9andget 1.9 16.47 = 31.29thevalue31.29isalmost2standarddeviationsawayfromthemeanofthe j y )]TJ/F58 9.9626 Tf 11.835 0.144 Td [( y j values. N OTE :Thenumber1.9 s isequalto 1.9standarddeviations .Itisameasurethatisalmost2 standarddeviations.Ifweweretomeasuretheverticaldistancefromanydatapointtothecorrespondingpointonthelineofbesttandthatdistancewasequalto1.9 s orgreater,thenwewould considerthedatapointtobe"toofar"fromthelineofbestt.Wewouldcallthatpointa potential outlier Fortheexample,ifanyofthe j y )]TJ/F58 9.9626 Tf 11.813 0.144 Td [( y j valuesare atleast 31.29,thecorresponding x y pointdata pointisapotentialoutlier. Mathematically,wesaythatif j y )]TJ/F58 9.9626 Tf 11.835 0.144 Td [( y j 1.9 s ,thenthecorrespondingpointisanoutlier.

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475 Forthethirdexam/nalexamproblem,allthe j y )]TJ/F58 9.9626 Tf 11.873 0.145 Td [( y j 'sarelessthan31.29exceptfortherstone whichis35. 35 > 31.29Thatis, j y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y j 1.9 s Thepointwhichcorrespondsto j y )]TJ/F58 9.9626 Tf 11.468 0.144 Td [( y j = 35is 65,175 Therefore,thepoint 65,175 isanoutlier. Forthisexample,wewilldeleteit.Remember,wedonotalwaysdeleteanoutlier.Thenextstep istocomputeanewbest-tlineusingthe10remainingpoints.Thenewlineofbesttandthe correlationcoefcientare: y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(355.19 + 7.39x and r = 0.9121 Ifyoucompare r = 0.9121toitscriticalvalue0.632,0.9121 > 0.632.Therefore, r issignicant.In fact, r = 0.9121isabetter r thantheoriginal.6631because r = 0.9121iscloserto1.Thismeans thatthe10pointstthelinebetter.Thelinecanbetterpredictthenalexamscoregiventhethird examscore. Example12.13 Usingthenewlineofbesttcalculatedwith10points,whatwouldastudentwhoreceivesa73 onthethirdexamexpecttoreceiveonthenalexam? Example12.14 FromTheConsumerPriceIndexesWebsite TheConsumerPriceIndexCPImeasurestheaveragechangeovertimeinthepricespaidbyurbanconsumersforconsumergoodsandservices.The CPIaffectsnearlyallAmericansbecauseofthemanywaysitisused.Oneofitsbiggestusesisas ameasureofination.ByprovidinginformationaboutpricechangesintheNation'seconomyto government,business,andlabor,theCPIhelpsthemtomakeeconomicdecisions.ThePresident, Congress,andtheFederalReserveBoardusetheCPI'strendstoformulatemonetaryandscal policies.

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476 CHAPTER12.LINEARREGRESSIONANDCORRELATION Data: x y 1915 10.1 1926 17.7 1935 13.7 1940 14.7 1947 24.1 1952 26.5 1964 31.0 1969 36.7 1975 49.3 1979 72.6 1980 82.4 1986 109.6 1991 130.7 1999 166.6 Problem Makeascatterplotofthedata. Calculatetheleastsquaresline.Writetheequationintheform y = a + bx Drawthelineonthescatterplot. Findthecorrelationcoefcient.Isitsignicant? WhatistheaverageCPIfortheyear1990? Solution Scatterplotandlineofbestt. y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(3204 + 1.662x istheequationofthelineofbestt. r = 0.8694 Thenumberofdatapointsis n = 14.Usethe95%CriticalValuesoftheSampleCorrelation CoefcienttableattheendofChapter12. n )]TJ/F58 9.9626 Tf 10.37 0 Td [(2 = 12.Thecorrespondingcriticalvalueis 0.532.Since0.8694 > 0.532, r issignicant. y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(3204 + 1.662 1990 = 103.4CPI

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477 Figure12.14

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478 CHAPTER12.LINEARREGRESSIONANDCORRELATION 12.1095%CriticalValuesoftheSampleCorrelationCoefcientTable 10 DegreesofFreedom: n )]TJ/F58 9.9626 Tf 10.131 0 Td [(2 CriticalValues: + and )]TJ/F63 9.9626 Tf 8.194 0 Td [( 1 0.997 2 0.950 3 0.878 4 0.811 5 0.754 6 0.707 7 0.666 8 0.632 9 0.602 10 0.576 11 0.555 12 0.532 13 0.514 14 0.497 15 0.482 16 0.468 17 0.456 18 0.444 19 0.433 20 0.423 21 0.413 22 0.404 23 0.396 24 0.388 25 0.381 continuedonnextpage 10 Thiscontentisavailableonlineat.

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479 26 0.374 27 0.367 28 0.361 29 0.355 30 0.349 40 0.304 50 0.273 60 0.250 70 0.232 80 0.217 90 0.205 100andover 0.195

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480 CHAPTER12.LINEARREGRESSIONANDCORRELATION 12.11Summary 11 BivariateData: Eachdatapointhastwovalues.Theformis x y LineofBestFitorLeastSquaresLineLSL: y = a + bx x =independentvariable; y =dependentvariable Residual: Actualyvalue )]TJ/F132 9.9626 Tf 10.132 0 Td [(predictedyvalue = y )]TJ/F58 9.9626 Tf 11.835 0.145 Td [( y CorrelationCoefcientr: 1.Usedtodeterminewhetheralineofbesttisgoodforprediction. 2.Between-1and1inclusive.Thecloser r isto1or-1,theclosertheoriginalpointsaretoastraightline. 3.If r isnegative,theslopeisnegative.If r ispositive,theslopeispositive. 4.If r = 0,thenthelineishorizontal. SumofSquaredErrorsSSE: Thesmallerthe SSE ,thebettertheoriginalsetofpointststhelineofbestt. Outlier: Apointthatdoesnotseemtottherestofthedata. 11 Thiscontentisavailableonlineat.

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481 12.12Practice:LinearRegression 12 12.12.1StudentLearningOutcomes Thestudentwillexplorethepropertiesoflinearregression. 12.12.2Given Thedatabelowarereal.Keepinmindthattheseareonlyreportedgures. Source:CentersforDisease ControlandPrevention,NationalCenterforHIV,STD,andTBPrevention,October24,2003 AdultsandAdolescentsonly,UnitedStates Year #AIDScasesdiagnosed #AIDSdeaths Pre-1981 91 29 1981 319 121 1982 1,170 453 1983 3,076 1,482 1984 6,240 3,466 1985 11,776 6,878 1986 19,032 11,987 1987 28,564 16,162 1988 35,447 20,868 1989 42,674 27,591 1990 48,634 31,335 1991 59,660 36,560 1992 78,530 41,055 1993 78,834 44,730 1994 71,874 49,095 1995 68,505 49,456 1996 59,347 38,510 1997 47,149 20,736 1998 38,393 19,005 1999 25,174 18,454 2000 25,522 17,347 2001 25,643 17,402 2002 26,464 16,371 Total 802,118 489,093 12 Thiscontentisavailableonlineat.

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482 CHAPTER12.LINEARREGRESSIONANDCORRELATION N OTE :Wewillusethecolumnsyearand#AIDScasesdiagnosedforallquestionsunless otherwisestated. 12.12.3Graphing Graphyearvs.#AIDScasesdiagnosed. Plotthepointsonthegraphlocatedbelowinthesectiontitled "Plot" .Donotincludepre-1981.Labelbothaxeswithwords.Scalebothaxes. 12.12.4Data Exercise12.12.1 Enteryourdataintoyourcalculatororcomputer.Thepre-1981datashouldnotbeincluded.Why isthatso? 12.12.5LinearEquation Writethelinearequationbelow,roundingto4decimalplaces: Exercise12.12.2 Solutiononp.508. Calculatethefollowing: a. a = b. b = c. corr. = d. n = #ofpairs Exercise12.12.3 Solutiononp.508. equation: y = 12.12.6Solve Exercise12.12.4 Solutiononp.508. Solve. a -When x = 1985 y = b -When x = 1990 y = 12.12.7Plot Plotthe2abovepointsonthegraphbelow.Then,connectthe2pointstoformtheregressionline.

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483 Obtainthegraphonyourcalculatororcomputer. 12.12.8DiscussionQuestions Lookatthegraphabove. Exercise12.12.5 Doesthelineseemtotthedata?Whyorwhynot? Exercise12.12.6 Doyouthinkalineartisbest?Whyorwhynot? Exercise12.12.7 Handdrawasmoothcurveonthegraphabovethatshowstheowofthedata. Exercise12.12.8 Whatdoesthecorrelationimplyabouttherelationshipbetweentimeyearsandthenumberof diagnosedAIDScasesreportedintheU.S.? Exercise12.12.9 Whyisyeartheindependentvariableand#AIDScasesdiagnosed.thedependentvariable insteadofthereverse? Exercise12.12.10 Solutiononp.508. Solve. a. When x = 1970 y = : b. Whydoesn'tthisanswermakesense?

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484 CHAPTER12.LINEARREGRESSIONANDCORRELATION 12.13Homework 13 Exercise12.13.1 Solutiononp.508. Foreachsituationbelow,statetheindependentvariableandthedependentvariable. a. Astudyisdonetodetermineifelderlydriversareinvolvedinmoremotorvehiclefatalitiesthanallotherdrivers.Thenumberoffatalitiesper100,000driversiscomparedto theageofdrivers. b. Astudyisdonetodetermineiftheweeklygrocerybillchangesbasedonthenumberof familymembers. c. Insurancecompaniesbaselifeinsurancepremiumspartiallyontheageoftheapplicant. d. Utilitybillsvaryaccordingtopowerconsumption. e. Astudyisdonetodetermineifahighereducationreducesthecrimerateinapopulation. Exercise12.13.2 In1990thenumberofdriverdeathsper100,000forthedifferentagegroupswasasfollows Source: TheNationalHighwayTrafcSafetyAdministration'sNationalCenterforStatisticsand Analysis : Age NumberofDriverDeathsper100,000 15-24 28 25-39 15 40-69 10 70-79 15 80+ 25 a. Foreachagegroup,pickthemidpointoftheintervalforthexvalue.Forthe80+group, use85. b. UsingagesastheindependentvariableandNumberofdriverdeathsper100,000as thedependentvariable,makeascatterplotofthedata. c. Calculatetheleastsquaresbesttline.Puttheequationintheformof: y = a + bx d. Findthecorrelationcoefcient.Isitsignicant? e. Picktwoagesandndtheestimatedfatalityrates. f. Usethetwopointsinetoplottheleastsquareslineonyourgraphfromb. g. Basedontheabovedata,istherealinearrelationshipbetweenageofadriveranddriver fatalityrate? h. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.3 Solutiononp.508. Theaveragenumberofpeopleinafamilythatreceivedwelfareforvariousyearsisgivenbelow. Source: HouseWaysandMeansCommittee,HealthandHumanServicesDepartment 13 Thiscontentisavailableonlineat.

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485 Year Welfarefamilysize 1969 4.0 1973 3.6 1975 3.2 1979 3.0 1983 3.0 1988 3.0 1991 2.9 a. Usingyearastheindependentvariableandwelfarefamilysizeasthedependent variable,makeascatterplotofthedata. b. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx c. Findthecorrelationcoefcient.Isitsignicant? d. Picktwoyearsbetween1969and1991andndtheestimatedwelfarefamilysizes. e. Usethetwopointsindtoplottheleastsquareslineonyourgraphfromb. f. Basedontheabovedata,istherealinearrelationshipbetweentheyearandtheaverage numberofpeopleinawelfarefamily? g. Usingtheleastsquaresline,estimatethewelfarefamilysizesfor1960and1995.Doesthe leastsquareslinegiveanaccurateestimateforthoseyears?Explainwhyorwhynot. h. Arethereanyoutliersintheabovedata? i. Whatistheestimatedaveragewelfarefamilysizefor1986?Doestheleastsquaresline giveanaccurateestimateforthatyear?Explainwhyorwhynot. j. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.4 UsetheAIDSdatafromthepracticeforthissectionSection12.12.2:Given,butthistimeusethe columnsyear#and#newAIDSdeathsinU.S.Answerallofthequestionsfromthepractice again,usingthenewcolumns. Exercise12.13.5 Solutiononp.508. TheheightsidewalktoroofofnotabletallbuildingsinAmericaiscomparedtothenumberof storiesofthebuildingbeginningatstreetlevel.Source: MicrosoftBookshelf Heightinfeet Stories 1050 57 428 28 362 26 529 40 790 60 401 22 380 38 1454 110 1127 100 700 46

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486 CHAPTER12.LINEARREGRESSIONANDCORRELATION a. Usingstoriesastheindependentvariableandheightasthedependentvariable,make ascatterplotofthedata. b. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables? c. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx d. Findthecorrelationcoefcient.Isitsignicant? e. Findtheestimatedheightsfor32storiesandfor94stories. f. Usethetwopointsinetoplottheleastsquareslineonyourgraphfromb. g. Basedontheabovedata,istherealinearrelationshipbetweenthenumberofstoriesin tallbuildingsandtheheightofthebuildings? h. Arethereanyoutliersintheabovedata?Ifso,whichpoints? i. Whatistheestimatedheightofabuildingwith6stories?Doestheleastsquareslinegive anaccurateestimateofheight?Explainwhyorwhynot. j. Basedontheleastsquaresline,addinganextrastoryaddsabouthowmanyfeettoa building? k. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.6 BelowisthelifeexpectancyforanindividualbornintheUnitedStatesincertainyears.Source: NationalCenterforHealthStatistics YearofBirth LifeExpectancy 1930 59.7 1940 62.9 1950 70.2 1965 69.7 1973 71.4 1982 74.5 1987 75 1992 75.7 a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable. b. Drawascatterplotoftheorderedpairs. c. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx d. Findthecorrelationcoefcient.Isitsignicant? e. Findtheestimatedlifeexpectancyforanindividualbornin1950andforonebornin1982. f. Whyaren'ttheanswerstopartethevaluesontheabovechartthatcorrespondtothose years? g. Usethetwopointsinetoplottheleastsquareslineonyourgraphfromb. h. Basedontheabovedata,istherealinearrelationshipbetweentheyearofbirthandlife expectancy? i. Arethereanyoutliersintheabovedata? j. Usingtheleastsquaresline,ndtheestimatedlifeexpectancyforanindividualbornin 1850.Doestheleastsquareslinegiveanaccurateestimateforthatyear?Explainwhy orwhynot. k. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope.

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487 Exercise12.13.7 Solutiononp.509. Thepercentoffemalewageandsalaryworkerswhoarepaidhourlyratesisgivenbelowforthe years1979-1992.Source: BureauofLaborStatistics,U.S.Dept.ofLabor Year Percentofworkerspaidhourlyrates 1979 61.2 1980 60.7 1981 61.3 1982 61.3 1983 61.8 1984 61.7 1985 61.8 1986 62.0 1987 62.7 1990 62.8 1992 62.9 a. Usingyearastheindependentvariableandpercentasthedependentvariable,make ascatterplotofthedata. b. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? c. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx d. Findthecorrelationcoefcient.Isitsignicant? e. Findtheestimatedpercentsfor1991and1988. f. Usethetwopointsinetoplottheleastsquareslineonyourgraphfromb. g. Basedontheabovedata,istherealinearrelationshipbetweentheyearandthepercentof femalewageandsalaryearnerswhoarepaidhourlyrates? h. Arethereanyoutliersintheabovedata? i. Whatistheestimatedpercentfortheyear2050?Doestheleastsquareslinegiveanaccurateestimateforthatyear?Explainwhyorwhynot? j. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.8 ThemaximumdiscountvalueoftheEntertainmentcardfortheFineDiningsection,Edition 10,forvariouspagesisgivenbelow.

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488 CHAPTER12.LINEARREGRESSIONANDCORRELATION Pagenumber Maximumvalue\$ 4 16 14 19 25 15 32 17 43 19 57 15 72 16 85 15 90 17 a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable. b. Drawascatterplotoftheorderedpairs. c. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx d. Findthecorrelationcoefcient.Isitsignicant? e. Findtheestimatedmaximumvaluesfortherestaurantsonpage10andonpage70. f. Usethetwopointsinetoplottheleastsquareslineonyourgraphfromb. g. Doesitappearthattherestaurantsgivingthemaximumvalueareplacedinthebeginning oftheFineDiningsection?Howdidyouarriveatyouranswer? h. Supposethattherewere200pagesofrestaurants.Whatdoyouestimatetobethemaximumvalueforarestaurantlistedonpage200? i. Istheleastsquareslinevalidforpage200?Whyorwhynot? j. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Thenexttwoquestionsrefertothefollowingdata: Thecostofaleadingliquidlaundrydetergentindifferentsizesisgivenbelow. Sizeounces Cost\$ Costperounce 16 3.99 32 4.99 64 5.99 200 10.99 Exercise12.13.9 Solutiononp.509. a. Usingsizeastheindependentvariableandcostasthedependentvariable,makea scatterplot. b. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? c. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx d. Findthecorrelationcoefcient.Isitsignicant? e. Ifthelaundrydetergentweresoldina40ouncesize,ndtheestimatedcost. f. Ifthelaundrydetergentweresoldina90ouncesize,ndtheestimatedcost. g. Usethetwopointsineandftoplottheleastsquareslineonyourgraphfroma.

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489 h. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot? i. Arethereanyoutliersintheabovedata? j. Istheleastsquareslinevalidforpredictingwhata300ouncesizeofthelaundrydetergent wouldcost?Whyorwhynot? k. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.10 a. Completetheabovetableforthecostperounceofthedifferentsizes. b. UsingSizeastheindependentvariableandCostperounceasthedependentvariable, makeascatterplotofthedata. c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Isitsignicant? f. Ifthelaundrydetergentweresoldina40ouncesize,ndtheestimatedcostperounce. g. Ifthelaundrydetergentweresoldina90ouncesize,ndtheestimatedcostperounce. h. Usethetwopointsinfandgtoplottheleastsquareslineonyourgraphfromb. i. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot? j. Arethereanyoutliersintheabovedata? k. Istheleastsquareslinevalidforpredictingwhata300ouncesizeofthelaundrydetergent wouldcostperounce?Whyorwhynot? l. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.11 Solutiononp.509. AccordingtoyerbyaPrudentialInsuranceCompanyrepresentative,thecostsofapproximate probatefeesandtaxesforselectednettaxableestatesareasfollows: NetTaxableEstate\$ ApproximateProbateFeesandTaxes\$ 600,000 30,000 750,000 92,500 1,000,000 203,000 1,500,000 438,000 2,000,000 688,000 2,500,000 1,037,000 3,000,000 1,350,000 a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable. b. Makeascatterplotofthedata. c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Isitsignicant? f. Findtheestimatedtotalcostforanettaxableestateof\$1,000,000.Findthecostfor \$2,500,000. g. Usethetwopointsinftoplottheleastsquareslineonyourgraphfromb. h. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot?

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490 CHAPTER12.LINEARREGRESSIONANDCORRELATION i. Arethereanyoutliersintheabovedata? j. Basedontheabove,whatwouldbetheprobatefeesandtaxesforanestatethatdoesnot haveanyassets? k. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.12 ThefollowingareadvertisedsalepricesofcolortelevisionsatAnderson's. Sizeinches SalePrice\$ 9 147 20 197 27 297 31 447 35 1177 40 2177 60 2497 a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable. b. Makeascatterplotofthedata. c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Isitsignicant? f. Findtheestimatedsalepricefora32inchtelevision.Findthecostfora50inchtelevision. g. Usethetwopointsinftoplottheleastsquareslineonyourgraphfromb. h. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot? i. Arethereanyoutliersintheabovedata? j. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.13 Solutiononp.509. BelowaretheaverageheightsforAmericanboys.Source: Physician'sHandbook,1990 Ageyears Heightcm birth 50.8 2 83.8 3 91.4 5 106.6 7 119.3 10 137.1 14 157.5 a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable.

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491 b. Makeascatterplotofthedata. c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Isitsignicant? f. Findtheestimatedaverageheightforaoneyearold.Findtheestimatedaverageheight foranelevenyearold. g. Usethetwopointsinftoplottheleastsquareslineonyourgraphfromb. h. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot? i. Arethereanyoutliersintheabovedata? j. Usetheleastsquareslinetoestimatetheaverageheightforasixtytwoyearoldman.Do youthinkthatyouranswerisreasonable?Whyorwhynot? k. Whatistheslopeoftheleastsquaresbest-tline?Interprettheslope. Exercise12.13.14 ThefollowingchartgivesthegoldmedaltimesforeveryotherSummerOlympicsforthewomen's 100meterfreestyleswimming. Year Timeseconds 1912 82.2 1924 72.4 1932 66.8 1952 66.8 1960 61.2 1968 60.0 1976 55.65 1984 55.92 1992 54.64 a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable. b. Makeascatterplotofthedata. c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Isthedecreaseintimessignicant? f. Findtheestimatedgoldmedaltimefor1932.Findtheestimatedtimefor1984. g. Whyaretheanswersfromfdifferentfromthechartvalues? h. Usethetwopointsinftoplottheleastsquareslineonyourgraphfromb. i. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot? j. UsetheleastsquareslinetoestimatethegoldmedaltimeforthenextSummerOlympics. Doyouthinkthatyouranswerisreasonable?Whyorwhynot?

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492 CHAPTER12.LINEARREGRESSIONANDCORRELATION Thenextthreequestionsusethefollowingstateinformation. State #lettersinname Yearenteredthe Union Rankforentering theUnion Areasquare miles Alabama 7 1819 22 52,423 Colorado 1876 38 104,100 Hawaii 1959 50 10,932 Iowa 1846 29 56,276 Maryland 1788 7 12,407 Missouri 1821 24 69,709 NewJersey 1787 3 8,722 Ohio 1803 17 44,828 SouthCarolina 13 1788 8 32,008 Utah 1896 45 84,904 Wisconsin 1848 30 65,499 Exercise12.13.15 Solutiononp.509. Weareinterestedinwhetherornotthenumberoflettersinastatenamedependsupontheyear thestateenteredtheUnion. a. Decidewhichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable. b. Makeascatterplotofthedata. c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Whatdoesitimplyaboutthesignicanceoftherelationship? f. Findtheestimatednumberofletterstothenearestintegerastatewouldhaveifitentered theUnionin1900.Findtheestimatednumberoflettersastatewouldhaveifitentered theUnionin1940. g. Usethetwopointsinftoplottheleastsquareslineonyourgraphfromb. h. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot? i. Usetheleastsquareslinetoestimatethenumberoflettersanewstatethatentersthe Unionthisyearwouldhave.Cantheleastsquareslinebeusedtopredictit?Whyor whynot? Exercise12.13.16 Weareinterestedinwhetherthereisarelationshipbetweentherankingofastateandtheareaof thestate. a. Letrankbetheindependentvariableandareabethedependentvariable. b. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata. c. Doesitappearfrominspectionthatthereisarelationshipbetweenthevariables?Whyor whynot? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx

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493 e. Findthecorrelationcoefcient.Whatdoesitimplyaboutthesignicanceoftherelationship? f. FindtheestimatedareasforAlabamaandforColorado.Aretheyclosetotheactualareas? g. Usethetwopointsinftoplottheleastsquareslineonyourgraphfromb. h. Doesitappearthatalineisthebestwaytotthedata?Whyorwhynot? i. Arethereanyoutliers? j. UsetheleastsquareslinetoestimatetheareaofanewstatethatenterstheUnion.Canthe leastsquareslinebeusedtopredictit?Whyorwhynot? k. DeleteHawaiiandsubstituteAlaskaforit.Alaskaisthefortiethstatewithanareaof 656,424squaremiles. l. Calculatethenewleastsquaresline. m. FindtheestimatedareaforAlabama.Isitclosertotheactualareawiththisnewleast squareslineorwiththepreviousonethatincludedHawaii?Whydoyouthinkthat's thecase? n. Doyouthinkthat,ingeneral,newerstatesarelargerthantheoriginalstates? Exercise12.13.17 Solutiononp.510. Weareinterestedinwhetherthereisarelationshipbetweentherankofastateandtheyearit enteredtheUnion. a. Letyearbetheindependentvariableandrankbethedependentvariable. b. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata. c. Whymusttherelationshipbepositivebetweenthevariables? d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Whatdoesitimplyaboutthesignicanceoftherelationship? f. Let'ssayafty-rststateenteredtheunion.Basedupontheleastsquaresline,when shouldthathaveoccurred? g. Usingtheleastsquaresline,howmanystatesdowecurrentlyhave? h. Whyisn'ttheleastsquareslineagoodestimatorforthisyear? Exercise12.13.18 BelowarethepercentsoftheU.S.laborforceexcludingself-employedandunemployedthat aremembersofaunion.Weareinterestedinwhetherthedecreaseissignicant.Source: Bureau ofLaborStatistics,U.S.Dept.ofLabor Year Percent 1945 35.5 1950 31.5 1960 31.4 1970 27.3 1980 21.9 1986 17.5 1993 15.8 a. Letyearbetheindependentvariableandpercentbethedependentvariable. b. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata. c. Whywilltherelationshipbetweenthevariablesbenegative?

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494 CHAPTER12.LINEARREGRESSIONANDCORRELATION d. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx e. Findthecorrelationcoefcient.Whatdoesitimplyaboutthesignicanceoftherelationship? f. Basedonyouranswertoe,doyouthinkthattherelationshipcanbesaidtobedecreasing? g. Ifthetrendcontinues,whenwilltherenolongerbeanyunionmembers?Doyouthink thatwillhappen?

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495 Thenexttwoquestionsrefertothefollowinginformation: Thedatabelowreectsthe1991-92Reunion ClassGiving.Source: SUNYAlbanyalumnimagazine ClassYear AverageGift TotalGiving 1922 41.67 125 1927 60.75 1,215 1932 83.82 3,772 1937 87.84 5,710 1947 88.27 6,003 1952 76.14 5,254 1957 52.29 4,393 1962 57.80 4,451 1972 42.68 18,093 1976 49.39 22,473 1981 46.87 20,997 1986 37.03 12,590 Exercise12.13.19 Solutiononp.510. Wewillusethecolumnsclassyearandtotalgivingforallquestions,unlessotherwisestated. a. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata. b. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx c. Findthecorrelationcoefcient.Whatdoesitimplyaboutthesignicanceoftherelationship? d. Fortheclassof1930,predictthetotalclassgift. e. Fortheclassof1964,predictthetotalclassgift. f. Fortheclassof1850,predictthetotalclassgift.Whydoesn'tthisvaluemakeanysense? Exercise12.13.20 Wewillusethecolumnsclassyearandaveragegiftforallquestions,unlessotherwisestated. a. Whatdoyouthinkthescatterplotwilllooklike?Makeascatterplotofthedata. b. Calculatetheleastsquaresline.Puttheequationintheformof: y = a + bx c. Findthecorrelationcoefcient.Whatdoesitimplyaboutthesignicanceoftherelationship? d. Fortheclassof1930,predicttheaverageclassgift. e. Fortheclassof1964,predicttheaverageclassgift. f. Fortheclassof2010,predicttheaverageclassgift.Whydoesn'tthisvaluemakeany sense? 12.13.1Trythesemultiplechoicequestions Exercise12.13.21 Solutiononp.510. Acorrelationcoefcientof-0.95meansthereisa____________betweenthetwovariables.

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496 CHAPTER12.LINEARREGRESSIONANDCORRELATION A. Strongpositivecorrelation B. Weaknegativecorrelation C. Strongnegativecorrelation D. NoCorrelation Exercise12.13.22 Solutiononp.510. AccordingtothedatareportedbytheNewYorkStateDepartmentofHealthregardingWestNile Virusfortheyears2000-2004,theleastsquareslineequationforthenumberofreporteddeadbirds x versusthenumberofhumanWestNileviruscases y is y = )]TJ/F132 9.9626 Tf 8.194 0 Td [(10.2638 + 0 .0491x .Ifthenumber ofdeadbirdsreportedinayearis732,howmanyhumancasesofWestNileviruscanbeexpected? A. 25.7 B. 46.2 C. -25.7 D. 7513 Thenextthreequestionsrefertothefollowingdata: showingthenumberofhurricanesbycategoryto directlystrikethemainlandU.S.eachdecadeobtainedfrom www.nhc.noaa.gov/gifs/table6.gif 14 Amajor hurricaneisonewithastrengthratingof3,4or5. Decade TotalNumberofHurricanes NumberofMajorHurricanes 1941-1950 24 10 1951-1960 17 8 1961-1970 14 6 1971-1980 12 4 1981-1990 15 5 1991-2000 14 5 20012004 9 3 Exercise12.13.23 Solutiononp.510. Usingonlycompleteddecades2000,calculatetheleastsquareslineforthenumberof majorhurricanesexpectedbaseduponthetotalnumberofhurricanes. A. y = )]TJ/F58 9.9626 Tf 8.195 0 Td [(1 .67x + 0 5 B. y = 0 5 x )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 .67 C. y = 0 .94x )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 .67 D. y = )]TJ/F58 9.9626 Tf 8.195 0 Td [(2 x + 1 Exercise12.13.24 Solutiononp.510. Thecorrelationcoefcientis0.942.Isthisconsideredsignicant?Whyorwhynot? A. No,because0.942isgreaterthanthecriticalvalueof0.707 B. Yes,because0.942isgreaterthanthecriticalvalueof0.707 C. No,because0942isgreaterthanthecriticalvalueof0.811 D. Yes,because0.942isgreaterthanthecriticalvalueof0.811 14 http://www.nhc.noaa.gov/gifs/table6.gif

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497 Exercise12.13.25 Solutiononp.510. Thedatafor2001-2004show9hurricaneshavehitthemainlandUnitedStates.Thelineofbestt predicts2.83majorhurricanestohitmainlandU.S.Cantheleastsquareslinebeusedtomakethis prediction? A. No,because9liesoutsidetheindependentvariablevalues B. Yes,because,infact,therehavebeen3majorhurricanesthisdecade C. No,because2.83liesoutsidethedependentvariablevalues D. Yes,becausehowelsecouldwepredictwhatisgoingtohappenthisdecade.

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498 CHAPTER12.LINEARREGRESSIONANDCORRELATION 12.14Lab1:RegressionDistancefromSchool 15 ClassTime: Names: 12.14.1StudentLearningOutcomes: Thestudentwillcalculateandconstructthelineofbesttbetweentwovariables. Thestudentwillevaluatetherelationshipbetweentwovariablestodetermineifthatrelationshipis signicant. 12.14.2CollecttheData Use8membersofyourclassforthesample.Collectbivariatedatadistanceanindividuallivesfromschool, thecostofsuppliesforthecurrentterm. 1.Completethetable. Distancefromschool Costofsuppliesthisterm Figure12.15 2.Whichvariableshouldbethedependentvariableandwhichshouldbetheindependentvariable? Why? 3.Graphdistancevs.cost.Plotthepointsonthegraph.Labelbothaxeswithwords.Scaleboth axes. 15 Thiscontentisavailableonlineat.

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499 Figure12.16 12.14.3AnalyzetheData Enteryourdataintoyourcalculatororcomputer.Writethelinearequationbelow,roundingto4decimal places. 1.Calculatethefollowing: a. a = b. b = c. correlation= d. n = e. equation: y = f. Isthecorrelationsignicant?Whyorwhynot?Answerin1-3completesentences. 2.Supplyananswerforthefollowingsenarios: a. Forapersonwholives8milesfromcampus,predictthetotalcostofsuppliesthisterm: b. Forapersonwholives80milesfromcampus,predictthetotalcostofsuppliesthisterm: 3.Obtainthegraphonyourcalculatororcomputer.Sketchtheregressionlinebelow.

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501 12.15Lab2:RegressionTextbookCost 16 ClassTime: Names: 12.15.1StudentLearningOutcomes: Thestudentwillcalculateandconstructthelineofbesttbetweentwovariables. Thestudentwillevaluatetherelationshipbetweentwovariablestodetermineifthatrelationshipis signicant. 12.15.2CollecttheData Survey10textbooks.Collectbivariatedatanumberofpagesinatextbook,thecostofthetextbook. 1.Completethetable. Numberofpages Costoftextbook Figure12.18 2.Whichvariableshouldbethedependentvariableandwhichshouldbetheindependentvariable? Why? 3.Graphdistancevs.cost.Plotthepointsonthegraph.Labelbothaxeswithwords.Scaleboth axes. 16 Thiscontentisavailableonlineat.

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502 CHAPTER12.LINEARREGRESSIONANDCORRELATION Figure12.19 12.15.3AnalyzetheData Enteryourdataintoyourcalculatororcomputer.Writethelinearequationbelow,roundingto4decimal places. 1.Calculatethefollowing: a. a = b. b = c. correlation= d. n = e. equation: y = f. Isthecorrelationsignicant?Whyorwhynot?Answerin1-3completesentences. 2.Supplyananswerforthefollowingsenarios: a. Foratextbookwith400pages,predictthecost: b. Foratextbookwith600pages,predictthecost: 3.Obtainthegraphonyourcalculatororcomputer.Sketchtheregressionlinebelow.

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504 CHAPTER12.LINEARREGRESSIONANDCORRELATION 12.16Lab3:RegressionFuelEfciency 17 ClassTime: Names: 12.16.1StudentLearningOutcomes: Thestudentwillcalculateandconstructthelineofbesttbetweentwovariables. Thestudentwillevaluatetherelationshipbetweentwovariablestodetermineifthatrelationshipis signicant. 12.16.2CollecttheData UsethemostrecentAprilissueofConsumerReports.Itwillgivethetotalfuelefciencyinmilesper gallonandweightinpoundsofnewmodelcarswithautomatictransmissions.Wewillusethisdatato determinetherelationship,ifany,betweenthefuelefciencyofacaranditsweight. 1.Whichvariableshouldbetheindependentvariableandwhichshouldbethedependentvariable? Explainyouranswerinoneortwocompletesentences. 2.Usingyourrandomnumbergenerator,randomlyselect20carsfromthelistandrecordtheirweights andfuelefciencyintothetablebelow. 17 Thiscontentisavailableonlineat.

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505 Weight FuelEfciency Figure12.21 3.Whichvariableshouldbethedependentvariableandwhichshouldbetheindependentvariable? Why? 4.Byhand,doascatterplotofweightvs.fuelefciency.Plotthepointsongraphpaper.Labelboth axeswithwords.Scalebothaxesaccurately.

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506 CHAPTER12.LINEARREGRESSIONANDCORRELATION Figure12.22 12.16.3AnalyzetheData Enteryourdataintoyourcalculatororcomputer.Writethelinearequationbelow,roundingto4decimal places. 1.Calculatethefollowing: a. a = b. b = c. correlation= d. n = e. equation: y = 2.Obtainthegraphoftheregressionlineonyourcalculator.Sketchtheregressionlineonthesameaxes asyourscatterplot. 12.16.4DiscussionQuestions 1.Isthecorrelationsignicant?Explainhowyoudeterminedthisincompletesentences.

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507 2.Istherelationshipapositiveoneoranegativeone?Explainhowyoucantellandwhatthismeansin termsofweightandfuelefciency. 3.Inoneortwocompletesentences,whatisthepracticalinterpretationoftheslopeoftheleastsquares lineintermsoffuelefciencyandweight? 4.Foracarthatweighs4000pounds,predictitsfuelefciency.Includeunits. 5.Canwepredictthefuelefciencyofacarthatweighs10000poundsusingtheleastsquaresline? Explainwhyorwhynot. 6.Questions.Answereachin1to3completesentences. a. Doesthelineseemtotthedata?Whyorwhynot? b. Whatdoesthecorrelationimplyabouttherelationshipbetweenfuelefciencyandweightof acar?Isthiswhatyouexpected? 7.Arethereanyoutliers?Ifso,whichpointisanoutlier? **ThislabwasdesignedandcontributedbyDianeMathios.

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508 CHAPTER12.LINEARREGRESSIONANDCORRELATION SolutionstoExercisesinChapter12 Example12.11,Problem2p.472 78isoutsideofthedomainofxvaluesindependentvariables,soyoucannotreliablypredictthenal examscoreforthisstudent. Example12.13p.475 184.28 SolutionstoPractice:LinearRegression SolutiontoExercise12.12.2p.482 a. a = -3,448,225 b. b = 1750 c. corr. = 0 .4526 d. n = 22 SolutiontoExercise12.12.3p.482 y = -3,448,225 + 1750x SolutiontoExercise12.12.4p.482 a. 25082 b. 33,831 SolutiontoExercise12.12.10p.483 a. -1164 SolutionstoHomework SolutiontoExercise12.13.1p.484 a. Independent:Age;Dependent:Fatalities d. Independent:PowerConsumption;Dependent:Utility SolutiontoExercise12.13.3p.484 b. y = 88.7206 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0 .0432x c. -0.8533,Yes g. No h. No. i. 2.97,Yes j. slope=-0.0432.Astheyearincreasesbyone,thewelfarefamilysizedecreasesby0.0432people. SolutiontoExercise12.13.5p.485 b. Yes c. y = 102.4287 + 11.7585x d. 0.9436;yes e. 478.70feet;1207.73feet g. Yes h. Yes; 57 1050 i. 172.98;No

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509 j. 11.7585feet k. slope=11.7585.Asthenumberofstoriesincreasesbyone,theheightofthebuildingincreasesby 11.7585feet. SolutiontoExercise12.13.7p.487 b. Yes c. y = )]TJ/F132 9.9626 Tf 8.194 0 Td [(266.8863 + 0 .1656x d. 0.9448;Yes e. 62.9206;62.4237 h. No i. 72.639;No j. slope=0.1656.Astheyearincreasesbyone,thepercentofworkerspaidhourlyratesincreasesby 0.1565. SolutiontoExercise12.13.9p.488 b. Yes c. y = 3 .5984 + 0 .0371x d. 0.9986;Yes e. \$5.08 f. \$6.93 i. No j. Notvalid k. slope=0.0371.Asthenumberofouncesincreasesbyone,thecostoftheliquiddetergentincreases by\$0.0371orabout4cents. SolutiontoExercise12.13.11p.489 c. Yes d. y = )]TJ/F132 9.9626 Tf 8.195 0 Td [(337 424.6478 + 0 .5463x e. 0.9964;Yes f. \$208,872.49;\$1,028,318.20 h. Yes i. No k. slope=0.5463.Asthenettaxableestateincreasesbyonedollar,theapproximateprobatefeesand taxesincreasesby0.5463dollarsabout55cents. SolutiontoExercise12.13.13p.490 c. Yes d. y = 65.0876 + 7 .0948x e. 0.9761;yes f. 72.2cm;143.13cm h. Yes i. No j. 505.0cm;No k. slope=7.0948.AstheageofanAmericanboyincreasesbyoneyear,theaverageheightincreases by7.0948cm. SolutiontoExercise12.13.15p.492 c. No d. y = 47.03 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0 .216x e. -0.4280

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510 CHAPTER12.LINEARREGRESSIONANDCORRELATION f. 6;5 SolutiontoExercise12.13.17p.493 d. y = )]TJ/F132 9.9626 Tf 8.194 0 Td [(480.5845 + 0 .2748x e. 0.9553 f. 1934 SolutiontoExercise12.13.19p.495 b. y = )]TJ/F132 9.9626 Tf 8.194 0 Td [(569 770.2796 + 296.0351 c. 0.8302 d. \$1577.48 e. \$11,642.68 f. -\$22,105.33 SolutiontoExercise12.13.21p.495 C SolutiontoExercise12.13.22p.496 A SolutiontoExercise12.13.23p.496 A SolutiontoExercise12.13.24p.496 D SolutiontoExercise12.13.25p.497 A

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Chapter13 FDistributionandANOVA 13.1FDistributionandANOVA 1 13.1.1StudentLearningObjectives Bytheendofthischapter,thestudentshouldbeableto: Discussbasicideasoflinearregressionandcorrelation. Createandinterpretalineofbestt. Calculateandinterpretthecorrelationcoefcient. Calculateandinterpretoutliers. 13.1.2Introduction Manystatisticalapplicationsinpsychology,socialscience,businessadministration,andthenaturalsciences involveseveralgroups.Forexample,anenvironmentalistisinterestedinknowingiftheaverageamountof pollutionvariesinseveralbodiesofwater.Asociologistisinterestedinknowingiftheamountofincomea personearnsvariesaccordingtohisorherupbringing.Aconsumerlookingforanewcarmightcompare theaveragegasmileageofseveralmodels. Forhypothesistestsinvolvingmorethantwoaverages,statisticianshavedevelopedamethodcalledAnalysisofVariance"abbreviatedANOVA.Inthischapter,youwillstudythesimplestformofANOVAcalled singlefactororone-wayANOVA.YouwillalsostudytheFdistribution,usedforANOVA,andthetestof twovariances.ThisisjustaverybriefoverviewofANOVA.Youwillstudythistopicinmuchgreaterdetail infuturestatisticscourses. N OTE :ANOVA,asitispresentedhere,reliesheavilyonacalculatororcomputer. 13.2ANOVA 2 13.2.1FDistributionandANOVA:PurposeandBasicAssumptionofANOVA Thepurposeofan ANOVA testistodeterminetheexistenceofastatisticallysignicantdifferenceamong severalgroupmeans.Thetestactuallyuses variances tohelpdetermineifthemeansareequalornot. 1 Thiscontentisavailableonlineat. 2 Thiscontentisavailableonlineat. 511

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512 CHAPTER13.FDISTRIBUTIONANDANOVA InordertoperformanANOVAtest,therearethreebasic assumptions tobefullled: Eachpopulationfromwhichasampleistakenisassumedtobenormal. Eachsampleisrandomlyselectedandindependent. Thepopulationsareassumedtohave equalstandarddeviationsorvariances. 13.2.2TheNullandAlternateHypotheses Thenullhypothesisissimplythatallthegrouppopulationmeansarethesame.Thealternatehypothesis isthatatleastonepairofmeansisdifferent.Forexample,ifthereare k groups: H o : m 1 = m 2 = m 3 = ... = m k H a :Atleasttwoofthegroupmeans m 1 m 2 m 3 ,..., m k arenotequal. 13.3TheFDistributionandtheFRatio 3 Thedistributionusedforthehypothesistestisanewone.ItiscalledtheFdistribution,namedafterSir RonaldFisher,anEnglishstatistician.TheFstatisticisaratioafraction.Therearetwosetsofdegreesof freedom;oneforthenumeratorandoneforthedenominator. Forexample,if F followsan F distributionandthedegreesoffreedomforthenumeratorare4andthe degreesoffreedomforthedenominatorare10,then F F 4,10 Tocalculatethe F ratio,twoestimatesofthevariancearemade. 1. Variancebetweensamples: Anestimateof s 2 thatisthevarianceofthesamplemeans.Ifthesamples aredifferentsizes,thevariancebetweensamplesisweightedtoaccountforthedifferentsamplesizes. Itisalsocalled variationduetotreatmentorexplainedvariation. 2. Variancewithinsamples: Anestimateof s 2 thatistheaverageofthesamplevariancesalsoknown asapooledvariance.Whenthesamplesizesaredifferent,thevariancewithinsamplesisweighted. Itisalsocalledthe variationduetoerrororunexplainedvariation. SS between = thesumofsquaresthatrepresentsthevariationamongthedifferentsamples. SS within = thesumofsquaresthatrepresentsthevariationwithinsamplesthatisduetochance. Tonda"sumofsquares"meanstoaddtogethersquaredquantitieswhich,insomecases,maybeweighted. WeusedsumofsquarestocalculatethesamplevarianceandthesamplestandarddeviationinChapter2. InthisverybriefoverviewofANOVA,wewillnotgointodetailexplaininghow SS between and SS within are calculated.Ifyoutakeanotherstatisticscourse,youwillcoversumofsquaresindetail.Yourcalculator caneasilydothecalculationsof SS between and SS within .Rememberthatthesetwoquantitiesaremeasuresof variabilityorvariation. df between = k )]TJ/F58 9.9626 Tf 10.166 0 Td [(1where k isthenumberofgroupssamples.Thisisthedegreesoffreedomforthenumerator. df within = N )]TJ/F132 9.9626 Tf 10.256 0 Td [(k where N isthetotalsamplesize.Thisisthedegreesoffreedomforthedenominator. MS means"meansquare." MS between isthevariancebetweengroupsand MS within isthevariancewithin groups. 3 Thiscontentisavailableonlineat.

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513 MS between = SS between df between = SS between k )]TJ/F58 7.5716 Tf 6.227 0 Td [(1 MS within = SS within df within = SS within N )]TJ/F132 7.5716 Tf 6.322 0 Td [(k TheANOVAtestdependsonthefactthat MS between canbeinuencedbypopulationdifferencesamong meansoftheseveralgroups.Since MS within comparesvaluesofeachgrouptoitsowngroupmean,thefact thatgroupmeansmightbedifferentdoesnotaffect MS within Thenullhypothesissaysthatallgroupsaresamplesfrompopulationshavingthesamenormaldistribution. Thealternatehypothesissaysthatatleasttwoofthesamplegroupscomefrompopulationswithdifferent normaldistributions.Ifthenullhypothesisistrue, MS between and MS within shouldbothestimatethesame value. TheF-statisticor F-ratio ,asitisoftencalled,is F = MS between MS within If MS between and MS within estimatethesamevaluefollowingthebeliefthat H o istrue,thentheF-ratio shouldbeapproximatelyequalto1.Onlysamplingerrorswouldcontributetovariationsawayfrom1.As itturnsout, MS between consistsofthepopulationvarianceplusavarianceproducedfromthedifferencesbetweenthesamples. MS within isanestimateofthepopulationvariance.Sincevariancesarealwayspositive, ifthenullhypothesisisfalse, MS between willbelargerthan MS within .TheF-ratiowillbelargerthan1. TheANOVAhypothesistestisalwaysright-tailed becauselargerF-valuesarewayoutintherighttailof theF-distributioncurveandtendtomakeusreject H o 13.3.1Notation ThenotationfortheFdistributionis F F dfnum dfdenom where dfnum = df between and dfdenom = df within ThemeanfortheFdistributionis m = df num df denom )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 13.4FactsAbouttheFDistribution 4 1.Thecurveisnotsymmetricalbutskewedtotheright. 2.Thereisadifferentcurveforeachsetof dfs 3.TheFstatisticisgreaterthanorequaltozero. 4.Asthedegreesoffreedomforthenumeratorandforthedenominatorgetlarger,thecurveapproximatesthenormal. 5.OtherusesfortheFdistributionincludecomparingtwovariancesandTwo-WayAnalysisofVariance. Comparingtwovariancesisdiscussedattheendofthechapter.Two-WayAnalysisismentionedfor yourinformationonly. 4 Thiscontentisavailableonlineat.

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514 CHAPTER13.FDISTRIBUTIONANDANOVA a b Figure13.1 Example13.1 One-WayANOVA: Foursororitiestookarandomsampleofsistersregardingtheirgradeaverages forthepastterm.Theresultsareshownbelow: GRADEAVERAGESFORFOURSORORITIES Sorority1 Sorority2 Sorority3 Sorority4 2.17 2.63 2.63 3.79 1.85 1.77 3.78 3.45 2.83 3.25 4.00 3.08 1.69 1.86 2.55 2.26 3.33 2.21 2.45 3.18 Problem Usingasignicancelevelof1%,isthereadifferenceingradeaveragesamongthesororities? Solution Let m 1 m 2 m 3 m 4 bethepopulationmeansofthesororities.Rememberthatthenullhypothesis claimsthatthesororitygroupsarefromthesamenormaldistribution.Thealternatehypothesis saysthatatleasttwoofthesororitygroupscomefrompopulationswithdifferentnormaldistributions.Noticethatthefoursamplesizesareeachsize5. H o : m 1 = m 2 = m 3 = m 4 H a :Notallofthemeans m 1 m 2 m 3 m 4 areequal. Distributionforthetest: F 3,16 where k = 4groups and N = 20samplesintotal df num = k )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 4 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = 3 df denom = N )]TJ/F132 9.9626 Tf 10.256 0 Td [(k = 20 )]TJ/F58 9.9626 Tf 10.131 0 Td [(4 = 16

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515 Calculatetheteststatistic: F = 2.23 Graph: Figure13.2 Probabilitystatement: p-value = P F > 2.23 = 0.1241 Compare a andthe p )]TJ/F132 9.9626 Tf 10.256 0 Td [(value : a = 0.01 p-value = 0.1242 a < p-value Makeadecision: Since a < p-value ,youcannotreject H o Thismeansthatthepopulationaveragesappeartobethesame. Conclusion: Thereisnotsufcientevidencetoconcludethatthereisadifferenceamongthegrade averagesforthesororities. TI-83+orTI84: PutthedataintolistsL1,L2,L3,andL4.Press STAT andarrowoverto TESTS Arrowdownto F:ANOVA .Press ENTER andEnter L1,L2,L3,L4 .TheFstatisticis2.2303andthe p-value is0.1241. dfnumerator=3 under "Factor" and dfdenominator=16 under Error Example13.2 Afourthgradeclassisstudyingtheenvironment.Oneoftheassignmentsistogrowbeanplants indifferentsoils.Tommychosetogrowhisbeanplantsinsoilfoundoutsidehisclassroommixed withdryerlint.Tarachosetogrowherbeanplantsinpottingsoilboughtatthelocalnursery. Nickchosetogrowhisbeanplantsinsoilfromhismother'sgarden.Nochemicalswereused ontheplants,onlywater.Theyweregrowninsidetheclassroomnexttoalargewindow.Each childgrew5plants.Attheendofthegrowingperiod,eachplantwasmeasured,producingthe followingdataininches: Tommy'sPlants Tara'sPlants Nick'sPlants 24 25 23 21 31 27 23 23 22 30 20 30 23 28 20

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516 CHAPTER13.FDISTRIBUTIONANDANOVA Problem1 Doesitappearthatthethreemediainwhichthebeanplantsweregrownproducethesame averageheight?Testata3%levelofsignicance. Solution Thistime,wewillperformthecalculationsthatleadtotheF'statistic.Noticethateachgrouphas thesamenumberofplants. First,calculatethesamplemeanandsamplevarianceofeachgroup. Tommy'sPlants Tara'sPlants Nick'sPlants SampleMean 24.2 25.4 24.4 SampleVariance 11.7 18.3 16.3 Next,calculatethevarianceofthethreegroupmeansCalculatethevarianceof24.2,25.4,and 24.4. Varianceofthegroupmeans=0.413 Then MS between = 5 0.413 wherethe5isthesamplesizenumberofplantseachchildgrew. CalculatetheaverageofthethreesamplevariancesCalculatetheaverageof11.7,11.3,and16.3. Averageofthesamplevariances=15.433 Then MS within = 15.433. The F statisticor F ratiois F = MS between MS within = 5 0.413 15.433 = 0.134 Thedfsforthenumerator= thenumberofgroups )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 3 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 2 Thedfsforthedenominator= thetotalnumberofsamples )]TJ/F132 9.9626 Tf 10.131 0 Td [(thenumberofgroups = 15 )]TJ/F58 9.9626 Tf 10.131 0 Td [(3 = 12 Thedistributionforthetestis F 2,12 andtheFstatisticis F = 0.134 Thep-valueis P F > 0.134 = 0.8759. Decision: Since a = 0.03andthe p-value = 0.8759,donotreject H o .Why? Conclusion: Witha3%thelevelofsignicance,fromthesampledata,theevidenceisnotsufcient toconcludethattheaverageheightsofthebeanplantsarenotdifferent.Ofthethreemediatested, itappearsthatitdoesnotmatterwhichonethebeanplantsaregrownin. Thisexperimentwasactuallydonebythreeclassmatesofthesonofoneoftheauthors. Anotherfourthgraderalsogrewbeanplantsbutthistimeinajelly-likemass.Theheightswere ininches24,28,25,30,and32. Problem2 DoanANOVAtestonthe4groups. Youmayuseyourcalculatororcomputertoperformthetest. Aretheheightsofthebeanplantsdifferent?UseasolutionsheetSection14.5.4.

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517 13.4.1OptionalClassroomActivity Randomlydividetheclassintofourgroupsofthesamesize.Haveeachmemberofeachgrouprecordthe numberofstatesintheUnitedStatesheorshehasvisited.RunanANOVAtesttodetermineiftheaverage numberofstatesvisitedinthefourgroupsarethesame.Testata1%levelofsignicance.Useoneofthe solutionsheetsSection14.5.4attheendofthechapterafterthehomework. 13.5TestofTwoVariances 5 AnotheroftheusesoftheFdistributionistestingtwovariances.Itisoftendesirabletocomparetwo variancesratherthantwoaverages.Forinstance,collegeadministratorswouldliketwocollegeprofessors gradingexamstohavethesamevariationintheirgrading.Inorderforalidtotacontainer,thevariation inthelidandthecontainershouldbethesame.Asupermarketmightbeinterestedinthevariabilityof check-outtimesfortwocheckers. InordertoperformaFtestoftwovariances,itisimportantthatthefollowingaretrue: 1.Thepopulationsfromwhichthetwosamplesaredrawnarenormallydistributed. 2.Thetwopopulationsareindependentofeachother. Supposewesamplerandomlyfromtwoindependentnormalpopulations.Let s 2 1 and s 2 2 bethepopulation variancesand s 2 1 and s 2 2 bethesamplevariances.Letthesamplesizesbe n 1 and n 2 .Sinceweareinterested incomparingthetwosamplevariances,weusetheFratio F = s 1 2 s 1 2 # s 2 2 s 2 2 # F hasthedistribution F F n 1 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1, n 2 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 where n 1 )]TJ/F58 9.9626 Tf 10.332 0 Td [(1arethedegreesoffreedomforthenumeratorand n 2 )]TJ/F58 9.9626 Tf 10.332 0 Td [(1arethedegreesoffreedomforthe denominator. Atestoftwovariancesmaybeleft,right,ortwo-tailed. Example13.3 Twocollegeinstructorsareinterestedinwhetherornotthereisanyvariationinthewaythey grademathexams.Theyeachgradethesamesetof30exams.Therstinstructor'sgradeshavea varianceof52.3.Thesecondinstructor'sgradeshaveavarianceof89.9. Problem Testtheclaimthattherstinstructor'svarianceissmaller.Inmostcolleges,itisdesirablefor thevariancesofexamgradestobenearlythesameamonginstructors.Thelevelofsignicanceis 10%. Solution Let1and2bethesubscriptsthatindicatetherstandsecondinstructor. n 1 = n 2 = 30. H o : s 2 1 = s 2 2 H a : s 2 1 < s 2 2 Calculatetheteststatistic: Bythenullhypothesis )]TJ/F134 9.9626 Tf 4.812 -8.074 Td [(s 2 1 = s 2 2 ,theFstatisticis 5 Thiscontentisavailableonlineat.

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518 CHAPTER13.FDISTRIBUTIONANDANOVA F = s 1 2 s 1 2 # s 2 2 s 2 2 # = s 1 2 s 2 2 = 52.3 89.9 = 0.6 Distributionforthetest: F 29,29 where n 1 )]TJ/F58 9.9626 Tf 10.132 0 Td [(1 = 29and n 2 )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 = 29. Graph:Thistestislefttailed. Drawthegraphlabelingandshadingappropriately. Figure13.3 Probabilitystatement: p-value = P F < 0.582 = 0.0755 Compare a andthep-value: a = 0.10 a > p-value Makeadecision: Since a > p-value ,reject H o Conclusion: Witha10%levelofsignicance,fromthedata,thereissufcientevidencetoconclude thatthevarianceingradesfortherstinstructorissmaller. TI-83+andTI-84: Press STAT andarrowoverto TESTS .Arrowdownto D:2-SampFTest .Press ENTER .Arrowto Stats andpress ENTER .For Sx1 n1 Sx2 ,and n2 ,enter p 52.3 30 p 89.9 ,and 30 .Press ENTER aftereach.Arrowto s 1: and < s 2 .Press ENTER .Arrowdownto Calculate and press ENTER F = 0.5818and p-value = 0.0753.Dotheprocedureagainandtry Draw insteadof Calculate .

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519 13.6Summary 6 An ANOVA hypothesistestdeterminesifseveralpopulationmeansareequal.Thedistributionfor thetestistheFdistributionwith2differentdegreesoffreedom. Assumptions: 1.Eachpopulationfromwhichasampleistakenisassumedtobenormal. 2.Eachsampleisrandomlyselectedandindependent. 3.Thepopulationsareassumedtohaveequalstandarddeviationsorvariances A TestofTwoVariances hypothesistestdeterminesiftwovariancesarethesame.Thedistribution forthehypothesistestistheFdistributionwith2differentdegreesoffreedom. Assumptions: 1.Thepopulationsfromwhichthetwosamplesaredrawnarenormallydistributed. 2.Thetwopopulationsareindependentofeachother. 6 Thiscontentisavailableonlineat.

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520 CHAPTER13.FDISTRIBUTIONANDANOVA 13.7Practice:ANOVA 7 13.7.1StudentLearningOutcome ThestudentwillexplorethepropertiesofANOVA. 13.7.2Given Supposeagroupisinterestedindeterminingwhetherteenagersobtaintheirdriverslicensesatapproximatelythesameaverageageacrossthecountry.Supposethatthefollowingdataarerandomlycollected fromveteenagersineachregionofthecountry.Thenumbersrepresenttheageatwhichteenagersobtainedtheirdriverslicenses. Northeast South West Central East 16.3 16.9 16.4 16.2 17.1 16.1 16.5 16.5 16.6 17.2 16.4 16.4 16.6 16.5 16.6 16.5 16.2 16.1 16.4 16.8 x = ________ ________ ________ ________ ________ s 2 = ________ ________ ________ ________ ________ 13.7.3Hypothesis Exercise13.7.1 Statethehypotheses. H o : H a : 13.7.4DataEntry Enterthedataintoyourcalculatororcomputer. Exercise13.7.2 Solutiononp.529. degreesoffreedom-numerator: df n = Exercise13.7.3 Solutiononp.529. degreesoffreedom-denominator: df d = Exercise13.7.4 Solutiononp.529. Fteststatistic= Exercise13.7.5 Solutiononp.529. p-value = 7 Thiscontentisavailableonlineat.

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521 13.7.5DecisionsandConclusions Statethedecisionsandconclusionsincompletesentencesforthefollowingpreconceivedlevelsof a Exercise13.7.6 a = 0 .05 Decision: Conclusion: Exercise13.7.7 a = 0 .01 Decision: Conclusion:

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522 CHAPTER13.FDISTRIBUTIONANDANOVA 13.8Homework 8 DIRECTIONS :Useasolutionsheettoconductthefollowinghypothesistests. Exercise13.8.1 Solutiononp.529. Threestudents,Linda,Tuan,andJavier,aregiven5laboratoryratseachforanutritionalexperiment.Eachrat'sweightisrecordedingrams.LindafeedsherratsFormulaA,Tuanfeedshisrats FormulaB,andJavierfeedshisratsFormulaC.Attheendofaspeciedtimeperiod,eachratis weighedagainandthenetgainingramsisrecorded.Usingasignicancelevelof10%,testthe hypothesisthatthethreeformulasproducethesameaverageweightgain. WeightsofStudentLabRats Linda'srats Tuan'srats Javier'srats 43.5 47.0 51.2 39.4 40.5 40.9 41.3 38.9 37.9 46.0 46.3 45.0 38.2 44.2 48.6 Exercise13.8.2 Agrassrootsgroupopposedtoaproposedincreaseinthegastaxclaimedthattheincrease wouldhurtworking-classpeoplethemost,sincetheycommutethefarthesttowork.Suppose thatthegrouprandomlysurveyed24individualsandaskedthemtheirdailyone-waycommutingmileage.Theresultsarebelow: working-class professionalmiddleincomes professionalwealthy 17.8 16.5 8.5 26.7 17.4 6.3 49.4 22.0 4.6 9.4 7.4 12.6 65.4 9.4 11.0 47.1 2.1 28.6 19.5 6.4 15.4 51.2 13.9 9.3 Exercise13.8.3 Solutiononp.529. RefertoExercise13.8.1.Determinewhetherornotthevarianceinweightgainisstatisticallythe sameamongJavier'sandLinda'srats. Exercise13.8.4 RefertoExercise13.8.2aboveExercise13.8.2.Determinewhetherornotthevarianceinmileage drivenisstatisticallythesameamongtheworkingclassandprofessionalmiddleincomegroups. Forthenexttwoproblems,refertothedatafromTerriVogel'sLogBook[linkpending]. 8 Thiscontentisavailableonlineat.

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523 Exercise13.8.5 Solutiononp.529. Examinethe7practicelaps.Determinewhethertheaveragelaptimeisstatisticallythesamefor the7practicelaps,orifthereisatleastonelapthathasadifferentaveragetimefromtheothers. Exercise13.8.6 Examinepracticelaps3and4.Determinewhetherornotthevarianceinlaptimeisstatistically thesameforthosepracticelaps. Forthenextfourproblems,refertothefollowingdata. Thefollowingtableliststhenumberofpagesinfourdifferenttypesofmagazines. homedecorating news health computer 172 87 82 104 286 94 153 136 163 123 87 98 205 106 103 207 197 101 96 146 Exercise13.8.7 Solutiononp.529. Usingasignicancelevelof5%,testthehypothesisthatthefourmagazinetypeshavethesame averagelength. Exercise13.8.8 Eliminateonemagazinetypethatyounowfeelhasanaveragelengthdifferentthantheothers. Redothehypothesistest,testingthattheremainingthreeaveragesarestatisticallythesame.Usea newsolutionsheet.Basedonthistest,aretheaveragelengthsfortheremainingthreemagazines statisticallythesame? Exercise13.8.9 Whichtwomagazinetypesdoyouthinkhavethesamevarianceinlength? Exercise13.8.10 Whichtwomagazinetypesdoyouthinkhavedifferentvariancesinlength? 13.9Review 9 Thenexttwoquestionsrefertothefollowingsituation: Supposethattheprobabilityofadroughtinanyindependentyearis20%.Outofthoseyearsinwhicha droughtoccurs,theprobabilityofwaterrationingis10%.However,inanyyear,theprobabilityofwater rationingis5%. Exercise13.9.1 Solutiononp.530. Whatistheprobabilityofbothadrought and waterrationingoccurring? Exercise13.9.2 Solutiononp.530. Outoftheyearswithwaterrationing,ndtheprobabilitythatthereisadrought. Thenextthreequestionsrefertothefollowingsurvey: 9 Thiscontentisavailableonlineat.

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524 CHAPTER13.FDISTRIBUTIONANDANOVA FavoriteTypeofPiebyGender apple pumpkin pecan female 40 10 30 male 20 30 10 Exercise13.9.3 Solutiononp.530. Supposethatoneindividualisrandomlychosen.Findtheprobabilitythattheperson'sfavorite pieisapple or thepersonismale. Exercise13.9.4 Solutiononp.530. Supposethatonemaleisrandomlychosen.Findtheprobabilityhisfavoritepieispecan. Exercise13.9.5 Solutiononp.530. Conductahypothesistesttodetermineiffavoritepietypeandgenderareindependent. Thenexttwoquestionsrefertothefollowingsituation: Let'ssaythattheprobabilitythatanadultwatchesthenewsatleastonceperweekis0.60. Exercise13.9.6 Solutiononp.530. Werandomlysurvey14people.Onaverage,howmanypeopledoweexpecttowatchthenews atleastonceperweek? Exercise13.9.7 Solutiononp.530. Werandomlysurvey14people.Ofinterestisthenumberthatwatchthenewsatleastonceper week.Statethedistributionof X X Exercise13.9.8 Solutiononp.530. Thefollowinghistogramismostlikelytobearesultofsamplingfromwhichdistribution? Figure13.4 A. Chi-Square B. Geometric C. Uniform D. Binomial Exercise13.9.9 TheagesofDeAnzaeveningstudentsisknowntobenormallydistributed.Asampleof6De Anzaeveningstudentsreportedtheiragesinyearsas:28;35;47;45;30;50.Findtheprobability thattheaverageof6agesofrandomlychosenstudentsislessthan35years. Thenextthreequestionsrefertothefollowingsituation:

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525 Theamountofmoneyacustomerspendsinonetriptothesupermarketisknowntohaveanexponential distribution.Supposetheaverageamountofmoneyacustomerspendsinonetriptothesupermarketis \$72. Exercise13.9.10 Solutiononp.530. Findtheprobabilitythatonecustomerspendslessthan\$72inonetriptothesupermarket? Exercise13.9.11 Solutiononp.530. Suppose5customerspooltheirmoney.Theyarepoorcollegestudents.Howmuchmoney altogetherwouldyouexpectthe5customerstospendinonetriptothesupermarketindollars? Exercise13.9.12 Solutiononp.530. Statethedistributiontouseisifyouwanttondtheprobabilitythatthe average amountspent by5customersinonetriptothesupermarketislessthan\$60. Exercise13.9.13 Solutiononp.530. AmathexamwasgiventoallthefthgradechildrenattendingCountrySchool.Tworandom samplesofscoresweretaken.Thenullhypothesisisthattheaveragemathscoresforboysand girlsinfthgradearethesame.Conductahypothesistest. n x s 2 Boys 55 82 29 Girls 60 86 46 Exercise13.9.14 Solutiononp.530. Inasurveyof80males,55hadplayedanorganizedsportgrowingup.Ofthe70femalessurveyed, 25hadplayedanorganizedsportgrowingup.Weareinterestedinwhethertheproportionfor malesishigherthantheproportionforfemales.Conductahypothesistest. Exercise13.9.15 Solutiononp.530. Whichofthefollowingispreferablewhendesigningahypothesistest? A. Maximize a andminimize b B. Minimize a andmaximize b C. Maximize a and b D. Minimize a and b Thenextthreequestionsrefertothefollowingsituation: 120peopleweresurveyedastotheirfavoritebeveragenon-alcoholic.Theresultsarebelow. PreferredBeveragebyAge 09 1019 2029 30+ Totals Milk 14 10 6 0 30 Soda 3 8 26 15 52 Juice 7 12 12 7 38 Totals 24 30 44 22 120 Exercise13.9.16 Solutiononp.530. Aretheeventsof milk and 30+ : a. Independentevents?Justifyyouranswer.

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526 CHAPTER13.FDISTRIBUTIONANDANOVA b. Mutuallyexclusiveevents?Justifyyouranswer. Exercise13.9.17 Solutiononp.530. Supposethatonepersonisrandomlychosen.Findtheprobabilitythatpersonis 1019 given thathe/she prefersjuice Exercise13.9.18 Solutiononp.530. Are PreferredBeverage and Age independentevents?Conductahypothesistest. Exercise13.9.19 Solutiononp.530. Giventhefollowinghistogram,whichdistributionisthedatamostlikelytocomefrom? Figure13.5 A. uniform B. exponential C. normal D. chi-square

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527 13.10Lab:ANOVA 10 ClassTime: Names: 13.10.1StudentLearningOutcome: ThestudentwillconductasimpleANOVAtestinvolvingthreevariables. 13.10.2CollecttheData Recordthepriceperpoundof8fruits,8vegetables,and8breadsinyourlocalsupermarket. 1.Recordthepriceperpoundof8fruits,8vegetables,and8breadsinyourlocalsupermarket. Fruits Vegetables Breads 2.Explainhowyoucouldtrytocollectthedatarandomly. 13.10.3AnalyzetheData 1.Computethefollowing: a. Fruit: i. x = ii. s x = iii. n = a. Vegetables: i. x = ii. s x = iii. n = a. Bread: i. x = ii. s x = iii. n = 10 Thiscontentisavailableonlineat.

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528 CHAPTER13.FDISTRIBUTIONANDANOVA 2.Findthefollowing: a. df num = b. df denom = 3.Statetheapproximatedistributionforthetest. 4.Teststatistic: F = 5.Sketchagraphofthissituation.CLEARLY,labelandscalethehorizontalaxisandshadetheregions correspondingtothep-value. 6.p-value= 7.Testat a = 0.05.Stateyourdecisionandconclusion. 8. a. Decision:Whydidyoumakethisdecision? b. Conclusionwriteacompletesentence. c. Basedontheresultsofyourstudy,isthereaneedtofurtherinvestigateanyofthefoodgroups' prices?Whyorwhynot?

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529 SolutionstoExercisesinChapter13 Example13.2,Problem2p.516 F =0.9496 p )]TJ/F132 9.9626 Tf 10.255 0 Td [(value =0.4401 Theheightsofthebeanplantsarethesame. SolutionstoPractice:ANOVA SolutiontoExercise13.7.2p.520 df 1 = 4 SolutiontoExercise13.7.3p.520 df 2 = 15 SolutiontoExercise13.7.4p.520 Teststatistic = F = 4.22 SolutiontoExercise13.7.5p.520 0.017 SolutionstoHomework SolutiontoExercise13.8.1p.522 a. H o : m L = m T = m J c. df n = 2; df d = 12 e. 0.67 f. 0.5305 h. Decision:Donotrejectnull;Conclusion:Meansaresame SolutiontoExercise13.8.3p.522 c. df n = 4; df d = 4 e. 3.00 f. 2 0 .1563 = 0 .3126 h. Decision:Donotrejectnull;Conclusion:Variancesaresame SolutiontoExercise13.8.5p.522 c. df n = 6; df d = 98 e. 1.69 f. 0.1319 h. Decision:Donotrejectnull;Conclusion:Averagelaptimesarethesame SolutiontoExercise13.8.7p.523 a. H o : m d = m n = m h = m c b. Atleastoneaverageisdifferent c. df n = 3; df d = 16 e. 8.69 f. 0.0012 h. Decision:Rejectnull;Conclusion:Atleastoneaverageisdifferent

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530 APPENDIX SolutionstoReview SolutiontoExercise13.9.1p.523 0.02 SolutiontoExercise13.9.2p.523 0.40 SolutiontoExercise13.9.3p.524 100 140 SolutiontoExercise13.9.4p.524 10 60 SolutiontoExercise13.9.5p.524 p-value = 0;Rejectnull;Concludedependentevents SolutiontoExercise13.9.6p.524 8.4 SolutiontoExercise13.9.7p.524 B 14 ,0 .60 SolutiontoExercise13.9.8p.524 D SolutiontoExercise13.9.10p.525 0.6321 SolutiontoExercise13.9.11p.525 \$360 SolutiontoExercise13.9.12p.525 N 72 72 p 5 SolutiontoExercise13.9.13p.525 p-value = 0.0006;Rejectnull;Concludeaveragesarenotequal SolutiontoExercise13.9.14p.525 p-value = 0;Rejectnull;Concludeproportionofmalesishigher SolutiontoExercise13.9.15p.525 D SolutiontoExercise13.9.16p.525 a. No b. Yes, P Mand30 + = 0 SolutiontoExercise13.9.17p.526 12 38 SolutiontoExercise13.9.18p.526 No; p-value = 0 SolutiontoExercise13.9.19p.526 A

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Appendix 14.1PracticeFinalExam1 11 Questions1-2refertothefollowing: Anexperimentconsistsoftossingtwo12-sideddicethenumbers1-12areprintedonthesidesofeach dice. LetEvent A =bothdiceshowanevennumber LetEvent B =bothdiceshowanumbermorethan8 Exercise14.1.1 Solutiononp.575. Events A and B are: A. Mutuallyexclusive. B. Independent. C. Mutuallyexclusiveandindependent. D. Neithermutuallyexclusivenorindependent. Exercise14.1.2 Solutiononp.575. Find P A j B A. 2 4 B. 16 144 C. 4 16 C. 2 144 Exercise14.1.3 Solutiononp.575. WhichofthefollowingareTRUEwhenweperformahypothesistestonmatchedorpairedsamples? A. Samplesizesarealmostneversmall. B. Twomeasurementsaredrawnfromthesamepairofindividualsorobjects. C. Twosampleaveragesarecomparedtoeachother. D. AnswerchoicesBandCarebothtrue. Questions4-5refertothefollowing: 118studentswereaskedwhattypeofcolortheirbedroomswerepainted:lightcolors,darkcolorsorvibrant colors.Theresultsweretabulatedaccordingtogender. 11 Thiscontentisavailableonlineat. 531

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532 APPENDIX Lightcolors Darkcolors Vibrantcolors Female 20 22 28 Male 10 30 8 Exercise14.1.4 Solutiononp.575. Findtheprobabilitythatarandomlychosenstudentismaleorhasabedroompaintedwithlight colors. A. 10 118 B. 68 118 C. 48 118 D. 10 48 Exercise14.1.5 Solutiononp.575. Findtheprobabilitythatarandomlychosenstudentismalegiventhestudent'sbedroomis paintedwithdarkcolors. A. 30 118 B. 30 48 C. 22 118 D. 30 52 Questions67refertothefollowing: Weareinterestedinthenumberoftimesateenagermustberemindedtodohis/herchoreseachweek.A surveyof40motherswasconducted.Thetablebelowshowstheresultsofthesurvey. X P x 0 2 40 1 5 40 2 3 14 40 4 7 40 5 4 40 Exercise14.1.6 Solutiononp.575. Findtheprobabilitythatateenagerisreminded2times. A. 8 B. 8 40 C. 6 40 D. 2 Exercise14.1.7 Solutiononp.575. Findtheexpectednumberoftimesateenagerisremindedtodohis/herchores. A. 15 B. 2.78

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APPENDIX 533 C. 1.0 D. 3.13 Questions89refertothefollowing: Onanygivenday,approximately37.5%ofthecarsparkedintheDeAnzaparkingstructureareparked crookedly.SurveydonebyKathyPlum.Werandomlysurvey22cars.Weareinterestedinthenumberof carsthatareparkedcrookedly. Exercise14.1.8 Solutiononp.575. Forevery22cars,howmanywouldyouexpecttobeparkedcrookedly,onaverage? A. 8.25 B. 11 C. 18 D. 7.5 Exercise14.1.9 Solutiononp.575. Whatistheprobabilitythatatleast10ofthe22carsareparkedcrookedly. A. 0.1263 B. 0.1607 C. 0.2870 D. 0.8393 Exercise14.1.10 Solutiononp.575. Usingasampleof15Stanford-BinetIQscores,wewishtoconductahypothesistest.Ourclaim isthattheaverageIQscoreontheStanford-BinetIQtestismorethan100.Itisknownthatthe standarddeviationofallStanford-BinetIQscoresis15points.Thecorrectdistributiontousefor thehypothesistestis: A. Binomial B. Student-t C. Normal D. Uniform Questions1113refertothefollowing: DeAnzaCollegekeepsstatisticsonthepassrateofstudentswhoenrollinmathclasses.Accordingtothe statisticskeptfromFall1997throughFall1999,1795studentsenrolledinMath1Astquartercalculus and1428passedthecourse.Inthesametimeperiod,ofthe856studentsenrolledinMath1Bndquarter calculus,662passed.Ingeneral,arethepassratesofMath1AandMath1Bstatisticallythesame?LetA= thesubscriptforMath1AandB=thesubscriptforMath1B. Exercise14.1.11 Solutiononp.575. Ifyouweretoconductanappropriatehypothesistest,thealternatehypothesiswouldbe: A. H a : p A = p B B. H a : p A > p B C. H o : p A = p B D. H a : p A 6 = p B

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534 APPENDIX Exercise14.1.12 Solutiononp.575. TheTypeIerroristo: A. believethatthepassrateforMath1AisthesameasthepassrateforMath1Bwhen,in fact,thepassratesaredifferent. B. believethatthepassrateforMath1AisdifferentthanthepassrateforMath1Bwhen,in fact,thepassratesarethesame. C. believethatthepassrateforMath1AisgreaterthanthepassrateforMath1Bwhen,in fact,thepassrateforMath1AislessthanthepassrateforMath1B. D. believethatthepassrateforMath1AisthesameasthepassrateforMath1Bwhen,in fact,theyarethesame. Exercise14.1.13 Solutiononp.575. Thecorrectdecisionisto: A. reject H o B. notreject H o C. notmakeadecisionbecauseoflackofinformation Kia,Alejandra,andIrisarerunnersonthetrackteamsatthreedifferentschools.Theirrunningtimes,in minutes,andthestatisticsforthetrackteamsattheirrespectiveschools,foraonemilerun,aregiveninthe tablebelow: RunningTime SchoolAverageRunningTime SchoolStandardDeviation Kia 4.9 5.2 .15 Alejandra 4.2 4.6 .25 Iris 4.5 4.9 .12 Exercise14.1.14 Solutiononp.575. Whichstudentisthebestwhencomparedtotheotherrunnersatherschool? A. Kia B. Alejandra C. Iris D. Impossibletodetermine Questions1516refertothefollowing: ThefollowingadultskisweaterpricesarefromtheGorsuchLtd.Wintercatalog: f \$212,\$292,\$278,\$199\$280,\$236 g Assumetheunderlyingsweaterpricepopulationisapproximatelynormal.Thenullhypothesisisthatthe averagepriceofadultskisweatersfromGorsuchLtd.isatleast\$275. Exercise14.1.15 Solutiononp.575. Thecorrectdistributiontouseforthehypothesistestis: A. Normal B. Binomial C. Student-t D. Exponential

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APPENDIX 535 Exercise14.1.16 Solutiononp.575. Thehypothesistest: A. istwo-tailed B. isleft-tailed C. isright-tailed D. hasnotails Exercise14.1.17 Solutiononp.575. Sara,astatisticsstudent,wantedtodeterminetheaveragenumberofbooksthatcollegeprofessors haveintheirofce.Sherandomlyselected2buildingsoncampusandaskedeachprofessorinthe selectedbuildingshowmanybooksareinhis/herofce.Sarasurveyed25professors.Thetype ofsamplingselectedisa: A. simplerandomsampling B. systematicsampling C. clustersampling D. stratiedsampling Exercise14.1.18 Solutiononp.575. Aclothingstorewouldusewhichmeasureofthecenterofdatawhenplacingorders? A. Mean B. Median C. Mode D. IQR Exercise14.1.19 Solutiononp.575. Inahypothesistest,thep-valueis A. theprobabilitythatanoutcomeofthedatawillhappenpurelybychancewhenthenull hypothesisistrue. B. calledthepreconceivedalpha. C. comparedtobetatodecidewhethertorejectornotrejectthenullhypothesis. D. AnswerchoicesAandBarebothtrue. Questions20-22refertothefollowing: Acommunitycollegeoffersclasses6daysaweek:MondaythroughSaturday.Mariaconductedastudy ofthestudentsinherclassestodeterminehowmanydaysperweekthestudentswhoareinherclasses cometocampusforclasses.Ineachofher5classessherandomlyselected10studentsandaskedthem howmanydaystheycometocampusforclasses.Theresultsofhersurveyaresummarizedinthetable below. NumberofDaysonCampus Frequency RelativeFrequency CumulativeRelativeFrequency 1 2 2 12 .24 3 10 .20 4 .98 5 0 6 1 .02 1.00

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536 APPENDIX Exercise14.1.20 Solutiononp.576. Combinedwithconveniencesampling,whatothersamplingtechniquedidMariause? A. simplerandom B. systematic C. cluster D. stratied Exercise14.1.21 Solutiononp.576. Howmanystudentscometocampusforclasses4daysaweek? A. 49 B. 25 C. 30 D. 13 Exercise14.1.22 Solutiononp.576. Whatisthe60thpercentileforthethisdata? A. 2 B. 3 C. 4 D. 5 Thenexttwoquestionsrefertothefollowing: Thefollowingdataaretheresultsofarandomsurveyof110Reservistscalledtoactivedutytoincrease securityatCaliforniaairports. NumberofDependents Frequency 0 11 1 27 2 33 3 20 4 19 Exercise14.1.23 Solutiononp.576. Constructa95%CondenceIntervalforthetruepopulationaveragenumberofdependentsof ReservistscalledtoactivedutytoincreasesecurityatCaliforniaairports. A. .85,2.32 B. .80,2.36 C. .97,2.46 D. .92,2.50 Exercise14.1.24 Solutiononp.576. The95%condenceIntervalabovemeans: A. 5%ofCondenceIntervalsconstructedthiswaywillnotcontainthetruepopulation aveagenumberofdependents.

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APPENDIX 537 B. Weare95%condentthetruepopulationaveragenumberofdependentsfallsintheinterval. C. Bothoftheaboveanswerchoicesarecorrect. D. Noneoftheabove. Exercise14.1.25 Solutiononp.576. X U 4,10 .Findthe30thpercentile. A. 0.3000 B. 3 C. 5.8 D. 6.1 Exercise14.1.26 Solutiononp.576. If X Exp 0.8 ,then P X < m = A. 0.3679 B. 0.4727 C. 0.6321 D. cannotbedetermined Exercise14.1.27 Solutiononp.576. Thelifetimeofacomputercircuitboardisnormallydistributedwithameanof2500hoursanda standarddeviationof60hours.Whatistheprobabilitythatarandomlychosenboardwilllastat most2560hours? A. 0.8413 B. 0.1587 C. 0.3461 D. 0.6539 Exercise14.1.28 Solutiononp.576. Asurveyof123ReservistscalledtoactivedutyasaresultoftheSeptember11,2001,attacks wasconductedtodeterminetheproportionthatweremarried.Eighty-sixreportedbeingmarried. Constructa98%condenceintervalforthetruepopulationproportionofreservistscalledtoactive dutythataremarried. A. .6030,0.7954 B. .6181,0.7802 C. .5927,0.8057 D. .6312,0.7672 Exercise14.1.29 Solutiononp.576. Winningtimesin26milemarathonsrunbyworldclassrunnersaverage145minuteswithastandarddeviationof14minutes.Asampleofthelast10marathonwinningtimesiscollected. Let x =averagewinningtimesfor10marathons. Thedistributionfor x is: A. N 145, 14 p 10 B. N 145,14 C. t 9

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538 APPENDIX D. t 10 Exercise14.1.30 Solutiononp.576. SupposethatPhiBetaKappahonorsthetop1%ofcollegeanduniversityseniors.Assumethat gradepointaveragesG.P.A.atacertaincollegearenormallydistributedwitha2.5averageand astandarddeviationof0.5.WhatwouldbetheminimumG.P.A.neededtobecomeamemberof PhiBetaKappaatthatcollege? A. 3.99 B. 1.34 C. 3.00 D. 3.66 ThenumberofpeoplelivingonAmericanfarmshasdeclinedsteadilyduringthiscentury.Herearedata onthefarmpopulationinmillionsofpersonsfrom1935to1980. Year 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 Population 32.1 30.5 24.4 23.0 19.1 15.6 12.4 9.7 8.9 7.2 Thelinearregressionequationisy-hat=1166.930.5868x Exercise14.1.31 Solutiononp.576. Whatwastheexpectedfarmpopulationinmillionsofpersonsfor1980? A. 7.2 B. 5.1 C. 6.0 D. 8.0 Exercise14.1.32 Solutiononp.576. InlinearregressionwhichSSEifpreferable? A. 13.46 B. 18.22 C. 24.05 D. 16.33 Exercise14.1.33 Solutiononp.576. Inregressionanalysis,ifthecorrelationcoefcientiscloseto1whatcanbesaidaboutthebestt line? A. Itisahorizontalline.Therefore,wecannotuseit. B. Thereisastronglinearpattern.Therefore,itismostlikelyagoodmodeltobeused. C. Thecoefcientcorrelationisclosetothelimit.Therefore,itishardtomakeadecision. D. Wedonothavetheequation.Therefore,wecannotsayanythingaboutit. Question34-36refertothefollowing: Astudyofthecareerplansofyoungwomenandmensentquestionnairestoall722membersofthesenior classintheCollegeofBusinessAdministrationattheUniversityofIllinois.Onequestionaskedwhich majorwithinthebusinessprogramthestudenthadchosen.Herearethedatafromthestudentswho responded.

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APPENDIX 539 Female Male Accounting 68 56 Administration 91 40 Ecomonics 5 6 Finance 61 59 Doesthedatasuggestthatthereisarelationshipbetweenthegenderofstudentsandtheirchoiceofmajor? Exercise14.1.34 Solutiononp.576. Thedistributionforthetestis: A. Chi 2 8 B. Chi 2 3 C. t 722 D. N 0,1 Exercise14.1.35 Solutiononp.576. TheexpectednumberoffemalewhochooseFinanceis: A. 37 B. 61 C. 60 D. 70 Exercise14.1.36 Solutiononp.576. Thep-valueis0.0127.Theconclusiontothetestis: A. Thechoiceofmajorandthegenderofthestudentareindependentofeachother. B. Thechoiceofmajorandthegenderofthestudentarenotindependentofeachother. C. StudentsndEconomicsveryhard. D. MorefemalespreferAdministrationthanmales. Exercise14.1.37 Solutiononp.576. Anagencyreportedthattheworkforcenationwideiscomposedof10%professional,10%clerical, 30%skilled,15%service,and35%semiskilledlaborers.Arandomsampleof100SanJoseresidents indicated15professional,15clerical,40skilled,10service,and20semiskilledlaborers.At a =.10 doestheworkforceinSanJoseappeartobeconsistentwiththeagencyreportforthenation? Whichkindoftestisit? A. Chi 2 goodnessoft B. Chi 2 testofindependence C. Independentgroupsproportions D. Unabletodetermine

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540 APPENDIX 14.2PracticeFinalExam2 12 Exercise14.2.1 Solutiononp.576. Astudywasdonetodeterminetheproportionofteenagersthatownacar.Thetrueproportion ofteenagersthatownacaristhe: A. statistic B. parameter C. population D. variable Thenexttwoquestionsrefertothefollowingdata: value frequency 0 1 1 4 2 7 3 9 6 4 Exercise14.2.2 Solutiononp.576. Theboxplotforthedatais: A. B. C. 12 Thiscontentisavailableonlineat.

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APPENDIX 541 D. Exercise14.2.3 Solutiononp.577. If6wereaddedtoeachvalue,the15thpercentilewouldbe: A. 6 B. 1 C. 7 D. 8 Thenexttwoquestionsrefertothefollowingsituation: Supposethattheprobabilityofadroughtinanyindependentyearis20%.Outofthoseyearsinwhicha droughtoccurs,theprobabilityofwaterrationingis10%.However,inanyyear,theprobabilityofwater rationingis5%. Exercise14.2.4 Solutiononp.577. Whatistheprobabilityofbothadroughtandwaterrationingoccurring? A. 0.05 B. 0.01 C. 0.02 D. 0.30 Exercise14.2.5 Solutiononp.577. Whichofthefollowingistrue? A. droughtandwaterrationingareindependentevents B. droughtandwaterrationingaremutuallyexclusiveevents C. noneoftheabove Thenexttwoquestionsrefertothefollowingsituation: Supposethatasurveyyieldedthefollowingdata: FavoritePieType gender apple pumpkin pecan female 40 10 30 male 20 30 10 Exercise14.2.6 Solutiononp.577. Supposethatoneindividualisrandomlychosen.Theprobabilitythattheperson'sfavoritepieis appleorthepersonismaleis: A. 40 60

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542 APPENDIX B. 60 140 C. 120 140 D. 100 140 Exercise14.2.7 Solutiononp.577. Suppose H o is:Favoritepietypeandgenderareindependent. The p-value is: A. 0 B. 1 C. 0.05 D. cannotbedetermined Thenexttwoquestionsrefertothefollowingsituation: Let'ssaythattheprobabilitythatanadultwatchesthenewsatleastonceperweekis0.60.Werandomly survey14people.Ofinterestisthenumberthatwatchthenewsatleastonceperweek. Exercise14.2.8 Solutiononp.577. WhichofthefollowingstatementsisFALSE? A. X B 14,0.60 B. Thevaluesfor x are: f 1,2,3,...,14 g C. m = 8.4 D. P X = 5 = 0.0408 Exercise14.2.9 Solutiononp.577. Findtheprobabilitythatatleast6adultswatchthenews. A. 6 14 B. 0.8499 C. 0.9417 D. 0.6429 Exercise14.2.10 Solutiononp.577. Thefollowinghistogramismostlikelytobearesultofsamplingfromwhichdistribution?

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APPENDIX 543 A. Chi-Square B. Exponential C. Uniform D. Binomial Theagesofcampusdayandeveningstudentsisknowntobenormallydistributed.Asampleof6campus dayandeveningstudentsreportedtheiragesinyearsas: f 18,35,27,45,20,20 g Exercise14.2.11 Solutiononp.577. Whatistheerrorboundforthe90%condenceintervalofthetrueaverageage? A. 11.2 B. 22.3 C. 17.5 D. 8.7 Exercise14.2.12 Solutiononp.577. Ifanormallydistributedrandomvariablehas m = 0and s = 1,then97.5%ofthepopulation valueslieabove: A. -1.96 B. 1.96 C. 1 D. -1 Thenextthreequestionsrefertothefollowingsituation: Theamountofmoneyacustomerspendsinonetriptothesupermarketisknowntohaveanexponential distribution.Supposetheaverageamountofmoneyacustomerspendsinonetriptothesupermarketis \$72. Exercise14.2.13 Solutiononp.577. Whatistheprobabilitythatonecustomerspendslessthan\$72inonetriptothesupermarket? A. 0.6321 B. 0.5000 C. 0.3714 D. 1 Exercise14.2.14 Solutiononp.577. Howmuchmoneyaltogetherwouldyouexpectnext5customerstospendinonetriptothe supermarketindollars? A. 72 B. 72 2 5 C. 5184 D. 360 Exercise14.2.15 Solutiononp.577. Ifyouwanttondtheprobabilitythattheaverageof5customersislessthan\$60,thedistribution touseis: A. N 72,72 B. N 72,72 p 5

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544 APPENDIX C. Exp 72 D. Exp 1 72 Thenextthreequestionsrefertothefollowingsituation: Theamountoftimeittakesafourthgradertocarryoutthetrashisuniformlydistributedintheinterval from1to10minutes. Exercise14.2.16 Solutiononp.577. Whatistheprobabilitythatarandomlychosenfourthgradertakesmorethan7minutestotake outthetrash? A. 3 9 B. 7 9 C. 3 10 D. 7 10 Exercise14.2.17 Solutiononp.577. Whichgraphbestshowstheprobabilitythatarandomlychosenfourthgradertakesmorethan6 minutestotakeoutthetrashgiventhathe/shehasalreadytakenmorethan3minutes? Exercise14.2.18 Solutiononp.577. Weshouldexpectafourthgradertotakehowmanyminutestotakeoutthetrash? A. 4.5 B. 5.5 C. 5 D. 10 Thenextthreequestionsrefertothefollowingsituation: Atthebeginningofthequarter,theamountoftimeastudentwaitsinlineatthecampuscafeteriaisnormallydistributedwithameanof5minutesandastandarddeviationof2minutes. Exercise14.2.19 Solutiononp.577. Whatisthe90thpercentileofwaitingtimesinminutes?

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APPENDIX 545 A. 1.28 B. 90 C. 8.29 D. 7.56 Exercise14.2.20 Solutiononp.577. Themedianwaitingtimeinminutesforonestudentis: A. 5 B. 50 C. 2.5 D. 2 Exercise14.2.21 Solutiononp.577. Asampleof10studentshasanaveragewaitingtimeof5.5minutes.The95%condenceinterval forthetruepopulationmeanis: A. 4.46,6.04 B. 4.26,6.74 C. 2.4,8.6 D. 1.58,9.42 Exercise14.2.22 Solutiononp.577. Asampleof80softwareengineersinSiliconValleyistakenanditisfoundthat20%ofthemearn approximately\$50,000peryear.ApointestimateforthetrueproportionofengineersinSilicon Valleywhoearn\$50,000peryearis: A. 16 B. 0.2 C. 1 D. 0.95 Exercise14.2.23 Solutiononp.577. If P Z < z a = 0.1587where Z N 0,1 ,then a isequalto: A. -1 B. 0.1587 C. 0.8413 D. 1 Exercise14.2.24 Solutiononp.578. Aprofessortested35studentstodeterminetheirenteringskills.Attheendoftheterm,after completingthecourse,thesametestwasadministeredtothesame25studentstostudytheir improvement.Thiswouldbeatestof: A. independentgroups B. 2proportions C. dependentgroups D. exclusivegroups Exercise14.2.25 Solutiononp.578. AmathexamwasgiventoallthethirdgradechildrenattendingABCSchool.Tworandom samplesofscoresweretaken.

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APPENDIX 547 Exercise14.2.29 Solutiononp.578. Isthereacorrelationbetweenthenumberofemployeesandthenumberofbathroomssignicant? A. Yes B. No C. Notenoughinformationtoanswerquestion Exercise14.2.30 Solutiononp.578. Thelinearregressionequationis: A. y = 0.0094 )]TJ/F58 9.9626 Tf 10.131 0 Td [(79.96 x B. y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(79.96 + 0.0094 x C. y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(79.96 )]TJ/F58 9.9626 Tf 10.132 0 Td [(0.0094 x D. y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(0.0094 + 79.96 x Exercise14.2.31 Solutiononp.578. Ifasitehas1150employees,approximatelyhowmanybathroomsshouldithave? A. 69 B. 121 C. 101 D. 86 Exercise14.2.32 Solutiononp.578. Supposethatasampleofsize10wascollected,with x =4.4and s =1.4. H o : s 2 =1.6vs. H a : s 2 6 = 1.6 Exercise14.2.33 Solutiononp.578. 64backpackerswereaskedthenumberofdaystheirlatestbackpackingtripwas.Thenumberof daysisgiveninthetablebelow: #ofdays 1 2 3 4 5 6 7 8 Frequency 5 9 6 12 7 10 5 10 Conductanappropriatetesttodetermineifthedistributionisuniform. A. The pvalue is > 0.10,thedistributionisuniform.

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548 APPENDIX B. The pvalue is < 0.01,thedistributionisuniform. C. The pvalue isbetween0.01and0.10,butwithout a thereisnotenoughinformation D. Thereisnosuchtestthatcanbeconducted. Exercise14.2.34 Solutiononp.578. Whichofthefollowingassumptionsismadewhenusingone-wayANOVA? A. Thepopulationsfromwhichthesamplesareselectedhavedifferentdistributions. B. Thesamplesizesarelarge. C. Thetestistodetermineifthedifferentgroupshavethesameaverages. D. Thereisacorrelationbetweenthefactorsoftheexperiment.

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APPENDIX 549 14.3DataSets 13 14.3.1LapTimes ThefollowingtablesprovidelaptimesfromTerriVogel'sLogBook.Timesarerecordedinsecondsfor 2.5-milelapscompletedinaseriesofracesandpracticeruns. RaceLapTimesinSeconds Lap1 Lap2 Lap3 Lap4 Lap5 Lap6 Lap7 Race1 135 130 131 132 130 131 133 Race2 134 131 131 129 128 128 129 Race3 129 128 127 127 130 127 129 Race4 125 125 126 125 124 125 125 Race5 133 132 132 132 131 130 132 Race6 130 130 130 129 129 130 129 Race7 132 131 133 131 134 134 131 Race8 127 128 127 130 128 126 128 Race9 132 130 127 128 126 127 124 Race10 135 131 131 132 130 131 130 Race11 132 131 132 131 130 129 129 Race12 134 130 130 130 131 130 130 Race13 128 127 128 128 128 129 128 Race14 132 131 131 131 132 130 130 Race15 136 129 129 129 129 129 129 Race16 129 129 129 128 128 129 129 Race17 134 131 132 131 132 132 132 Race18 129 129 130 130 133 133 127 Race19 130 129 129 129 129 129 128 Race20 131 128 130 128 129 130 130 13 Thiscontentisavailableonlineat.

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550 APPENDIX PracticeLapTimesinSeconds Lap1 Lap2 Lap3 Lap4 Lap5 Lap6 Lap7 Practice1 142 143 180 137 134 134 172 Practice2 140 135 134 133 128 128 131 Practice3 130 133 130 128 135 133 133 Practice4 141 136 137 136 136 136 145 Practice5 140 138 136 137 135 134 134 Practice6 142 142 139 138 129 129 127 Practice7 139 137 135 135 137 134 135 Practice8 143 136 134 133 134 133 132 Practice9 135 134 133 133 132 132 133 Practice10 131 130 128 129 127 128 127 Practice11 143 139 139 138 138 137 138 Practice12 132 133 131 129 128 127 126 Practice13 149 144 144 139 138 138 137 Practice14 133 132 137 133 134 130 131 Practice15 138 136 133 133 132 131 131

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APPENDIX 551 14.3.2StockPrices ThefollowingtablelistsinitialpublicofferingIPOstickpricesforall1999stocksthatatleastdoubledin valueduringtherstdayoftrading. IPOOfferPrices \$17.00 \$23.00 \$14.00 \$16.00 \$12.00 \$26.00 \$20.00 \$22.00 \$14.00 \$15.00 \$22.00 \$18.00 \$18.00 \$21.00 \$21.00 \$19.00 \$15.00 \$21.00 \$18.00 \$17.00 \$15.00 \$25.00 \$14.00 \$30.00 \$16.00 \$10.00 \$20.00 \$12.00 \$16.00 \$17.44 \$16.00 \$14.00 \$15.00 \$20.00 \$20.00 \$16.00 \$17.00 \$16.00 \$15.00 \$15.00 \$19.00 \$48.00 \$16.00 \$18.00 \$9.00 \$18.00 \$18.00 \$20.00 \$8.00 \$20.00 \$17.00 \$14.00 \$11.00 \$16.00 \$19.00 \$15.00 \$21.00 \$12.00 \$8.00 \$16.00 \$13.00 \$14.00 \$15.00 \$14.00 \$13.41 \$28.00 \$21.00 \$17.00 \$28.00 \$17.00 \$19.00 \$16.00 \$17.00 \$19.00 \$18.00 \$17.00 \$15.00 \$14.00 \$21.00 \$12.00 \$18.00 \$24.00 \$15.00 \$23.00 \$14.00 \$16.00 \$12.00 \$24.00 \$20.00 \$14.00 \$14.00 \$15.00 \$14.00 \$19.00 \$16.00 \$38.00 \$20.00 \$24.00 \$16.00 \$8.00 \$18.00 \$17.00 \$16.00 \$15.00 \$7.00 \$19.00 \$12.00 \$8.00 \$23.00 \$12.00 \$18.00 \$20.00 \$21.00 \$34.00 \$16.00 \$26.00 \$14.00 N OTE : DatacompiledbyJayR.RitterofUniv.ofFloridausingdatafromSecuritiesDataCo.and Bloomberg.

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552 APPENDIX 14.4GroupProjects 14.4.1GroupProject:UnivariateData 14 14.4.1.1StudentLearningObjectives Thestudentwilldesignandcarryoutasurvey. Thestudentwillanalyzeandgraphicallydisplaytheresultsofthesurvey. 14.4.1.2Instructions Asyoucompleteeachtaskbelow,checkitoff.Answerallquestionsinyoursummary. ____ Decidewhatdatayouaregoingtostudy. E XAMPLES :Herearetwoexamples,butyoumay NOT usethem:numberofM&M'sper smallbag,numberofpencilsstudentshaveintheirbackpacks. ____ Isyourdatadiscreteorcontinuous?Howdoyouknow? ____ Decidehowyouaregoingtocollectthedataforinstance,buy30bagsofM&M's;collectdata fromtheWorldWideWeb. ____ Describeyoursamplingtechniqueindetail.Usecluster,stratied,systematic,orsimplerandom usingarandomnumbergeneratorsampling.Donotuseconveniencesampling.Whatmethod didyouuse?Whydidyoupickthatmethod? ____ Conductyoursurvey. Yourdatasizemustbeatleast30. ____ Summarizeyourdatainachartwithcolumnsshowing datavalue,frequency,relativefrequency andcumulativerelativefrequency. ____ Answerthefollowingroundedto2decimalplaces: 1. x = 2. s = 3. Firstquartile= 4. Median= 5. 70thpercentile= ____ Whatvalueis2standarddeviationsabovethemean? ____ Whatvalueis1.5standarddeviationsbelowthemean? ____ Constructahistogramdisplayingyourdata. ____ Incompletesentences,describetheshapeofyourgraph. ____ Doyounoticeanypotentialoutliers?Ifso,whatvaluesarethey?Showyourworkinhowyou usedthepotentialoutlierformulainChapter2sinceyouhaveunivariatedatatodetermine whetherornotthevaluesmightbeoutliers. ____ Constructaboxplotdisplayingyourdata. ____ Doesthemiddle50%ofthedataappeartobeconcentratedtogetherorspreadapart?Explain howyoudeterminedthis. ____ Lookingatboththehistogramandtheboxplot,discussthedistributionofyourdata. 14.4.1.3AssignmentChecklist Youneedtoturninthefollowingtypedandstapledpacket,withpagesinthefollowingorder: ____ Coversheet :name,classtime,andnameofyourstudy 14 Thiscontentisavailableonlineat.

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APPENDIX 553 ____ Summarypage :Thisshouldcontainparagraphswrittenwithcompletesentences.Itshould includeanswerstoallthequestionsabove.Itshouldalsoincludestatementsdescribingthe populationunderstudy,thesample,aparameterorparametersbeingstudied,andthestatistic orstatisticsproduced. ____ URL fordata,ifyourdataarefromtheWorldWideWeb. ____ Chartofdata,frequency,relativefrequencyandcumulativerelativefrequency. ____ Pagesofgraphs: histogramandboxplot.

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554 APPENDIX 14.4.2GroupProject:ContinuousDistributionsandCentralLimitTheorem 15 14.4.2.1StudentLearningObjectives Thestudentwillcollectasampleofcontinuousdata. Thestudentwillattempttotthedatasampletovariousdistributionmodels. ThestudentwillvalidatetheCentralLimitTheorem. 14.4.2.2Instructions Asyoucompleteeachtaskbelow,checkitoff.Answerallquestionsinyoursummary. 14.4.2.3PartI:Sampling ____ Decidewhat continuous datayouaregoingtostudy.Herearetwoexamples,butyoumayNOT usethem:theamountofmoneyastudentspendsoncollegesuppliesthistermorthelengthof alongdistancetelephonecall. ____ Describeyoursamplingtechniqueindetail.Usecluster,stratied,systematic,orsimplerandom usingarandomnumbergeneratorsampling.Donotuseconveniencesampling.Whatmethod didyouuse?Whydidyoupickthatmethod? ____ Conductyoursurvey.Gather atleast150piecesofcontinuousquantitativedata ____ Deneinwordstherandomvariableforyourdata. X =_______ ____ Create2listsofyourdata:unordereddata,inorderofsmallesttolargest. ____ Findthesamplemeanandthesamplestandarddeviationroundedto2decimalplaces. 1. )]TJ/F132 9.9626 Tf 0.561 -6.809 Td [(x = 2. s = ____ Constructahistogramofyourdatacontaining5-10intervalsofequalwidth.Thehistogram shouldbearepresentativedisplayofyourdata.Labelandscaleit. 14.4.2.4PartII:PossibleDistributions ____ Supposethat X followedthetheoreticaldistributionsbelow.Setupeachdistributionusingthe appropriateinformationfromyourdata. ____ Uniform: X U ____________Usethelowestandhighestvaluesas a and b ____ Exponential: X Exp ____________Use x toestimate m ____ Normal: X N ____________Use x toestimatefor m and s toestimatefor s ____ Must yourdatatoneoftheabovedistributions?Explainwhyorwhynot. ____ Could thedatat2or3oftheabovedistributionsatthesametime?Explain. ____ Calculatethevalue k an X valuethatis1.75standarddeviationsabovethesamplemean. k = _________roundedto2decimalplacesNote: k = x + 1.75 s ____ Determinetherelativefrequencies RF roundedto4decimalplaces. 1. RF = frequency totalnumbersurveyed 2. RF X < k = 3. RF X > k = 4. RF X = k = UseaseparatepieceofpaperforEACHdistributionuniform,exponential,normaltorespondtothe followingquestions. 15 Thiscontentisavailableonlineat.

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APPENDIX 555 N OTE :Youshouldhaveonepagefortheuniform,onepagefortheexponential,andonepagefor thenormal ____ Statethedistribution: X _________ ____ Drawagraphforeachofthethreetheoreticaldistributions.Labeltheaxesandmarkthem appropriately. ____ Findthefollowingtheoreticalprobabilitiesroundedto4decimalplaces. 1. PX < k = 2. PX > k = 3. PX=k = ____ Comparetherelativefrequenciestothecorrespondingprobabilities.Arethevaluesclose? ____ Doesitappearthatthedatatthedistributionwell?Justifyyouranswerbycomparingthe probabilitiestotherelativefrequencies,andthehistogramstothetheoreticalgraphs. 14.4.2.5PartIII:CLTExperiments ______ Fromyouroriginaldatabeforeordering,usearandomnumbergeneratortopick40samples ofsize5.Foreachsample,calculatetheaverage. ______ Onaseparatepage,attachedtothesummary,includethe40samplesofsize5,alongwiththe 40sampleaverages. ______ Listthe40averagesinorderfromsmallesttolargest. ______ Denetherandomvariable, X ,inwords. X = ______ Statetheapproximatetheoreticaldistributionof X X ______ Basethisonthemeanandstandarddeviationfromyouroriginaldata. ______ Constructahistogramdisplayingyourdata.Use5to6intervalsofequalwidth.Labeland scaleit. Calculatethevalue k an X valuethatis1.75standarddeviationsabovethesamplemean. k =_____ roundedto2decimalplaces DeterminetherelativefrequenciesRFroundedto4decimalplaces. 1. RF X < k = 2. RF X > k = 3. RF X = k = Findthefollowingtheoreticalprobabilitiesroundedto4decimalplaces. 1 -P X < k = 2 -P X > k = 3 -P X = k = ______ Drawthegraphofthetheoreticaldistributionof X ______ Answerthequestionsbelow. ______ Comparetherelativefrequenciestotheprobabilities.Arethevaluesclose? ______ Doesitappearthatthedataofaveragestthedistributionof )]TJ/F132 9.9626 Tf -0.725 -8.882 Td [(X well?Justifyyouranswer bycomparingtheprobabilitiestotherelativefrequencies,andthehistogramtothetheoretical graph. ______ In3-5completesentencesforeach,answerthefollowingquestions.Givethoughtfulexplanations. ______ Insummary,doyouroriginaldataseemtottheuniform,exponential,ornormaldistributions?Answerwhyorwhynotforeachdistribution.Ifthedatadonottanyofthose distributions,explainwhy.

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556 APPENDIX ______ Whathappenedtotheshapeanddistributionwhenyouaveragedyourdata? Intheory, what shouldhavehappened?Intheory,woulditalwayshappen?Whyorwhynot? ______ Weretherelativefrequenciescomparedtothetheoreticalprobabilitiescloserwhencomparing the X or )]TJ/F132 9.9626 Tf -0.724 -8.882 Td [(X distributions?Explainyouranswer. 14.4.2.6AssignmentChecklist Youneedtoturninthefollowingtypedandstapledpacket,withpagesinthefollowingorder: ____ Coversheet :name,classtime,andnameofyourstudy ____ Summarypages :Theseshouldcontainseveralparagraphswrittenwithcompletesentencesthat describetheexperiment,includingwhatyoustudiedandyoursamplingtechnique,aswellas answerstoallofthequestionsabove. ____ URL fordata,ifyourdataarefromtheWorldWideWeb. ____ Pages,oneforeachtheoreticaldistribution ,withthedistributionstated,thegraph,andthe probabilityquestionsanswered ____ Pagesofthedatarequested ____ Allgraphsrequired

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APPENDIX 557 14.4.3PartnerProject:HypothesisTesting-Article 16 14.4.3.1StudentLearningObjectives Thestudentwillidentifyahypothesistestingprobleminprint. Thestudentwillconductasurveytoverifyordisputetheresultsofthehypothesistest. Thestudentwillsummarizethearticle,analysis,andconclusionsinareport. 14.4.3.2Instructions Asyoucompleteteachtaskbelow,checkitoff.Answerallquestionsinyoursummary. ____ Findanarticle inanewspaper,magazineorontheinternetwhichmakesaclaimabout ONE populationmeanor ONE populationproportion.Theclaimmaybebaseduponasurveythat thearticlewasreportingon.Decidewhetherthisclaimisthenulloralternatehypothesis. ____ Copyorprintoutthearticle andincludeacopyinyourproject,alongwiththesource. ____ Statehowyouwillcollectyourdata. Conveniencesamplingisnotacceptable. ____ Conductyoursurvey.Youmusthavemorethan50responsesinyoursample. Whenyouhand inyournalproject,attachthetallysheetorthepacketofquestionnairesthatyouusedtocollect data.Yourdatamustbereal. ____ Statethestatistics thatarearesultofyourdatacollection:samplesize,samplemean,andsample standarddeviation,ORsamplesizeandnumberofsuccesses. ____ Make2copiesoftheappropriatesolutionsheet. ____ Recordthehypothesistest onthesolutionsheet,basedonyourexperiment. DoaDRAFTsolution rstononeofthesolutionsheetsandcheckitovercarefully.Haveaclassmatecheck yoursolutiontoseeifitisdonecorrectly.Makeyourdecisionusinga5%levelofsignicance. Includethe95%condenceintervalonthesolutionsheet. ____ Createagraphthatillustratesyourdata. Thismaybeapieorbarchartormaybeahistogram orboxplot,dependingonthenatureofyourdata.Produceagraphthatmakessenseforyour dataandgivesusefulvisualinformationaboutyourdata.Youmayneedtolookatseveraltypes ofgraphsbeforeyoudecidewhichisthemostappropriateforthetypeofdatainyourproject. ____ Writeyoursummary incompletesentencesandparagraphs,withpropergrammarandcorrect spellingthatdescribestheproject.Thesummary MUST include: 1. Briefdiscussionofthearticle,includingthesource. 2. Statementoftheclaimmadeinthearticleoneofthehypotheses. 3. Detaileddescriptionofhow,where,andwhenyoucollectedthedata,includingthesampling technique.Didyouusecluster,stratied,systematic,orsimplerandomsamplingusinga randomnumbergenerator?Asstatedabove,conveniencesamplingisnotacceptable. 4. Conclusionaboutthearticleclaiminlightofyourhypothesistest.Thisistheconclusion ofyourhypothesistest,statedinwords,inthecontextofthesituationinyourprojectin sentenceform,asifyouwerewritingthisconclusionforanon-statistician. 5. Sentenceinterpretingyourcondenceintervalinthecontextofthesituationinyourproject. 14.4.3.3AssignmentChecklist Turninthefollowingtypedpointandstapledpacketforyournalproject: ____ Coversheet containingyournames,classtime,andthenameofyourstudy. ____ Summary ,whichincludesallitemslistedonsummarychecklist. ____ Solutionsheet neatlyandcompletelylledout.Thesolutionsheetdoesnotneedtobetyped. 16 Thiscontentisavailableonlineat.

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558 APPENDIX ____ Graphicrepresentationofyourdata ,createdfollowingtheguidelinesdiscussedabove.Include onlygraphswhichareappropriateanduseful. ____ RawdatacollectedANDatablesummarizingthesampledata n,xbarands;orx,n,andp', asappropriateforyourhypotheses.Therawdatadoesnotneedtobetyped,butthesummarydoes.Handinthedataasyoucollectedit.Eitherattachyourtallysheetoranenvelope containingyourquestionnaires.

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APPENDIX 559 14.4.4PartnerProject:HypothesisTesting-WordProblem 17 14.4.4.1StudentLearningObjectives Thestudentwillwrite,edit,andsolveahypothesistestingwordproblem. 14.4.4.2Instructions Writeanoriginalhypothesistestingproblemforeither ONE populationmeanor ONE populationproportion.Asyoucompleteeachtask,checkitoff.Answerallquestionsinyoursummary.Lookatthehomework fortheHypothesisTesting:SingleMeanandSingleProportionchapterforexamplespoems,twoactsofa play,aworkrelatedproblem.Theproblemswithnamesattachedtothemareproblemswrittenbystudents inpastquarters.Someotherexamplesthatarenotinthehomeworkinclude:asoccerhypothesistesting poster,acartoon,anewsreports,achildren'sstory,asong. ____ Yourproblemmustbeoriginalandcreative.ItalsomustbeinproperEnglish.IfEnglishis difcultforyou,havesomeoneedityourproblem. ____ Yourproblemmustbeatleastpage,typedandsingledspaced.This DOESNOT includethe data.Datawillmaketheproblemlongerandthatisne.Forthisproblem,thedataandstory mayberealorctional. ____ Inthenarrativeoftheproblem,makeitveryclearwhatthenullandalternativehypothesesare. ____ Yoursamplesizemustbe LARGERTHAN50 evenifitisctional. ____ Stateinyourproblemhowyouwillcollectyourdata. ____ Includeyourdatawithyourwordproblem. ____ Statethestatisticsthatarearesultofyourdatacollection:samplesize,samplemean,andsample standarddeviation,ORsamplesizeandnumberofsuccesses. ____ Createagraphthatillustratesyourproblem.Thismaybeapieorbarchartormaybeahistogramorboxplot,dependingonthenatureofyourdata.Produceagraphthatmakessensefor yourdataandgivesusefulvisualinformationaboutyourdata.Youmayneedtolookatseveral typesofgraphsbeforeyoudecidewhichisthemostappropriateforyourproblem. ____ Make2copiesoftheappropriatesolutionsheet. ____ Recordthehypothesistestonthesolutionsheet,basedonyourproblem.Doa DRAFT solution rstononeofthesolutionsheetsandcheckitovercarefully.Makeyourdecisionusinga5% levelofsignicance.Includethe95%condenceintervalonthesolution 14.4.4.3AssignmentChecklist Youneedtoturninthefollowingtypedpointandstapledpacketforyournalproject: ____ Coversheet containingyourname,thenameofyourproblem,andthedate ____ Theproblem ____ Datafortheproblem ____ Solutionsheet neatlyandcompletelylledout.Thesolutionsheetdoesnotneedtobetyped. ____ Graphicrepresentationofthedata ,createdfollowingtheguidelinesdiscussedabove.Include onlygraphsthatareappropriateanduseful. ____ Sentencesinterpretingtheresultsofthehypothesistestandthecondenceinterval inthecontextofthesituationintheproject. 17 Thiscontentisavailableonlineat.

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560 APPENDIX 14.4.5GroupProject:BivariateData,LinearRegression,andUnivariateData 18 14.4.5.1StudentLearningObjectives Thestudentswillcollectabivariatedatasamplethroughtheuseofappropriatesamplingtechniques. Thestudentwillattempttotthedatatoalinearmodel. Thestudentwilldeterminetheappropriatenessoflineartofthemodel. Thestudentwillanalyzeandgraphunivariatedata. 14.4.5.2Instructions 1.Asyoucompleteeachtaskbelow,checkitoff.Answerallquestionsinyourintroductionorsummary. 2.Checkyourcoursecalendarforintermediateandnalduedates. 3.Graphsmaybeconstructedbyhandorbycomputer,unlessyourinstructorinformsyouotherwise. Allgraphsmustbeneatandaccurate. 4.Allotherresponsesmustbedoneonthecomputer. 5.Neatnessandqualityofexplanationsareusedtodetermineyournalgrade. 14.4.5.3PartI:BivariateData Introduction ____ Statethebivariatedatayourgroupisgoingtostudy. E XAMPLES :Herearetwoexamples,butyoumay NOT usethem:heightvs.weightandage vs.runningdistance. ____ Describehowyourgroupisgoingtocollectthedataforinstance,collectdatafromtheweb, surveystudentsoncampus. ____ Describeyoursamplingtechniqueindetail.Usecluster,stratied,systematic,orsimplerandomsamplingusingarandomnumbergeneratorsampling.Conveniencesamplingis NOT acceptable. ____ Conductyoursurvey.Yournumberofpairsmustbeatleast30. ____ Printoutacopyofyourdata. Analysis ____ Onaseparatesheetofpaperconstructascatterplotofthedata.Labelandscalebothaxes. ____ Statetheleastsquareslineandthecorrelationcoefcient. ____ Onyourscatterplot,inadifferentcolor,constructtheleastsquaresline. ____ Isthecorrelationcoefcientsignicant?Explainandshowhowyoudeterminedthis. ____ Interprettheslopeofthelinearregressionlineinthecontextofthedatainyourproject.Relate theexplanationtoyourdata,andquantifywhattheslopetellsyou. ____ Doestheregressionlineseemtotthedata?Whyorwhynot?Ifthedatadoesnotseemtobe linear,explainifanyothermodelseemstotthedatabetter. ____ Arethereanyoutliers?Ifso,whatarethey?Showyourworkinhowyouusedthepotential outlierformulaintheLinearRegressionandCorrelationchaptersinceyouhavebivariatedata todeterminewhetherornotanypairsmightbeoutliers. 18 Thiscontentisavailableonlineat.

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APPENDIX 561 14.4.5.4PartII:UnivariateData Inthissection,youwillusethedatafor ONE variableonly.Pickthevariablethatismoreinterestingto analyze.Forexample:ifyourindependentvariableissequentialdatasuchasyearwith30yearsandone pieceofdataperyear,yourx-valuesmightbe1971,1972,1973,1974,...,2000.Thiswouldnotbeinteresting toanalyze.Inthatcase,choosetousethedependentvariabletoanalyzeforthispartoftheproject. _____ Summarizeyourdatainachartwithcolumnsshowingdatavalue,frequency,relativefrequency,andcumulativerelativefrequency. _____ Answerthefollowing,roundedto2decimalplaces: 1. Samplemean= 2. Samplestandarddeviation= 3. Firstquartile= 4. Thirdquartile= 5. Median= 6. 70thpercentile= 7. Valuethatis2standarddeviationsabovethemean= 8. Valuethatis1.5standarddeviationsbelowthemean= _____ Constructahistogramdisplayingyourdata.Groupyourdatainto610intervalsofequal width.Pickregularlyspacedintervalsthatmakesenseinrelationtoyourdata.Forexample, doNOTgroupdatabyageas20-26,27-33,34-40,41-47,48-54,55-61...Instead,maybeuseage groups19.5-24.5,24.5-29.5,...or19.5-29.5,29.5-39.5,39.5-49.5,... _____ Incompletesentences,describetheshapeofyourhistogram. _____ Arethereanypotentialoutliers?Whichvaluesarethey?Showyourworkandcalculationsas tohowyouusedthepotentialoutlierformulainchapter2sinceyouarenowusingunivariate datatodeterminewhichvaluesmightbeoutliers. _____ Constructaboxplotofyourdata. _____ Doesthemiddle50%ofyourdataappeartobeconcentratedtogetherorspreadout?Explain howyoudeterminedthis. _____ LookingatboththehistogramANDtheboxplot,discussthedistributionofyourdata.For example:howdoesthespreadofthemiddle50%ofyourdatacomparetothespreadofthe restofthedatarepresentedintheboxplot;howdoesthiscorrespondtoyourdescriptionofthe shapeofthehistogram;howdoesthegraphicaldisplayshowanyoutliersyoumayhavefound; doesthehistogramshowanygapsinthedatathatarenotvisibleintheboxplot;arethereany interestingfeaturesofyourdatathatyoushouldpointout.

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APPENDIX 563 14.5SolutionSheets 14.5.1SolutionSheet:HypothesisTestingforSingleMeanandSingleProportion 19 ClassTime: Name: a. H o : b. H a : c. Inwords, CLEARLY statewhatyourrandomvariable X or P 'represents. d. Statethedistributiontouseforthetest. e. Whatistheteststatistic? f. Whatisthe p -value?In12completesentences,explainwhatthe p -valuemeansforthisproblem. g. Usethepreviousinformationtosketchapictureofthissituation.CLEARLY,labelandscalethe horizontalaxisandshadetheregionscorrespondingtothe p -value. Figure14.1 h. Indicatethecorrectdecisionrejectordonotrejectthenullhypothesis,thereasonforit,and writeanappropriateconclusion,using completesentences i. Alpha: ii. Decision: iii. Reasonfordecision: iv. Conclusion: i. Constructa95%CondenceIntervalforthetruemeanorproportion.Includeasketchofthegraph ofthesituation.LabelthepointestimateandthelowerandupperboundsoftheCondence Interval. Figure14.2 19 Thiscontentisavailableonlineat.

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564 APPENDIX 14.5.2SolutionSheet:HypothesisTestingforTwoMeans,PairedData,andTwo Proportions 20 ClassTime: Name: a. H o :_______ b. H a :_______ c. Inwords, clearly statewhatyourrandomvariable X 1 )]TJETq1 0 0 1 344.106 588.705 cm[]0 d 0 J 0.398 w 0 0 m 7.811 0 l SQBT/F132 9.9626 Tf 344.38 580.421 Td [(X 2 P 1 )]TJ/F132 9.9626 Tf 10.455 0 Td [(P 2 '-or X d represents. d. Statethedistributiontouseforthetest. e. Whatistheteststatistic? f. Whatisthe p -value?In12completesentences,explainwhatthep-valuemeansforthisproblem. g. Usethepreviousinformationtosketchapictureofthissituation. CLEARLY labelandscalethe horizontalaxisandshadetheregionscorrespondingtothe p -value. Figure14.3 h. Indicatethecorrectdecisionrejectordonotrejectthenullhypothesis,thereasonforit,and writeanappropriateconclusion,using completesentences i. Alpha: ii. Decision: iii. Reasonfordecision: iv. Conclusion: i. Incompletesentences,explainhowyoudeterminedwhichdistributiontouse. 20 Thiscontentisavailableonlineat.

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APPENDIX 565 14.5.3SolutionSheet:TheChi-SquareDistribution 21 ClassTime: Name: a. H o :_______ b. H a : c. Whatarethedegreesoffreedom? d. Statethedistributiontouseforthetest. e. Whatistheteststatistic? f. Whatisthe p -value?In12completesentences,explainwhatthe p -valuemeansforthisproblem. g. Usethepreviousinformationtosketchapictureofthissituation. Clearly labelandscalethehorizontalaxisandshadetheregionscorrespondingtothe p -value. Figure14.4 h. Indicatethecorrectdecisionrejectordonotrejectthenullhypothesisandwriteappropriate conclusions,using completesentences. i. Alpha: ii. Decision: iii. Reasonfordecision: iv. Conclusion: 21 Thiscontentisavailableonlineat.

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566 APPENDIX 14.5.4SolutionSheet:FDistributionandANOVA 22 ClassTime: Name: a. H o : b. H a : c. df n = d. df d = e. Statethedistributiontouseforthetest. f. Whatistheteststatistic? g. Whatisthe p -value?In12completesentences,explainwhatthe p -valuemeansforthisproblem. h. Usethepreviousinformationtosketchapictureofthissituation. Clearly labelandscalethehorizontalaxisandshadetheregionscorrespondingtothe p -value. Figure14.5 i. Indicatethecorrectdecisionrejectordonotrejectthenullhypothesisandwriteappropriate conclusions,using completesentences i. Alpha: ii. Decision: iii. Reasonfordecision: iv. Conclusion: 22 Thiscontentisavailableonlineat.

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APPENDIX 567 14.6EnglishPhrasesWrittenMathematically 23 14.6.1EnglishPhrasesWrittenMathematically WhentheEnglishsays: Interpretthisas: X isatleast4. X 4 X Theminimumis4. X 4 X isnolessthan4. X 4 X isgreaterthanorequalto4. X 4 X isatmost4. X 4 X Themaximumis4. X 4 X isnomorethan4. X 4 X islessthanorequalto4. X 4 X doesnotexceed4. X 4 X isgreaterthan4. X > 4 X Therearemorethan4. X > 4 X exceeds4. X > 4 X islessthan4. X < 4 X Therearefewerthan4. X < 4 X is4. X = 4 X isequalto4. X = 4 X isthesameas4. X = 4 X isnot4. X 6 = 4 X isnotequalto4. X 6 = 4 X isnotthesameas4. X 6 = 4 X isdifferentthan4. X 6 = 4 23 Thiscontentisavailableonlineat.

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568 APPENDIX 14.7SymbolsandtheirMeanings 24 SymbolsandtheirMeanings Chapterstused Symbol Spoken Meaning SamplingandData p Thesquarerootof same SamplingandData p Pi 3.14159...aspecic number DescriptiveStatistics Q 1 Quartileone therstquartile DescriptiveStatistics Q 2 Quartiletwo thesecondquartile DescriptiveStatistics Q 3 Quartilethree thethirdquartile DescriptiveStatistics IQR inter-quartilerange Q3-Q1=IQR DescriptiveStatistics x x-bar samplemean DescriptiveStatistics m mu populationmean DescriptiveStatistics ss x sx s samplestandarddeviation DescriptiveStatistics s 2 s x 2 s-sqaured samplevariance DescriptiveStatistics ss x s x sigma populationstandard deviation DescriptiveStatistics s 2 s x 2 sigma-squared populationvariance DescriptiveStatistics S capitalsigma sum ProbabilityTopics fg brackets setnotation ProbabilityTopics S S samplespace ProbabilityTopics A EventA eventA ProbabilityTopics P A probabilityofA probabilityofAoccurring ProbabilityTopics P A j B probabilityofAgivenB prob.ofAoccurring givenBhasoccurred ProbabilityTopics P AorB prob.ofAorB prob.ofAorBorboth occurring continuedonnextpage 24 Thiscontentisavailableonlineat.

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APPENDIX 569 ProbabilityTopics P AandB prob.ofAandB prob.ofbothAandB occurringsametime ProbabilityTopics A A-prime,complement ofA complementofA,notA ProbabilityTopics P A prob.ofcomplementof A same ProbabilityTopics G 1 greenonrstpick same ProbabilityTopics P G 1 prob.ofgreenonrst pick same DiscreteRandomVariables PDF prob.distributionfunction same DiscreteRandomVariables X X therandomvariableX DiscreteRandomVariables X thedistributionofX same DiscreteRandomVariables B binomialdistribution same DiscreteRandomVariables G geometricdistribution same DiscreteRandomVariables H hypergeometricdist. same DiscreteRandomVariables P Poissondist. same DiscreteRandomVariables l Lambda averageofPoissondistribution DiscreteRandomVariables greaterthanorequalto same DiscreteRandomVariables lessthanorequalto same DiscreteRandomVariables = equalto same DiscreteRandomVariables 6 = notequalto same continuedonnextpage

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570 APPENDIX ContinuousRandom Variables f x fofx functionofx ContinuousRandom Variables pdf prob.densityfunction same ContinuousRandom Variables U uniformdistribution same ContinuousRandom Variables Exp exponentialdistribution same ContinuousRandom Variables k k criticalvalue ContinuousRandom Variables f x = fofxequals same ContinuousRandom Variables m m decayrateforexp. dist. TheNormalDistribution N normaldistribution same TheNormalDistribution z z-score same TheNormalDistribution Z standardnormaldist. same TheCentralLimitTheorem CLT CentralLimitTheorem same TheCentralLimitTheorem X X-bar therandomvariableXbar TheCentralLimitTheorem m x meanofX theaverageofX TheCentralLimitTheorem m x meanofX-bar theaverageofX-bar TheCentralLimitTheorem s x standarddeviationofX same TheCentralLimitTheorem s x standarddeviationof X-bar same TheCentralLimitTheorem S X sumofX same continuedonnextpage

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APPENDIX 571 TheCentralLimitTheorem S x sumofx same CondenceIntervals CL condencelevel same CondenceIntervals CI condenceinterval same CondenceIntervals EBM errorboundforamean same CondenceIntervals EBP errorboundforaproportion same CondenceIntervals t student-tdistribution same CondenceIntervals df degreesoffreedom same CondenceIntervals t a 2 student-twitha/2area inrighttail same CondenceIntervals p p p-hat sampleproportionof success CondenceIntervals P P distributionofp-hat dist.ofsampleproportions CondenceIntervals q q q-hat sampleproportionof failure HypothesisTesting H 0 H-naught,H-sub0 nullhypothesis HypothesisTesting H a H-a,H-suba alternatehypothesis HypothesisTesting H 1 H-1,H-sub1 alternatehypothesis HypothesisTesting a alpha probabilityofTypeIerror HypothesisTesting b beta probabilityofTypeII error HypothesisTesting X 1 )]TJETq1 0 0 1 220.261 323.177 cm[]0 d 0 J 0.398 w 0 0 m 12.792 0 l SQBT/F132 9.9626 Tf 220.535 314.893 Td [(X 2 X1-barminusX2-bar differenceinsample means m 1 )]TJ/F134 9.9626 Tf 10.256 0 Td [(m 2 mu-1minusmu-2 differenceinpopulationmeans P 1 )]TJ/F132 9.9626 Tf 10.455 0 Td [(P 2 P1-hatminusP2-hat differenceinsample proportions p 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p 2 p1minusp2 differenceinpopulationproportions continuedonnextpage

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572 APPENDIX Chi-SquareDistribution X 2 Ky-square Chi-square O Observed Observedfrequency E Expected Expectedfrequency LinearRegressionand Correlation y = a + bx yequalsaplusb-x equationofaline y y-hat estimatedvalueofy r correlationcoefcient same e error same SSE SumofSquaredErrors same 1.9 s 1.9timess cut-offvalueforoutliers F-Distributionand ANOVA F Fratio Fratio

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APPENDIX 573 14.8Formulas 25 Formula14.1: Factorial n = n n )]TJ/F58 9.9626 Tf 10.131 0 Td [(1 n )]TJ/F58 9.9626 Tf 10.131 0 Td [(2 ... 1 0! = 1 Formula14.2: Combinations )]TJ/F132 7.5716 Tf 5.977 -3.652 Td [(n r = n n )]TJ/F132 7.5716 Tf 6.246 0 Td [(r r Formula14.3: BinomialDistribution X B n p P X = x = )]TJ/F132 7.5716 Tf 5.978 -3.652 Td [(n x p x q n )]TJ/F132 7.5716 Tf 6.451 0 Td [(x ,for x = 0,1,2,..., n Formula14.4: GeometricDistribution X G p P X = x = q x )]TJ/F58 7.5716 Tf 6.227 0 Td [(1 p ,for x = 1,2,3,... Formula14.5: HypergeometricDistribution X H r b n P X = x = r x b n )]TJ/F132 5.9776 Tf 5.093 0 Td [(x r + b n Formula14.6: PoissonDistribution X P m P X = x = m x e )]TJ/F134 5.9776 Tf 4.991 0 Td [(m x Formula14.7: UniformDistribution X U a b f X = 1 b )]TJ/F132 7.5716 Tf 6.436 0 Td [(a a < x < b Formula14.8: ExponentialDistribution X Exp m f x = me )]TJ/F132 7.5716 Tf 6.322 0 Td [(mx m > 0, x 0 Formula14.9: NormalDistribution X N )]TJ/F134 9.9626 Tf 4.811 -8.075 Td [(m s 2 f x = 1 s p 2 p e )]TJ/F142 6.2286 Tf 4.991 -0.062 Td [( x )]TJ/F134 5.9776 Tf 4.991 0 Td [(m 2 2 s 2 )]TJ/F59 9.9626 Tf 10.255 0 Td [( < x < Formula14.10: GammaFunction G z = R 0 x z )]TJ/F58 7.5716 Tf 6.227 0 Td [(1 e )]TJ/F132 7.5716 Tf 6.451 0 Td [(x dxz > 0 G 1 2 = p p G m + 1 = m !for m ,anonnegativeinteger otherwise: G a + 1 = a G a Formula14.11: Student-tDistribution X t df 25 Thiscontentisavailableonlineat.

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574 APPENDIX f x = 1 + x 2 n )]TJ/F142 6.2286 Tf 4.992 -0.063 Td [( n + 1 2 G n + 1 2 p n p G n 2 X = Z q Y n Z N 0,1 Y X 2 df n =degreesoffreedom Formula14.12: Chi-SquareDistribution X X 2 df f x = x n )]TJ/F58 5.9776 Tf 4.917 0 Td [(2 2 e )]TJ/F132 5.9776 Tf 5.093 0 Td [(x 2 2 n 2 G n 2 x > 0, n =positiveintegeranddegreesoffreedom Formula14.13: FDistribution X F df n df d df n = degreesoffreedomforthenumerator df d = degreesoffreedomforthedenominator f x = G u + v 2 G u 2 G v 2 )]TJ/F132 7.5716 Tf 5.977 -4.15 Td [(u v u 2 x u 2 )]TJ/F58 7.5716 Tf 6.228 0 Td [(1 h 1 + )]TJ/F132 7.5716 Tf 5.978 -4.15 Td [(u v x )]TJ/F58 7.5716 Tf 6.227 0 Td [(.5 u + v i X = Y u W v Y W arechi-square

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APPENDIX 575 SolutionstoExercisesinChapter14 SolutionstoPracticeFinalExam1 SolutiontoExercise14.1.1p.531 B:Independent. SolutiontoExercise14.1.2p.531 C: 4 16 SolutiontoExercise14.1.3p.531 B:Twomeasurementsaredrawnfromthesamepairofindividualsorobjects. SolutiontoExercise14.1.4p.532 B: 68 118 SolutiontoExercise14.1.5p.532 D: 30 52 SolutiontoExercise14.1.6p.532 B: 8 40 SolutiontoExercise14.1.7p.532 B:2.78 SolutiontoExercise14.1.8p.533 A:8.25 SolutiontoExercise14.1.9p.533 C:0.2870 SolutiontoExercise14.1.10p.533 C:Normal SolutiontoExercise14.1.11p.533 D: H a : p A 6 = p B SolutiontoExercise14.1.12p.534 B:believethatthepassrateforMath1AisdifferentthanthepassrateforMath1Bwhen,infact,thepass ratesarethesame. SolutiontoExercise14.1.13p.534 B:notreject H o SolutiontoExercise14.1.14p.534 C:Iris SolutiontoExercise14.1.15p.534 C:Student-t SolutiontoExercise14.1.16p.535 B:isleft-tailed SolutiontoExercise14.1.17p.535 C:clustersampling SolutiontoExercise14.1.18p.535 C:Mode SolutiontoExercise14.1.19p.535 A:theprobabilitythatanoutcomeofthedatawillhappenpurelybychancewhenthenullhypothesisis true.

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576 APPENDIX SolutiontoExercise14.1.20p.536 D:stratied SolutiontoExercise14.1.21p.536 B:25 SolutiontoExercise14.1.22p.536 C:4 SolutiontoExercise14.1.23p.536 A:.85,2.32 SolutiontoExercise14.1.24p.536 C:Bothabovearecorrect. SolutiontoExercise14.1.25p.537 C:5.8 SolutiontoExercise14.1.26p.537 C:0.6321 SolutiontoExercise14.1.27p.537 A:0.8413 SolutiontoExercise14.1.28p.537 A:.6030,0.7954 SolutiontoExercise14.1.29p.537 A: N 145, 14 p 10 SolutiontoExercise14.1.30p.538 D:3.66 SolutiontoExercise14.1.31p.538 B:5.1 SolutiontoExercise14.1.32p.538 A:13.46 SolutiontoExercise14.1.33p.538 B:Thereisastronglinearpattern.Therefore,itismostlikelyagoodmodeltobeused. SolutiontoExercise14.1.34p.539 B: Chi 2 3 SolutiontoExercise14.1.35p.539 D:70 SolutiontoExercise14.1.36p.539 B:Thechoiceofmajorandthegenderofthestudentarenotindependentofeachother. SolutiontoExercise14.1.37p.539 A: Chi 2 goodnessoft SolutionstoPracticeFinalExam2 SolutiontoExercise14.2.1p.540 B:parameter SolutiontoExercise14.2.2p.540 A

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APPENDIX 577 SolutiontoExercise14.2.3p.541 A:6 SolutiontoExercise14.2.4p.541 C:0.02 SolutiontoExercise14.2.5p.541 C:noneoftheabove SolutiontoExercise14.2.6p.541 D: 100 140 SolutiontoExercise14.2.7p.542 A: 0 SolutiontoExercise14.2.8p.542 B:Thevaluesfor x are: f 1,2,3,...,14 g SolutiontoExercise14.2.9p.542 C:0.9417 SolutiontoExercise14.2.10p.542 D:Binomial SolutiontoExercise14.2.11p.543 D:8.7 SolutiontoExercise14.2.12p.543 A:-1.96 SolutiontoExercise14.2.13p.543 A:0.6321 SolutiontoExercise14.2.14p.543 D:360 SolutiontoExercise14.2.15p.543 B: N 72,72 p 5 SolutiontoExercise14.2.16p.544 A: 3 9 SolutiontoExercise14.2.17p.544 D SolutiontoExercise14.2.18p.544 B:5.5 SolutiontoExercise14.2.19p.544 D:7.56 SolutiontoExercise14.2.20p.545 A:5 SolutiontoExercise14.2.21p.545 B:.26,6.74 SolutiontoExercise14.2.22p.545 B:0.2 SolutiontoExercise14.2.23p.545 A:-1

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578 APPENDIX SolutiontoExercise14.2.24p.545 C:dependentgroups SolutiontoExercise14.2.25p.545 D:Reject H o .Thereisadifferenceinthemeanscores. SolutiontoExercise14.2.26p.546 C:Theproportionformalesishigherthantheproportionforfemales. SolutiontoExercise14.2.27p.546 B:No SolutiontoExercise14.2.28p.546 C:Notenoughinformationgiventosolvetheproblem SolutiontoExercise14.2.29p.547 B:No SolutiontoExercise14.2.30p.547 C: y = )]TJ/F58 9.9626 Tf 8.194 0 Td [(79.96 x )]TJ/F58 9.9626 Tf 10.131 0 Td [(0.0094 SolutiontoExercise14.2.31p.547 A:69 SolutiontoExercise14.2.32p.547 C SolutiontoExercise14.2.33p.547 B:The pvalue is < 0.01,thedistributionisuniform. SolutiontoExercise14.2.34p.548 C:Thetestistodetermineifthedifferentgroupshavethesameaverages.

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GLOSSARY 589 Glossary A AnalysisofVariance AlsoreferredtoasANOVA.Amethodoftestingwhetherornotthemeansofthreeormore populationsareequal.Themethodisapplicableif: Allpopulationsofinterestarenormallydistributed. Thepopulationshaveequalstandarddeviations. Samplesnotnecessarilyofthesamesizearerandomlyandindependentlyselectedfrom eachpopulation. TheteststatisticforanalysisofvarianceistheF-ratio. Average Anumberthatdescribesthecentraltendencyofthedata.Thereareanumberofspecialized averages,includingthearithmeticmean,weightedmean,median,mode,andgeometricmean. B BernoulliTrials Anexperimentwiththefollowingcharacteristics: Thereareonly2possibleoutcomescalledsuccessandfailureforeachtrial. Theprobabilities p ofsuccessand q = 1 )]TJ/F132 9.9626 Tf 10.804 0 Td [(p forfailurearethesameforanytrial. BinomialDistribution AdiscreterandomvariableRVwhicharisesfromtheBernoullitrialswiththenextadditional requirements.Therearexednumber, n ,ofindependenttrials.Independentmeansthatthe resulttoanytrialforexample,trial1innowayaffectstheanswertoallthefollowingtrials, andalltrialsareconductedunderthesameconditions.Underthesecircumstancesthebinomial RV X isdenedasthenumberofsuccessinntrials.Thenotationis: X B n p ;thedomainis themeanis m = np ,andthevarianceis s 2 = df .Theprobabilitytohaveexactly x successesin n trialsis P X = x = )]TJ/F132 7.5716 Tf 5.978 -3.651 Td [(n x p x q n )]TJ/F132 7.5716 Tf 6.451 0 Td [(x BinomialDistribution AdiscreterandomvariableRVwhicharisesfromtheBernoullitrialswiththenextadditional requirements.Therearexednumber,n,ofindependenttrials.Independentmeansthatthe resulttoanytrialforexample,trial1innowayaffectstheanswertoallthefollowingtrials, andalltrialsareconductedunderthesameconditions.Underthesecircumstancesthebinomial RV X isdenedasthenumberofsuccessinntrials.Thenotationis: X B n p ;thedomainis themeanis m = np ,andthevarianceis s 2 = df .Theprobabilitytohaveexactly x successesin n trialsis P X = x = )]TJ/F132 7.5716 Tf 5.978 -3.651 Td [(n x p x q n )]TJ/F132 7.5716 Tf 6.451 0 Td [(x C CentralLimitTheorem GivenarandomvariableRVwithknownmean m andknownvariance s 2 ,wearesampling withsizenandweareinterestedintwonewRV-samplemean, X ,andsamplesum, S X .Ifthe sizenofthesampleissufcientlylarge,then X N n m s 2 n and S X N )]TJ/F132 9.9626 Tf 4.812 -8.075 Td [(n m n s 2 .In words,ifthesizenofthesampleissufcientlylarge,thenthedistributionofthesamplemeans andthedistributionofthesamplesumswillapproximateanormaldistributionregardlessof

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590 GLOSSARY theshapeofthepopulation.Andevenmore,themeanofthesamplingdistributionwillequal thepopulationmeanandmeanofsamplingsumswillequalntimesthepopulationmean.The standarddeviationofthedistributionofthesamplemeans, s p n ,iscalledstandarderrorofthe mean. CoefcientofCorrelation AmeasuredevelopedbyKarlPearsonearly1900sthatgivesthestrengthofassociation betweentheindependentvariableandthedependentvariable.Theformulais: r = n XY )]TJ/F142 10.3811 Tf 10.256 -0.105 Td [( X Y r h n X 2 )]TJ/F142 10.3811 Tf 10.256 -0.105 Td [( X 2 ih n Y 2 )]TJ/F142 10.3811 Tf 10.256 -0.105 Td [( Y 2 i ,.6 wherenisthenumberofdatapoints.Thecoefcientcannotbemorethen1andlessthen-1. Thecloserthecoefcientisto 1,thestrongertheevidenceofasignicantlinearrelationship between X and Y ConditionalProbability Thelikelihoodthataneventwilloccurgiventhatanothereventhasalreadyoccurred. CondenceLevel Thepercentexpressionfortheprobabilitythatthecondenceintervalcontainsthetrue populationparameter.Thatis,forex.,ifCL=90%,thenin90outof100samplestheinterval estimatewillenclosethetruepopulationparameter. CondentialInterval Anintervalestimateforunknownpopulationparameter.Thisdependson: Thedesiredcondencelevel. Whatisknownforthedistributioninformationforex.,knownvariance. Gatheringfromthesamplinginformation. ContingencyTable Themethodofdisplayingafrequencydistributionincaseofdependablecontingentvariables; thetableprovidestheeasywaytocalculateconditionalprobabilities. ContinuousRV ARVwithcontinuousdomain.Ex.:heightoftreesintheforest. CumulativeRelativeFrequency Theconceptappliestoanorderedsetofobservationsfromsmallesttolargest,orviseversa. Cumulativerelativefrequencyisthesumofrelativefrequenciesforallvaluesthatarelessthan orequaltothegivenvalue. D Data Asetofobservationsasetofpossibleoutcomes.Mostdatacanbeputintotwogroups: qualitative haircolor,ethnicgroupsandmanyother attributes ofpopulationand quantitative distancetraveledtocollege,numberofchildreninafamily,etc..Initsturnquantitativedata canbeseparatedintotwosubgroups: discrete and continuous .Roughlyspeaking,datais discreteifitisresultofcountinganumberofstudentofthegivenethnicgroupinaclass,a numberofbooksonashelf,etc.,anddataiscontinuousifitisresultofmeasuringdistance traveled,weightofluggage,etc.

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GLOSSARY 591 DegreesofFreedomdf Thenumberofobjectsinasamplethatarefreetovary. DiscreteRV ARVthatcanassumeonlycountablesetofvalues.Ex.'s.:.Facenominationsofcubicdie = f 1,2,3,4,5,6 g ,.anumberofaccidentsonHW280atThanksgivingHolidays. E EquallyLikely Eachoutcomeofanexperimenthasthesameprobability. ErrorBoundforaPopulationMeanEBM Themarginoferror.Dependsonthecondencelevel,samplesize,andknownorestimated populationstandarddeviation. Event Asubsetinthesetofalloutcomesofanexperiment.Thesetofalloutcomesofanexperimentis calleda samplespace anddenoted,asarule,by S .Aneventisanyarbitrarysubsetin S :itcan containoneoutcome,twooutcomes,andevennooutcomesemptysubsetorallofthem samplespace.Standardnotationsforeventsarecapitalletterssuchas A B C etc ExpectedValue Expectedarithmeticaveragewhenanexperimentisrepeatedmanytimes.Calledalsomean. Notations: E x m FordiscreterandomvariableRVwithprobabilitydistributionfunction P x = P X = x thedenitionalsocanbewrittenintheform E x = m = xP x Experiment Aplannedactivitycarriedoutundercontrolledconditions. ExponentialDistribution ContinuousrandomvariableRVthatappearswhenweareinterestedinintervalsoftime betweensomerandomevents,forexample,thelengthoftimebetweenemergencyarrivalsata hospital.Notation: X Exp m ;themeanis m = 1 m ,andthevarianceis s 2 = 1 m 2 ,theprobability densityfunctionis f x = me )]TJ/F132 7.5716 Tf 6.228 0 Td [(mx x 0andcumulativedistributionis P X x = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.227 0 Td [(mx ExponentialDistribution ContinuousrandomvariableRVthatappearswhenweareinterestedinintervalsoftime betweensomerandomevents,forexample,thelengthoftimebetweenemergencyarrivalsata hospital.Notation: X~Exp m ;themeanis m = 1 m ,andthevarianceis s 2 = 1 m 2 ,theprobability densityfunctionis f x = me )]TJ/F132 7.5716 Tf 6.228 0 Td [(mx x 0andcumulativedistributionis P X x = 1 )]TJ/F132 9.9626 Tf 10.256 0 Td [(e )]TJ/F132 7.5716 Tf 6.227 0 Td [(mx F Frequency Anumberoftimesavalueofthedataisoccurredinthesetofalldata. H HypergeometricProbability AdiscreterandomvariableRVwithcharacteristics: Thereisaxednumberoftrials. Theprobabilityofsuccessisnotthesamefromtrialtotrial,soitisnotBernoullitrials. Thetypicalexampleissamplingfromamixtureoftwogroupsofitems,whenweareinterested intheonlyone. X isdenedasthenumberofsuccessesoutofthetotalnumberchosen.The notationis: X~H r b n ,where r =numberofitemsinthegroupofinterest, b =numberof itemsinthegroupnotofinterest,and n =numberofitemschosen.

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592 GLOSSARY Hypothesis Astatementaboutthevalueofapopulationparameter.Incaseoftwohypotheses,thestatement assumedtobetrueiscallednullhypothesisnotation H 0 andcontradictorystatementiscalled alternatehypothesisnotation H a HypothesisTesting Basedonsampleevidenceproceduretodeterminewhetherthehypothesisstatedisareasonable statementandcannotberejected,orisunreasonableandshouldberejected. I IndependentEvents Theoccurrenceofoneeventhasnoeffectontheprobabilityoftheoccurrenceofanyotherevent. EventsAandBareindependentifoneofthefollowingistrue:. P )]TJ/F132 9.9626 Tf 5.31 -8.075 Td [(A 2 B = P A ; P )]TJ/F132 9.9626 Tf 5.011 -8.075 Td [(B 2 A = P B ; P AandB = P A P B InterquartileRangeIQR Thedistancebetweenthethirdquartileandtherstquartile. M Mean Anumbertomeasurethecentraltendencyaverage,shorteningfromarithmeticmean.By denition,themeanforasampleusuallydenotedby X is X = Sumofallvaluesinthesample Numberofvaluesinthesample ,andthe meanforapopulationusuallydenotedby m is m = Sumofallvaluesinthepopulation Numberofvaluesinthepopulation Median Anumberthatseparatesordereddataintohalves:halfthevaluesarethesamenumberor smallerthanthemedianandhalfthevaluesarethesamenumberorlargerthanthemedian. Themedianmayormaynotbepartofthedata. Mode Thevaluethatappearsmostfrequentlyinasetofdata. MutuallyExclusive Anobservationcannotfallintomorethanoneclasscategory.Beinginonecategoryprevents beinginamutuallyexclusivecategory. N NormalDistribution AcontinuousrandomvariableRVwith pdf = 1 s p 2 p e )]TJ/F142 7.8896 Tf 6.323 -0.079 Td [( x )]TJ/F134 7.5716 Tf 6.323 0 Td [(m 2 /2 s 2 ,where m isthemeanofthe distributionand s isitsstandarddeviation.Notation: X N )]TJ/F134 9.9626 Tf 4.812 -8.075 Td [(m s 2 .If m = 0and s = 1,theRV iscalled standardnormaldistribution ,or z-score O Outcomeobservation Aparticularresultofanexperiment. Outlier Anobservationthatdoesnotttherestofthedata.

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GLOSSARY 593 P p-value Theprobabilitythateventwillhappenpurelybychanceassumingthenullhypothesisistrue. Thesmallerp-value,thestrongertheevidenceisagainstthenullhypothesis. Parameter Anumericalcharacteristicofthepopulation. Example: Themeanpricetorenta1-bedroomapartmentinCalifornia. Percentile Anumberthatseparates 1 100 ofthedata. Example: Letadatasetcontain200orderedobservationsstartingwith f 2 3,2 7,2 8,2 9,2 9,3 0 ... g Thentherstpercentileis 2 7 + 2 8 2 = 2 .75 ,because1%ofthedataistotheleftofthispointon thenumberlineand99%ofthedataisonitsright.Thesecondpercentileis 2 9 + 2 9 2 = 2 9, separating2%ofthedata.Percentilesmayormaynotbepartofthedata.Inthisexample,the rstpercentileisnotinthedata,butthesecondpercentileis..Themedianofthedataisthe secondquartileandisthe50-thpercentileatthesametime.Therstandthirdquartilesare25th and75thpercentiles,respectively. PoissonDistribution AdiscreterandomvariableRVisthenumberoftimesacertaineventwilloccurinaspecic periodoftime,orinspecicarea,oranyotherunitsofmeasurement.Thecharacteristicsofthe variableare:theprobabilitythataneventoccursinagivenunitisthesameforallunitsand doesn'tdependonthenumberofeventthatoccursintheotherunits.Thedistributionis completelydenedbythemeannumber m ofeventintheunitintervalofmeasurement.The notationis: X~P m ;thedomainiswholenumbers, f 0,1,2, ... g ;themeanis m = np ,andthe varianceis s 2 = m 2 ,theprobabilitytohaveexactly x successesin r trialsis P X = x = e )]TJ/F134 7.5716 Tf 6.322 0 Td [(m m x x ThePoissondistributionoftenusedtoapproximatethebinomialdistributionwhennislarge andpissmallageneralruleisthatnshouldbeequaltoorgreaterthan20and p equaltoor lessthan.05. Population Thecollection,orset,ofallindividuals,objects,ormeasurementswhosepropertiesarebeing studied. Probability Anumberbetween0and1,inclusive,thatgivesthelikelihoodthataspeciceventwilloccur. Moreexact,thefoundationofstatisticsaregivenbythefollowing3axiomsbyA.N. Kolmogorov,1930's:Let S denotethesamplespace, A and B areanytwoeventsin S .Then: .0 P A 1;.If A and B areanytwomutuallyexclusiveevents,then P AorB = P A + P B ;. P S = 1. ProbabilityDistributionFunctionPDF AmathematicaldescriptionofadiscreterandomvariableRV,giveneitherintheformofthe equationbyformula,orintheformofatablelistingallthepossibleoutcomesofan experimentandtheprobabilityassociatedwitheachoutcome. Example: Abiasedcoinwithprobability0.7ofheadistossed5times.Weareinterestedinthe numberofheadsmeans,theRV X =thenumberofheads. X isBinomialRV,so X B 5,.7 and P X = x = 0 @ 5 x 1 A .7 x .3 5 )]TJ/F132 7.5716 Tf 6.451 0 Td [(x orintheformofthetable.

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594 GLOSSARY x P X = x 0 0.0024 1 0.0284 2 0.1323 3 0.3087 4 0.3602 5 0.1681 Proportion GivenabinomialrandomvariableRV, X B n p ,let'sconsidertheratioofnumber X of successinnBernoulitrialstothenumber n oftrials, P = X n .ThisnewRViscalledaproportion, andifthenumberoftrials, n ,islargeenough, P N )]TJ/F132 9.9626 Tf 5.36 -8.075 Td [(p pq n Q QualitativeData See Data Quantitative Quartiles Thenumbersthatseparatethedataintoquarters.Quartilesmayormaynotbepartofthedata. Thesecondquartileisthemedianofthedata. Quartiles R RandomVariableRV see Variable RelativeFrequency Theratioofanumberoftimesavalueofthedataisoccurredinthesetofalloutcomestothe numberofalloutcomes. S Sample Aportionofthepopulationunderstudy.Asampleisrepresentativeifitcharacterizesthe populationbeingstudied. SampleSpace Thesetofallpossibleoutcomesofanexperiment. Sampling Aprocedureforgatheringinformationaboutentirepopulation.themorepopularprocedures are:systematicsampling,simplerandomsampling,stratiedsampling,clusteredsampling. StandardDeviation Anumberthatisequaltothesquarerootofthevarianceandmeasureshowfardatavaluesare fromtheirmean.Notations:sforsamplestandarddeviationand s forpopulationstandard deviation. StandardDeviation Anumberthatisequaltothesquarerootofthevarianceandmeasureshowfardatavaluesare fromtheirmean.Notations:sforsamplestandarddeviationand s forpopulationstandard deviation.

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GLOSSARY 595 StandardErroroftheMean Thestandarddeviationofthedistributionofthesamplemeans, s p n StandardNormalDistribution AcontinuousrandomvariableRV X~N 0,1 .WhenXfollowsthestandardnormal distribution,itisoftennotedas Z~N 0,1 Statistic Anumericalcharacteristicofthesample.Statisticestimatesthecorrespondingpopulation parameter.Forexample,theaveragenumberoffull-timestudentsina7:30a.m.classforthis termstatisticisanestimatefortheaveragenumberoffull-timestudentsinanyclassthisterm parameter. Student-tDistribution InvestigatedandreportedbyWilliamS.Gossettin1908andpublishedunderthepseudonym Student.ThemajorcharacteristicsoftherandomvariableRVare: Itisacontinuousandassumesanyrealvalues. Thepdfissymmetricalaboutitsmeanofzero.However,itismorespreadoutandatterat theapexthanthenormaldistribution. Itapproachesthestandardnormaldistributionasngetslarger. Thereisa"family"oftdistributions:everyrepresentativeoffamilyiscompletelydened bythenumberofdegreesoffreedomwhichisonelessthanthenumberofdata. T TreeDiagram Theusefulvisualrepresentationofasamplespaceandeventsintheformoftreewith branchesmarkedbypossibleoutcomessimultaneouslywithassociatedprobabilities frequencies,relativefrequencies. Type1Error ThedecisionistorejectNullhypothesis,when,infact,Nullhypothesisistrue. Type2Error ThedecisionisnottorejectNullhypothesis,when,Nullhypothesisisfalse. U UniformDistribution ContinuousrandomvariableRVthatappearstohaveequallylikelyoutcomesoverthe domain, a < x < b .Oftenreferredas Rectangulardistribution becausegraphofitspdfhasform ofrectangle.Notation: X~U a b .Themeanis m = a + b 2 ,andthevarianceis s 2 = b )]TJ/F132 7.5716 Tf 6.436 0 Td [(a 2 12 ,the probabilitydensityfunctionis f x = 1 b )]TJ/F132 7.5716 Tf 6.436 0 Td [(a a X b ,andcumulativedistributionis P X x = x )]TJ/F132 7.5716 Tf 6.435 0 Td [(a b )]TJ/F132 7.5716 Tf 6.436 0 Td [(a V VariableRandomVariable Acharacteristicofinterestinapopulationbeingstudied.Commonnotationforvariablesare uppercaseLatinletters X Y Z ,...;commonnotationforspecicvaluefromthedomainsetof allpossiblevaluesofavariablearelowercaseLatinletters x y z ,....Forexample,if X isa numberofchildreninafamily,thendomainisand x representsanyintegerfrom0to20. Variableinstatisticsdiffersfromvariableinintermediatealgebraintwofollowingways. ThedomainofrandomvariableRVisnotnecessarilynumericalset;itcanbesome wordingset;forexample,if X =haircolorthenthedomainis{black,blond,gray,green, orange}.

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596 GLOSSARY Wecantellwhatspecicvalueof x doesthevariable X takeonlyafterperformingthe experiment. Beforetheexperimentanyvaluefromdomainispossible.Forexample,withoutultrasoundwe cannottellthegenderofababythatshouldbedelivered,butafterdeliverythegenderis evident.Moreexact,everyvaluefromthedomainisaccompaniedwithsomenumber p 0 p 1,thatcharacterizesthechancetohavethisvalueasanoutcomeoftheexperiment.In theexamplewithgender, p = 1 2 .That'swhystatisticiansusemoreexactname Random variableRV insteadofvariable.Evenmore,theyuseworddistributionhavinginthemind theRV,thatisthepairingvalue,probabilityofthevalue. Variance Meanofthesquareddeviationsfromthemean.Squareofthestandarddeviation. VennDiagram Theusefulvisualrepresentationofasamplespaceandeventsintheformofcirclesorovals showingtheirintersections. Z z-score Let'sconsiderthelineartransformationoftheform z = x )]TJ/F132 7.5716 Tf 6.322 0 Td [(m s .Ifthistransformationisappliedto anynormaldistribution X~N )]TJ/F134 9.9626 Tf 4.812 -8.075 Td [(m s 2 ,theresultisthestandardnormaldistribution Z~N 0,1 Ifthistransformationisappliedtoanyspecicvalue x ofRVwithmean m andstandard deviation s ,theresultiscalledz-scoreof x .z-scoreallowstocomparedatathatarenormally distributedbutscaleddifferently.

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INDEX 597 IndexofKeywordsandTerms Keywords arelistedbythesectionwiththatkeywordpagenumbersareinparentheses.Keywords donotnecessarilyappearinthetextofthepage.Theyaremerelyassociatedwiththatsection. Ex. apples,1.1 Terms arereferencedbythepagetheyappearon. Ex. apples,1 "hypothesistesting.",321 "inferentialstatistics.",277 A AANDB,3.2 AORB,3.2 addition,3.4 alternatehypothesis,13.2,13.3 ANOVA,13.1,13.2,511, 13.3,13.4,13.5, 14.5.4 answer,1.8 appendix,14.3 area,5.2 article,14.4.3 average,1.4,10,4.3,244,246,248 B bar,2.4 Bernoulli,4.5,4.15 BernoulliTrial,139 binomial,4.4,4.5,4.9, 4.15 binomialdistribution,325 binomialprobabilitydistribution,138 bivariate,14.4.5 box,2.5,2.11,2.13,5.10 boxes,2.4 C cards,4.17 categorical,1.4 center,2.6 central,14.4.2 CentralLimitTheorem,7.2,7.3, 7.10,330 chance,3.2,3.3 chi,11.4,11.5 chi-square,14.5.3 CLT,248 cluster,1.10,1.14 clustersample,1.6 collaborative,,14.9 ?? collection,1 condition,3.3 conditional,3.2,3.11,190 conditionalprobability,95 condenceinterval,278,283 condenceintervals,284,321 condencelevel,279,285 contingency,3.5,3.9 contingencytable,100,424 Continuous,1.5,11,1.10,1.12, 5.1,5.2,5.3,5.4, 5.5,5.6,5.7,5.8, 5.9,5.10,14.4.2 continuousrandomvariable,191 convenience,1.10 Conveniencesampling,1.6 Counting,1.5 criticalvalue,221 cumulative,1.9,1.10,1.12, 1.13 cumulativedistributionfunctionCDF,192 Cumulativerelativefrequency,19 curve,13.4 D Data,1.1,1.2,7,1.4,10,1.5, 1.7,1.10,1.11,1.12, 1.13,2.1,2.2,2.4, 14.3,14.4.1,14.4.5, 14.5.2 degreesoffreedom,283,13.3, 13.4,13.5 degreesoffreedomdf,375 descriptive,1.2,2.2,2.3, 2.6,2.11,2.13 deviation,2.11,2.13 diagram,3.6,3.7 dice,4.18 Discrete,1.5,11,1.10,1.12, 4.1,4.2,4.3,4.4, 4.5,4.6,4.7,4.8, 4.9,4.10,4.15,4.16, 4.17,4.18 display,2.2

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598 INDEX distribution,4.1,4.2,4.5, 4.6,4.7,4.8,4.10, 4.16,4.17,4.18,5.1, 5.2,5.3,5.4,5.5, 5.6,5.7,5.8,5.9, 5.10,11.5,14.4.2, 14.5.3 distributionisbinomial,285 dotplot,1.2 E elementary,,,2.1,2.2, 2.3,2.6,2.7,2.8, 2.9,2.10,2.11,2.12, 2.14,3.4,3.5,3.6, 3.7,3.8,3.9,3.10, 3.11,3.12,3.13,4.1, 4.2,4.3,4.4,4.5, 4.6,4.7,4.8,4.9, 4.10,4.12,4.13,4.14, 4.15,4.16,4.17,4.18, 5.1,5.2,5.3,5.4, 5.5,5.6,5.7,5.8, 5.9,5.10,6.1,6.2, 6.3,6.4,6.5,6.6, 6.7,6.8,6.9,6.10, 6.11,7.1,7.4,7.5, 7.6,7.7,7.8,7.9, 8.1,8.2,8.3,8.4, 8.5,8.6,8.7,8.8, 8.9,8.10,8.11,8.12, 8.13,9.1,9.2,9.3, 9.4,9.5,9.6,9.7, 9.8,9.9,9.10,9.11, 9.12,9.13,9.14,9.15, 9.16,9.17,9.18,10.1, 10.2,10.3,10.4,10.5, 10.6,10.7,10.8,10.9, 10.10,10.11,11.1, 11.2,11.3,11.4,11.5, 11.6,11.7,11.8,11.9, 11.10,11.11,11.12, 11.13,11.14,12.1, 12.2,12.3,12.4,12.5, 12.6,12.7,12.8,12.9, 12.10,12.11,12.12, 12.13,12.14,12.15, 12.16,13.6,13.7, 13.8,13.9,13.10, 14.1,14.2,14.3, 14.4.1,14.4.2,14.4.3, 14.4.4,14.4.5,14.5.1, 14.5.2,14.5.3,14.5.4, 14.6,14.7,14.8,14.9 ?? elementarystatistics,,2.13 empirical,5.10 equallylikely,3.2,94 errorboundforapopulationmean,279,283 errorbound285 event,3.2,94,3.11 Excel,14.9 ?? exclusive,3.3,3.10,3.11 exercise,1.14,2.13,3.9, 3.13,4.15,4.16,4.17, 4.18,5.6,5.7,5.9, 5.10 exercises,3.12 expected,4.3 expectedvalue,135 experiment,3.2,94,4.1,4.5, 4.7,4.8,4.17,4.18 exponential,5.1,183,5.4,190, 5.5,5.7 exponentialdistribution,250 F FDistribution,13.1,13.2, 13.3,13.4,13.5, 14.5.4 FRatio,13.3 t,11.4 formula,3.8,4.9,5.5 frequency,1.9,19,1.10,1.11, 1.12,1.13,1.14,45,2.11, 2.13,3.2 function,4.2,4.4,4.5, 4.6,4.7,4.9,4.16, 5.1,5.3,5.4,5.6, 5.7,5.8,5.9 functions,5.2 G geometric,4.4,4.6,4.9, 4.15 good,11.4 graph,2.2,2.3,5.1,5.6, 5.7,5.8,14.4.1 H histogram,2.4,2.11,2.13, 5.10 Homework,1.12,2.11,2.13, 3.9,3.12,3.13,4.15, 4.16,4.17,4.18,5.6, 5.7,5.8,5.9,5.10 hypergeometric,4.4,4.7, 4.15 hypergeometricprobability,144

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INDEX 599 hypergeometrical,4.9 hypotheses,322 hypothesis,14.4.3,14.4.4, 14.5.1,14.5.2 hypothesistest,325,327,328,13.2, 13.3 I independence,11.5 independent,3.3,95,97,3.10, 3.11 inferential,1.2 interquartilerange,52 Introduction,1.1,3.1,4.1 IQR,2.6 K keyterms,3.2 L lab,1.14,3.13,4.17, 4.18,5.10,7.10, 14.4.1,14.4.2,14.4.5 large,4.3 law,4.3 leaf,2.3 likelihood,1.3 limit,14.4.2 location,2.6 longterm,3.2 long-term,3.13,4.3 M mean,2.6,54,2.11,2.13, 4.3,135,248,14.5.1 means,14.5.2 meanssquare,13.3 measurement,1.7 Measuring,1.5 median,2.1,2.5,49,2.6,54, 2.11,2.13 Minitab,14.9 ?? mode,2.6,55,2.11,2.13 modules,1 multiplication,3.4 mutually,3.3,3.10,3.11 mutuallyexclusive,95,98 N nonsamplingerrors,1.6 normaldistribution,218,283,285,324, 13.5 normallydistributed,244,246,325 nullhypothesis,325,326,327,13.2, 13.3 numbers,4.3 numerical,1.4 O One-WayAnalysisofVariance,13.1, 13.2,13.3 outcome,3.2,94,3.3 outlier,44,52 outliers,52,473 P p-value,325,327,327,329 pair,14.5.2 parameter,1.4,9,1.10 PDF,4.2 percentile,2.6,2.11,2.13 percentiles,51 plot,2.11,2.13,5.10 pointestimate,277 Poisson,4.4,4.8,4.9, 4.15 Poissonprobabilitydistribution,146 population,1.4,9,17,1.10, 2.13,5.9,13.2,13.3 populations,13.5 practice,1.11,2.11,3.9, 3.10,3.12,4.16,5.6, 5.7,5.9 probability,1.3,9,1.10,3.1, 3.2,94,3.3,3.4,3.5, 3.6,3.7,3.8,3.9, 3.10,3.11,3.12,3.13, 4.2,4.3,4.4,4.5, 4.6,4.7,4.8,4.9, 4.10,4.16,4.17,4.18, 5.1,5.2,5.3,5.5, 5.6,5.7,5.9 probabilitydistributionfunction,134 problem,2.13,4.15,14.4.4 project,14.4.1,14.4.2, 14.4.5 proportion,1.4,10,14.5.1, 14.5.2 Q Qualitative,1.5,1.10,1.12 Qualitativedata,11 Quantitative,1.5,1.10,1.12 Quantitativedata,11 quartile,2.6,2.11,2.13 quartiles,2.5,49,51 R random,1.3,1.10,1.12, 1.14,4.1,4.2,4.3, 4.4,4.5,4.6,4.7, 4.8,4.9,4.10,4.15, 4.16,4.17,4.18,5.1, 5.2,5.3,5.4,5.5,

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600 INDEX 5.6,5.7,5.8,5.9 randomsampling,1.6 randomvariable,133,375,378 randomness,1.3 relative,1.9,1.10,1.12, 1.13,2.13,3.2 relativefrequency,19,45 replacement,1.10,3.13 representative,1.4 review,3.12,4.16,5.9 round,1.8 rounding,1.8 rule,3.4 S sample,1.4,9,1.6,1.7, 1.10,1.12,1.14,2.13, 13.2,13.3,13.5, 14.4.5 SampleMeans,7.2 samplespace,3.2,94,97,106 samples,17 Sampling,1.1,1.4,9,1.6, 1.7,1.10,1.11,1.12, 1.13,1.14 samplingerrors,1.6 set,14.3 sheet,14.5.1,14.5.2,14.5.3 simple,1.10 simplerandomsampling,1.6 single,14.5.1 SirRonaldFisher,13.3 size,1.7 skew,2.6,13.4 solution,14.5.1,14.5.2, 14.5.3,14.5.4 spread,2.6 square,11.4,11.5 standard,2.11,2.13 standarddeviation,57,282,324,325,326,329 standarderror,374 standarderrorofthemean.,245 standardnormaldistribution,218 statistic,1.4,9,1.10 statistics,,,1.1,1.2,7, 1.3,1.5,1.6,1.7,1.8, 1.9,1.10,1.11,1.12, 1.13,1.14,2.1,2.2, 2.3,2.6,2.7,2.8, 2.9,2.10,2.11,2.12, 2.13,2.14,3.1,3.2, 3.3,3.4,3.5,3.6, 3.7,3.8,3.9,3.10, 3.11,3.12,3.13,4.1, 4.2,4.3,4.4,4.5, 4.6,4.7,4.8,4.9, 4.10,4.12,4.13,4.14, 4.15,4.16,4.17,4.18, 5.1,5.2,5.3,5.4, 5.5,5.6,5.7,5.8, 5.9,5.10,6.1,6.2, 6.3,6.4,6.5,6.6, 6.7,6.8,6.9,6.10, 6.11,7.1,7.2,7.3, 7.4,7.5,7.6,7.7, 7.8,7.9,7.10,8.1, 8.2,8.3,8.4,8.5, 8.6,8.7,8.8,8.9, 8.10,8.11,8.12,8.13, 9.1,9.2,9.3,9.4, 9.5,9.6,9.7,9.8, 9.9,9.10,9.11,9.12, 9.13,9.14,9.15,9.16, 9.17,9.18,10.1,10.2, 10.3,10.4,10.5,10.6, 10.7,10.8,10.9, 10.10,10.11,11.1, 11.2,11.3,11.4,11.5, 11.6,11.7,11.8,11.9, 11.10,11.11,11.12, 11.13,11.14,12.1, 12.2,12.3,12.4,12.5, 12.6,12.7,12.8,12.9, 12.10,12.11,12.12, 12.13,12.14,12.15, 12.16,13.1,13.2, 13.3,13.4,13.5,13.6, 13.7,13.8,13.9, 13.10,14.1,14.2, 14.3,14.4.1,14.4.2, 14.4.3,14.4.4,14.4.5, 14.5.1,14.5.2,14.5.3, 14.5.4,14.6,14.7, 14.8,14.9 ?? stem,2.3 stemplot,2.3 stratied,1.10,1.14 stratiedsample,1.6 Student-tdistribution,283,325 student-tdistribution.,324 sumofsquares,13.3 summary,5.5 Sums,7.3 survey,1.12,14.4.1,14.4.3

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INDEX 601 systematic,1.10,1.13,1.14 systematicsample,1.6 T table,3.5,3.9 technology,14.9 ?? Terminology,3.2 test,11.4,11.5,14.4.3, 14.4.4,14.5.1,14.5.2 TestofTwoVariances,13.1,13.5 TheCentralLimitTheorem,243,244 theorem,14.4.2 TI-83,14.9 ?? TI-84,14.9 ?? tree,3.7 treediagram,105 trial,4.5 two,14.5.2 Two-WayAnalysisofVariance,13.4 TypeIerror,323,327 TypeIIerror,323 U uniform,5.1,183,5.3,5.5, 5.6,5.10 uniformdistribution,186 univariate,14.4.1,14.4.5 V variability,1.7 variable,1.4,9,1.10,4.1, 4.2,4.3,4.4,4.5, 4.6,4.7,4.8,4.9, 4.10,4.15,4.16,4.17, 4.18,5.1,5.3,5.4, 5.5,5.6,5.7,5.9 variables,5.8 variance,58,13.2,13.3, 13.5 variances,511 variation,1.7 Venn,3.6 Venndiagram,103 W withreplacement,1.6 withoutreplacement,1.6 word,14.4.4 Z z-score,283 z-scores,218

TRACE ROUTE

Total Execution Time: 65 Milliseconds

MILLISECOND   CLASS.METHODMESSAGE
0sobekcm_page_globals.constructor
0sobekcm_page_globals.constructorApplication State validated or built
0sobekcm_database.verify_item_lookup_object
0sobekcm_page_globals.constructorNavigation Object created from URI query string
0sobekcm_database.verify_item_lookup_object
0sobekcm_page_globals.display_itemRetrieving item or group information
0sobekcm_page_globals.get_entire_collection_hierarchyRetrieving hierarchy information
0sobekcm_assistant.get_entire_collection_hierarchy
0cached_data_manager.retrieve_item_aggregation
0cached_data_manager.retrieve_item_aggregationFound item aggregation on local cache
0item_aggregation_builder.get_item_aggregationFound 'all' item aggregation in cache