The measurement of inspector accuracy

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Title:
The measurement of inspector accuracy
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xvii, 131 leaves. : ill. ; 28 cm.
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Weaver, Lee Allen, 1935-
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Subjects / Keywords:
Quality control   ( lcsh )
Sampling (Statistics)   ( lcsh )
Management and Business Law thesis Ph. D   ( lcsh )
Dissertations, Academic -- Management and Business Law -- UF   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 128-130.
Additional Physical Form:
Also available online.
Statement of Responsibility:
By Lee A. Weaver.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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aleph - 022665887
oclc - 13979788
System ID:
AA00025720:00001

Full Text















THE MEASUREMENT OF INSPECTOR ACCURACY


By

LEE ALLEN WEAVER












A DISSERTATION PRESENTED TO THE GRADUATE C'jU; THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREP.ME]TS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA
1972
















ACKNOWLEDGMENTS


The writer would like to thank Dr. Warren Menke for his

assistance and guidance in the preparation of this disserta-

tion. He would also like to thank the other members of his

committee.

















TABLE OF CONTENTS


ACKNOWLEDGMENTS ...

LIST OF TABLES ......

LIST OF FIGURES . ..

KEY TO ABBREVIATIONS. .

ABSTRACT. . . . .

CHAPTER


INTRODUCTION . . . . . . ..

A SURVEY OF THE LITERATURE . . . ..

MEASURES OF INSPECTOR ACCURACY . . ..

EXPECTED VALUES FOR THE SAMPLING DISTRI-
BUTION OF THE RATIO OF DEFECTIVE PRODUCT
REJECTED . . . . . . . ..

CALCULATION OF EXPECTED VALUES . . ..

VERIFICATION OF DERIVED EXPECTED VALUES
BY SIMULATION . . . . . . .

SINGLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY . . . . . .

DOUBLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY . . . . . .

CONCLUSION . . . . . . ..


iii


I

II

III

IV


V

VI



VII



VIII



IX


Page


. . . . . . .ii

V
. .. . . . . . v

. . . . . . . xi

. . . . . . xii

. . . . o . xv


1

4

18




30

46



62



73



94

114









TABLE OF CONTENTS (CONTINUED)

Page


APPENDIX . . . . . . . . . . 117

REFERENCES . . . . . . . . . .. .128

BIOGRAPHICAL SKETCH . . . . . . . .. 131
















LIST OF TABLES

TABLE Paqe

2.1 INSPECTOR ACCURACY IN THE USE OF PRECISION
INSTRUMENTS . . . . . . .. 13

2.2 DEFECTS FOUND IN FOUR SUCCESSIVE VISUAL AND
GAGING INSPECTIONS OF 30,000 UNITS. . 15

3.1 SUMMARY OF ACCURACY MEASURES OBTAINED FROM
TWO SETS OF SAMPLE DATA . . . .. 29

4.1 SUMMARY OF EXPECTED VALUE FUNCTIONS . .. 39

5.1 EXAMPLES OF ACCURACY EXPECTED VALUES
OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION
SAMPLE SIZE = 100, INSPECTION FRACTION
DEFECTIVE = 0.05 . . . . . .. 50

5.2 EXAMPLES OF ACCURACY EXPECTED VALUES
OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION
SAMPLE SIZE = 100, INSPECTION FRACTION
DEFECTIVE = 0.10 . . . . . . 51

5.3 EXAMPLES OF ACCURACY EXPECTED VALUES
OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION
SAMPLE SIZE = 100, INSPECTION FRACTION
DEFECTIVE = 0.25 . . . . . .. 52

5.4 EXAMPLES OF ACCURACY EXPECTED VALUES
OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION
SAMPLE SIZE = 1000, INSPECTION FRACTION
DEFECTIVE = 0.01 . . . . . ... 54









LIST OF TABLES (CONTINUED)

TABLE Page

5.5 EXAMPLES OF ACCURACY EXPECTED VALUES
OBTAINED FROM DIFFERENT INSPECTION
AND REINSPECTION TEST RESULTS, IN-
SPECTION SAMPLE SIZE = 1000, INSPECTION
FRACTION DEFECTIVE = 0.05 . . . .. .55

5.6 EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED
FROM DIFFERENT INSPECTION AND REINSPECTION
TEST RESULTS, INSPECTION SAMPLE SIZE = 1000,
INSPECTION FRACTION DEFECTIVE = 0.10. . 57

5.7 EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED
FROM DIFFERENT INSPECTION AND REINSPECTION
TEST RESULTS, INSPECTION SAMPLE SIZE =
1000, INSPECTION FRACTION DEFECTIVE =
0.25 . . . . . . . . . 59

6.1 RESULTS OF SIMULATION ANALYSIS BASED ON 1000
RUNS FOR CASE 1, NUMBER INSPECTED = 1000,
NUMB3ER AUDITED = 500 . . . . ... 68

6.2 RESULTS OF SIMULATION ANALYSIS BASED ON 1000
RUNS FOR CASE 2, NUMBER INSPECTED = 1000,
NUMBER AUDITED = 500

6.3 RESULTS OF SIMULATION ANALYSIS BASED ON 1000
RUNS FOR CASE 1, NUMBER INSPECTED = 1000,
NUMBER AUDITED = 200 . . . . ... 70

6.4 RESULTS OF SIMULATION ANALYSIS BASED ON 1000
RUNS FOR CASE 2, UM?3FR INSPECTED = 1000,
NUMBER AUDITED = 200 . . . . ... 71

7.1 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, I2P'UCTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 250 . . . .. 78

7.2 MINIMUtl NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION S.-;1PLE SIZE = 500,
AUDIT SAMPLE SIZE = 100 . . . . .. 79










LIST OF TABLES (CONTINUED)


TABLE Page

7.3 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 50 . . . .. 80

7.4 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 10 . . . .. 81

7.5 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE = 500 . . . .. 82

7.6 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE = 200 . . . .. 83

7.7 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE = 100 . . . .. *84

7.8 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE = 20 . . . ... 85

7.9 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION S.W'iPLE SIZE = 5000,
AUDIT SAMPLE SIZE = 2500 . . . .. 86

7.10 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE = 1000 . . . ... 87


vii









LIST OF TABLES (CONTINUED)


TABLE Page

7.11 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE = 500 . . . .. 88

7.12 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE = 100 . . . .. 89

7.13 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 10000,
AUDIT SAMPLE SIZE = 5000 . . . ... 90

7.14 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 10000,
AUDIT SAMPLE SIZE = 2000 . . . ... 91

7.15 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 10000,
AUDIT SAMPLE SIZE = 1000 . . . .. 92

7.16 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 10000,
AUDIT SAMPLE SIZE = 200 ...... . 93

8.1 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 100 . . . . . .. 99

8.2 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.50, INSPECTION SAMPLE
SIZE = 500 . . . . . . . .. .100


viii









LIST OF TABLES (CONTINUED)


TABLE Page

8.3 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 1000 . . . . ... 101

8.4 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.50, INSPECTION SAMPLE
SIZE 5000 . . . . . . .. 102

8.5 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 10000 . . . . .. 103

8.6 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 100 . . . . . . ... .104

8.7 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 500 . . . . . . . .. .105

8.8 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90
AND ACCEPT ACCURACY = 0.60, INSPECTION
SAMPLE SIZE = 1000 . . . . ... 106

8.9 AUDIT SAMPLE SIZE AND MIN.ITrU NUMBER OF
DETECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 5000 . . . . . . . 107

8.10 AUDIT SAMPLE SIZE AND MnrIj'.-iM NUtilBER OF
DETECTIVES TO REJECT ACCURACY = 0.90
AND ACCEPT ACCURACY = 0.60, INSPECTION
SAMPLE SIZE = 10000 . . . . .. 108









LIST OF TABLES (CONTINUED)


TABLE Page

8.11 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 100 . . . . . . . .. .109

8.12 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 500 . . . . . . . . 110

8.13 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 1000 . . . . . . . i

8.14 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 5000 . . . . . . . 112

8.15 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DETECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 10000 . . . . . . . 113
















LIST OF FIGURES


Figure Page

4.1 Sample Production Flow for Case 1 with No
Replacement, Audit Accuracy = 100
Percent . . . . . . . .. 42

4.2 Sample Production Flow for Case 2 with No
Replacement, Audit Accuracy = Inspection
Accuracy . . . . . . . . 43

4.3 Sample Production Flow for Case 1 with
Replacement, Audit Accuracy = 100
Percent . . . . . . . .. 44

4.4 Sample Production Flow for Case 2 with
Replacement, Audit Accuracy . . ... 45

6.1 Histogram of Simulation Results . . .. 66

7.1 Graphical Presentation of a Single Hypothe-
sis Statistical Test for Inspection
Accuracy . . . . . . . .. .74

8.1 Graphical Presentation of a Double Hypothe-
sis Test for Inspection Accuracy .... . 95
















KEY TO ABBREVIATIONS


ACI A measure of inspection accuracy based on the ratio
of correct inspection.

ADR A measure of inspection accuracy based on the ratio
of defective product rejected.

AGA A measure of inspection accuracy based on the ratio
of good product accepted.

API A measure of inspection accuracy based on the reduc-
tion in the defect rate resulting from inspection.

AU A measure of inspection accuracy based on utility
theory.

D The absolute number of detectives produced by manu-
facturing (DGI + DDI)

DDI Number of actual defective items observed to be
defective by the inspector while performing an
initial inspection.

DGI Number of actual defective items observed to be good
by the inspector while performing an initial
inspection.

DI The number of items observed by the inspector to be
defective while performing an initial inspection
(GDI + DDI).

DR The number of items observed to be defective by the
auditor while performing a reinspection.

DR* The absolute number of detectives in a reinspection
sample.


xii








GDI Number of actual good items observed to be defective
by the inspector while performing an initial
inspection.

GGI Number of actual good items observed to be good by
the inspector while performing an initial inspection.

GI The number of items observed by the inspector to be
good while performing an initial inspection (GGI +
DGI).

GR The number of items observed to be good by the auditor
while performing a reinspection.

IOQL The inspection outgoing quality level is the ratio of
good product in the production line after initial
inspection but prior to reinspection by an auditor.

MQL The manufacturing quality level is the ratio of good
product in the production line prior to inspection.

NI The inspection sample size during an initial inspec-
tion.

NR The number of items reinspected by the auditor.

OQL The outgoing quality level is the ratio of good prod-
uct that. is shipped to the customer after all inspec-
tions have been performed.

PI A population parameter of the fraction defective
found by the inspector during an initial inspection.

PR A population parameter representing the fraction
defective found by the auditor while performing a
reinspection.

V(DDI) Value to the company of inspection determining an
actual defective item to be defective.

V(DGI) Value to the company of inspection determining an
actual defective item to be good.


xiii









V(GDI) Value to the company of inspection determining an
actual good item to be defective.

V(GGI) Value to the company of inspection determining an
actual good item to be good.


xiv










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

THE MEASUREMENT OF INSPECTOR ACCURACY

By

Lee Allen Weaver

December, 1972

Chairman: Warren Menke
Major Department: Management

The purpose of this study is to derive methods of

determining inspector accuracy during the production process.

The inspection function is viewed as the action taken by

an inspector in his role as a decision maker. This basic

function consists of examining a product and then deciding

whether or not it conforms to the specification. Since it is

imperative that defective material not be shipped to the

customer, it is necessary to be concerned with inspector

accuracy.

A search of the literature has revealed that very few

studies involving inspector accuracy have been published;

however, there may be studies on file in many industrial

inspection departments. Those studies that have been

published involve, for the most part, controlled experimental

conditions which attempt to determine causes of inspector

inaccuracy and are not concerned with a quantitative measure

of inspector accuracy. A survey of the literature is in-

cluded to show the need for quantitative measures of in-

spector accuracy that are obtained during the production

process.










The dissertation discusses five possible measures

of inspector accuracy: the ratio of correct inspection, a

utility theory approach, the ratio of good product accepted,

the ratio of defective product rejected, and the accuracy

of product improvement. The advantages and disadvantages of

each measure are reviewed. Sample calculations for each

measure are included based on two sets of data obtained from

the literature.

Based on current methods of data collection by indus-

trial inspection departments and their application in the

studies found in the literature, the ratio of defective

product rejected is further examined. The ratio of defec-

tive product rejected is determined by dividing the total

number of detectives found by the inspector by the total

number of detectives in the lot. During the actual pro-

duction process the only way to determine the total number

of detectives in the lot is by an audit reinspection of the

lot to determine how many detectives were missed by the

original inspector. Two cases are explored. In the first

case the auditor is perfect and has an accuracy of 100

percent. In the second case the auditor has the same accur-

acy as the initial inspector.

For each case an accuracy function is derived, as well

as the mean and standard deviation of the function.

Two types of sampling plans are derived. Single

hypothesis plans determine whether an acceptable inspection


xvi









accuracy is not being met based on the number of detectives

found during the audit inspection. Double hypothesis plans

determine whether a preselected acceptable inspection ac-

curacy level or a preselected unacceptable inspection ac-

curacy level is being attained. The double sampling plans

require that the audit sample size as well as the number of

audit detectives be stated in the sampling plan.

An effective tool for determining inspector accuracy

has been developed for use by industry. The sampling plans

result in estimates of inspector accuracy which can be used

to determine the actual manufacturing quality level and the

actual outgoing quality level. Good estimates of the out-

going quality level are required to determine future

warranty and customer liability costs.

Examples of the two types of sampling plans have been

included in the.dissertation. Computer programs have been

included in the appendix which can be modified to meet any

user's specific needs.


xvii
















CHAPTER I


INTRODUCTION


The rise of consumerism, resulting in increased

liability by the manufacturer, has caused control of the

production process to become increasingly important. While

in the past, manufacturing errors that would cause product

malfunction often were not detected prior to consumer use,

it is now imperative that these errors be detected before

the product is shipped to the consumer. The detection of

such defects is the responsibility of the quality inspector,

and since no process is perfect, defects will occur.

The present approach taken by manufacturers is based

upon the assumption of accurate inspection. This dependence

will exist as long as the number of defective units produced

is less than the allowable number of defective products

permitted to reach the customer. It is obvious that the

percentage of defective units will vary depending upon the

product and the production process involved. During the

past 15 years production facilities with which the author

has been associated have had control charts that have

indicated defect rates of from 5 percent to 70 percent for

subassemblies. With these types of defect rates,










considerable reliance needs to be placed on the ability of

the inspector to prevent them from being shipped to the

customer. To determine "typical' perfect defective values

would require a separate study in itself.

The above considerations lead us to a concern for

inspector accuracy. A search of the literature reveals

that few studies on inspector accuracy have been published,

although there may be such studies in the files of inspec-

tion departments throughout the United States.

Results of studies that are published are taken from

experiments performed under controlled conditions. A

typical experiment involves taking a product with a known

number of defects and submitting them to inspectors to

determine how many defects they can find. No papers were

found which gave specific procedures to measure the in-

spector's accuracy during the production process based on

accept/reject decisions involving the product currently

produced. Many journal articles and books on quality con-

trol do mention two possible methods, "salting" the

assembly line with known detectives or using an audit

inspector. Required sample sizes and the calculation of

quantitative accuracy measures are left unanswered.

Top management will obtain information on the quality

of their product without any special effort on their part.

These data will include consumer complaints of defective

products, financial data through the cost accounting system









concerning dollar losses of excess scrap and rework, and

complaints from the manufacturing personnel concerning

unreasonable rejections. Even if all this data is gathered

together and organized, it is not sufficient to give

management adequate control of quality problems. These

data do not provide a measure of the quality of the final

product as seen by the customer.

Companies have long recognized that financial per-

formance cannot be measured effectively without a system of

cost accounting. In a similar manner, companies will need

to recognize that quality performance cannot be measured

effectively without a system of quality accounting.
















CHAPTER II


A SURVEY OF THE LITERATURE


The majority of the literature findings were based on

investigations of causal factors under controlled experi-

mental conditions. Management and production engineers

need to know their relationship to job performance and

ultimately need a method of measuring their own inspector's

accuracy during the production process. A number of leading

books on statistical quality control (18, 19, 20, 21, 27)

do not even consider the effects of inspector accuracy

when discussing the application of acceptance sampling

plans.

Juran (9), as early as 1935, published evidence of

inspector biases. He felt that the serious study of in-

spector accuracy should be the analysis of the occurrence

of systematic, rather than random,errors. Inspectors,

being human, do not behave randomly. Rather, when they

make errors a pattern emerges. In his influential 1951

book, Juran (22) reproduced the same data and added rather

picturesque names to two types of inspector bias,"censor-

ship" and "flinching."






5


In censorship the inspector excludes unacceptable

findings. For instance in an inspection plan, three was the

maximum number of allowable defects. While accepting a lot

was a simple matter, rejecting one involved a great deal of

disliked paper work and trouble with the production people.

As a result, the inspector censored his findings so that an

unbelievably large number of lots "just happened" to have

the maximum number of allowable defects. Very few of the

lots contained four or more defects. A more Poisson-looking

distribution would probably portray better the actual number

of defects per lot.

Censorship also occurs when the inspector "finds" de-

fects, rather than "ignores,"them. Juran (22) gives an

example where the sampling plan was to take a sample of 100;

if no defects were found the lot was accepted, but if one or

more defects were found an additional sample of 165 was

taken. Since the time allowance for the second sample was

so liberal, the inspector could increase his personal

efficiency by finding sufficient defects to reject the first

sample. Very few lots contained no defects precluding the

taking of the second sample.

In flinching, the inspector accepts items which are

only slightly outside acceptance limits. Juran (22) found

two biases apparent from this study. The experiment con-

sisted of asking the inspector to read a needle meter with

digital numerals. The scale was from 0 to 50. First the









meter was graduated every two units, and the human tendency

to read to the nearest graduated unit showed itself in the

excessive number of even-numbered readings. Second, the

flinching occurred at a meter reading of 30. Although no

observations were recorded for 31, 32, or 33, it seems most

reasonable that some product occurred there. In effect, the

inspector had changed the acceptable maximum limit from 30

to 33. In several places Juran speaks of "honest inspecting"

in contrast to inspection involving censorship or flinching.

The implication is that these biases are deliberate and that

the inspector is therefore being dishonest. This may be

true, but a psychologist would be quick to point out that

it need not be deliberate. The inspectors could be completely

unaware of their biases and might be quite as shocked as

anyone upon being shown what they had been doing. Juran

recommends possible measures of accuracy which are discussed

in the next chapter.

McKenzie (14) feels that the main causes of inspector

inaccuracy fall into the following categories: basic indi-

vidual abilities, environmental factors, the formal organiza-

tion, and social relationships. He describes many situations

which can lead to inspector inaccuracy for each of the cate-

gories.

The ultimate limit of inspector accuracy is his indi-

vidual capability. The reading of instruments is dependent

upon the eye. The checking of noise level is dependent on










pitch. Micrometer accuracy is dependent upon touch and the

eye.

Environmental factors discussed by McKenzie include

light, temperature, noise, and work position. Formal or-

ganization factors included training, ill-defined standards,

repetitive boredom, and gauges and tools supplied. Social

relationships included relationships with production per-

sonnel, inspection supervision and management.

McKenzie apparently ran controlled experiments in

support of his conclusions; however, no data were presented.

He points out that when the experiments were run, the

inspectors knew they were run, and therefore the results did

not represent their every-day rates.

McKenzie offers three solutions to the controlled

experiment problem. One way is the introduction of known

detectives, examples of which are given later in this

chapter. Inspection supervision check on inspector accuracy

is rejected since his job is not to check not the product but

to supervise the inspection of them. An audit inspection

performed by a separate organization is recommended as the

best solution.

A recommendation that inspection accuracy should be a

design consideration was found in two papers (3, 28). The

argument is that proper operation of a piece of equipment

is dependent upon detectives being detected by inspection.










Sampling techniques have been developed which can be

used in quality system audits (12, 15). The auditing func-

tion is limited to determining adherence to policies, pro-

cedures, and instructions and not to hardware reinspection.

Schwartz (16) discusses a technique to determine

whether an inspector has developed inconsistent biases by

looking for non-random runs of accepted or rejected lots.

His technique does not result in a measure of inspection

accuracy.

The following paragraphs give the results of specific

studies performed in the area of inspector accuracy. The

results of most of the findings lead to causes of inspector

inaccuracy and can all be assigned to one of McKenzie's

groupings.

Jacobson (8) became concerned with quality control

inspectors. Plant opinion was that these inspectors were

95 to 98 percent effective, partly because of the non-

routine type of work they performed.

A unit was built with 1,000 soldered connections with

20 defects deliberately built in. Ten consisted of a wire

wrapped around the terminal but not soldered. The other 10

were poorly soldered; they were so insecure or loose that the

wire would move in the solder joint. Some 39 inspectors of

"all grades" were given one and one-half hours to inspect the

unit, the average inspection time on the regular inspection

line. Jacobson reported as a result of his experiment that










the inspector accuracy was 82.8 percent. This data is

further used in the next chapter in the derivation of

possible accuracy measures.

In defense of the reasonableness of the task put to

the inspectors, Jacobson cited two facts, that no defect was

found by all inspectors and no defect was missed by a ma-

jority of the inspectors.

Jacobson investigated many aspects of his data. The

average inspector identified 83 percent of the defects,

four found 100 percent of the defects, while one found 45

percent. There were very sizable differences among the

inspectors. Unfortunately, not even the four inspectors

who found 100 percent of the defects had a perfect record.

Two of them produced two defects each in finding the 20

defects, one erroneously found two extra defects, and the

fourth found one extra defect and produced six defects

himself.

It would seem that the insecure or loose connections

would be more difficult to find than those in which the wire

was simply wrapped around the terminal. This was not the

case, however, at least not to any significant degree. The

solderless connections were found 84 percent of the time,

while the loose connections were found 82 percent of the

time.

Jacobson found that age was not related to accuracy.

The age of the inspectors varied from 18 to 59. There










was a slight relationship with visual acuity as measured

by the Orthorater.

Hayes (7) reported on the inspection of piston rings for

surface defects. The foundry defects consisted of sand and

gas holes and about half of the rings produced were usually

scrapped. Hayes took 40 defective rings, selected to cover

the range of defects typically found. The rings were sub-

mitted to seven inspectors. Sixty-seven percent of the

defective rings were correctly classified as such. Only 10

percent, four rings, were judged by all seven inspectors to

be defective. Hayes also submitted the rings twice to the

inspectors, getting 7(40) = 280 pairs of decisions. Of

these 280 pairs of decisions, 23 percent were reversed.

In a second study, Hayes sent a lot of rejected rings

back through inspection and found that 67 percent were then

accepted. The inspectors were led to believe that the lot

had been reworked, though never told so. After this poor

showing, a second rejected lot was resubmitted. This time

the inspectors were told that the lot had not been reworked,

and 64 percent were correctly rejected. Of great interest

in this is the tremendous effect of the attitudes held by

the inspector; if he believes the product is good, he will

miss a good many defects.

Tiffin and Rogers (17) studied the accuracy of tin

plate inspectors. Some 150 inspectors judged 150 plates,

of which 61 were defective. The defects were surface









blemishes, unevenness in the coating of tin, and heaviness or

lightness of the coating. The inspectors were required to

classify each plate as satisfactory or defective, and if

defective, to identify the type of defect. The average

inspector made 78.5 percent correct identifications. The

150 inspectors ranged from 55 to 96 percent. The four

classes of defects were appearance defects I, II and III

and defective weight of the tinning.

Accuracy scores were very little related to visual

acuity, height, weight, age, or experience of the inspec-

tors. For example, the correlations between amount of

experience and accuracy in identifying each type of defect

were -0.05, -0.07, 0.00 and 0.06. Contrary to the expec-

tations of the supervisors in the department, the more ex-

perienced inspectors were not the better ones.

Ayers (1) studied 45 inspectors of rayon yarn cones.

He does not report any accuracy values; however, he found

very low correlations between accuracy and vision tests,

age, amount of production, and job experience.

Marien (13), in the earliest study located, studied the

introduction of a wage incentive system on the performance

of inspectors. While primarily concerned with the incentive

system, he notes that 1,700 tinned disks previously passed

by inspection were given a reinspection by two general

foremen and the chief inspector. Only 2 percent were found

at all questionable, that is, with black spots the size of










pin points. Day-working inspectors were never known to have

so high an accuracy. The incentive system was claimed to

be the cause of the high inspection accuracy.

Lawsche and Tiffin (10) made a study of inspector

accuracy in the use of precision instruments in two plants;

one was a manufacturer of variable pitch propellers for

aircraft and the other was a manufacturer of precision parts

for aircraft and automobile engines. In all cases the true

values were determined by ultra-precision instruments and

Johansen blocks. Every inspector was well trained and was

tested only on the instruments he used daily on his own

job. The inspectors were given separate booths and were

encouraged to take their time. It was suggested that they

take five measurements and then record their best judgment

as to the correct dimension. All instruments were properly

calibrated.

Between 113 and 162 inspectors were studied, using a

variety of precision instruments. Tolerances for each

instrument, established by the engineering department,

were identical with those encountered in the shops them-

selves. Table 2.1 summarizes the results.

Table 2.1 bears close inspection. From 9 to 64 percent

of the inspectors could read within the tolerances expected

of them. With Vernier micrometers, not even half of the

inspectors could read within 0.0001, and accuracy decreased

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scarcely one inspector in 10. A 6-inch micrometer com-

bined with an inside caliper was the most difficult to

read within the established tolerance.

One incidental finding was that micrometer reading

accuracy did not correlate with age, amount of experience

with the company, or length of time on the present job.

Kennedy (26) briefly mentions some data obtained on a

series of visual and gaging inspections. No fuller de-

scription is given, nor is the number of inspectors men-

tioned. Some 30,000 units were submitted, of which 100 were

defective. Four groups of inspectors were used; three

squads of regular inspectors under normal incentive speed,

and one selected squad of experts. The only measure of

accuracy that can be computed from the data given by Kennedy

is the proportion of defects correctly rejected. These data

are given in Table 2.2.

Apparently the "experts" did no better than the regular

inspectors. This fits in well with most other studies that

investigate the relation between accuracy and seniority or

experience.

Harris (5) performed an experiment to determine whether

inspection accuracy could be correlated with the defect

rate. He chose four samples containing 0.25, 1, 4, and 16

percent defective. The samples were inspected by 80 in-

spectors, 20 per condition, and inspection accuracies of

.58, .71, .74, and .82 were obtained. The results showed a










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high positive correlation, the higher the defect rate the

higher the inspection accuracy.

In a similar study Harris (6) performed an experiment to

determine whether inspection accuracy could be correlated

with equipment complexity. He chose 10 equipment items of

increasing complexity and selected 62 inspectors who regu-

larly inspected these items. He obtained a linear relation-

ship between complexity and inspection accuracy, with the

least complex item showing an inspection accuracy of .70 and

the most complex item showing an inspection accuracy of .20.

Mackworth (11) showed that vigilance deteriorates con-

siderably with time for tasks requiring intense attention.

In a visual watching of signals, 16 percent were missed the

first half hour, but 26 percent in the second.

Previously it was noted that McKenzie (14) recommended

that one approach to "building in" the check is to put

through from time to time a batch of work with deliberately

introduced detectives. The proportion of errors discovered

would give a measure of accuracy of inspection. The de-

fectives must be easily identifiable by the experimenter,

without spoiling the test. One method used for example by

Belbin (2) was to stain them with invisible dye that

fluoresces under ultra-violet light, thus the inspector's

decisions can be quickly checked. Similarly to determine

the accuracy of inspecting brass screw inserts Forster-

Cooper (4) had blind inserts deliberately introduced into










each batch. But these were made of steel and brass-plated:

thus mis-sorts could easily be picked out by means of a

magnet.

An overall review of the available studies shows that

considering the importance of inspection to the industrial

community, the lack of studies is lamentable. The experi-

mental designs are often naive, and the incomplete reporting

of results and methodology would often cause the careful

analyst to reject their results. The studies often do not

state how inspection accuracy was calculated. The next

chapter indicates that this could be a major fault since five

possible methods of calculating inspection accuracy are

discussed. There were no studies found involving inspection

accuracy during the industrial process and recommending

procedures of measuring this accuracy. The purpose of this

dissertation is to explore possible methods of determining

inspection accuracy in the industrial environment.
















CHAPTER III


MEASURES OF INSPECTOR ACCURACY


The present section will consider the question of how

inspector accuracy can be quantified. The topic is not as

simple and straightforward as it might first appear. Before

deriving possible measures the following symbols need to

be defined.

DDI Number of defective items observed to be

defective by the inspector.

DGI Number of defective items observed to be good

by the inspector.

GGI Number of good items observed to be good by

the inspector.

GDI Number of good items observed to be defective

by the inspector.

The accuracy measures discussed will be illustrated

by the following two examples of data obtained from the

literature. Both examples represent results obtained under

controlled experiments; however, they are two of the few

found in the literature which contained all four pieces of

data defined above.










Jacobson (8) became concerned with the ability of

inspectors to detect poor solder connections. For the ex-

periment a small wired unit with 1,000 wires soldered to

terminals was built with 20 deliberate defects. Thirty-nine

inspectors were given one and one-half hours to inspect the

unit, the average inspection time on the regular inspection

line. Out of a possible 780 defects and 38,220 good joints,

the following sample values were obtained:

DDI = 646

GGI = 38,195

DGI = 134

GDI = 25.

Jacobson investigated many aspects of his data. No

defect was found by all inspectors. Similarly, no defect

was missed by all of the inspectors. Only four of the in-

spectors found 'all 100 percent of the defects; however, two

of them called some of the good solder connections defective.

One inspector found only nine of the 20 defects indicating

a sizable difference in the accuracy of the inspectors.

Kelly (24) was interested in the inspection of tele-

vision panels for appearance defects. Ten panels were

selected, of which four were defective. Her main purpose

was to evaluate a new method of inspection by comparing it

with the inspection method in current use. The 10 panels

were inspected by 14 inspectors, yielding 140 inspections;

however, one good panel was lost when evaluating the new

inspection method, yielding 126 inspections.










Kelly obtained the following results:

Old method of inspection:

DDI = 16

DGI = 40

GGI = 55

GDI = 29.

New method of inspection:

DDI = 51

DGI = 5

GGI = 63

GDI = 8.

Further comments on the above data will be made in the

following discussion on possible measures of inspector

accuracy.


Ratio of Correct Inspections

If the interest is to maximize the total number of

correct inspection decisions or minimize all errors of

misclassification, the following measure would be of inter-

est:

ACI = GGI + DDI
NI

In terms of the example data previously noted, we would

obtain for the solder connection inspectors:

ACI = 38,195 + 646 = .9959
39,000










In other words 99.59 percent of the inspection de-

cisions were correct. The assumption is that both errors,

rejection of good and acceptance of bad, are equally

important. For the other example involving television

panels we would obtain:

ACI (old method) = 55 + 16 = .507
140

and
ACI (new method) = 62 + 51 = .897
126

This measure of accuracy shows a 39 percent increase

in inspection accuracy involving correct inspections.


Utility Approach to Accuracy

Many times the assumption that both types of inspection

errors are equally important does not fit the practical

situation, and a utility theory approach might be considered

appropriate. If for each of the four possible inspection

outcomes, a dollar value could be determined, we can use the

following function to determine the expected value.
AU = GG I V(GGI)D+ GDI
AU = I V(GGI) + V(GDI) + DDI V(DDI) + DGI V(DGI)


where

AU An accuracy measure based on a utility approach,

V(GGI) Value to the company of inspection determining

a good unit to be good,

and

V(GDI), V(DGI), V(DDI) are defined as V(GGI) above.










Consider the Jacobson data and assume the following

utility values:

V(GGI) = +1

V(DDI) = +1

V(GDI) = -1

V(DGI) = -6.

Considerations involved in the above allocations would

include that rejection of good material would only involve

reinspection costs and, therefore, was given a value of

minus one; however, the acceptance of defective material

would result in failures by the customer resulting in loss

of customer good will and possible liability claims. We

obtain

AU = 1(38,195) -1(25) -6(134) +1(646)
39,000

= .975.

The .975 would express the utility to the company of the*

inspection process; however, it does not provide a measure

of the satisfactory product being shipped to the consumer

and, therefore, will not be considered any further in this

paper.


Ratio of Good Product Accepted

In some situations, one might be interested in maxi-

mizing the probability of accepting good product or minimizing

the probability of rejecting good. In this case, accuracy

would be measured by the following equation:









AGA = GGI
GGI + GDI

For the Jacobson data:

AGA = 38,195 = .9993
38,195 + 25

For the Kelly data:

AGA (old method) = 55 = .655
55 + 29

and

AGA (new method) = 62 = .886
62 + 8

The question is, when would a situation arise that would

allow the ratio of good product accepted be an appropriate

measure of inspector accuracy? Presumably, in cases where

the cost of the product is very high and one would want to

avoid the rejection of good product at all costs.

Juran (22) feels that any measure involving the per-

centage of good. pieces identified is not a measure of the

accuracy of the inspector. His argument is that because the

majority of product submitted to inspection consists of good

pieces, the inspector does not exert much effort to identify

correctly good pieces. Effort, however, is not the real

concern, the search is for a measure of accuracy and not how

hard the inspector is working.

A more reasonable argument against this measure would

be to consider a batch of 100 items of which five are

defective. The inspector could call all the pieces good

without inspection and claim a 100 percent accuracy in










identifying good pieces. Rather than being absurd, it is

the most intelligent thing for the inspector to do under

the circumstances. If the company's purpose is best served

by maximizing the probability of accepting good product

this is what the inspector should do, and his performance

should be evaluated accordingly.

This measure, however, does not involve any measures

of the quality of the product reaching the consumer. Only

measures involving the correct identification of defective

units will be sufficient for this purpose. The two measures

in the following section each have this characteristic.


Ratio of Defective Product Rejected

In some situations it is desirable to maximize the

probability of rejecting defective product or minimize the

probability of accepting defective product. The following

accuracy function would be appropriate:

ADR = DDI
DDI + DGI

For the Jacobson data:

ADR = 646 = .828
646 + 134

For the Kelly data:

ADR (old method) = 16 = .286
16 + 40

and

ADR (new method) = 51 = .911
51 + 5










Therefore 82.8 percent of the defective soldered joints

were rejected in the Jacobson example, while the percent

defective rejected in the Kelly data was increased by 62.5

percent.

This measure of accuracy appears to be one that is use-

ful in many common circumstances; however, it concentrates

on defects and ignores the rejection of good items. This

may not be serious, as most rejects are returned for rework

and,if found not to be defective, are returned to the line.

This is the measure of accuracy used in most of the published

research located. Most inspectors feel that the accurate

identification of defects is the key inspection problem.

Sampling plans for this measure are developed in subsequent

chapters.

Juran and Gryna (23) discuss this measure of accuracy;

however, they worry about the event of an inspector classi-

fying good pieces as being defective and recommend sub-

tracting GDI from both the numerator and denominator. For

the Jacobson data we would obtain the following:

Accuracy = 646 25 = .8225
646 + 134 25

It is felt that this corrects for type 1 errors;

however, even if this is the case, the correction destroys

the simple interpretation of ADR. Accuracy is not 82.25

percent of anything simple or clear, a characteristic which

makes the measure unusable.










Accuracy of Product Improve. *-t

As noted earlier the purpose of the ij actionn depart-

ment is to screen out defective material. The amount of

product improvement resulting from the screening process

could be used as a measure of inspector accuracy. The ratio

of good material received from manufacturing to the total

amount received prior to inspection can be called the manu-

facturing quality level (MQL) and can be determined from the

following:

MQL = GGI + GDI
NI

After the inspection process the ratio of good items

to the number of items determined to be good by the inspector

can be called the outgoing quality level (OQL) and can be

determined from the following:

OQL= GGI .
GGI + DGI

The maximum amount of quality improvement is

1 MQL.

The observed amount of quality improvement is

OQL MQL.

The accuracy of product improvement can be given as

the ratio of the above differences:

API = OQL MQL
1 MQL










For the Jacobson data the following results are ob-

tained:

MQL = 38,195 + 25 = .9800
39,000

OQL = 38,195 = .9965
38,195 + 134

API = .9965 .9800 = .8250
1 .9800

In this case the inspectors were 82.50 percent accurate

in the improvement of the product,or they made 82.50 per-

cent of the maximum amount of improvement possible.

For the Kelly data the following results are obtained:

MQL (old method) = 55 + 29 = .600
140

OQL (old method) = 55 = .579
55 + 40

API (old method) = .579 .600 = -.053
1 .600

MQL (new method) = 62 + 8 = .556
126

OQL (new method) = .925 .556 = .831
1 .556

API (new method) = .831 .556 = .619
1 .556

The most interesting result is the negative value of

API obtained under the old method of inspection meaning that

the company would have been better off if no inspection

had been performed. While the percent of detectives

rejected for the Kelly data for the old method was 28.5

percent, it was not zero, leading to the conclusion that the

inspectors may be doing some good. This incorrect










conclusion would not be reached if API had been calculated.


Summary

Several measures of inspector accuracy were derived

and discussed. Each has its special merits and limitations,

but all of them do lead to a better understanding of ac-

curacy.

Table-3.1 summarizes the accuracy measures derived

and the quantitative results obtained from the two examples.

An interesting result is shown in this table for the new

inspection method of television panels; the ratio of good

product accepted is less than the ratio of bad product

rejected, refuting Juran's statement noted earlier that

less effort is required to determine good units as good.

The table shows the wide variety in quantitative measures

of inspector accuracy that are possible from the same set

of data.
















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CHAPTER IV


EXPECTED VALUES FOR THE SAMPLING DISTRIBUTION
OF THE RATIO OF DEFECTIVE PRODUCT REJECTED


This section will develop the expected values for the

accuracy measure based on the ratio of defective product

rejected. The two expected values to be derived for the

sampling distribution of the ratio of defective product

rejected are the mean and the variance. The expected values

to be derived are based on a reinspection by an auditor who

will either reinspect 100 percent of the product or a sample

of the product that was previously accepted by the inspector.

The following terms need to be defined.

NI the number of items inspected by the inspector.

DI the number of items observed to be defective by
the inspector.

GI the number of items observed to be good by the
inspector.

PI the inspection fraction defective.

NR the number of items reinspected by the auditor.

DR the number of items observed to be defective when
reinspected by the auditor.

GR the number of items observed to be good when re-
inspected by the auditor.

PR the auditor fraction defective.

MQL manufacturing quality level.









IOQL the product quality level after initial inspection.

OQL product quality level after audit inspection.

The derivations will assume that the number of good

items observed to be defective (GDI) is zero or is negligible.

When a unit is rejected by inspection it is usually returned

to the manufacturing area for disposition; scrap, rework, or

resubmit to inspection. If a good unit is rejected it will

be resubmitted to inspection and re-enter the flow of good

product. Whatever the value of GDI it will not affect the

outgoing quality level.

Two different cases will be developed for the situation

where manufacturing is responsible for the lot size, by the

submission of a predetermined NI to be submitted to inspec-

tion.

Case 1: The auditor performing the reinspection is
perfect, or auditor accuracy = 100 percent.

Case 2: The auditor performing the reinspection has the
same accuracy as the initial inspector.

The two cases should represent the extremes in the

capability of the auditor. It is unreasonable to assume that

a company would select for the audit function a person whose

accuracy is less than that of the initial inspector.

Many manufacturing companies predetermine the lot size as

the number of units required after the initial inspection, or

GI. The necessary modifications to the functions derived

will be discussed for this situation.










The following expected values will be determined.

p(ADR) = mean value for the sampling distribution of
the ratio of defective product rejected.
2
a (ADR)= variance for the sampling distribution of the
ratio of defective product rejected.

Tables of comparative values for each of the expected

values for different assumed values of NI, PI, NR, and PR

are included in the next chapter.


Case 1 Perfect Auditor

From the previous chapter, the equation for inspection

accuracy based on the ratio of defective product rejected

was given as

ADR = DDI
DDI + DGI

Since we have assumed that GDI equals zero,

DDI = DI.

An estimate of DGI, the number of defective items de-

termined to be good by the initial inspector would be given

by

DGI = (GI) (DR)
NR

therefore

ADR= DI
DI + (GI) (DR)
NR

Dividing both the numerator and denominator by NI, we

have
DI
ADR -= NI
NDI + ~DI (DR









We can define the following expected values




E [DR
NJ]I = PI



[ERJ PR .

If DI/NI and DR/NR are independent variables we can
write the expected value of ADR as (25, p. 52)

p(ADR) = PI PR
PI + (l-PI) PR

Intuitively the number of detectives found by two different
inspectors should vary independently as a function of his own

capability. The assumption of independence is further
strengthened by the results of the simulation analyses given

in Chapter VI.
Since ADR is a function of the two random variables
PI and PR, it is necessary to use the following equation

(1, p. 232) for determining the variance of ADR.

+2 (g) (= 22 (Dq)2 2 3 pxy.


For our problem the above equation would be written
as

2 (ADR 2 ) + ( 2 2ADR ADR
a (ADR) = PI-- / (PI) + )-PR (PR) + 2 DPI DPR

pa (PI) a (PR).
It can be argued that the number of defects found by the
initial inspector and the reinspector should be independent
and therefore p = 0. If PI and PR are correlated it should










be in a positive direction. As shown below the partial

derivative with respect to PI is positive while the partial

derivative with respect to PR is negative. Therefore, the

last term in o2 (ADR) is either 0 or negative. The variances

derived below will exclude the last term and at worst case

will be a conservative approximation of the true variance.

Since PI and PR follow binomial distributions we have

a2 (PI) = PI(l-PI)
NI

and

G2(PR) = PR(l-PR)
NR

The partial derivatives of ADR are given as

9ADR [PI + PR(l-PI)] (1) -PI(1-PI)
aPI [PI + PR(l-PI)]2


= PR
[PI + PR(l-PI)]2 ,

and

DADR [PI + PR(l-PI)] (O)-PI(l-PI)
@PR [PI + PR(l-PI) ]

PI (l-PI)
[PI + PR(l-PI)]2

Combining the above terms we obtain

a(ADR) = (PR) (PI) (1-PI) / 1 I -PR
[PI + PR(l-PI)] 2 (NI) (PI) (1-PI) + (NR) (PR)


To determine the expected value of the manufacturing

quality level we apply the following modification to the

equation given in Chapter III.









MQL = GGI + GDI = DDI + DGI
NI NI

since

DDI = DI,

and
DGI (GI) (DR)
NR

Using the expected values for NI/DI and NR/DR we obtain the

following

id (MQL) = 1-PI -(l-PI)PR.

The true fraction defective in the inspection sample is

1-MQL.

The outgoing quality level can be determined at two

points in the production flow, immediately after the initial

inspection (IOQL) or after the reinspection (OQL) is per-

formed. If the reinspection is not performed on every lot

but is performed on an item only periodically then the IOQL

would be more representative of the outgoing quality level.

If reinspection is a normal part of the production process

then the OQL would be more representative of the outgoing

quality level.

The general equation for the outgoing quality level was

given as

OL= GGI
SGGI + DGI

For the IOQL we have that

GGI + DGI = GI,










and

GGI = GI 1-DR
NR

therefore

IOQL =1 DR
NR

or

V(IOQL) = 1- PR.

For the OQL we have that

GGI + DGI = GI DR

and

GGI = GI DR

Solving we obtain
DR)
OQL = GI (l NR
GI DR

Substituting the appropriate expected values in order to

have OQL as a function of NI, NR, PI and PR we obtain

i(OQL) -1 (NI) (PR)(1-PI) -(NR) (PR)
(OQL) 1- (NI) (1-PI) (NR) (PR)

Using the manufacturing quality level and the two

measures of outgoing quality level, we can obtain two meas-

ures of the accuracy of product improvement, one before

reinspection and one after reinspection.

IAPI = IOQL MQL
1 MQL

and

API = OQL MQL
1 MQL

The above equations will hold for all the cases and, there-

fore, will not be discussed in the following paragraphs.










Case 2 Auditor Accuracy Equal to ADR

The following expected values are derived on the as-

sumption that the accuracy of the reinspector is equal to

that of the initial inspector.

Let

DR* = absolute number of defects submitted to
reinspection,

and

DR = the observed number of defects found by rein-
spection.

By the definition of ADR,
DR
ADR -
DR*

and

ADR = DDI
DDI + DGI

Since

DDI=DI,

and

DGI = (GI) (DR*) (GI)(DR)
NR (NR)(ADR)

we obtain
DI
ADR = DI + (GI) (D
(ADR) (NR)

Solving for ADR and substituting the expected values

for DI/NI and DR/NR we obtain

i(ADR) = PI (I-PI)PR
PI










Since this function is also non-linear it is necessary

to obtain the standard deviation using the same method as

used in Case 1. The partial derivatives of ADR are given

as

3ADR PI(1+PR) [PI-PR(l-PI)](l) PR
-pT = pi2 PI2

DADR PI(l-PI) [PI-PR(l-PI)](0) = (1-PI)
PR PI2 PI

Combining the above terms with U2(PI) and 2 (PR) we

obtain


OADR PR(1-PI) / 1 1-PR
PI / (NI) (PI) (1-PI) (NR) (PR)"


Using the methods previously discussed for Case 1

we obtain the following additional expected values for Case

2.

(MQL) = 1 PI PI)PR
ADR "


p(IOQL) = 1 PR
ADR

p(OQL) = 1 (NI) (PR) (l-PI)-(NR) (PR) (ADR)
ADR [NI (1-PI)-(NR) (PR) ]I

The equations derived for Cases 1 and 2 are summarized

in Table 4.1.















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Sampling Statistics with Replacement

The preceding cases can be considered to be non-

replacement situations since the manufacturing unit produces

NI items and the defective items are not replaced during the

production process.

If the number of items to be produced by the manu-

facturing facility is GI, meaning that all defective items

need to be replaced with good items until GI items complete

inspection, we have a case of inspection with replacement.

Since GI is specified, it is necessary to determine NI, by

SGI
NI = I-PI "

This equation will give the number of items that need to

be produced by the manufacturing facility, in order to

obtain GI items when the inspection reject rate is PI.

This value of NI can be substituted into the equations

previously derived. Since inspection with replacement is

similar to inspection with no-replacement it will not be

discussed separately in the remainder of the dissertation.


Summary
I
Table 4.1 lists the equations for each of the expected

values derived in this chapter. Figure 4 1 through 4.4

graphically show typical flow diagrams of product for the

two sets of equations as well as the two special cases.

Figure 4.1 shows a typical production flow for the

case where the auditor is assumed to be perfect. The

following values have been assumed:










PI = .200

PR = .125

NI = 1000

NR = 160.

The expected values are calculated at the various points in

the flow diagram. The number of actual good units and bad

units are noted at each point on the flow diagram.

Figure 4.2 uses the same assumed values as Figure 4.2,

the only difference being that the accuracy of the rein-

spector is equal to that of the initial inspector.

Figure 4.3 is a flow diagram where GI is specified

instead of NI, therefore, the following values have been

assumed:

PI = .200

PR = .125

GI = 1000

NR = 160.

The assumption of a perfect inspector is made in Figure 4.3,

while the assumption of the accuracy of the auditor equal to

the initial inspection is made for Figure 4.4.











MANUFACTURING NI = 1000


MQL = .700




INSPECTION


IOQL = .875

DI = 200
<------


GI = 800

7 00D


REINSPECTION ,NR = 160

-- 4 = ---
E~n06i


DR = 20
< ----


GR = 140


GI NR = 64
EwaWB! 01 Cl)i,


->


CUSTOMER


OQL = ( )


PI = .200 NR = 160

PR = .125 NI = 1000


C= = Number of true good units

CD = Number of true bad units


ADR = .667


Figure 4.1

Sample Production Flow for Case 1 with No Replacement,
Audit Accuracy = 100 Percent


U 3 (2, )








MANUFACTURING NI = 1000
6Q (00 4 0
MQL = .600


600 (


GI NR = 640


120 OQL = 0


PI = .200 NR = 160
PR = .125 NI = 1000
ADR = .500


I = Number of true good units
Q = Number of true bad units


Figure 4.2
Sample Production Flow for Case 2 with No Replacement,
Audit Accuracy = Inspection Accuracy































GI NR = 840


CUSTOMER


OQL = 9


PI = .200 NR = 160 I = Number of true

PR = .125 GI = 1000 = Number of true

ADR = .667


good units

bad units


Figure 4.3

Sample Production Flow for Case 1 with Replacement,
Audit Accuracy = 100 Percent


(Z)~













MANUFACTURING NI = 1250



MQL = .600





INSPECTION GI =1000
i750


IOQL = .750

DI = 250


NR = 160


GI NR = 840


GR = 140


CUSTOMER



OQL = .765
.Dll 111Il'"


PI = .200 NR = 160 = Number of true good units

PR = .125 GI = 1000 = Number of true bad units

ADR = .500



Figure 4.4

Sample Production Flow for Case 2 with Replacement,
Audit Accuracy = Inspection Accuracy


DR = 20


Eomft.
















CHAPTER V


CALCULATION OF EXPECTED VALUES


This chapter evaluates the expected value functions given

in Table 4.1 of the previous chapter for various inspection

and audit fraction defective. The following expected values

are calculated for each of the two cases:

i(ADR) = accuracy

o(ADR) = accuracy standard deviation

V(MQL) = manufacturing quality level

p(OQL) = outgoing quality level

Seven tables are given at the end of this chapter.

Each table is determined by the inspection sample size and

the inspection fraction defective (PI) which is noted at the

top of the table. Within each table the audit sample size

and the audit fraction defective are assigned different

values in order to see the effects on the calculated ex-

pected values over a wide range of values. It is to be

noted that the inspection fraction defective (PI) and the

audit fraction defective (PR) are expected values and are

the parameters of their respective binomial distributions.

For example Table 5.1 is based on an inspection sample

size of 100 and a PI of .05. If the auditor reinspects










100 percent of the product that passes inspection the audit

sample size would be 95, which is listed as the first group

of numbers in the first column. Within an audit sample size

an assumed set of increasing values for the audit fraction

defective is used to calculate v(ADR), o(ADR), l (MQL), and

P(OQL) for both Case 1 and Case 2. In Table 5.1 a similar

set of calculations is performed for audit sample sizes

of 50, 20, and 10.

The equation for p(ADR) for Case 2 given below

P(ADR) = PI (l-PI)PR
PI

will result in negative values for some values of the audit

fraction defective greater than the inspection fraction

defective. Those cases which result in values of negative

accuracy are denoted by "Invalid for Case 2."

The situations which are invalid for Case 2 arise from

the assumption that the auditors accuracy is equal to that of

the initial inspector. Consider the following example for

Case 2.

Inspection sample size = 100

Inspection fraction defective = .10

Audit sample size = 50

Audit fraction defective = .10

Then

p(ADR) = .10 (.90)(.10) .10
10










If the accuracy of the inspector is .10 this means the whole

inspection sample is defective, which is verified by the

equation for p(MQL) for Case 2

p(MQL) = 1 PI (l-PI) (PR)
ADR

= 1 .10 (.90)(.10) = o.
.10

If the audit fraction defective is greater than the inspec-

tion fraction defective, negative values for p(MQL) will

result,which is an impossibility. If in the accumulation of

actual sampling data the above situations arise, the assump-

tion regarding the accuracy of the auditor should be re-

examined.


Summary

The first observation that is apparent is that sampling

plans for the accuracy will involve inspection sample sizes

that are fairly large. To have a reasonable sampling plan to

test whether inspection accuracy is equal to .90, the ex-

pected value of the accuracy should be greater than .90 and

the standard deviation should be small, less than .05. In

Table 5.1 with NI = 100 and PI = .05, no calculated value of

the accuracy is greater than .8333 and the standard deviations

are fairly large. In Tables 5.2 and 5.3 NI = 100, and PI =

.10 and .25, the standard deviations are all fairly large.

In developing the sampling plans of Chapters VII and VIII, the

difficulty involving small lot sizes was readily apparent.










The following paragraphs summarize the effect of changes

in the accuracy and the accuracy standard deviation as a

function of the inspection and reinspection sample sizes and

the inspection fraction defective. Only the noted charac-

teristic changes while all others remain constant. The

effects of changes in two or more of the above characteris-

tics would be difficult to evaluate.

The effect of increasing the inspection sample size can

be seen by comparing Table 5.1 with Table 5.5 For audit

sample sizes that are the same percentage of the inspection

sample size and the same audit fraction defective, the ex-

pected value for the accuracy is the same; however, the

accuracy standard deviation decreases considerably as the

inspection sample size increases.

The effect of increasing inspection fraction defective

can be seen by comparing Table 5.1 with Table 5.2. In-

creasing inspection fraction defective causes the expected

value for the accuracy to increase and the accuracy standard

deviation to decrease.

The effect of increasing audit sample size and con-

stant audit fraction defective can be determined by in-

specting any of the tables. In this case the expected value

for the accuracy remains constant while the accuracy standard

deviation decreases.

The Appendix contains the computer program which can be

readily modified to calculate the expected values for any

inspection sample size and inspection fraction defective.
















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CHAPTER VI


VERIFICATION OF DERIVED EXPECTED
VALUES BY SIMULATION


The purpose of this chapter is to verify the previously

derived equations for the expected values of the mean and

the standard deviation for the distribution of the ratio of

defective product rejected by the method of simulation.

A computer program utilizing random numbers was written

to determine the reasonableness of the five following as-

sumptions.

1. Is the expected value of the accuracy equal to

the function given for p(ADR) in Table 4.1?

2. Is the standard deviation of the accuracy equal to

the function given for a(ADR) given in Table 4.1?

3. Is the assumption of zero correlation between PI

and PR justifiable?

4. Is the sampling distribution of ADR unimodal?

5. Does the sampling distribution of ADR follow a

normal distribution?

The purpose of the simulation was to follow as closely

as possible the flow of events in a typical production line.

The following steps were used in the development of the

computer program. Each run through the simulation represents

one lot.










1. A value for the manufacturing quality level (MQL)

was predetermined. Since the actual number of detectives in

a lot of fixed size will vary from lot to lot, the number

of detectives in a lot was determined by simulation. The

MQL follows a binomial distribution which for large lot

sizes can be approximated by a normal distribution. For

each lot the number of detectives in the lot (D) was de-

termined by generating a normal random deviate (Z) and solving

for D in the following equation

D = NI (l-MQL) + Z VNI (MQL) (I-MQL).

2. A value for the accuracy (ADR) of the inspectors

was also preselected. Since the accuracy of an inspector

will vary from lot to lot, the number of detectives found by

the inspector was also determined by simulation. Since ADR

is the probability that an inspector will find a defective

when a lot contains D detectives, ADR also follows a binomial.

distribution which can be approximated by a normal distribu-

tion. For each lot the number of detectives found by in-

spection (DI) was determined by generating a normal random

deviate (Z) and solving for DI in the following equation

DI = (D) (ADR) + Z/(D) (ADR)(1-ADR) .

3. The lot PI was calculated.

4. The absolute number of detectives in the sample

submitted to reinspection is also generated by the use of a

random normal deviate and using the difference of the pre-

viously determined values of D and DI.









5. The number of detectives found by the inspector

for Case 1 is equal to that found in Step 4 since the auditor

has 100 percent accuracy. For Case 2 the number of defec-

tives found by the auditor is a function of his accuracy and

is determined in a manner similar to Step 2.

6. The lot PR is calculated.

7. For Case 1 and Case 2 the observed accuracy is

determined for each lot.

8. The following statistics are calculated from the

simulation where K equals the number of runs.


Average PI = -P
K

EPR
Average PR -
K


Average Accuracy -
K

/ZADR2 (EADR)2
Accuracy Standard Deviation= K
K-1


Correlation of PI and PR for K lots.

9. The number of ADR values generated in intervals of

.01 were calculated to determine if the distribution is

unimodal.

10. The number of ADR values generated more than one and

two standard deviations from the mean are also determined as

a rough check on normality.

The above steps require only the following values as

initial conditions:










p(MQL) = population manufacturing quality level

V(ADR) = assumed population accuracy

NI = inspection sample size

NR = reinspection sample size.

For example the following results were obtained after

1000 runs for the following initial conditions for Case 1

where the accuracy of the auditor is assumed to be 100

percent.

For the initial conditions,

p(MQL) = .75

pj(ADR) = .90

NI = 1000

NR = 500,

the simulation results can be compared to the expected

results based on the equations given in Table 4.1.

Expected Value Simulation Value

p(ADR) .9000 .9002

PI .2250 .2252

PR .0323 .0323

o(ADR) .0231 .0225

The table of frequencies for the 1000 ADR values is

unimodal as shown in Figure 6.1.

For the 1000 runs 44 observations exceeded two standard

deviations while 320 observations exceeded one standard

deviation. This compares with 45.6 and 317.4 expected

observations under the assumption of normality.








200




150



100




50


Initial Conditions
ji(DR) = .90
p(MQL)= .75
NI = 1000
NR = 500


I


Observed Accuracy


Figure 6.1
Histogram of Simulation Results


FT


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'-










The results of the simulation are very close to the

expected results from the equations that were derived in

Chapter IV.

Table 6.1 gives the results of simulations where the

initial conditions are NI = 1000, NR = 100, and all combina-

tions of p(MQL) = .50, .75, and .90 with V(ADR) = .50,

.75, and .90 for Case 1. The table lists the initial con-

ditions, the expected values, and the observed values from

the simulations for ADR a(ADR), PI, and PR. In addition the

correlation of PI and PR and the number of observations

exceeding one and two standard deviations are listed.

Table 6.2 is similar to Table 6.1 except the results

are for Case 2. Tables 6.3 and 6.4 are similar except that

the audit sample size is 200.


Conclusions

The conclusions of this chapter based on the results of

the simulations are the answers to the questions that were

raised at the beginning of the chapter.

1. The expected value for the ADR is equal to the

function given in Table 4.1.

2. The standard deviation of the accuracy based on

the simulation results is equal to the function given in

Table 4.1.

3. The correlation between PI and PR can be assumed

to be equal to zero.

4. The sampling distribution for ADR appears to be

unimodal.












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5. Based on the unimodal appearance of the observed

sampling distribution and the observed number of samples

exceeding one and two standard deviations being approximately

equal to that under the assumption of normality,it may be

reasonable to assume that ADR follows a normal distribution.
















CHAPTER VII


SINGLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY


It is desirable to determine whether inspection per-

sonnel have an accuracy of less than some minimum value at

some predetermined confidence level. This chapter develops

the audit sampling plans equivalent to the above hypothesis.

The null hypothesis for these plans is that the in-

spection accuracy is equal to or greater than some minimum

acceptable value while the alternate hypothesis is that the

inspection accuracy is less than some minimum acceptable

accuracy with a fixed alpha error.

Figure 7.1 is a graphic presentation of the above

described classifical statistical one-tail test. If the

observed inspection accuracy plus K standard deviations is

less than minimum acceptable accuracy, the alternate hypothe-

sis would be accepted. The observed inspection accuracy and

the standard deviation need to be calculated from the in-

spection and reinspection results and are therefore sampling

statistics. The sampling statistics are obtained by sub-

stituting






74










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DI
NI = PI

DR
NR PR

into the equations given in Table 4.1. For example the

following equations for the sampling statistics result for

the accuracy and accuracy standard deviation for Case 1.

ADR = (NR)(DI)
(NR) (DI) + (NI-DI)(DR)

S(ADR) (NR) (DR) (DI) (NI-DI) / NI (NR-DR)
S (D [(NR)(DI) + (DR) (NI-DI)] 2 DI (NI-DI) (NR) (DR)

Similar equations can be determined for Case 2. The value of

K is selected on the basis of the desired alpha error.

Before sampling plans can be determined it is necessary

to know something about the form of the distribution. Since

we are dealing with a non-linear mathematical function con-

sisting of two variables, the exact form of the distribution

is difficult to obtain analytically.

The simulation studies of Chapter VI verified the

assumption of a unimodal distribution and also lend support

to the assumption of normality. Since there is no theoreti-

cal basis for the assumption of normality the fact that the

distribution is unimodal permits us to use a special case

of Chebyshev's inequality known as the Camp Meidel in-

equality (25, p. 89)

P[jx > Ka] <1
For a one-tail test and for the accuracy function we can
For a one-tail test and for the accuracy function we can










determine an alpha error value
1
P[ (ADR (ADR)) > K (ADR) ] < (2) (225) K2
--(2) (2.2T) -K2

where

11(ADR) = minimum acceptable accuracy

ADR = observed accuracy.

The alpha error for the Camp Meidel inequality when K =

1.645, the normal variate for a = .05, is
1
a(CAMP MEIDEL) = (2.25) (2) (1.645)2 = .082.

Both alpha values are given on the following tables.

The sampling plans on the following tables were obtained

by selecting inspection and audit sample sizes, and for

various observed inspection fraction defective test results

the number of detectives found by the auditor necessary to

reject accuracies of .75 and .90 are given.

For example for an inspection sample size of 500, and an

audit sample size of 250, if the inspector has an observed

fraction defective of .05, the hypothesis of .75 accuracy for

Case 1 would be rejected if the auditor finds 9 or more

detectives.

The tables at the end of the chapter give sampling plans

for inspection sample sizes of 500, 1000, 5000, and 10,000

and for audit sample sizes of 1/2, 1/5, 1/10, and 1/50 of

the inspection sample sizes.

The Appendix lists the computer program for the single-

hypothesis sampling plans. The computer program can be





77



easily modified for other alpha error values or sample sizes

by modifying the data cards. If a sample size is selected

that is too small, which results, in the problems discussed

in Chapter V, the computer program will print "no test"

instead of the required number of detectives necessary to

be found by the auditor.








TABLE 7>l

MINIMUM NUMBER OF DEFECTIVE ITEMS FUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5^0#
AUDIT SAMPLE SIZE = 250


ALPHA ERROR
ALPHA ERROR

OBSERVED
INSPECTION
FRACTION
DEFECTIVE

0o310

Oo020

0o030

Oo040

0050

o0060

0,080

02100

0O120

Oo 140

0160

o0 180

0.200

03250

Oo300

o00350

0430

0*450

0O500


=0o053 (NORMAL)
=0o082 (CAMP MEIDEL)

CASE I

ACCURACY ACCURACY
=0o75 =0o90

4 2

5 3

7 4

8 5

9 5

11 6

13 7

16 8

19 9

22 10

24 11

27 12

30 13

39 16

48 20

59 23

71 28

85 33

103 39


CASE 2

47CURACY. ACCURACY
=0:75 =0,90

9 7

7 6

8 6

9 6

10 6

10 6

12 7

14 8

16 9

18 10

21 11

23 12

25 13

31 16

38 19

46 22

56 26

67 31

80 36








TABLE 7,2

MINIMUM NUMBER OF DEFECTIVE ITEMS F3UNDO BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =500,
AUDIT SAMPLE SIZE =. 100


ALPHA ERROR
ALPHA ERROR

OBSERVED
INSPECTION
FRACTION
DEFECTIVE

00010

0.020

09030

0*040

0*050

0.060

Oa,080

0*100

0 120

0 140

00 160

00180

0,200

0O250

Oo300

00350

o0400

0,450

Oo500


=0o050 (NORMAL)
=0o082 (CAMP MEIDEL)

CASE I

ACCURACY ACCURACY
=Do75 =0090

2 2

3 2

4 3

4 3

5 3

6 3

7 4

8 4

9 5

10 5

12 6

13 6

14 7

18 8

21 10

26 11

31 13

37 15

43 18


CASE 2

A:CU(ACY ACCURACY
=0,75 =090

7 7

5 5

5 4

6 4

6 5

6 5

7 5

8 5

9 6

10 6

11 6

12 7

13 7

15 8

18 10

21 11

25 13

30 15

35 17








TABLE 7,3

MINIMUM NUMBER OF DEFECTIVE ITEMS F3UND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE 50


ALPHA ERROR
ALPHA ERROR

OBSERVED
INSPECTION
FRACTION
DEFECTIVE

00310

0020

0,030

o0040

00050

0060

00080

o00100

Oo120

O0 140

Oo 160

00180

0O200

Oo250

Oo300

Oo350

Oo400

Oo450

Oo500


=0o050 (NORMAL)
=OoOB2 (CAMP MEIDEL)

CASE I

ACCURACY ACCURACY
=0o75 =0o90

1 1

2 2

2 2

3 2

3 2

4 2

4 3

5 3

6 3

6 4

7 4

8 4

8 5

10 5

12 6

14 7

17 8

20 9

23 11


CASE 2

4CURACY ACCURACY
=0,75 =0,90

6 6

5 4

4 4

5 4

5 4

5 4

5 4

6 4

6 4

7 5

7 5

7 5

8 5

9 6

11 7

13 7

14 8

17 9

19 10








TABLE 714

MINIMUM NUMBER OF DEFECTIVE ITEMS FUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE % 10


ALPHA ERROR
ALPHA ERROR

OBSERVED
INSPECTION
FRACTION
DEFECTIVE

00010

0a020

0.030

O,040

0o050

0060

00080

0,100

0120

0 140

O0160

Oo 18O

03200

Oo250

0,300

0o350

00400

09450

0o500


=00050 (NOR4AL
=0o082 (CAMP MEIDEL)

CASE 1

ACCURACY ACCURACY
=0o75 =0090

I I

I I

1 1

1 I

1 I

2 1

2 1

2 2

2 2

2 2

3 2

3 2

3 2

3 2

4 3

4 3

5 3

5 3

6 4


CASE 2

ACURACY ACCURACY
=0,75 =0,90

4 4

3 3

3 3

3 3

3 3

3 3

3 3

3 3

3 3

3 3

3 3

4 3

4 3

4 3

4 3

4 3

5 4

5 4

6 4








TABLE 7*5

MINIMUM NUMBER OF DEFECTIVE ITEMS FUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE = 500


ALPHA ERROR =0a050 (NORMAL)
ALPHA ERROR =0o082 (CAMP MEIDEL)


OBSERVED
INSPECTION
FRACTION
DEFECTIVE

00010

0020

0,030

0040

0o050

0o060

008O

0,100

0,120

Oo 140

0160

o0 180

0,200

0O250

03300

0350

O400

03450

0500


CASE I

ACCURACY ACCURACY
=0o 75 =0, 90

5 3

8 5

11 6

13 7

15 7

18 8

23 10

28 12

33 14

38 16

43 18

49 23

55 22

70 28

88 34

108 41

132 49

160 59

193 70


CASE 2

ACURACY ACCURACY
=0;75 =0090

7 6

9 6

10 6

12 7

14 8

16 9

19 10

23 12

27 14

31 15

35 17

39 19

43 21

55 26

69 31

84 38

102 45

123 54

148 64








TABLE 7x6

MINIMUM NUMBER OF DEFECTIVE ITEMS FUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY#
INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE =m 200


ALPHA ERROR
ALPHA ERROR

OBSERVED
INSPECTION
FRACTION
DEFECTIVE

0010

0O020

0,030

O0040

0,050

O0060

0O080

0o100

O0 120

Oo 140

o0160

O2 180

0*200

0O250

O 300

0,350

Oo40O

Oo450

0500


=0o050 (NOR4AL)
=0082 (CAMP MEIDEL)

CASE I

ACCURACY ACCURACY
=0075 =0090

3 2

4 3

5 3

7 4

8 4

9 5

11 6

13 7

15 7

17 8

20 9

22 10

25 11

31 13

38 16

47 19

56 23

67 27

81 31


CASE 2

4CCUIACY ACCURACY
=0,75 =0)90

5 5

6 4

6 5

7 5

8 5

9 6

10 6

12 7

13 8

15 8

17 9

19 10

20 11

25 13

31 15

37 18

45 21

53 25

63 29