64
5. The number of defectives 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 =
K
Average PR =
K
Average Accuracy =
ZADR
K
Accuracy Standard Deviation=
/ZADR2 (IADR)2
K
Kl
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
tvvo 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:
TABLE 2.1
INSPECTOR
Instrument
1" Vernier Micrometer
2" Vernier Micrometer
6" Vernier Micrometer
3" Regular Micrometer
Depth Micrometer
Inside Micrometer
Inside Caliper & 2" Mic
Inside Caliper & 6" Mic
Outside Caliper & 6" Rule
Inside Vernier Caliper
Outside Vernier Caliper
ACCURACY IN THE USE OF PRECISION INSTRUMENTS
Number of
Established
inspectors
tolerance
162
0.0001
138
0.0001
131
0.0001
146
0.001
142
0.001
117
0.001
127
0.001
112
0.002
117
0.01563
113
0.001
117
0.001
Percentage of inspectors
reading within tolerance
43
17
11
64
53
66
46
9
49
42
51
25
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.
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.
ii
100
TABLE 8a 2
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,50,
INSPECTION SAMPLE SIZE = 500
ALPHA ERROR =
ALPHA ERROR =
BETA ERROR
BETA ERROR
= 0,050
= 0,082
I NORMAL)
(CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SIZE
2
AUDIT
DEFECTIV
OoOlO
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Oa 020
293
3
LOT SIZE
TOO
SMALL
0,030
175
3
LOT SIZE
TOO
SMALL
0,040
123
3
LOT SIZE
TOO
SMALL
Oe 050
95
3
456
6
0,060
76
3
354
5
0,080
54
3
241
5
0,100
41
3
180
5
0,120
33
3
141
5
0,140
27
2
115
5
0,160
23
2
96
5
0, 180
19
2
82
5
0,200
17
2
71
4
0,250
12
2
51
4
0,300
9
2
37
4
0,350
7
2
28
4
0,400
5
2
21
4
0,450
3
2
16
4
0,500
2
2
12
3
TABLE 5,3
AUOIT
AUDIT
1 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDeDEVo
MQL
20
0,1500
0,6897
0,1242
0,6375
20
0,2000
Oo 6250
0,1180
0,6000
20
0, 2500
Oo 5 714
0,1104
0,5625
20
0,3500
Oo 487 8
0,0955
0,4875
20
0,4500
Oo 4255
0,0827
0,4125
20
Oo 7500
Oo 3077
0,0564
0,1875
20
10 0000
Oo 2500
0,0433
0,0
10
Oo1000
Oo 7 692
0,1733
0,6750
10
0,2000
Oo 6250
0,1578
0,6000
10
Oo 3000
Oo 5263
0,1335
0,5250
10
0,4000
Oo 4545
0,1118
0,4500
10
0,5000
Oo 4000
0,0940
0,3750
10
0,7000
Oo 3226
0,0678
0,2250
10
0,9000
Oo 2703
0,0501
0,0750
1
1,0000
Oo 2500
0,0433
o
o
O
ON TINUED
  CASE 2   
OQL ACCURACY STDnDEV, MQL OQL
0,8854
0,5500
0,2611
0
5455
0,7576
0,8451
0,4000
0,3020
0
3750
0,5282
0,8036
INVALID
FOR
CASE
2
0,7169
INVALID
FOR
CASE
2
0,6250
INVALID
FOR
CASE
2
0,3125
INVALID
FOR
CASE
2
",0
INVALID
FOR
CASE
2
na 9122
0> 7000
0,2929
0,
6429
0,8687
0,8219
0,4000
0,4040
0,
3750
0,5137
0,7292
INVALID
FOR
CASE
2
0,6338
INVALID
FOR
CASE
2
0,5357
INVALID
FOR
CASE
2
0,3309
INVALID
FOR
CASE
2
0,1136
INVALID
FOR
CASE
2
o
a
INVALID
FOR
CASE
2
328X,AJDlT SAMPLE SIZE =',I4,///,
4L5X,'ALPHA ERROR = F 5=, 3, 2 X ( NORMAL ) / ,
515X,'ALPHA E K O R = F 5, 3, 2X ( C AMP ME I DE L) //
615X,OBSERVED',13X,'CASE l',19X,'CASE 2' /15X ' INSPECTION' /
715X,FRACTION',7X,'ACCURACY' 2 X ,'ACCURACY', 7X ACCURACY*, 2X,
81 ACCURACY ',/,
915X,'DEFECTIVE',8X,'=0375',5X,'=0>90',10X,'=0375',5X,'=0>93',/)
823 FORMAT!17X,F53 3,8X, 'NO TEST',3X,'NO TEST',8X,'MO TEST',3Xt
l'MO TEST',/)
833 FORMAT!L7X,F5,3,8X, 'MO TEST*,3X,I 7,8X,NO TEST',3X'NO TEST',/)
843 FORMAT!17X,F533,8X,I7,3X,I7,8X,'NO TEST',3X,'NO TEST',/)
853 FORMAT(17XF5338X, 'NO TEST',3X,'NO TEST',8X,'NO TEST 3X I 7 / )
863 FORMAT!17X,F53 3,8X, 'NO TEST',3X,'NO TEST*,8X,I 7,3X,I 7,/)
873 FORMAT!17X,F53 3,8X,'NO TEST,3X,I 7,8X,NO TEST,3X,I 7,/)
883 FORMAT!17X,F533,8X,I7,3X,17,8X,'NO TEST,3X I 7,/)
893 FORMAT!17X,F533,8X,'NO TEST,3X,I 7,8X,I 7,3X,I 7,/)
903 FORMAT!17X,F533,8X,I7,3X,I7,8X,I7,3XI7,/)
913 FORMAT (1H1,////,35X,'TABLE 7',II,/)
923 FORMAT 11H1,////,35X,'TABLE 7,',12,/i
603 CONTINJE
613 CONTINJE
STOP
END
. Figure 8.1
Graphical Presentation of a Double Hypothesis
Test for Inspection Accuracy
f
/
VO
U1
76
determine an alpha error value
P [ (ADR y (ADR) )
> K a (ADR) ]
 (2) (2.25) K2
where
y (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
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 defectives 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
defectives.
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
19
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. Thirtynine
inspectors were given one and onehalf 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.
86
TABLE 7,9
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUNO BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =2500
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0,082 (CAMP ME I DEL)
OBSERVED CASE I CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0,75
ACCURACY
= 0,90
0o0I0
15
7
13
8
Oa 020
26
11
22
ll
0,030
36
15
29
15
Oo 040
47
19
37
18
0a050
57
23
45
21
0,060
68
27
53
25
0,080
89
34
70
32
0,100
111
42
86
39
0, 120
134
50
104
46
0,140
158
58
122
54
0,160
183
67
140
61
0, 180
209
76
160
69
0,200
236
85
180
78
0,250
309
111
236
101
0,300
393
139
299
127
0,350
489
172
371
156
0,400
600
210
456
191
0,450
732
255
555
231
0,500
890
309
674
279
75
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) (PI)
(NR) (DI) + (NIDI) (DR)
c jnRl = (NR) (DR) (DI) (NIDI) f
K ; [ (NR) (DI) + (DR) (NIDI) 1 2/
NI
(NRDR)
[(NR)(DI) + (DR)(NIDI)]V DI(NIDI) (NR)(PR)
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 nonlinear 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[x ul > Ka] < r:5 T .
For a onetail test and for the accuracy function we can
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.
Xlll
97
z ADR y ^ (ADR)
a (ADR)
ADR is a sampling statistic which was derived in the previous
chapter as
ADR = (NR) (PI) _
(NR) (DI) + (NIDI) (DR) *
Since a(ADR) is unknown it is necessary to use an estimate
obtained from the sample, which was also derived in the
previous chapter as
= (NR)(DR)(PI)(NIDI) / NI ~ NRDR
1 [(NR)(DI) + DR(NIDI)]V DI(NIDI) (NR)(DR)*
Substituting the functions for the sampling statistics and
selecting values for the inspection sample size and the ob
served inspection defective into the pair of Z equations, we
have two equations in two unknowns, the audit sample size
(NR) and the number of audit defectives (DR).
The sampling plans developed in this chapter show that
large sample sizes are necessary to adequately determine
values of inspector accuracy, especially if the observed
inspection fraction defective is very low.
Table 8.11, which is included only for illustrative
purposes,shows that no statistical tests on inspection ac
curacy are possible for inspection sample sizes of 100 and
the following hypotheses.
Hq: y0(ADR) = .90
Hi :
y^(ADR) = .75 .
28
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.
Table3.1 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.
oooooooonooooooo
THIS PROGRAM DETERMINES AUDIT SAMPLE SIZES AND THE MINIMUM NUMBER
OF DEFECTIVES NECESSARY TO REJ rC T THE NULL HYPOTHESIS FOR A
DOUBLE HYPOTHESIS TEST ON INSPECTION ACCURACY. THE NULL
HYPOTHESIS IS EQUAL TO AN ACCEPTABLE VALUE OF INSPECTION ACCURACY.
THE ALTERNATE HYPOTHESIS IS EQUAL TO AN UNACCEPTABLE VALUE OF
INSPECTION ACCURACY. THE DATA CARD INCLUDES THE FOLLOWING
INFORMATION AND CAN BE MODIFIED FOR OTHER VALUES.
NI IS THE INSPECTION SAMPLE SIZE, NNI IS AN INDEX NUMBER FOR NI IF
MORE THAN ONE VALUE IS USED PI IS THE INSPECTION FRACTION
DEFECTIVE, NPI IS ITS ASSOCIATED INDEX NUMBER.
ADRAC IS THE VALUE OF ACCEPTABLE INSPECTION ACCURACY.
ADRUN ARE THE VALUES OF UNACCEPTABLE INSPECTION ACCURACY, NUN
IS ITS INDEX NUMBER.
ALNORM IS THE ALPHA ERROR DESIRED AS9UMING A NORMAL DISTRIBUTION
AND CONST IS THE ASSOCIATED NORMALIZED Z STATISTIC.
DIMENSION NI(5),PI(19),ADRU{3)
DATA NI/lOO,500,1000,5000,lOOOO /,PI/.01,.02,.03,.04,.05,.06,
1.08,.10,.12,.14,.16,.18,.20,.25,.30,.35,.40,.45,.50/,NNI/5/,NPI/19
2/,CONST/1.645/,ALNORM/.05/,ADRU/.50,.60,.75/,ADR AC/.90/,NUN/3/
500 FORMAT!19X,F5.3,8X,16 ,5X,I6 ,5X,I6 ,5X,I6,/)
510 FORMAT!19X,F5.3,8X,'LOT SIZE TOO SMALL',4X,'LOT SIZE TOO SMALL',/)
520 FORMAT!19X,F5.3,8X,'LOT SIZE TOO SMALL',4X,16 ,5X,I6 ,/)
530 FORMAT! 19X.F5.3,8X,16 ,5X,I6 ,5X,'L0T SIZE TOO SMALL',/)
540 FORMAT! 19X,AU0IT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES
1T0*,/,22X,'REJECT ACCURACY =,F4.2, 1 X,AND ACCEPT ACCURACY =',
2F4.2,','/,30X,INSPECTION SAMPLE SIZE = I 6 ,///,17X,
3 ALPHA ERROR = BETA ERROR = ,F5.3,2X,!NORMAL) ',/, 17X,ALPHA ERROR
4 = BETA ERROR = ,F5.3,2X, {CAMn MEIDEL) ,//,
517X,'OBSERVED',12X,
6 CASE l',16X,"CASE 2, /,17X,'INSPECTION',5X,'AUDIT',6X,'AUDIT',
7 6X 'AUDIT' 7XAUDIT,/,17X,FRACTI ON,7X,'SAMPLE* ,3X,'DEFE
8CT I VES',3X,'SAMPLE',4X,* DEFECTI VES,/,17X,DEFECTIVE,6X,SIZE,18
9X 'SIZE*,/)
550 FORMAT!1H1,////,40X,'TABLE 8.',II,/)
560 FORMATl1H1,////,40X,'TABLE 8.',12,/)
125
oooonoooo
THIS PROGRAM CALCULATES ACCURACY EXPECTEO VALUES FOR VARIOUS
INSPECTION AND REINSPECTION TEST RESULTS. THE DATA CARD INCLUDES
THE FOLLCWING INFORMATION AND GAN RE MODIFIED FOR OTHER VALUES.
NI IS THE INSPECTION SAMPLE SI*E, NNI IS AN INDEX NUM8ER FOR NI IF
MORE THAN ONE VALUE IS USED PI IS THE INSPECTION FRACTION
DEFECTIVE, NPI IS ITS ASSOCIATED INDEX NUMBER NRFAC IS A
FACTOR TO DETERMINE THE AUDIT SAMPLE SIZE AND IS DIVIDED INTO NI,
NNR IS ITS ASSOCIATED INDEX NUMBER.
REAL NI!2),PI(4),NRFAC(5) NR,MQL1,MQL2
DATA NI/1.0E2, 1.0E3/, PI/.Ol, .05,.10,.25/,NRFAC/1.0,2.0,5.O,I 0.0,
1100.0/,NNI/2/,NPI/4/,NNR/5/
ITAB =L
DO 100 INI = 1,NNI
DO 100 IPI = 1,NP I
KOUNT =0
KONTIN=0
IT AB=IT AB + 1
J N I =. N I ( I N I )
WRITE!6,530)ITAB
WRITE!6,500)JNI,PI(IPI)
WRITE!6,505)
DO 100 I NR = 1,NNR
IF! INR.NE.1) GO TO 130
NR=NI( INI )*(1.0PI( IPI))
GO TO 140
130 NR=NI(I NI)/NRFAC!I NR)
WRITE !6,560)
KOUNT =KOUNT + 1
140 IF INI(INI)*(1.0PI(IPI) ).LT.NR) GO TO 110
SADR=1.0
DR=0
160 IF(DR.GE.IO) GO TO 200
DR=DR+1
GO TO 220
200 IF!DR.GE.50) GO TO 210
oo
3
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.
Aa L To o 9) GO TO 240
AaLTaa75) GO TO 220
GO TO 110
160 IF(DR9GToNR) GO TO 230
IF(ADR 2 A + CON ST*S IG 2
GO TO 120
230 DR290 = 1
NRUN = NRUN +1
GO TO no
240 DR290=DR
NRUN = NRUN +1
GO TO 110
150 I F ( DR a GTo NR ) GO TO 210
IF(ADR 2 A*CONST*SIG 2
GO TO 120
210 DR275="1
GO TO 580
220 DR?75 = DR
580 IF(0R190oLTaQaANDaDR290aLTi0) WRITE(6,820) PI(IPI)
IF(DR175oLTa0a ANDaDR190>a0aAND:>DR290aLTa0) WRITE(6,830) PKIPI),
1 DR 190
[F(DR175oGT00,AND3DR190aGT500AND3DR2903LTa0) WRITE(6,840) PKIPI)
1 DRL75DR190
lF(DR190oLTa0,AND,DR2750LTsO0ANDaDR290aGTa0) WRITE(6,850) PI(IPI),
1 DR290
IF(DR190cLTo0sANDaDR275oGTo0&ANDaDR290aGTa0) WRITE(6,850) PKIPI),
1NR,DR275,DR290
IF(DR175oLTa0> AND,DR 190oGTa0aANDaDR275,LTa0,AND,DR290,ST,0) WRITE
1(6,870) PKIPI), DR190,DR290
I F(DR 17 5o GTa 0aAND,DRl90aGTa0aAND,DR275,LTa0AND, DR2 90aGTa0) WRITE
1(6,880) PI(IPI), DR175,DR190,DR290
IF(DR17 5oLT,0,AND,DR190aGTaOaAND,DR275,GTa0aANO,DR290a GTa 0) WRITE
1( 6,890) PKIPI), DR190DR275DR290
IF(DR175.aGTo0aANDaDR190aGTa0oANDaDR275,GTa0aAN0, 0R29 0aGTa 0)
IWRITE( 6,900) PKIPI) D R 175, OR 190, DR275 ,DR290
810 FQRMAK17X,'MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE*,/,
116X,AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,*,/*
225X,INSPECTION SAMPLE SIZE =*,15,*,*,/,
123
16
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 defectives. 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 ultraviolet 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
47
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 y(ADR), a(ADR), y(MQL), and
y(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 y(ADR) for Case 2 given below
U (ADR) = PI ~ (1PI) 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
_ .10 (.90) (.10) _
.10
y (ADR)
.10 .
TABLE 5,>3
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION SAMPLE SIZE 103,
INSPECTION FRACTION DEFECTIVE = 0,25
AUDIT
AUDIT
1 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MQL
OQL
ACCURACY
STD,DEV
, MQL
OQL
75
0,0133
Oo 9 615
0,0377
0,7400
1,0000
0,9600
0,0408
0,7396
0,9994
75
0,0267
0o 9259
0,0504
0,7300
1,0000
0,9200
0,0588
0,7233
0,9976
75
0,0533
Oo 8621
0,0640
0,7100
1,0000
0, 8400
0,0862
0,7024
0,9893
75
0,0800
Oo 8065
0,0710
0,6900
1,0000
0,7600
0,1091
3,6711
0,9725
75
0,1200
0,7353
0,0757
0,6600
1,0000
0,6400
0,1399
0,5094
0,9233
75
0,2000
0,6250
0,0765
0, 6000
1,0000
0,4000
0,1950
3,3750
0,6253
75
0,2667
Oo 5556
0,0741
0,5500
1,0000
INVALID
FOR CASE
2
75
0,3333
Oo 5000
0,0707
0,5000
1,0000
INVALID
FDR CASE
2
75
0,4667
Oo 416 7
0,0636
0, 40 0 0
1,0000
INVALID
FDR CASE
2
75
0= 6000
Oo 3 571
0,0573
0, 3000
1,0000
INVALID
FDR CASE
2
75
0,8000
Oo 2941
0,0494
Oo1500
1,0000
INVALID
FOR CASE
2
50
0,0200
Oo 9434
0,0543
0, 7350
0,9932
0,9400
0,0610
0,7343
0,9919
50
0,0400
0,8929
0,0699
0, 7200
0,9863
0> 8800
0,0876
3,7159
0,9837
50
0,0800
Oo 8 065
0,0831
Oo 6900
0,9718
0, 7600
0,1277
0,5711
0,9451
50
0,1200
Oo 7353
0,0870
0,6600
9565
0,6400
0,1610
3,6394
0,8832
50
0,1600
0,6757
0,0672
0,6300
0,9403
0,5200
0, 1910
0,5192
0,7750
50
0,2000
Oo 6250
0,0856
0,6000
0,9231
0,4000
0,2191
3, 3750
0,5769
50
0,3000
Oo 5263
0,0788
0,5250
0,8750
INVALID
FOR CASE
2
50
0,4000
Oo 4545
0,0716
0,4500
0,8182
INVALID
FDR CASE
2
50
0,5000
Oo 4000
0,0650
0,3750
0,7500
INVALID
FOR CASE
2
50
0,7000
Oo 3226
0,0544
O,2250
0,5625
INVALID
FOR CASE
2
50
0,9000
Oo 2703
0,0465
0, 0750
0,2500
INVALID
FOR CASE
2
20
0,0500
Oo 8696
0,1136
0,7125
0,9628
0,8500
0,1502
0,7059
0,9539
20
0,1000
Oo 7692
0,1259
0,6750
0,9247
0,7000
0,2128
3,5429
0,3806
ui
ooooooooooooo
THIS PROGRAM DETERMINES THE MINIMUM NUMDER OF DEFECTIVES TO BE
FOUND BV THE AUDITOR TO REJECT A SPECIFIED VALUE OF INSPECTION
ACCURACY,, THIS IS A SINGLE HYPOTHESIS TEST, THE DATA CARD
INCLUDES THE FOLLOWING INFORMATION AND CAN BE MODIFIED FOR OTHER
VALUES,
NI IS THE INSPECTION SAMPLE SIZE, NNI IS AN INDEX NUMBER FOR NI IF
MORE THAN ONE VALUE IS USED"PI IS THE INSPECTION FRACTION
DEFECTIVE, NPI IS ITS ASSOCIATED INDEX NUMBER NRFAC IS A
FACTOR TO DETERMINE THE AUDIT SAMPLE SIZE AND IS DIVIDED INTO NI,
NNR IS ITS ASSOCIATED INDEX NJMBER,
ALNORM IS THE ALPHA ERROR DESIRED ASSUMING A NORMAL DISTRIBUTION
AND CONST IS THE ASSOCIATED NORMALIZED Z STATISTIC,
DIMENSION NI(4),PI(19),NRFAC(4)
INTEGER DR175,DR190,DR275,3*290
DATA NI/53D,IDCO,5003,10000/
DATA PI/oDl,,02,,03,,04,305,,36,608,<>10,,12,>l4,>16,13f,23,,25,,3
10,o35,,40,,45,,50/
DATA NRFAC/2,5,10,50/
DATA NN 1/4/,NP1/19/,NNR/4/,CONCT/1,645/,ALNORM/,35/
IT AB= 3
DO 613 INI = 1,NNI
DO 613 INR = 1,NNR
133 NR = NI( INII/NRFACI INR)
IT AB=IT AB+l
IF(ITAB,LT,13) WRITE (6,910)IT*B
I F ( ITABoGE, 10) WRITE (6,920)ITAB
ALCAMP = 10 0 / ( 2, 25<=C0NST**2)/2, 3
WRITE(6,810) Nil INI I ,NR,ALNORM,ALCAMP
DO 613 IPI = 1,NPI
NRUN =1
113 IF(IPI,EQ,1) DR = l
IF{NRJN,E381,AND,IPIaNEol)DR=3190
IF(NRJN,E3o2,AND,IP I,NEo1)DR = 3?175
IF(NRJN,EQ,3aAND,IPI,NEol)DR=3290
IF(NRJN,EQ,4,AND0IPI,NEal)DR=D>275
121
TABLE rÃ‚Â¡> 5
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION SAMPLE SIZE = 1003,
INSPECTION FRACTION DEFECTIVE 005
AUDIT
AUDIT
1 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STD0DEV0
MQL
DQL
ACCURACY
STD,OEV
, MOL
OQL
950
OoOOlI
Oo 9804
Os 0194
Os 9490
Is 0000
0,9800
0,0202
0,9490
1,0000
950
O0 002 I
Oo 9 61 5
O,0267
0,9480
i0 0000
0,9600
0,0288
0, 9479
0,9999
950
0,0032
Oo 9434
Os 0317
0,9470
1,0000
0,9400
0,0357
0,9468
0,9998
950
0,0042
0o 9259
Os 0356
0, 9460
1,0000
0,9200
0,0416
0,9457
0,9996
950
0, 0053
Oo 909l
0,0388
O0 9450
1,0000
0,9000
0,0469
0, 9444
0,9994
950
0, 0095
Oo 8 475
0,0468
0o9410
1,0000
0,8200
0,0652
0,9390
0,9979
950
0, 0158
Oc 7692
0,0523
Oo 9350
1,0000
0,7000
0,0883
0,9286
0,9931
950
Oo 0211
Oo 7143
0,0540
0,9300
1,0000
0,6000
0,1058
0,9157
0,9857
950
Os 0316
Oo 6250
Os 0541
0,9200
1,0000
0,4000
0,1386
0,8750
0,9511
950
0,0421
Oo 5556
0,0524
0,9100
1,0000
0,2000
0,1697
0,7500
0,8242
950
Oo 0526
Oo 5000
0,0500
0,9000
1,0000
INVALID
FOR CASE
2
950
Os 0737
Oo 4167
0,0450
0,8800
1,0000
INVALID
FOR CASE
2
950
Os 0947
Oo 3571
0,0405
0, 8600
1,0000
INVALID
FOR CASE
2
950
0S 1263
Oo 2941
0,0349
0,8300
1,0000
INVALID
FOR CASE
2
500
Os 0020
Oo 9634
0,0356
Os 9481
",9991
0, 9620
0,0384
0,9430
0,9990
500
Os 0040
0o 9294
0,0473
0,9462
",9981
0,9240
0,0548
0,9459
0;9978
500
Os 0080
O0868I
0,0594
0,9424
",9962
0,8480
0,0788
0,9410
0,9948
500
0 0120
0o8143
0,0652
0,9386
", 9943
0,7720
0,0983
0,9352
0,9907
500
Os 0180
Oo 7 452
0,0685
Oo 9329
",9914
0,6580
0,1234
0,9240
0,9819
500
Os 0300
Oo 6369
0,0677
0,9215
0,9856
0,4300
0,1669
0,8837
0,9452
500
0 0400
Oo 5682
0,0645
0,9120
0,9806
0,2400
0,1997
0,7917
0,8513
500
0,0500
Oo 5128
Os 0607
0,9025
0,9757
INVALID
FOR CASE
2
500
0 07 00
Oo 4292
0,0535
0,8835
", 9656
INVALID
FOR CASE
2
500
Os 0900
Oo 3690
0,0473
Oo 8645
0,9552
INVALID
FOR CASE
2
500
Os 1200
Oo 3049
0,0401
0,8360
0,9393
INVALID
FOR CASE
2
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,
NUMBER 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, NUMBER INSPECTED = 1000,
NUMBER AUDITED = 2 00 71
7.1 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 2 50 78
7.2 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 100 79
vi
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
80
TABLE 7,3
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY! THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 50
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
INSPECTION
CASE
1
CASE
2
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
= 0,75
= 0,90
0,010
1
l
6
6
0,020
2
2
5
4
0,030
2
2
4
4
0,040
3
2
5
4
0,050
3
2
5
4
0,060
4
2
5
4
0,080
4
3
5
4
0,100
5
3
6
4
0, 120
6
3
6
4
0, 140
6
4
7
5
0, 160
7
4
7
5
0,180
8
4
7
5
0,200
8
5
8
5
0,250
10
5
9
6
0, 300
12
6
11
7
Oo 3 50
14
7
13
7
0,400
17
8
14
8
0,450
20
9
17
9
0,500
23
11
19
10
106
TABLE 80 8
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT
ACCURACY =0,
INSPECTION
90 AND ACCEPT ACCURACY =0,
SAMPLE SIZE = 1000
60
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
= 0,050 (NORMAL)
= 0, 082 (CAMP ME I DEL).
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE 2
AUDIT AUDIT
SAMPLE DEFECTIVES
SIZE
0o010
LOT SIZE
TOO SMALL
LOT SIZE TOO
SMALL
0,020
575
4
LOT SIZE TOO
SMALL
0,030
351
4
LOT SIZE TOO
SMALL
0,040
251
4
811
7
Do 050
194
4
610
7
Oo 060
157
4
486
7
0,080
112
4
341
6
Or, 100
86
4
259
6
0, 120
69
4
207
6
Oo 140
57
4
170
6
O0 160
48
3
143
6
Oo 180
41
3
122
6
Oo 200
36
3
106
6
Oo 250
26
3
77
6
Oo 300
19
3
58
6
0,350
15
3
44
5
0,40.0
11
3
34
5
Oo 450
8
3
26
5
0,500
6
3
20
5
45
OQL = (. 765
PI =
.200
NR = 160
PR =
.125
GI = 1000
ADR =
.500
O
= Number of true good units
Figure 4.4
Sample Production Flow for Case 2 with Replacement/
Audit Accuracy = Inspection Accuracy
72
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.
10
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. Sixtyseven 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
84
TABLE 7,7
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE 100
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
CASE
1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
= 0,75
= 0,90
0,010
2
2
5
4
0,020
3
2
5
4
0,030
4
2
5
4
0, 040
4
3
5
4
0,050
5
3
6
4
0,060
5
3
6
4
0,080
7
4
7
5
0, 100
8
4
8
5
0,120
9
5
9
6
0, 140
10
5
9
6
0,160
11
6
10
6
0,180
13
6
11
7
0,200
14
7
12
7
0,250
17
3
15
8
0,300
21
10
18
10
0,350
25
11
21
11
0,400
30
13
25
13
0,450
36
15
29
15
0,500
42
18
34
17
TABLE 6.3
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 1
NUMBER INSPECTED = 1000 NUMBER AUDITED = 200
MQL Initial
.90
o
<7\
.90
.75
.75
.75
.50
.50
.50
Initial
Accuracy
Observed
.90
.9010
.75
.7562
.50
.5089
.90
. 9008
.75
.7519
.50
. 5027
.90
.9002
.75
.7506
.50
.5008
Expected
a (ADR)
Observed
.0612
. 0565
. 0827
.0821
.0833
. 0851
. 0355
. 0347
.0484
. 0472
. 0495
.0482
.0209
. 0204
. 0292
.0282
.0310
. 0298
Expected
PI
Observed
. 0900
. 0901
.0750
. 0751
.0500
. 0501
.2250
.2252
.1875
.1877
.1250
.1251
.4500
.4502
.3750
. 3752
.2500
.2501
Expected
PR
Observed
. 0110
. 0112
.0270
. 0271
.0526
. 0528
.0323
. 0324
.0769
.0771
.1429
.1431
.0909
.0911
.2000
.2002
. 3333
. 3336
Correlation
.0311
.0394
.0389
. 0381
. 0395
.0390
.0389
.0400
. 0387
Number Greater
than 1 a
303
314
299
317
318
311
316
310
316
Number Greater
than 2 a
28
45
38
55
44
42
46
40
47
j
o
THE MEASUREMENT OF INSPECTOR ACCURACY
By
LEE ALLEN WEAVER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS 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.
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
LIST OF TABLES V
LIST OF FIGURES xi
KEY TO ABBREVIATIONS xii
ABSTRACT XV
CHAPTER
I INTRODUCTION 1
II A SURVEY OF THE LITERATURE 4
III MEASURES OF INSPECTOR ACCURACY 18
IV EXPECTED VALUES FOR THE SAMPLING DISTRI
BUTION OF THE RATIO OF DEFECTIVE PRODUCT
REJECTED 30
V CALCULATION OF EXPECTED VALUES 46
VI VERIFICATION OF DERIVED EXPECTED VALUES
BY SIMULATION 62
VII SINGLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY 73
VIII DOUBLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY 94
IX CONCLUSION 114
iii
TABLE OF CONTENTS (CONTINUED)
Page
APPENDIX 117
REFERENCES 128
BIOGRAPHICAL SKETCH 131
iv
LIST OF TABLES
TABLE Page
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
/
v
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,
NUMBER 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, NUMBER INSPECTED = 1000,
NUMBER AUDITED = 2 00 71
7.1 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 2 50 78
7.2 MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY
THE AUDITOR TO REJECT SPECIFIED INSPECTION
ACCURACY, INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 100 79
vi
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 SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE = 2 500 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
vxi
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
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 100 99
8.2 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.50, INSPECTION SAMPLE
SIZE = 500 100
viii
LIST OF TABLES (CONTINUED)
TABLE
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
Page
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 1000 101
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.50, INSPECTION SAMPLE
SIZE 5000 102
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 10000 103
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 100 104
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 500 105
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.60, INSPECTION
SAMPLE SIZE = 1000 106
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 5000 107
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.60, INSPECTION
SAMPLE SIZE = 10000 108
IX
LIST OF TABLES (CONTINUED)
TABLE
Paqe
8.11
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE 100
109
8.12
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE 500
110
8.13
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE 1000
111
8.14
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 5000
112
8.15
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 10000
113
x
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
xi
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 defectives 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 defectives 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.
Xlll
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.
xv
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 defectives found by the inspector by the total
number of defectives in the lot. During the actual pro
duction process the only way to determine the total number
of defectives in the lot is by an audit reinspection of the
lot to determine how many defectives 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 defectives
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 defectives 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.
xvxi
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,
1
2
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 defectives or using an audit
inspector. Required sample sizes and the calculation of
quantitative accuracy measures are left unansv/ered.
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
3
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."
4
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 Poissonlooking
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
6
meter was graduated every two units, and the human tendency
to read to the nearest graduated unit showed itself in the
excessive number of evennumbered 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
7
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, illdefined 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 everyday rates.
McKenzie offers three solutions to the controlled
experiment problem. One way is the introduction of known
defectives, 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 defectives being detected by inspection.
8
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 nonrandom 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 onehalf hours to inspect the
unit, the average inspection time on the regular inspection
line. Jacobson reported as a result of his experiment that
9
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
10
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. Sixtyseven 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
11
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
12
pin points. Dayworking 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 ultraprecision 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 v/ith 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
dramatically with the large 6inch Vernier micrometer to
TABLE 2.1
INSPECTOR
Instrument
1" Vernier Micrometer
2" Vernier Micrometer
6" Vernier Micrometer
3" Regular Micrometer
Depth Micrometer
Inside Micrometer
Inside Caliper & 2" Mic
Inside Caliper & 6" Mic
Outside Caliper & 6" Rule
Inside Vernier Caliper
Outside Vernier Caliper
ACCURACY IN THE USE OF PRECISION INSTRUMENTS
Number of
Established
inspectors
tolerance
162
0.0001
138
0.0001
131
0.0001
146
0.001
142
0.001
117
0.001
127
0.001
112
0.002
117
0.01563
113
0.001
117
0.001
Percentage of inspectors
reading within tolerance
43
17
11
64
53
66
46
9
49
42
51
14
scarcely one inspector in 10. A 6inch 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 22.
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
TABLE 2.2
DEFECTS FOUND IN FOUR
SUCCESSIVE VISUAL
AND GAGING
INSPECTIONS OF 30,000 UNITS
Submitted
Regular I
Regular II
Regular III
Experts
Good 29900
Defective 100
v Found 68
Left 32
> Found 18
Left 14
> Found 8
Left 6
* Found 4
Left 2
Accuracy
0.6800
0.5625
0.5714
0.6667
16
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 defectives. 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 ultraviolet 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
17
each batch. But these were made of steel and brassplated:
thus missorts 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.
18
19
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. Thirtynine
inspectors were given one and onehalf 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.
20
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:
AC I = 38,195 + 646 = .9959
39,000
21
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
GGI
NI
V (GGI) +
GDI
NI
V (GDI) +
DDI
NI
V (DDI) +
DGI
NI
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.
22
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:
23
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
24
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
= 51
ADR (new method)
51 + 5
.911
25
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.
26
Accuracy of Product Improver,'t
As noted earlier the purpose of the ii ection 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 *
27
For the Jacobson data the following results are ob
tained:
MQL = 38,195 +25
.9800
39,000
i
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 defectives
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
28
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.
Table3.1 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.
TABLE 3.1
Accuracy Measure
ACI = Ratio of Correct
Inspections
AGA = Ratio of Good
Product Accepted
ADR = Ratio of Defective
Product Rejected
API = Accuracy of
Product Improvement
SUMMARY OF ACCURACY MEASURES OBTAINED FROM
TWO SETS OF SAMPLE DATA
Solder
Connection
Example
Television
Old Inspection
Method
Panels
New Inspection
Method
.996
.507
.897
.999
.654
. 886
.828
.286
.911
. 825
.053
.831
831
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.
30
31
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 reenter 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.
32
The following expected values will be determined.
y(ADR) = mean value for the sampling distribution of
the ratio of defective product rejected.
2
c (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
therefore
ADR =
(GI) (DR)
NR
DI
DI + (GI) TORT
NR
Dividing both the numerator and denominator by NI, we
have
DI
NI
ADR
33
We can define the following expected values
E Iff] PI
If DI/NI and DR/NR are independent variables we can
write the expected value of ADR as (25, p. 52)
y (ADR)
PI
PI + (1PI)PR
Intuitively the number of defectives 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.
o2 (g)
3g
37
p axay.
For our problem the above equation would be written
as
a2 (ADR)
(3 ADrY
\3 PI )
3ADRV(PI)
+
/3ADR\2
\3 PR J
O2 (PR)
3 ADR 3 ADR
3 PI 3PR
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
34
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 a2 (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) =
NI
and
a2(PR) = PRdPR).
NR
The partial derivatives of ADR are given as
9ADR [PI + PR(1PI)] (1) PI(lPI)
8PI [PI + PR(1PI)]2
= PR
[PI + PR (1PI) ] 2 '
and
3 ADR [PI + PR(lPI)] (O)PI(lPI)
3 PR [PI + PR (1PI) ]
PI (1PI)
[PI + PR(1PI)]2 '
Combining the above terms we obtain
a (ADR) =
(PR) (PI) (1PI)
[PI + PR (1PI) ] 2
(NI)(PI)(1PI)
+
1PR
(NR) (PR) *
To determine the expected value of the manufacturing
quality level we apply the following modification to the
equation given in Chapter III.
35
mot. = 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
y(MQL) = 1PI (1PI)PR.
The true fraction defective in the inspection sample is
1MQL.
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
OQL
GGI
GGI + DGI
For the IOQL we have that
= GI,
GGI + DGI
36
and
GGI =
GI [lDR\,
NR
therefore
IQL = 1 gg .
or
y(IOQL) = 1 PR
For the OQL we have that
GGI + DGI = GI DR
and
GGI = GI
Solving we obtain
OQL = GI
fl 
\ NR/
:ain
ljd .
GI DR
Substituting the appropriate expected values in order to
have OQL as a function of NI, NR, PI and PR we obtain
U(OQL)
(NI) (PR) (1PI) (NR) (PR)
(NI) (1PI) (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 = QL ~ 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,
ADR =
DR
DR*
and
ADR =
DDI
Since
DDI + DGI
DDI=DI,
and
DGI =
. (GI)(DR*) (GI)(DR)
NR (NR) (ADR) '
we obtain
DI
ADR =
nr 4. (GD'CdRT *
(ADR) (NR)
Solving for ADR and substituting the expected values
for DI/NI and DR/NR we obtain
. PI (1PI)PR
y (ADR)
PI
38
Since this function is also nonlinear 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
Sadr pi(1+pr) [pipr(ipi)](l) pr
s pi ~ pi"2 pi2
Sadr _ pi(ipi) [pipr(ipi)](0) = (1pi)
5 PR PI2 PI
Combining the above terms with
obtain
o ADR = PR(1PI) / 1 !pR .
PI J (NI) (PI) (1PI) (NR) (PR) '
Using the methods previously discussed for Case 1
we obtain the following additional expected values for Case
2.
U(MQL, = 1 P! .
pdOQL) = 1 .
u(OQL) = 1 (NI) (PR) (1PI) ~(NR) (PR) (ADR)
ADR[NI (1PI) (NR) (PR) ] '
The equations derived for Cases 1 and 2 are summarized
in Table 4.1.
TABLE 4.1
SUMMARY OF EXPECTED VALUE FUNCTIONS
Expected
Value
CASE 1
Auditor Accuracy = 100%
CASE 2
Auditor Accuracy = Inspection Accuracy
PI
PI (1PI)PR
U (ADR)
PI + (1PI)PR
PI
a (ADR)
(PR) (PI) (1PI) / 1 1PR
PR(1PI) / 1 1PR
[PI+PR(1PI)]2 / (NI) (PI) (1PI) (NR) (PR)
PI / (NI) (PI) (1PI) (NR) (PR)
U(MQL)
1PR(1PI)PR
1.PT (IPI)PR
x ADR
U(IOQL)
1PR
1 PR
ADR
V(OQL)
(NI) (PR) (1PI)(NR) (PR)
(NI) (PR) (1PI)(NR) (PR) (ADR)
1 (NI) (1PI) (NR) (PR)
1 "ADR [NI (1PI)(NR) (PR)]
40
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
NI
GI
1PI *
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 noreplacement 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:
41
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.
42
CUSTOMER
PI =
.200
NR = 160
PR =
.125
NI = 1000
ADR =
.667
[= Number of true good units
CD = Number of true bad units
Figure 4.1
Sample Production Flow for Case 1 with No Replacement,
Audit Accuracy = 100 Percent
43
CUSTOMER
OQL = ^769)
PI = .200
PR = .125
ADR = .500
NR = 160
NI = 1000
= Number of
= Number of
true good units
true bad units
Figure 4.2
Sample Production Flow for Case 2 with No Replacements
Audit Accuracy = Inspection Accuracy
44
PI =
.200
NR = 160
PR =
.125
GI = 1000
ADR = .667
Number of true good units
Number of true bad units
Figure 4.3
Sample Production Flow for Case 1 with Replacement,
Audit Accuracy = 100 Percent
45
OQL = (. 765
PI =
.200
NR = 160
PR =
.125
GI = 1000
ADR =
.500
O
= Number of true good units
Figure 4.4
Sample Production Flow for Case 2 with Replacement/
Audit Accuracy = Inspection Accuracy
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:
y(ADR) = accuracy
a(ADR) = accuracy standard deviation
y(MQL) = manufacturing quality level
y(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
46
47
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 y(ADR), a(ADR), y(MQL), and
y(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 y(ADR) for Case 2 given below
U (ADR) = PI ~ (1PI) 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
_ .10 (.90) (.10) _
.10
y (ADR)
.10 .
48
If the accuracy of the inspector is .10 this means the whole
inspection sample is defective, which is verified by the
equation for y(MQL) for Case 2
y (MQL) = 1 PI (1 PI.MPJll
ADR
= 1 .10 
(.90) (.10)
.10
= 0.
If the audit fraction defective is greater than the inspec
tion fraction defective, negative values for y(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.
49
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.
i/0 O')
TABLE 501
EXAMPLES
OF ACCURACY EXPECTED VALUES
STAINED
FROM DIFFERENT INSPEC
TIGN
AND
RE iMSPECT Z ON TEST
RESULTS,
INSPECTION
SAMPLE
SIZE = 100,
INSPECTION
1 FRACTION
DEFECTIVE
= 0> 05
AUDIT
AUDIT
IU ct KJ U
CASE
1
ca w* w ps
 CASE 2
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MQL
DQL
ACCURACY
STDs DEV
> MQL
OQL
95
0> 0105
Oo 3 333
Os 1521
Os 9400
IsOOOO
Os 8000
Os 2191
0,
9375
05 9973
95
Os 0211
Oc 7143
Os 1707
Oo 9300
1o 0000
O,6000
0,3347
0
9167
0,9857
95
Os 0316
Oo 6250
Os 1712
Os 9200
1o 0000
0,4000
0,4332
Os
3750
Os 9511
95
Os 0421
Oo 5556
Os 1656
Os 9100
IsOOOO
0,2000
0,5367
0,
7500
Os 8242
95
03 0526
Oo 5 000
Os 1581
Os 90 00
lo 0000
INVALID
FDR
CASE
2
95
Os 0737
Oo 4167
Os 1423
Os 88 0 0
IsOOOO
INVAL D
FDR
CASE
2
95
0s 0947
Oo 3571
Os 1281
0o 86 0 0
IsOOOO
INVALID
FOR
CASE
2
95
Os 1579
Oo 2500
0,0968
Oo 8000
1,0000
INVALID
FOR
CASE
2
50
Os 0200
Oo 7246
Os 2177
Oo 9310
Os 9904
0> 6200
O,4146
Os
9194
0,9780
50
Os 0400
Oo 5682
Os 2039
Os 9120
O0 9806
0,2400
0,6315
0,
7917
0,8513
50
Os 0600
Oo 4673
Os 1802
Oo 8930
0,9707
INVALID
FDR
CASE
2
50
Os 0800
Oo 3968
Os 1589
Os 8740
Os 9604
INVALID
FOR
CASE
2
50
Os 1000
Oo 3448
Os 1412
Os 8550
Os 9500
INVALID
FDR
CASE
2
50
Os 1400
Oc 2732
Os 1147
Oo 8170
00 92 84
INVALID
FOR
CASE
2
20
Os 0500
Oo 5128
Os 2691
Os 9025
ns 9601
INVALID
FOR
CASE
2
20
Os 1000
Oo 3 448
Os 1836
Oo 8550
*39194
INVALID
FDR
CASE
2
20
Os 1500
Oo 2597
Os 1351
Oo 8075
*s 8777
INVALID
FOR
CASE
2
10
Os 1000
Oo 3448
0 2381
Oo 8550
*s 9096
INVALID
FDR
CASE
2
TABLE
EXAMPLES
OF ACCURACY EXPECTED VALUES
OBTAINED
FROM DIFFERENT INSPECTION
AND
REINSPECTION TEST
RESULTS,
INSPECTION
SAMPLE
SIZE 100,
INSPECTION
1 FRACTION
DEFECTIVE
* 0,10
AUDIT
AUDIT
1  
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
ST Do DEVo
MQL
OQL
ACCURACY
STD,DEV
MOL
OQL
90
0,0111
Oo 9091
0,0867
0,8900
l,0000
0,9000
0,1049
0,3889
0,9988
90
0,0222
Oo 8333
Oo1076
Oo 8800
loOOOO
0,8000
0,1549
0,8750
0,9943
90
0,0333
Oo 7692
0,1169
0o 8700
1,0000
0,7000
0,1975
0,8571
0,9852
90
0o 0444
Oo 7143
0,1207
Oo 8600
loOOOO
0,6000
0,2366
0,3333
3,9690
90
0,0667
Oo 6250
0#1210
Oo 8400
1,0000
0,4000
0,3098
3,7530
0,8929
90
0,0889
Oo 5556
0,1171
0,8200
1,0000
0,2000
0,3795
0,5000
0,6098
90
0,1111
Oo 5000
0,1118
0,8000
loOOOO
INVALID
FDR CASE
2
90
0,1667
Oo 4030
0,0980
Oo 7500
loOOOO
INVALID
FDR CASE
2
90
0,2222
Oo 3333
0,0861
0,7000
loOOOO
INVALID
FDR CASE
2
50
0,0200
Oo 8475
0,1350
Oo 8820
0,9910
0, 8200
0, 1833
0,8780
3,9866
50
0,0400
Oo 7353
0,1496
Oo 8640
0 9818
0,6400
0,2758
3,8438
0,9583
50
0,0600
0o 6494
Oo1483
0,8460
0,9724
0,4600
0,3518
3,7826
0,8996
50
0,0800
Oo 5 814
0,1421
Oo 8280
Oo 9628
0,2800
0,4205
0,6429
0,7475
50
0,1000
Oo 5263
Oo1345
0, 8100
0,9529
INVALID
FOR CASE
2
50
0,1400
Oo 4425
0,1193
Oo 7740
0,9325
INVALID
FDR CASE
2
50
0,1800
Oo 3817
0,1061
Oo 73 8 0
Oo 9111
INVALID
FDR CASE
2
50
0,3000
Oo 2703
Oo 0733
Oo 6300
", 8400
INVALID
FDR CASE
2
20
0,0500
Oo 6 397
0,2205
0,8550
0,9607
0,5500
0,4635
0,3182
0,9193
20
0,1000
Oo 5263
Oo1867
Oo 8100
0,9205
INVALID
FDR CASE
2
20
0,1500
Oo 4255
Oo1535
0,7650
0,8793
INVALID
FOR CASE
2
20
0,2000
Oo 3571
0,1281
0,7200
0,8372
INVALID
FDR CASE
2
20
0, 3000
Oo 2703
0,0941
Oo 6300
",7500
INVALID
FDR CASE
2
10
0,1000
Oo 5263
Oo 2507
0, 8100
0,9101
INVALID
FOR CASE
2
TABLE 5,>3
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION SAMPLE SIZE 103,
INSPECTION FRACTION DEFECTIVE = 0,25
AUDIT
AUDIT
1 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MQL
OQL
ACCURACY
STD,DEV
, MQL
OQL
75
0,0133
Oo 9 615
0,0377
0,7400
1,0000
0,9600
0,0408
0,7396
0,9994
75
0,0267
0o 9259
0,0504
0,7300
1,0000
0,9200
0,0588
0,7233
0,9976
75
0,0533
Oo 8621
0,0640
0,7100
1,0000
0, 8400
0,0862
0,7024
0,9893
75
0,0800
Oo 8065
0,0710
0,6900
1,0000
0,7600
0,1091
3,6711
0,9725
75
0,1200
0,7353
0,0757
0,6600
1,0000
0,6400
0,1399
0,5094
0,9233
75
0,2000
0,6250
0,0765
0, 6000
1,0000
0,4000
0,1950
3,3750
0,6253
75
0,2667
Oo 5556
0,0741
0,5500
1,0000
INVALID
FOR CASE
2
75
0,3333
Oo 5000
0,0707
0,5000
1,0000
INVALID
FDR CASE
2
75
0,4667
Oo 416 7
0,0636
0, 40 0 0
1,0000
INVALID
FDR CASE
2
75
0= 6000
Oo 3 571
0,0573
0, 3000
1,0000
INVALID
FDR CASE
2
75
0,8000
Oo 2941
0,0494
Oo1500
1,0000
INVALID
FOR CASE
2
50
0,0200
Oo 9434
0,0543
0, 7350
0,9932
0,9400
0,0610
0,7343
0,9919
50
0,0400
0,8929
0,0699
0, 7200
0,9863
0> 8800
0,0876
3,7159
0,9837
50
0,0800
Oo 8 065
0,0831
Oo 6900
0,9718
0, 7600
0,1277
0,5711
0,9451
50
0,1200
Oo 7353
0,0870
0,6600
9565
0,6400
0,1610
3,6394
0,8832
50
0,1600
0,6757
0,0672
0,6300
0,9403
0,5200
0, 1910
0,5192
0,7750
50
0,2000
Oo 6250
0,0856
0,6000
0,9231
0,4000
0,2191
3, 3750
0,5769
50
0,3000
Oo 5263
0,0788
0,5250
0,8750
INVALID
FOR CASE
2
50
0,4000
Oo 4545
0,0716
0,4500
0,8182
INVALID
FDR CASE
2
50
0,5000
Oo 4000
0,0650
0,3750
0,7500
INVALID
FOR CASE
2
50
0,7000
Oo 3226
0,0544
O,2250
0,5625
INVALID
FOR CASE
2
50
0,9000
Oo 2703
0,0465
0, 0750
0,2500
INVALID
FOR CASE
2
20
0,0500
Oo 8696
0,1136
0,7125
0,9628
0,8500
0,1502
0,7059
0,9539
20
0,1000
Oo 7692
0,1259
0,6750
0,9247
0,7000
0,2128
3,5429
0,3806
ui
TABLE 5,3
AUOIT
AUDIT
1 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDeDEVo
MQL
20
0,1500
0,6897
0,1242
0,6375
20
0,2000
Oo 6250
0,1180
0,6000
20
0, 2500
Oo 5 714
0,1104
0,5625
20
0,3500
Oo 487 8
0,0955
0,4875
20
0,4500
Oo 4255
0,0827
0,4125
20
Oo 7500
Oo 3077
0,0564
0,1875
20
10 0000
Oo 2500
0,0433
0,0
10
Oo1000
Oo 7 692
0,1733
0,6750
10
0,2000
Oo 6250
0,1578
0,6000
10
Oo 3000
Oo 5263
0,1335
0,5250
10
0,4000
Oo 4545
0,1118
0,4500
10
0,5000
Oo 4000
0,0940
0,3750
10
0,7000
Oo 3226
0,0678
0,2250
10
0,9000
Oo 2703
0,0501
0,0750
1
1,0000
Oo 2500
0,0433
o
o
O
ON TINUED
  CASE 2   
OQL ACCURACY STDnDEV, MQL OQL
0,8854
0,5500
0,2611
0
5455
0,7576
0,8451
0,4000
0,3020
0
3750
0,5282
0,8036
INVALID
FOR
CASE
2
0,7169
INVALID
FOR
CASE
2
0,6250
INVALID
FOR
CASE
2
0,3125
INVALID
FOR
CASE
2
",0
INVALID
FOR
CASE
2
na 9122
0> 7000
0,2929
0,
6429
0,8687
0,8219
0,4000
0,4040
0,
3750
0,5137
0,7292
INVALID
FOR
CASE
2
0,6338
INVALID
FOR
CASE
2
0,5357
INVALID
FOR
CASE
2
0,3309
INVALID
FOR
CASE
2
0,1136
INVALID
FOR
CASE
2
o
a
INVALID
FOR
CASE
2
TABLE 5,4
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST
RESULTS,
INSPECTION
SAMPLE
SIZE 1000,
INSPECTION FRACTION
1 DEFECTIVE
 0,01
AUDIT
AUDIT
L  
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCJRACY
STDoDEV,
MOL
OQL
ACCURACY
STD,DEV
', MOL
OQL
990
Oo 0010
Oo 909 1
0,0867
0,9890
1,0000
0,9000
0,1049
0,9889
0,9999
990
0,0020
0,8333
0,1076
0,9880
1,0000
0,8000
0, 1549
0,9875
05 9995
990
0,0030
Oo 7692
0,1169
0,9870
1,0000
0,7000
0,1975
0,9857
0, 9987
990
0,0040
Oo 7 14 3
0,1207
0,9860
1,0000
0,6000
0,2366
0,9333
0,9973
990
0,0061
0,6250
0,1210
0,9840
1,0000
0,4000
0,3098
0,9750
0,9909
990
0,0081
0,5556
0,1171
0,9820
1,0000
0,2000
0,3795
0,9500
0,9674
990
0,0101
Oo 5000
0,1118
0,9800
1,0000
INVALID
FOR CASE
2
990
0,0152
Oo 4000
0,0980
0, 9750
1,0000
INVALID
FOR CASE
2
990
0,0202
Oo 3333
0,0861
0,9700
1,00 00
INVALID
FOR CASE
2
500
0,0020
Oo 8347
0,1446
Oo 9880
0,9990
0,8020
0,2076
0,9875
0,9985
500
0,0040
Oo 716 3
0,1573
Oo 9860
0,9980
0 6040
0,3065
0,9334
0,9954
500
0,0060
Oo 6274
0,1537
Oo 98 41
0,9970
0,4060
0,3906
0,9754
0,9882
500
0,0080
Oo 5580
0,1457
Oo 98 2 1
0,9960
0,2080
0,4679
0,9519
0,9654
500
0,0100
0,5025
0,1367
Oo 9801
0,9950
INVALID
FOR CASE
2
500
0,0140
Oo 4191
0,1197
0,9761
0,9930
INVALID
FOR CASE
2
500
0,0180
0,3595
0,1055
0o 9722
0,9910
INVALID
FOR CASE
2
500
0,0300
Oo 2519
0,0767
Oo 9603
0,9849
INVALID
FOR CASE
2
200
0,0050
Oo 6689
0,2319
0,9851
0,9960
0,5050
0,5182
0,9302
0,9911
200
0,0100
Oo 502 5
0,1930
Oo 98 01
",9920
INVALID
FOR CASE
2
200
0,0150
Oo 4024
0,1576
0,9752
",9880
INVALID
FOR CASE
2
200
0,0200
O,3356
0,1312
0,9702
",9840
INVALID
FOR CASE
2
200
0,0300
Oo 2519
0,0966
0,9603
",9759
INVALID
FDR CASE
2
100
0,0100
Oo 5025
0,2611
Oo 9801
0,9910
INVALID
FOR CASE
2
TABLE rÃ‚Â¡> 5
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION SAMPLE SIZE = 1003,
INSPECTION FRACTION DEFECTIVE 005
AUDIT
AUDIT
1 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STD0DEV0
MQL
DQL
ACCURACY
STD,OEV
, MOL
OQL
950
OoOOlI
Oo 9804
Os 0194
Os 9490
Is 0000
0,9800
0,0202
0,9490
1,0000
950
O0 002 I
Oo 9 61 5
O,0267
0,9480
i0 0000
0,9600
0,0288
0, 9479
0,9999
950
0,0032
Oo 9434
Os 0317
0,9470
1,0000
0,9400
0,0357
0,9468
0,9998
950
0,0042
0o 9259
Os 0356
0, 9460
1,0000
0,9200
0,0416
0,9457
0,9996
950
0, 0053
Oo 909l
0,0388
O0 9450
1,0000
0,9000
0,0469
0, 9444
0,9994
950
0, 0095
Oo 8 475
0,0468
0o9410
1,0000
0,8200
0,0652
0,9390
0,9979
950
0, 0158
Oc 7692
0,0523
Oo 9350
1,0000
0,7000
0,0883
0,9286
0,9931
950
Oo 0211
Oo 7143
0,0540
0,9300
1,0000
0,6000
0,1058
0,9157
0,9857
950
Os 0316
Oo 6250
Os 0541
0,9200
1,0000
0,4000
0,1386
0,8750
0,9511
950
0,0421
Oo 5556
0,0524
0,9100
1,0000
0,2000
0,1697
0,7500
0,8242
950
Oo 0526
Oo 5000
0,0500
0,9000
1,0000
INVALID
FOR CASE
2
950
Os 0737
Oo 4167
0,0450
0,8800
1,0000
INVALID
FOR CASE
2
950
Os 0947
Oo 3571
0,0405
0, 8600
1,0000
INVALID
FOR CASE
2
950
0S 1263
Oo 2941
0,0349
0,8300
1,0000
INVALID
FOR CASE
2
500
Os 0020
Oo 9634
0,0356
Os 9481
",9991
0, 9620
0,0384
0,9430
0,9990
500
Os 0040
0o 9294
0,0473
0,9462
",9981
0,9240
0,0548
0,9459
0;9978
500
Os 0080
O0868I
0,0594
0,9424
",9962
0,8480
0,0788
0,9410
0,9948
500
0 0120
0o8143
0,0652
0,9386
", 9943
0,7720
0,0983
0,9352
0,9907
500
Os 0180
Oo 7 452
0,0685
Oo 9329
",9914
0,6580
0,1234
0,9240
0,9819
500
Os 0300
Oo 6369
0,0677
0,9215
0,9856
0,4300
0,1669
0,8837
0,9452
500
0 0400
Oo 5682
0,0645
0,9120
0,9806
0,2400
0,1997
0,7917
0,8513
500
0,0500
Oo 5128
Os 0607
0,9025
0,9757
INVALID
FOR CASE
2
500
0 07 00
Oo 4292
0,0535
0,8835
", 9656
INVALID
FOR CASE
2
500
Os 0900
Oo 3690
0,0473
Oo 8645
0,9552
INVALID
FOR CASE
2
500
Os 1200
Oo 3049
0,0401
0,8360
0,9393
INVALID
FOR CASE
2
TABLE 5 5 CONTIMUED
AUDIT
AUDIT
1  .
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MOL
QL
ACCURACY
STDs DEV
s MOL
OQL
200
Os 0050
Oo 9132
Os 0799
Os 9453
% 9960
Os 9050
Os 0958
0,9448
Os 9955
200
OoOlOO
Oo 8403
Os 0964
Os 9405
"s 9921
0,8100
Os 1365
0,9383
0,9897
200
Oo 0150
Oo 7782
01020
Os 93 58
''o 9881
0=7150
Os 1685
0,9301
Os 9821
200
Os 0200
Oo 7246
0S1029
Os 9310
"Ã‚Â¡s 9841
0,6200
0>1960
0,9194
Os 9718
200
Os 0300
Oa 6369
Os 0988
Oo 9215
"o 9762
Os 4300
Os 2437
Os 3B37
0,9361
200
Oo 0400
Oo 5682
Os 0921
Oo 912 0
^o 9682
Os 2400
0,2854
0,7917
Os 8404
200
0s 0500
Oo 5128
Os 0851
Os 9025
no 9601
INVALID
FOR CASE
2
200
0 07 50
Oo 4124
Os 0697
Os 8788
Os 9398
INVALID
FOR CASE
2
200
Oo1000
Oo 3448
0B 0581
Oo 8550
Os 9194
INVALID
FOR CASE
2
200
Or, 1500
Oo 2 597
Os 0427
Oo 8075
Os 8777
INVALID
FOR CASE
2
100
OoOlOO
Oo 8403
0>1349
Os 9405
0=9910
Os 8100
019L 0
Os 9 38 3
0=9337
100
0a0200
Oo 7 246
Os 1426
Os 9310
Oo 9821
0> 6200
Os 2717
0=9194
0=9698
100
Oo 0300
Oo 6369
Os 1357
Oo 9215
0 973 1
Os 4300
0, 3345
0=3837
0,9332
v
100
0a 0400
Oo 5682
Os 1254
Oo 912 0
0s 9641
0 2400
0s 38 8 3
0=7917
Os 8369
100
Oo 0500
Oo 5128
0,1148
Oo 9025
"=9550
INVALID
FOR CASE
2
100
Oo 0700
Oo 4292
Os 0961
Oo 8835
"d 9369
INVALI0
FOR CASE
2
100
Oo 0900
Oo 3690
Os 0314
Oo 8645
Oo 9187
INVALID
FOR CASE
2
100
01500
Oo 2597
Os 0536
Os 8075
Oo 8636
INVALID
FOR CASE
2
10
Oo1000
Oo 3448
0 2168
Go 8550
Os 9009
INVALID
FOR CASE
2
\
ui
TABLE S6
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION SAMPLE SIZE 1000,
INSPECTION FRACTION DEFECTIVE = 0,10
AUDIT
AUDIT
m n U M
CASE
1  
M M M
m *a as ma
 CASE 2 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCJRACY
STDoDEVo
MQL
DQL
ACCURACY
STD,DEV
, MOL
OQL
900
0,0011
0o 9901
Os 0099
Oo 8990
1,0000
Os 9900
0,0100
0,3990
1,0000
900
Od 0033
0o 9709
Os 0166
Os 8970
1,0000
0,9700
0,0176
0,3969
0,9999
900
0, 0056
Oo 9524
Os 0208
Os 8950
,0000
0,9500
0,0229
0,3947
Os 9997
900
Os 0078
Oo 9346
Os 0239
Os 8930
I,0000
Os 9300
0,0274
0,3925
0,9994
900
OsOlOO
Oo 9174
Os 0264
0,8910
I,0000
0,9100
0,0313
0,3901
0,9990
900
Os 0222
Oo 8 3 3 3
Os 0340
Os 8800
,0000
0,8000
0,0490
0,3750
0,9943
900
Os 0333
Oo 7692
Os 0370
Os 8700
,0000
0,7000
0,0625
Os 8571
0,9852
900
Os 0444
Oo 7143
0> 0382
Oo 8600
IsOOOO
0,6000
0,0748
0,3333
0,9690
900
Os 0667
Oo 62 50
Os 038 3
Os 8400
,0000
0,4000
0,0980
0,7500
0,8929
900
Os 0889
Oo 5556
Os 0370
Os 8200
1,0000
0,2000
0, 1200
0,5000
0,6098
900
Os 1111
Oo 5000
0,0354
Os 8000
I,0000
INVALID
FOR CASE
2
900
Os 1444
Oo 4 348
0,0327
0,7700
l,0000
INVALID
FDR CASE
2
900
0,1778
Oo 3846
O,0302
0 7400
1,0000
INVALID
FDR CASE
2
900
Os 2222
Oo 3333
Oo 0272
Os 7000
Is oooo
INVALID
FOR CASE
2
900
Os 2889
Oo 277d
0,0236
Os 6400
1,0000
INVALID
FOR CASE
2
500
Os 0020
Oo 9823
0,0174
0,8982
0,9991
0,9820
0,0181
0,8982
0,9991
500
Os 0040
Oo 9 653
0,0239
Os 8964
0,9982
0,9640
0,0257
0, 3963
On 9981
500
Os 0060
Oo 9483
0,0284
Os 8946
0,9973
Os 9460
0,0316
0,3943
0,9970
500
Os 0080
Oo 9328
Os 0319
0,8928
0,9964
Os 9280
0,0367
0,3922
0,9958
500
OsOlOO
Oo 9174
0s 0346
Os 8910
0,9955
0, 9100
0,0412
0,3901
0,9945
500
Os 0120
Oo 9025
0,0369
O,8892
*,9946
0,8920
0,0453
0,8879
0,9932
500
Os 0200
Oo B475
0,0427
Os 8820
*,9910
Os 8200
0,0595
0,8780
0,9866
500
Os 0300
Oo 7874
0,0461
On 8730
fts9864
0,7300
0,0743
0,8630
0,9752
500
Os 0400
Oo 7353
0,0473
On 8640
0,9818
0,6400
0,0875
0,8438
0,9588
500
Os 0600
Os 6494
0,0469
Os 8460
0,9724
0,4600
0, 1113
0,7826
0,8996
ui
TABLE 5s 6
AUDIT
AUDIT
1 
SAMPLE
FRACTION!
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MQL
500
0o 0800
Oo 5 814
Os 0449
0, 8280
500
Os 1000
0 o 5 2 6 3
Oo 0425
08100
500
Os 1400
Oo 4425
Os 0377
Os 7740
500
0,1800
Oo 3817
0,0336
Os 7380
500
Os 2400
Oo 3165
Os 0286
Os 6840
500
Os 3200
Oo 2 577
0,0237
0,6120
200
Os 0050
Oo 9569
Oo 0413
Oo 8955
200
OsOlOO
Oo 9 174
0,0539
Os 8910
200
Os 0200
Oo 8 47 5
Os 0654
0,8820
200
Os 0300
Oo 7874
0,0696
Os 8730
200
Os 0400
Oo 7353
0,0705
0,8640
200
Os 0750
Oo 5970
0,0649
Os 8325
200
Os 1000
Oo 5263
0s 0591
0, 8100
200
Os 1250
Oo 4706
0,0535
0,7875
200
Os 1750
Oo 3883
0,0442
0,7425
200
0s 2250
Oo 3306
0,0372
Os 6975
200
Os 3000
0 2703
Oo 0298
Oo 6300
100
0,0100
Oo 9174
Os 0758
0,8910
100
Os 0200
Oo 8475
0,0915
0,8820
100
0 0300
Oo 7874
0,0968
0,8730
100
Os 0400
Oo 7353
0,0975
Oo 8640
100
0,0600
Oo 6494
0,0933
0,8460
100
Os 0800
Oo 5814
0,0864
0,8280
100
Os 1000
Oo 5263
Os 0793
Oo 8100
100
Os 1500
Oo 4255
Os 0636
Oo 7650
100
Os 2000
Oo 3571
0,0519
0, 7200
100
Oo 3000
Oo 2 703
0,0366
Oo 6300
10
Os 1000
Oo 5263
0,2380
0,8100
3MTIMUE0
CASE 2
DQL
ACCURACY
STD,DEV
, MQL
OQL
0,9628
0, 2800
Os 1330
0,5429
0,7475
Oo 9529
INVALID
FDR CASE
2
0,9325
INVALID
FOR CASE
2
0,9111
INVALID
FOR CASE
2
"a 8769
INVALID
FOR CASE
2
"a 82 70
INVALID
FOR CASE
2
0,9961
0,9550
0>0451
Os 3953
0,9959
0,9922
0,9100
0,0640
0,8901
0,9912
0,9844
0> 8200
0,0911
0,8780
0,9800
0,9765
0,7300
0,1122
0,3630
0,9653
'',9686
0> 6400
0,1304
0, 8438
Os 9459
", 9407
0,3250
0,1821
0,5923
0,7823
", 9205
INVALID
FOR CASE
2
", 9000
INVALID
FOR CASE
2
",8584
INVALID
FOR CASE
2
", 8158
INVALID
FOR CASE
2
",7500
INVALID
FOR CASE
2
",9911
0> 9100
0,0900
0,3901
0,9901
", 9822
0> 8200
0,1274
0,8780
0,9778
",9732
0,7300
0,1551
0,8530
0,9521
",9643
0,6400
0,1804
0,3438
0,9417
",9463
0,4600
0,2212
0,7826
0,8754
",9283
0,2800
0,2557
0,6429
0,7207
",9101
INVALID
FOR CASE
2
", 8644
INVALID
FOR CASE
2
",8182
INVALID
FOR CASE
2
Oa 7241
INVALID
FOR CASE
2
0,9010
INVALID
FOR CASE
2
TABLE 5.7
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
RE I NS PcCTI ON TEST
RESULTS,
INSPECTION
SAMPLE
SIZE = 1000,
INSPECTION FRACTION
1 DEFECTIVE
= 0.25
AUDIT
AUOI T
1
SAMPLE
FRACTION
SI ZE
DEFECTIVE
ACCURACY
STD.CEV.
MQL
OQL
ACCURACY
STO.OEV
. MQL
OQL
750
0.0013
0.996C
0.0040
0.7490
1.0000
0.9960
0.0040
0.7490
1.0000
750
0.0053
0.9843
0.0078
0.7460
1.0000
0.9840
0.0081
0.7459
0.9999
750
0.0093
0.9728
0.0102
0.7430
1.0000
0.9720
0.0107
0.7428
0.9997
750
0.0133
0.9615
0.0119
0.7400
1.0000
0.9600
0.0129
0.7396
0.9994
750
0.0200
0.9434
0.0142
C.7350
1.0000
0.9400
0.0159
0.7340
0.9987
750
0.0267
0.9259
0.0159
0.7300
1.0000
0.9200
0.0186
0.7283
0.9976
750
0.0333
0.9091
0.0173
0.7250
1.0000
0.9000
0.0210
0.7222
0.9962
750
0.0600
0.8475
0.0209
0.7050
1.0000
0.8200
0.0291
0.6951
0.9860
750
0.0933
0.7813
0.0231
0.6800
1.0000
0.7200
0.0379
0.6528
0.9600
750
0.1333
0.7143
0.0241
0.6500
1.0000
0.6000
0.0473
0.5833
0.8974
750
0.1733
0.6579
0.0243
0.6200
1.0000
0.4800
0.0562
0.4792
0.7728
750
0.2267
0.5952
0.0240
0.5800
1.0000
0.3200
0.0676
0.2188
0.3772
750
0.2800
0.5435
0.0232
0.5400
1.0000
INVALID
FCR CASE
2
750
0.3467
0.4902
0.0221
0.4900
1.0000
INVALID
FOR CASE
2
750
0.4267
0.4386
0.0208
0.4300
1.0000
INVALID
FCR CASE
2
750
0.5333
0.3846
0.0191
0.3500
1.0000
INVALID
FOR CASE
2
750
0.6667
0.3333
0.0172
0.2500
1.0000
INVALID
FOR CASE
2
750
0.8 533
0.2809
0.0151
0.1100
1.0000
INVALID
FUR CASE
2
500
0.0020
0.9940
0.0059
0.7485
0.9993
0.9940
0.0060
0.7485
0.9993
500
0.0060
0.9823
0.0101
0.7455
0.9980
0.9820
0.0104
0.7454
0.9979
5C0
0.0100
0.9709
0.0128
0.7425
0.9966
0.9700
0.0135
0.7423
0.9963
500
0.0140
0.9597
0.0143
0.7395
0.9953
0.9580
0.0161
0.7390
0.9947
500
0.0180
0.9488
0.0164
0.7365
0.9939
0.9460
0.0183
0.7357
0.9929
500
0.0300
0.9174
0. 0200
0.7275
0.9898
0.9100
0.0238
0.7253
0.9868
500
0.0600
0.8475
0.0248
0.7050
0.9792
0.8200
0.0345
0.6951
0.9654
Ln
VO
TABLE 5,7
AUDIT
AUDIT
1 
SAMPLE
FRACTIQM
SIZE
DEFECTIVE
ACCJRACY
S T Do 0 EVo
MQL
500
0,0900
Oo 7874
0,0268
0,6825
500
0,1200
Oo 7353
0,0275
0,6600
500
0,1600
Oo 6757
0,0276
0,6300
500
0,2000
Oo 6250
0,0271
0,6000
500
0,2600
Oo 5 618
0,0258
Oo 5550
500
0,3200
Oo 5102
0,0245
0,5100
500
0,4000
Oo 4545
0,0226
0,4500
500
0,5000
Oo 4000
0,0206
0,3750
500
0,6200
Oo 3497
0,0184
0,2850
500
0,7300
Oo 2 994
0,0161
0,1650
200
0, 0050
Oo 9852
0,0146
0,7463
200
0,0100
Oo 9709
0,0200
0,7425
200
0,0150
Oo 9569
0,0238
0,7388
200
0,0200
Oo 9434
0,0267
0,7350
200
0,0250
Oo 9302
0,0290
0,7313
200
0,0300
Oo 9174
0,0310
0,7275
200
0,0350
Oo 9050
0,0325
0,7238
200
0,0750
Oo 8163
0,0388
Oo 6938
200
0,1250
0r> 7273
0,0398
Oo 65 63
200
0,1750
Oo 6557
0,0384
0,6188
200
0,2250
Oo 5970
0,0361
Oo 5313
200
0,3000
Oo 5263
0,0325
0,5250
200
0,4000
Oo 4545
0,0281
0,4500
200
0,5000
Oo 4000
0,0244
0,3750
200
0,6500
Oo 3390
0,0201
O,2625
200
0,8500
Oo 2 817
0,0160
0, 1125
100
0,0100
Oo 9709
0,0282
0,7425
100
0,0200
Oo 9434
0,0376
0,7350
100
0,0300
Oo 9174
0,0434
0, 7275
OMTINUED
EASE 2
OQL
AEEURAEY
STD,DEV
, MQL
OQL
0,9681
0,7300
0,0432
0,6575
0,9327
0,9565
0,6400
0,0509
0,5094
0,8832
0,9403
0,5200
0,0604
0,5192
0,7750
0,9231
0,4000
0,0693
0,3750
0,5769
0,8952
INVALID
FDR EASE
2
0,8644
INVALID
FOR EASE
2
0,8182
INVALID
FOR EASE
2
^,7500
INVALID
FOR EASE
2
0,6477
INVALID
FOR EASE
2
^34583
INVALID
FOR EASE
2
0,9963
0,9850
0,0150
0,7462
0,9963
0,9926
0,9700
0,0212
0,7423
0,9923
0,9890
0,9550
0,0250
0,7332
0,9882
0,9353
0,9400
0,0300
0,7340
0,9840
0,9815
0,9250
0,0336
0,7297
0,9795
0,9778
0,9100
0,0358
0,7253
0,9748
n,9741
0,8950
0,0397
0,7207
0:9699
9439
0,7750
0,0582
0,6774
0,9217
0,9052
0,6250
0,0753
0,6000
0,8276
0,8654
0,4750
0,0893
0,4737
0,6625
0,8245
0,3250
0, 1014
0,2308
0,3273
9,7609
INVALID
FOR EASE
2
0,6716
INVALID
FOR EASE
2
0,5769
INVALID
FOR EASE
2
0,4234
INVALID
FOR EASE
2
0,1940
INVALID
FOR EASE
2
0,9913
0> 9700
0,0299
0,7423
0,9910
0,9826
0,9400
0,0422
0,7340
0,9813
0,9739
0,9100
0,0516
0,7253
0,9709
v
TABLE 5s 7
AUDIT
AUDIT
1 .
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
ST Do DEVo
MQL
100
Os 0600
Oo 847 5
Os 0520
Os 7050
100
Os 0900
Oo 7 8 74
Os 0546
Os 6825
100
Os 1500
0 o 6 3 9 7
Os 0533
Os 63 7 5
100
0,2000
Oo 6250
Os 0499
Os 6000
100
0 o 2 5 0 0
Oo 5 714
Os 0460
Os 5625
100
Os 3500
Oo 4878
Os 0386
Oo 4875
100
Os 4500
Oo 4255
Os 032 4
Os 4125
100
Os 6000
Oo 3 5 7 1
Os 0252
Os 3000
100
Os 8000
Oo 2 941
Os 0184
Op 1500
10
Os 1000
0o 7692
Os 1689
Os 6750
10
Os 2000
Oo 6250
Os 1492
Os 6000
10
Os 3000
Oo 5263
Os 1218
Os 5250
10
Os 4000
Oo 4545
Os 0977
Oo 4500
10
Os 5000
Oo 4000
Os 0779
Oo 3750
10
Os 700.0
Oo 3226
Os 0480
Os 2250
10
Os 9000
Oo 2 703
Os 0253
Oo 0750
3NTINUED
OQL
ACCURACY
STD,OEV
, MOL
OQL
Os 9476
0, 8200
0,0724
3 >
6951
0,9343
Os 9211
0 7300
0,0881
0,
6575
Os 8874
Oo 8673
Os 5500
O,1120
0,
5455
Oo 7421
Oo 8219
Os 4000
0,1277
0,
3 750
0,5137
Oo 7759
INVALID
FDR
CASE
2
Oo 6818
INVALID
FDR
CASE
2
Os 5851
INVALID
FOR
CASE
2
Oo 4348
INVALID
FOR
CASE
2
Os 2239
INVALID
FOR
CASE
2
*59012
0,7000
0,2854
0,
5429
0,B583
*s 8021
0,4000
0,3820
3 D
3750
0,5013
*b 7028
INVALID
FDR
CASE
2
*s 6032
INVALID
FOR
CASE
2
*s 5034
INVALID
FOR
CASE
2
*s 3028
INVALID
FOR
CASE
2
*s1012
INVALID
FOR
CASE
2
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 y (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.
62
63
1. A value for the manufacturing quality level (MQL)
was predetermined. Since the actual number of defectives in
a lot of fixed size will vary from lot to lot, the number
of defectives 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 defectives in the lot (D) was de
termined by generating a normal random deviate (Z) and solving
for D in the following equation
D = NI(lMQL) + Z /NI (MQL) (IMOL)".
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 defectives 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 defectives, ADR also follows a binomial
distribution which can be approximated by a normal distribu
tion. For each lot the number of defectives 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) (1ADR) .
3. The lot PI was calculated.
4. The absolute number of defectives 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.
64
5. The number of defectives 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 =
K
Average PR =
K
Average Accuracy =
ZADR
K
Accuracy Standard Deviation=
/ZADR2 (IADR)2
K
Kl
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
tvvo 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:
65
y(MQL) = population manufacturing quality level
y(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,
y(MQL) = .75
y (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
y(ADR)
.9000
.9002
PI
.2250
.2252
PR
.0323
.0323
0 (ADR)
. 0231
.0225
The table of
frequencies for
the 1000 ADR values
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.
Frequency
66
200
150
100
50
80 85 90 95 100
Observed Accuracy
Initial Conditions
Figure 6.1
Histogram of
Simulation Results
67
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 y(MQL) = .50, .75, and .90 with y(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.
TABLE 6.1
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 1
NUMBER INSPECTED = 1000 NUMBER AUDITED = 500
MQL Initial
.90
.90
.90
.75
.75
.75
.50
.50
. 50
Initial
Accuracy
Observed
.90
. 9006
.75
.7516
.50
.5021
.90
.9002
.75
.7504
.50
.5006
.90
.9000
.75
.7501
.50
.5002
Expected
a (ADR)
Observed
. 0395
. 0386
. 0551
.0538
.0597
. 0580
. 0231
.0225
.0328
.0317
.0364
.0350
. 0140
.0136
.0208
.0200
.0242
.0232
Expected
PI
Observed
. 0900
.0901
.0750
. 0751
. 0500
.0501
.2250
.2252
.1875
.1877
.1250
.1251
.4500
.4502
.3750
. 3752
.2500
.2501
Expected
PR
Observed
. 0110
.0110
. 0270
.0271
.0526
. 0527
.0323
. 0323
. 0769
.0770
.1429
.1430
.0909
. 0910
.2000
.2001
.3333
.3335
Correlation
. 0378
.0389
. 0388
.0379
. 0396
.0391
.0390
.0404
. 0391
Number Greater
than 1 a
321
317
315
320
319
310
315
313
309
Number Greater
than 2 a
49
41
42
44
40
45
39
41
45
CTl
03
TABLE 6.2
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 2
NUMBER INSPECTED = 1000 NUMBER AUDITED = 500
MQL Initial
. 90
.90
.90
.75
.75
.75
.50
.50
.50
Initial
. 90
.75
.50
VO
o
.75
.50
.90
.75
. 50
Accuracy
Observed
.8993
.7484
.4943
.8999
.7496
.4985
.9000
.7500
. 4996
Expected
. 0461
. 0833
.1542
.0269
.0495
. 0937
. 0163
.0312
.0619
a (ADR)
Observed
. 0456
.0832
.1562
.0266
.0489
.0933
.0160
.0307
.0616
Expected
. 0900
. 0750
.0500
.2250
.1875
.1250
.4500
.3750
.2500
PI
Observed
. 0901
. 0751
. 0501
.2252
.1877
.1251
..4502
. 3752
.2501
Expected
. 0099
. 0203
. 0263
. 0290
.0577
.0714
. 0818
.1500
.1667
PR
Observed
. 0099
.0202
. 0262
.0290
.0576
.0712
. 0818
.1498
.1663
Correlation
. 0408
.0375
.0290
. 0389
. 0361
.0272
. 0387
.0351
.0242
Number Greater
than 1 a
309
300
307
306
309
318
313
311
316
Number Greater
than 2 o
40
42
43
41
42
43
35
45
43
cr\
vo
TABLE 6.3
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 1
NUMBER INSPECTED = 1000 NUMBER AUDITED = 200
MQL Initial
.90
o
<7\
.90
.75
.75
.75
.50
.50
.50
Initial
Accuracy
Observed
.90
.9010
.75
.7562
.50
.5089
.90
. 9008
.75
.7519
.50
. 5027
.90
.9002
.75
.7506
.50
.5008
Expected
a (ADR)
Observed
.0612
. 0565
. 0827
.0821
.0833
. 0851
. 0355
. 0347
.0484
. 0472
. 0495
.0482
.0209
. 0204
. 0292
.0282
.0310
. 0298
Expected
PI
Observed
. 0900
. 0901
.0750
. 0751
.0500
. 0501
.2250
.2252
.1875
.1877
.1250
.1251
.4500
.4502
.3750
. 3752
.2500
.2501
Expected
PR
Observed
. 0110
. 0112
.0270
. 0271
.0526
. 0528
.0323
. 0324
.0769
.0771
.1429
.1431
.0909
.0911
.2000
.2002
. 3333
. 3336
Correlation
.0311
.0394
.0389
. 0381
. 0395
.0390
.0389
.0400
. 0387
Number Greater
than 1 a
303
314
299
317
318
311
316
310
316
Number Greater
than 2 a
28
45
38
55
44
42
46
40
47
j
o
TABLE 6.4
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 2
NUMBER INSPECTED = 1000 NUMBER AUDITED = 200
MQL Initial
.90
.90
.90
.75
.75
.75
.50
.50
.50
Initial
Accuracy
Observed
.90
.8978
.75
.7487
.50
.4961
.90
. 8999
.75
.7500
.50
.4996
.90
.9000
.75
.7502
.50
.5004
Expected
a (ADR)
Observed
.0716
. 0683
.1265
.1265
. 2270
.2311
.0416
.0410
. 0743
.0737
.1361
.1365
.0245
.0242
.0451
.0446
. 0871
. 0869
Expected
PI
Observed
. 0900
. 0901
.0750
.0751
.0500
. 0501
.2250
.2252
.1875
.1877
.1250
.1251
.4500
.4502
. 3750
. 3752
.2500
.2501
Expected
PR
Observed
.0099
.0101
. 0203
.0202
. 0263
. 0261
. 0290
.029 0
. 0577
.0575
.0714
.0710
.1818
.1817
. 1500
.1497
.1667
.1660
Correlation
.0361
.0404
.0307
. 0407
. 0373
.0280
.0393
.0354
.0245
Number Greater
than 1 a
331
304
301
311
308
317
309
308
319
Number Greater
than 2 o
35
36
44
46
44
46
41
43
49
72
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 onetail 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
73
I
5
!
Ã‚Â¡y (ADR)Ka(ADR)
I
f
!
S
3 M
i /
y
V
/
i /
\
1/
3
y (ADR)
Figure 7.1
Graphical Presentation of a Single Hypothesis
Statistical Test for Inspection Accuracy
75
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) (PI)
(NR) (DI) + (NIDI) (DR)
c jnRl = (NR) (DR) (DI) (NIDI) f
K ; [ (NR) (DI) + (DR) (NIDI) 1 2/
NI
(NRDR)
[(NR)(DI) + (DR)(NIDI)]V DI(NIDI) (NR)(PR)
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 nonlinear 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[x ul > Ka] < r:5 T .
For a onetail test and for the accuracy function we can
76
determine an alpha error value
P [ (ADR y (ADR) )
> K a (ADR) ]
 (2) (2.25) K2
where
y (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
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 defectives 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
defectives.
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 defectives necessary to
be found by the auditor.
78
TABLE 7,1
MINIMUM NUMBER OF DEFECTIVE ITEMS F^UND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5*0,
AUDIT SAMPLE SIZE = 250
ALPHA ERROR =0oC50 (NORMAL)
ALPHA ERROR =0o0B2 (CAMP ME I DEL)
OBSERVED
CASE
1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0 75
= 0o 90
= 0,75
= 0,90
OoOlO
4
2
9
7
Oo 0 20
5
3
7
6
Oo 0 30
7
4
8
6
0,040
8
5
9
6
Oo 050
9
5
10
6
Oo 060
11
6
10
6
0,080
13
7
12
7
Oo 100
16
8
14
8
Oo 120
19
9
16
9
Oo 140
22
10
18
10
Oo 160
24
ll
21
11
o
o
co
o
27
12
23
12
Oo 200
30
13
25
13
Oo 2 50
39
16
31
16
Oo 300
43
20
38
19
Oo 350
59
23
46
22
Oo 4 00
71
28
56
26
Oo 450
85
33
67
31
Oo 500
103
39
80
36
79
TABLE 7,2
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE =, 100
ALPHA ERROR =0,050
ALPHA ERROR =0,082
(NORMAL)
(CAMP ME I DEL)
OBSERVED
CASE
1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
Oj 75
= 0,90
0,010
2
2
7
7
0,020
3
2
5
5
0,030
4
3
5
4
0,040
4
3
6
4
0,050
5
3
6
5
0,060
6
3
6
5
0,080
7
4
7
5
0,100
8
4
8
5
0, 120
9
5
9
6
0, 140
10
5
10
6
0,160
12
6
11
6
0, 180
13
6
12
7
0,200
14
7
13
7
0,250
18
8
15
8
0,300
21
10
18
10
0,350
26
11
21
11
0,400
31
13
25
13
0,450
37
15
30
15
0,500
43
18
35
17
80
TABLE 7,3
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY! THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE = 50
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
INSPECTION
CASE
1
CASE
2
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
= 0,75
= 0,90
0,010
1
l
6
6
0,020
2
2
5
4
0,030
2
2
4
4
0,040
3
2
5
4
0,050
3
2
5
4
0,060
4
2
5
4
0,080
4
3
5
4
0,100
5
3
6
4
0, 120
6
3
6
4
0, 140
6
4
7
5
0, 160
7
4
7
5
0,180
8
4
7
5
0,200
8
5
8
5
0,250
10
5
9
6
0, 300
12
6
11
7
Oo 3 50
14
7
13
7
0,400
17
8
14
8
0,450
20
9
17
9
0,500
23
11
19
10
81
TABLE 7,4
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE =t 10
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0,082 (CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
0,010
0,020
0,030
0,040
0,050
0,060
0,080
0, 100
0,120
0, 140
0,160
0,180
0,200
0,250
0, 300
0,350
0,400
0,450
0,500
CASE l
ACCURACY ACCURACY
= 0o 75 =0,90
l
l
I
l
1
2
2
2
2
2
3
3
3
3
4
4
5
5
6
1
l
I
I
1
L
1
2
2
2
2
2
2
2
3
3
3
3
4
CASE 2
ACCURACY ACCURACY
=0,75 =0,90
4
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
5
5
6
4
3
3
3
3
3
3
3
3
3
3
3
Y
3
3
3
3
4
4
4
82
TABLE 7s 5
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 1000,
AUOIT SAMPLE SIZE = 500
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0o082 (CAMP ME I DEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
*0o 90
ACCURACY
= 0, 75
ACCURACY
=0,90
0,010
5
3
7
6
Os 0 20
8
5
9
6
Os 0 30
11
6
10
6
Oo 040
13
7
12
7
Os 0 50
15
7
14
8
Oo 060
18
8
16
9
Os 080
23
10
19
10
OslOO
28
12
23
12
Os 120
33
14
27
14
Oo 140
38
16
31
15
Oo 160
43
18
35
17
Oo 180
49
20
39
19
0,200
55
22
43
21
Oo 250
70
28
55
26
0,300
88
34
69
31
0,350
108
41
84
38
0,400
132
49
102
45
Os 450
160
59
123
54
Os 500
193
70
148
64
83
TABLE 7,6
MINIMUM NUMBER DF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE =. 200
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0,75
ACCURACY
=0,90
0,010
3
2
5
5
0,020
4
3
6
4
0,030
5
3
6
5
0,040
7
4
7
5
0,050
8
4
8
5
0,060
9
5
9
6
0,080
11
6
10
6
0, 100
13
7
12
7
0, 120
15
7
13
8
0,140
17
8
15
3
0,160
20
9
17
9
0, 180
22
10
19
10
0,200
25
11
20
11
0,250
31
13
25
13
0,300
38
16
31
15
0,350
47
19
37
18
0,400
56
23
45
21
0,450
67
27
53
25
0,500
81
31
63
29
84
TABLE 7,7
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE 100
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
CASE
1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
= 0,75
= 0,90
0,010
2
2
5
4
0,020
3
2
5
4
0,030
4
2
5
4
0, 040
4
3
5
4
0,050
5
3
6
4
0,060
5
3
6
4
0,080
7
4
7
5
0, 100
8
4
8
5
0,120
9
5
9
6
0, 140
10
5
9
6
0,160
11
6
10
6
0,180
13
6
11
7
0,200
14
7
12
7
0,250
17
3
15
8
0,300
21
10
18
10
0,350
25
11
21
11
0,400
30
13
25
13
0,450
36
15
29
15
0,500
42
18
34
17
85
TABLE 7,8
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = IOOO,
AUDIT SAMPLE SIZE =. 20
ALPHA ERROR
ALPHA ERROR
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
0,010
0,020
0,030
0,040
0,050
0,060
0,080
0, 100
0,120
0, 140
0, 160
0,180
0,200
0,250
0,300
0,350
0,400
0,450
0,500
= Qo 050 (NORMAL)
= 0,082 (CAMP ME I DEL)
CASE l
ACCURACY ACCURACY
= 0o 75 =0,90
l l
1 l
2 1
2 l
2 2
2 2
2 2
3 2
3 2
3 2
4 3
4 3
4 3
5 3
6 4
7 4
8 5
9 5
11 6
CASE 2
ACCURACY ACCURACY
=0,75 =0,90
4
3
3
3
3
4
4
4
4
4
4
5
5
5
6
7
7
8
9
4
3
3
3
3
3
3
3
3
3
4
4
4
4
4
5
5
5
6
86
TABLE 7,9
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUNO BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =2500
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0,082 (CAMP ME I DEL)
OBSERVED CASE I CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0,75
ACCURACY
= 0,90
0o0I0
15
7
13
8
Oa 020
26
11
22
ll
0,030
36
15
29
15
Oo 040
47
19
37
18
0a050
57
23
45
21
0,060
68
27
53
25
0,080
89
34
70
32
0,100
111
42
86
39
0, 120
134
50
104
46
0,140
158
58
122
54
0,160
183
67
140
61
0, 180
209
76
160
69
0,200
236
85
180
78
0,250
309
111
236
101
0,300
393
139
299
127
0,350
489
172
371
156
0,400
600
210
456
191
0,450
732
255
555
231
0,500
890
309
674
279
87
TABLE 7a10
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =1000
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0o082 (CAMP ME I DEL )
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0a 75
ACCURACY
= 0a 90
0 010
8
4
8
5
0a020
12
6
11
7
0,030
17
8
15
8
0a 040
21
10
18
10
0,050
26
11
21
ll
0,060
30
13
25
13
0,080
39
16
32
16
O0 100
48
20
39
19
Oa 1 20
58
23
46
22
Oa 140
68
27
53
25
Oo 160
78
30
61
28
0, 180
89
34
69
32
Oo 200
100
38
78
35
Oo 250
130
49
100
45
O0 300
164
61
126
56
Oa 350
202
74
155
68
Oo 400
247
90
189
82
Oa 450
300
108
229
98
Oo 500
363
130
277
118
88
TABLE 7, 1 i
MINIMUM NUMBER OF DEFECTIVE ITEMS FBUND BY. THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =. 500
ALPHA ERROR
ALPHA ERROR
OBSERVED
=0,050 (NORMAL)
=0,082 (CAMP MEIDEL)
CASE 1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o75
= 0,90
= 0,75
= 0,90
0,010
5
3
6
4
0 020
7
4
8
5
0,030
10
5
9
6
Oo 040
12
6
11
7
Oo 050
15
7
13
8
Os 060
17
8
15
8
0,080
22
10
18
10
0,100
27
12
22
12
0,120
32
14
26
13
0,140
37
16
30
15
0, 160
42
17
34
17
0,180
47
19
38
18
0,200
53
22
42
20
0,250
68
27
54
25
0,300
86
33
67
31
0,350
105
40
82
37
0,400
128
48
99
44
0,450
155
58
119
53
0,500
186
69
143
63
89
TABLE 7,12
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND 8Y THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =. 100
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
INSPECTION
CASE
1
CASE
2
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
= 0,75
= 0,90
0,010
2
2
4
3
0,020
3
2
4
4
0,030
3
2
5
4
0,040
4
3
5
4
0,050
5
3
5
4
0,060
5
3
6
4
0,080
7
4
7
5
o
o
O
O
8
4
8
5
0,120
9
5
8
5
0, 140
10
5
9
6
0,160
11
6
10
6
0, 180
12
6
11
7
0,200
14
7
12
7
0,250
17
8
15
8
0,300
21
10
17
10
0,350
25
11
21
11
0,400
30
13
24
13
0,450
35
15
28
14
0,500
42
18
33
17
90
TABLE 7, 13
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE =5000
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0o082 (CAMP MEIDEL)
OBSERVED CASE 1 CASE 2
INSPECT ION
FRACTION
DEFECTIVE
ACCURACY
= 0, 75
ACCURACY
= 0, 90
ACCURACY
= 0, 75
ACCURACY
=0,90
OoOlO
26
11
21
11
0 020
46
19
37
18
0 0 30
66
26
52
24
Oa 040
86
33
67
31
0050
106
40
82
37
0,060
126
47
98
44
Oa 080
168
62
129
57
Oo 100
211
77
162
70
0,120
256
92
196
84
0,140
302
108
231
99
Oo 160
351
125
267
114
Oo 180
401
142
306
130
Oo 200
455
161
346
146
0,250
599
210
455
191
Oo 300
764
266
579
241
0,350
953
330
721
299
0,400
1174
405
887
367
0,450
1434
493
1083
446
0,500
1746
598
1318
541
91
TABLE 7,14
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE =2000
ALPHA ERRDR =0o050 (NORMAL)
ALPHA ERROR =0o082 (CAMP ME I DEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0o 90
ACCURACY
= 0, 75
ACCURACY
=0 j 90
OoOlO
12
6
11
7
Oo 020
21
10
18
10
Oa 030
29
13
24
12
0 040
38
16
31
15
Oo 050
46
19
37
18
Oo 060
55
22
43
21
Oo 080
72
28
57
26
OolOO
90
35
70
32
Oo 120
108
41
84
38
Oo 140
127
48
98
44
Oo 160
147
55
113
50
Oo L80
168
62
129
57
Oo 200
189
69
145
64
Oo 250
248
90
190
82
Oo 300
315
113
240
103
Oo 350
391
139
298
127
Oo 400
480
170
365
154
0,450
584
205
443
186
Oo 500
709
248
538
225
92
TABLE 715
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE =1000
ALPHA ERRDR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP ME I DEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0,75
ACCURACY
= 0,90
0,010
7
4
8
5
0, 020
12
6
11
7
0,030
17
8
14
8
Oo 040
21
10
18
10
0,050
26
11
21
11
0,060
30
13
25
13
0,080
39
16
31
16
0,100
48
20
38
19
0, 120
58
23
46
22
0, 140
67
27
53
25
0,160
78
30
61
28
0, 180
88
34
69
32
0,200
99
38
77
35
0,250
129
48
100
45
0,300
163
60
125
55
0,350
201
74
155
68
0,400
246
89
188
82
0,450
298
108
228
98
0,500
361
129
275
118
93
TA8LE 7s16
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE ^ 200
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0o082 (CAMP MEIDEL)
OBSERVED
INSPECTION
CASE
1
CASE
2
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0,75
= 0o 90
= 0,75
= 0,90
0,010
3
2
4
4
Oo 0 20
4
3
5
4
09 0 30
5
3
6
4
Oo 0 40
6
4
7
5
0,050
7
4
7
5
Oo 060
9
5
8
5
Oo 080
11
6
10
6
0,100
13
6
11
7
Oo 120
15
7
13
8
0,140
17
8
15
8
Oo 160
19
9
16
9
Oo 180
22
10
18
10
Oo 200
24
11
20
11
0,2 50
30
13
25
13
0, 300
38
16
30
15
Oo 350
46
19
36
18
0,400
55
22
43
21
Oo 450
66
26
52
24
0,500
78
31
61
29
CHAPTER VIII
DOUBLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY
The sampling plans developed in this chapter are based
on a null hypothesis which states that the accuracy is equal
to some acceptable value and an alternate hypothesis which
states that the accuracy is equal to some unacceptable value.
To develop these sampling plans both the alpha and beta errors
are fixed. Figure 8.1 is a graphic presentation of the above
described classical statistical test. In tests of hypotheses
of this type two values need to be determined. The minimum
sample size necessary to reduce the standard deviation to
meet the alpha and beta errors and the actual decision value
need to be calculated.
The sampling plans developed are for a fixed inspection
sample size. Based on an observed inspection fraction de
fective, the audit sample size and the number of defects in
the audit sample to reject the null hypothesis are given.
For example in Table
8.1,
the hypotheses are:
Null hypotheses
^o
(ADR) =
.90
Alternate hypothesis
(ADR) =
.50
94
. Figure 8.1
Graphical Presentation of a Double Hypothesis
Test for Inspection Accuracy
f
/
VO
U1
96
If the inspection sample size is 100 and the observed in
spection fraction defective is .20, the auditor should take
a sample of 19 items. If 3 or more defectives are found the
alternate hypothesis is accepted, if 2 or less are found the
null hypothesis is accepted. Situations where the audit
sample size required to reduce the standard deviation are
greater than the original inspection sample size are noted
by the remark that the "lot size is too small."
Tables in the back of the chapter are given for in
spection sample sizes of 100, 500, 1000, and 5000, and for
alternate hypotheses of y ]_ (ADR) = .50, .60, and .75.
The computer program for these sampling plans is given
in the Appendix and can be modified for different hypotheses,
alpha and beta errors, and inspection sample sizes by modify
ing the data card.
The equations used to determine the audit sample sizes
are based on the assumption of normality; however, the equi
valent value for the alpha and beta errors for the Camp Meidel
inequality were calculated and are printed in the tables.
The following procedure for Case 1 was used to deter
mine the minimum audit sample sizes and decision criteria.
For each hypothesis the following two equations are deter
mined:
ADR y (ADR)
Z
a (ADR)
97
z ADR y ^ (ADR)
a (ADR)
ADR is a sampling statistic which was derived in the previous
chapter as
ADR = (NR) (PI) _
(NR) (DI) + (NIDI) (DR) *
Since a(ADR) is unknown it is necessary to use an estimate
obtained from the sample, which was also derived in the
previous chapter as
= (NR)(DR)(PI)(NIDI) / NI ~ NRDR
1 [(NR)(DI) + DR(NIDI)]V DI(NIDI) (NR)(DR)*
Substituting the functions for the sampling statistics and
selecting values for the inspection sample size and the ob
served inspection defective into the pair of Z equations, we
have two equations in two unknowns, the audit sample size
(NR) and the number of audit defectives (DR).
The sampling plans developed in this chapter show that
large sample sizes are necessary to adequately determine
values of inspector accuracy, especially if the observed
inspection fraction defective is very low.
Table 8.11, which is included only for illustrative
purposes,shows that no statistical tests on inspection ac
curacy are possible for inspection sample sizes of 100 and
the following hypotheses.
Hq: y0(ADR) = .90
Hi :
y^(ADR) = .75 .
98
Table 8.14 shows that with an inspection sample size of 5000
and observed fraction defective of .010 no test is possible
for the above hypotheses. However, higher inspection ob
served fraction defective will allow for adequate audit
inspection to determine inspection accuracy.
99
TABLE 8,1
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,50,
INSPECTION! SAMPLE SIZE = 100
ALPHA ERROR
ALPHA ERROR
= BETA
= BETA
ERROR
ERROR
 Oo
= 3o
050 (NORMAL)
082 (CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
i
AUDIT
DEFECTIVES
CASE
AUDIT
sample
SIZE
2
AUDIT
DEFECTIVES
0,010
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,020
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 030
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,040
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,050
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,060
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
OoOBO
74
3
LOT
SIZE
TOO
SMALL
0, 100
53
3
LOT
SIZE
TOO
SMALL
0,120
41
3
LOT
SIZE
TOO
SMALL
0, 140
33
3
LOT
SIZE
TOO
SMALL
0,160
27
3
LOT
SIZE
TOO
SMALL
0, 180
23
3
LOT
SIZE
TOO
SMALL
0,200
19
3
LOT
SIZE
TOO
SMALL
0,250
13
2
72
6
0,300
10
2
51
5
0,350
7
2
38
5
0,400
5
2
28
5
0,450
4
2
21
4
0,500
2
2
16
4
100
TABLE 8a 2
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,50,
INSPECTION SAMPLE SIZE = 500
ALPHA ERROR =
ALPHA ERROR =
BETA ERROR
BETA ERROR
= 0,050
= 0,082
I NORMAL)
(CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SIZE
2
AUDIT
DEFECTIV
OoOlO
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Oa 020
293
3
LOT SIZE
TOO
SMALL
0,030
175
3
LOT SIZE
TOO
SMALL
0,040
123
3
LOT SIZE
TOO
SMALL
Oe 050
95
3
456
6
0,060
76
3
354
5
0,080
54
3
241
5
0,100
41
3
180
5
0,120
33
3
141
5
0,140
27
2
115
5
0,160
23
2
96
5
0, 180
19
2
82
5
0,200
17
2
71
4
0,250
12
2
51
4
0,300
9
2
37
4
0,350
7
2
28
4
0,400
5
2
21
4
0,450
3
2
16
4
0,500
2
2
12
3
101
TA8LE 8a 3
AUDIT SAMPLE SIZE AND MINIMUM NUMBER 3F DEFECTIVES TO
REJECT
ACCURACY =0o
INSPECTION
90 AND
sample
ACCEPT
SIZE =
ACCURACY
1000
= 0
>50,
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
= 0o 050
= 0c 082
i (NORMAL)
(CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
S IZE
l
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
OaOlO
592
3
LOT SIZE
TOO
SMALL
Oo 020
253
3
LOT SIZE
TOO
SMALL
Oo 030
159
3
732
5
0 a 040
115
3
506
5
Oo 050
90
3
384
5
Oo 060
73
3
307
5
Oo 080
52
3
217
5
Oo 100
40
2
165
5
Oo 120
32
2
132
5
Oo 140
27
2
109
4
Oo 160
22
2
91
4
Oo 180
19
2
78
4
Oo 200
17
2
67
4
Oo 250
12
2
49
4
Oo 300
9
2
36
4
Oo 350
6
2
27
4
Oo 400
5
2
21
4
0 a 450
3
2
16
4
O 500
2
2
12
3
102
TABLE 8 4
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCJRACV = 0,90 AND ACCEPT ACCURACY =0,50,
INSPECTION SAMPLE SIZE = 5000
ALPHA ERROR = BETA ERROR =0,050 (NORMAL)
ALPHA ERROR =. BETA ERROR =0,082 (CAMP MEIDEL),
OBSERVED
CASE
1
CASE
2
INSPECTION
AUDIT
AUDIT
AUDIT
AUDIT
FRACTION
DEFECTIVE
SAMPLE
SIZE
DEFECTIVES
SAMPLE
SIZE
DEFECTIVES
0,010
474
3
2017
5
0,020
228
3
924
5
0,030
148
3
594
5
0,040
109
3
434
5
O, 050
86
3
340
4
0,060
70
3
278
4
0,080
51
2
201
4
0, 100
39
2
156
4
0,120
32
2
125
4
0, 140
26
2
104
4
0, 160
22
2
88
4
0,180
19
2
75
4
0,200
16
2
65
4
0,250
12
2
47
4
0,300
9
2
35
4
0,350
6
2
27
4
0,400
5
2
20
4
0,450
3
2
15
3
0,500
2
2
11
3
103
TABLE 80 5
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0o90 AND ACCEPT ACCURACY =0,50,
INSPECTION
SAMPLE SIZE
= 10000
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
=0,050 (NORMAL)
= 0& 082 (CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
3 010
462
3
1873
5
Oo 020
225
3
892
5
0,030
147
3
580
5
Oo 040
108
3
426
4
Oo 050
85
3
335
4
0,060
70
3
275
4
Oo 080
51
2
199
4
Oo 100
39
2
154
4
Oo 120
32
2
124
4
Oo 140
26
2
103
4
Oo 160
22
2
87
4
Oo 180
19
2
75
4
0,200
16
2
65
4
Oo 250
12
2
47
4
Oo 300
9
2
35
4
Oo 350
6
2
27
4
0o400
5
2
20
4
0 o 450
3
2
15
3
Oo 500
2
2
11
3
104
TA8LE 8o6
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0C90 AND ACCEPT ACCURACY =0,60t
INSPECTION SAMPLE SIZE = 100
ALPHA ERROR =
ALPHA ERROR =
BETA
BETA
ERROR
ERROR
n a
O O
ti II
050
082
(NORMAL)
(CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
l CASE
AUDIT AUDIT
DEFECTIVES SAMPLE
SIZE
2
AUDIT
DEFECTIV
0,010
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Do 020
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,030
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,040
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0o050
LOT
SIZE
TOO
SMALL
LOT
SI ZE
TOO
SMALL
0,060
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 080
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,100
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,120
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 140
80
4
LOT
SIZE
TOO
SMALL
Oo 160
65
4
LOT
SIZE
TOO
SMALL
Oo 180
54
4
LOT
SIZE
TOO
SMALL
0,200
46
4
LOT
SIZE
TOO
SMALL
Oo 250
32
4
LOT
SIZE
TOO
SMALL
Oo 300
23
4
LOT
SIZE
TOO
SMALL
Oo 350
17
3
LOT
SIZE
TOO
SMALL
Oo 400
13
3
49
7
0o450
10
3
38
7
Oo 500
7
3
29
6
105
TABLE 8, 7
AUOIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT
ACCURACY =3o
INSPECTION
90 AND ACCEPT ACCURACY.
SAMPLE SIZE = 500
=0,
60
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
= 0,050 (NORMAL)
=0,082 (CAMP MEIDEL).
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
sample
SIZE
2
AUDIT
DEFECTIVI
OoOlO
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,020
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,030
411
4
LOT SIZE
TOO
SMALL
0 3 040
281
4
LOT SIZE
TOO
SMALL
Os 050
212
4
LOT SIZE
TOO
SMALL
Os 060
169
4
LOT SIZE
TOO
SMALL
Os 080
119
4
388
7
OsIOO
90
4
287
7
Os 120
72
4
225
7
0,140
59
4
183
6
Os 160
50
4
153
6
Oa 180
42
3
130
6
Oa 200
37
4
112
6
0,250
26
3
80
6
Os 300
20
3
60
6
0,350
15
3
46
6
Oo 400
11
3
35
5
Os 450
8
3
27
5
0,500
6
3
21
5
106
TABLE 80 8
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT
ACCURACY =0,
INSPECTION
90 AND ACCEPT ACCURACY =0,
SAMPLE SIZE = 1000
60
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
= 0,050 (NORMAL)
= 0, 082 (CAMP ME I DEL).
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE 2
AUDIT AUDIT
SAMPLE DEFECTIVES
SIZE
0o010
LOT SIZE
TOO SMALL
LOT SIZE TOO
SMALL
0,020
575
4
LOT SIZE TOO
SMALL
0,030
351
4
LOT SIZE TOO
SMALL
0,040
251
4
811
7
Do 050
194
4
610
7
Oo 060
157
4
486
7
0,080
112
4
341
6
Or, 100
86
4
259
6
0, 120
69
4
207
6
Oo 140
57
4
170
6
O0 160
48
3
143
6
Oo 180
41
3
122
6
Oo 200
36
3
106
6
Oo 250
26
3
77
6
Oo 300
19
3
58
6
0,350
15
3
44
5
0,40.0
11
3
34
5
Oo 450
8
3
26
5
0,500
6
3
20
5
107
TABLE 8,9
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY =0e90 AND ACCEPT ACCURACY =0,60t
INSPECTION SAMPLE SIZE = 5000
ALPHA ERROR = BETA ERROR = 0,050 (NORMAL)
ALPHA ERROR =
BETA ERROR
=0,082 (C
AMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
0 a 0 10
1022
4
3190
7
Oo 020
484
4
1437
6
0 a 030
314
4
918
6
Da 040
231
4
670
6
0 a 050
181
4
524
6
0 a 060
148
4
429
6
0,080
108
4
310
6
0,100
83
4
240
6
0,120
67
3
194
6
0, 140
56
3
161
6
0,160
47
3
136
6
0,180
40
3
117
6
0,200
35
3
102
6
0,250
25
3
74
6
0,300
19
3
56
5
0,350
14
3
43
5
0,400
11
3
33
5
0,450
8
3
25
5
0,500
6
3
19
5
108
TABLE 8910
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0o90 AND ACCEPT ACCURACY =0,60
INSPECTION SAMPLE SIZE = 10000
ALPHA ERROR
ALPHA ERROR
BETA ERROR =0,050 (NORMAL)
BETA ERROR =0,082 (CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE 1
AUDIT AUDIT
SAMPLE DEFECTIVES
SIZE
CASE 2
AUDIT AUDIT
sample defectives
SIZE
010
981
4
2909
020
474
4
1375
030
310
4
892
040
228
4
656
050
180
4
515
060
147
4
422
080
107
4
307
100
83
3
238
120
67
3
192
140
56
3
160
160
*7
3
135
180
40
3
116
200
35
3
101
250
25
3
74
300
19
3
56
350
14
3
43
400
11
3
33
450
8
3
25
500
6
3
19
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
109
TABLE 8011
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCJRACY =0o90 AND ACCEPT ACCURACY =0>75,
INSPECTION SAMPLE SIZE = 100
ALPHA ERROR
ALPHA ERROR
BETA ERROR = 03050 (NORMAL)
BETA ERROR =0o082 (CAMP MEIDEL)
08SERVED
CASE 1
CASE 2
INSPECTION
FRACTION
DEFECTIVE
AUDIT
SAMPLE
SIZE
AUDIT
DEFECTIVES
AUDIT
SAMPLE
SI ZE
AUDIT
DEFECTIV
OcOlO
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 020
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SHALL
0o030
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 040
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 050
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 060
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 080
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 100
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 120
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 140
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 160
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0 a 18 0
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0a200
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 250
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0a300
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 350
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 400
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0 o 450
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 500
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
110
TABLE 8o12
AUDIT SAMPLE SIZE AND MINIMUM NUMBER 3F DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,75,
INSPECTION
SAMPLE SIZE
= 500
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
=0,050 (NORMAL)
= 0o 082 (CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
l
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SIZE
2
AUDIT
DEFECTIVi
0,010
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 020
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 030
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,040
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 050
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Oo 060
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 080
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
OolOO
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,120
372
11
LOT SIZE
TOO
SMALL
O, 140
301
10
LOT SIZE
TOO
SMALL
0,160
251
10
LOT SIZE
TOO
SMALL
0,180
213
10
LOT SIZE
TOO
SMALL
0,200
183
10
378
15
0,250
132
9
270
15
Oo 300
99
9
201
14
0,350
76
9
155
14
0,400
59
9
121
13
0,450
46
8
95
13
0,500
36
8
74
12
Ill
TABLE 8013
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY =0e90 AND ACCEPT ACCURACY =0,75,
INSPECTION SAMPLE SIZE = 1000
ALPHA ERROR =. BETA ERROR =0,050 (NORMAL)
ALPHA ERROR = BETA ERROR =0,082 (CAMP ME I DEL)
OBSERVED CASE l CASE 2
INSPECTION
FRACTION
DEFECTIVE
AUDIT
SAMPLE
SIZE
AUDIT
DEFECTIVES
AUDIT
SAMPLE
SI ZE
AUDIT
DEFECTIV
0,010
LOT SIZE
TOO
SMALL
L3T SIZE
TOO
SMALL
0,020
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
Os 030
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
Oo 040
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
0,050
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
0o060
795
11
LOT SIZE
TOO
SMALL
0,080
554
10
LOT SIZE
TOO
SMALL
0, LOO
419
10
851
15
0, 120
334
10
670
15
0, 140
274
9
547
14
Os 160
231
9
458
14
0,180
198
9
391
14
Os 200
171
9
338
14
Oa 250
125
9
246
13
Os 300
94
9
186
13
Os 350
73
9
144
13
0,400
57
8
112
12
0o450
44
8
88
12
0,500
34
8
69
12
112
TABLE 8o14
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY =0C90 AND ACCEPT ACCURACY =0,75,
INSPECTION SAMPLE SIZE = 8000
ALPHA ERROR
ALPHA ERROR
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
0,010
0o020
Oa 030
Oo 040
Oo 050
Oa060
Oo 080
Oo 100
Oo 120
0 a 140
Oo 160
Oo 180
0,200
0,250
0a300
0a350
Oo 400
0,450
BETA ERROR = 0o050 (NORMAL)
BETA ERROR =0,082 (CAMP ME I DEL)
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
LOT SIZE
TOO SMALL
LOT SIZE
TOO SMALL
2307
10
4634
15
1465
10
2879
14
1066
9
2074
14
834
9
1612
14
681
9
1312
14
493
9
946
13
382
9
731
13
308
9
590
13
256
9
490
13
217
9
415
13
187
9
358
13
163
9
312
13
119
9
229
13
91
8
175
12
70
8
136
12
55
8
107
12
43
8
84
12
33
7
66
11
Oo 500
113
TABLE 8015
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER DF DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,75,
INSPECTION SAMPLE SIZE = 10000
ALPHA ERROR = BETA ERROR =0,050 (NORMAL)
ALPHA ERROR = BETA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDI T
sample
SIZE
2
AUDIT
DEFECTI
0,010
4666
10
9364
15
0,020
2185
9
4243
14
0,030
1414
9
2719
14
0,040
1038
9
1988
14
0,050
816
9
1558
13
0,060
669
9
1275
13
0,080
486
9
926
13
0,100
378
9
719
13
0,120
305
9
581
13
0,140
254
9
483
13
0,160
216
9
410
13
0, 180
186
9
354
13
0,200
162
9
309
13
O0 250
119
9
227
13
0,300
90
8
173
12
0,350
70
8
135
12
0,400
55
8
106
12
0,450
43
8
83
11
0,500
33
7
66
11
CHAPTER IX
CONCLUSION
Five possible measures of inspection accuracy were
investigated: the ratio of correct inspections, 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 have been reviewed.
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 was selected as the measure that should
be further examined. The ratio of defective product re
jected is determined by dividing the total number of de
fectives found by the inspector by the total number of de
fectives in the lot. During the actual production process
the only way to determine the total number of defectives in
the lot is by an audit reinspection of the lot to determine
how many defectives were missed by the original inspector.
Two types of sampling plans were derived. Single
hypothesis plans determine whether an acceptable inspection
accuracy is not being met, based on the number of defectives
114
115
found during the audit inspection. Doublehypothesis plans
determine whether a preselected acceptable inspection accur
acy level or a preselected unacceptable inspection accuracy
level is being attained. The double sampling plans require
that the audit sample size as well as the number of audit
defectives be stated in the sampling plan.
The sampling plans are affected by a major assumption
regarding the accuracy of the audit inspector. Two cases
are explored, one where the audit inspector is 100 percent
accurate and the other where the audit inspector accuracy is
equal to that of the initial inspector.
A study of the sampling plans reveals that large lot
sizes are generally required if an adequate measure of
inspector accuracy is to be determined. As the observed
inspection defect rate goes down, the lot size requirements
become fairly large. The audit sample size also must be
fairly large with respect to the original lot size as the
observed inspection defect rate goes down. For example, in
a doublehypothesis test with the acceptable accuracy of .90
and an unacceptable accuracy of .50, for an observed inspec
tion fraction defective of .01 the inspection sample size
should be 1000. Table 8.3 also shows that the audit sample
size should be 592 if a 100 percent accurate audit inspector
is assumed. For the case where the audit inspector has the
same accuracy as the initial inspector an even larger in
spection sample size would be required.
116
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
outgoing 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.
APPENDIX
oooonoooo
THIS PROGRAM CALCULATES ACCURACY EXPECTEO VALUES FOR VARIOUS
INSPECTION AND REINSPECTION TEST RESULTS. THE DATA CARD INCLUDES
THE FOLLCWING INFORMATION AND GAN RE MODIFIED FOR OTHER VALUES.
NI IS THE INSPECTION SAMPLE SI*E, NNI IS AN INDEX NUM8ER FOR NI IF
MORE THAN ONE VALUE IS USED PI IS THE INSPECTION FRACTION
DEFECTIVE, NPI IS ITS ASSOCIATED INDEX NUMBER NRFAC IS A
FACTOR TO DETERMINE THE AUDIT SAMPLE SIZE AND IS DIVIDED INTO NI,
NNR IS ITS ASSOCIATED INDEX NUMBER.
REAL NI!2),PI(4),NRFAC(5) NR,MQL1,MQL2
DATA NI/1.0E2, 1.0E3/, PI/.Ol, .05,.10,.25/,NRFAC/1.0,2.0,5.O,I 0.0,
1100.0/,NNI/2/,NPI/4/,NNR/5/
ITAB =L
DO 100 INI = 1,NNI
DO 100 IPI = 1,NP I
KOUNT =0
KONTIN=0
IT AB=IT AB + 1
J N I =. N I ( I N I )
WRITE!6,530)ITAB
WRITE!6,500)JNI,PI(IPI)
WRITE!6,505)
DO 100 I NR = 1,NNR
IF! INR.NE.1) GO TO 130
NR=NI( INI )*(1.0PI( IPI))
GO TO 140
130 NR=NI(I NI)/NRFAC!I NR)
WRITE !6,560)
KOUNT =KOUNT + 1
140 IF INI(INI)*(1.0PI(IPI) ).LT.NR) GO TO 110
SADR=1.0
DR=0
160 IF(DR.GE.IO) GO TO 200
DR=DR+1
GO TO 220
200 IF!DR.GE.50) GO TO 210
oo
DR=DR+5
GO TO 220
210 DR =DR +1 0
220 PR=DR/NR
J DR = DR
JNR = NR
IF(PR.GT.l.O) GO TO 110
T 1 = ( 1.0PI(IPI ))*PR
T2= PI( IPI J+Tl
T3= PI{ IPI JTl
RAD=SQRT((1.0/NI!INI)*PI
ADR 1 =P I( IPI )/T2
IF(DR.LE.l.O) GO TO 150
IF ISADRADR 1 LT.0.01) GO TO 160
IF(ADR 1 .GT..90) GO TO 150
IF(SADRADR i .LT.0.05JG0 TO 1*0
150 SADR =ADR 1
IF(ADR 1 .LT.0.25) GO TO 110
KOUNT =K0UNT+1
IFIKONTIN.EQ.O.AND.KOUNT.LT.271 GO TO 180
IF(KONTIN.GT.O.AND.KOUNT.LT.311 GO TO 180
WRITE!6 540) I TAB
WRITE!6,505)
K0NTIN=K0NTIN+1
KOUNT = 0
180 ADR 2 =T3/PI(IPI )
SIG1 =((PI(IPI)*T1>/
MQL1 = 10T 2
OQL1=1.0{NI(INI)*TlDR)/(NI(INi)*ci.oPI(IPI))DR)
KADR2=1000*ADR 2
KPI = 1000*PI
IFtKADR2.LE.KPI ) GO TO 170
SIG2 =(T1/PI( IPI) )*RAD
MQL2=1.OPIIIPI)Tl/ADR 2
00L2=1.0!NI(INI)*T1DR*ADR 2)/( !NI!IN I)*(l.OPIIPI))DR)*ADR 2)
WRITE!6,510)JNR,PR ,ADR 1,SIG1,MOL 1,00L1,ADR 2,SIG2,MQL2,0QL2
119
GO TO 160
170 WRITE( 6 520)JNRPR ADR 1,SIG1 MQL 1 OQL l
GO TO 160
500 FORMAT2CX,'EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIF
1FERENT INSPECTION AND',/,29X,
2REINSPECT ION TEST RESULTS INSPECTION SAMPLE SIZE =,I5,*,,
3/,37X,INSPECTION FRACTION DEFECTIVE =,F5.2,/)
505 FORMAT!10X,AUDIT*,2X,'AUDIT*,7X,* CASE 1  
1 6X * CASE 2  ,/,10X, SAMPLE FRACTION',/,
210X,'SIZE DEFECTIVE ACCURACY STD.DEV MQL 0QL,6X,
3 ACCURACY STD.DEV. MQL' ^QL,/>
510 FORMAT(10X,15,F9.4 2X, 4F9.4,2X,4F9.4)
520 FORMAT(10X,15,F9.4 ,2X,4F9.413X,INVALID FOR CASE 2)
530 FORMAT!IH1,//////,50X,TABLE 5.',II,/)
540 FORMAT!1H1,//////,45X,TABLE 5.,11' CONTINUED',/)
560 FORMAT(IX, )
110 CONTINUE
100 CONTINUE
STOP
END
ooooooooooooo
THIS PROGRAM DETERMINES THE MINIMUM NUMDER OF DEFECTIVES TO BE
FOUND BV THE AUDITOR TO REJECT A SPECIFIED VALUE OF INSPECTION
ACCURACY,, THIS IS A SINGLE HYPOTHESIS TEST, THE DATA CARD
INCLUDES THE FOLLOWING INFORMATION AND CAN BE MODIFIED FOR OTHER
VALUES,
NI IS THE INSPECTION SAMPLE SIZE, NNI IS AN INDEX NUMBER FOR NI IF
MORE THAN ONE VALUE IS USED"PI IS THE INSPECTION FRACTION
DEFECTIVE, NPI IS ITS ASSOCIATED INDEX NUMBER NRFAC IS A
FACTOR TO DETERMINE THE AUDIT SAMPLE SIZE AND IS DIVIDED INTO NI,
NNR IS ITS ASSOCIATED INDEX NJMBER,
ALNORM IS THE ALPHA ERROR DESIRED ASSUMING A NORMAL DISTRIBUTION
AND CONST IS THE ASSOCIATED NORMALIZED Z STATISTIC,
DIMENSION NI(4),PI(19),NRFAC(4)
INTEGER DR175,DR190,DR275,3*290
DATA NI/53D,IDCO,5003,10000/
DATA PI/oDl,,02,,03,,04,305,,36,608,<>10,,12,>l4,>16,13f,23,,25,,3
10,o35,,40,,45,,50/
DATA NRFAC/2,5,10,50/
DATA NN 1/4/,NP1/19/,NNR/4/,CONCT/1,645/,ALNORM/,35/
IT AB= 3
DO 613 INI = 1,NNI
DO 613 INR = 1,NNR
133 NR = NI( INII/NRFACI INR)
IT AB=IT AB+l
IF(ITAB,LT,13) WRITE (6,910)IT*B
I F ( ITABoGE, 10) WRITE (6,920)ITAB
ALCAMP = 10 0 / ( 2, 25<=C0NST**2)/2, 3
WRITE(6,810) Nil INI I ,NR,ALNORM,ALCAMP
DO 613 IPI = 1,NPI
NRUN =1
113 IF(IPI,EQ,1) DR = l
IF{NRJN,E381,AND,IPIaNEol)DR=3190
IF(NRJN,E3o2,AND,IP I,NEo1)DR = 3?175
IF(NRJN,EQ,3aAND,IPI,NEol)DR=3290
IF(NRJN,EQ,4,AND0IPI,NEal)DR=D>275
121
122
i + rum = NnN
d0=061d0 C02
OIT 01 00
1+ NP.dN = NPbN
I=06IdQ C6T
021 01 00
002 01 00 li6coil'V I 0IS*1SI\03 + V 1 vavidi
061 oi as
011 01 00
1+ NOUN = NOUN
d(] = 511ba C91
011 01 00
1+ NPdN = NPdN
1=511bO cn
021 01 00
081 01 00 (i.leoneV 1 0IS*1SN03 + V 1 dOVIdl
on oi os nÃ‚Â¡Niscya)di coi
051 01 00 <^c035NrbN)dI
091 01 00 (Ã‚Â£cC3cNrbN)dI
001 01 00 (2 C3 eNfdN)dI
OM 01 CO (!c03cNrbN)dI
VCVd*(( Idl ) Id/VIl )=V 2 OIS
vavb*((v21*V21)/(VIi*(IdI )Id) )=V 1 OIS
((vdd*bN) / mdi )( (((i
(idi) id/vei =v 2 yav
V21 / ( Idl ) 1 <5 = V 1 dav
Vll(Idl ) ld=VÃ‚Â£i
V11+( Idl ) Id =V 2 1
Vdd*( ( Idl )IdI )=V11
dN/dO =Vdd
dN=dNV
009 01 00 ldNclle( < Idl )IdOeI )*( IM )IN)dI CS2
I*dO = da C2I
052 01 00
I = da (0o000Ie3Tc(INI )IN)dI
i = ba (crnfdO)di
Aa L To o 9) GO TO 240
AaLTaa75) GO TO 220
GO TO 110
160 IF(DR9GToNR) GO TO 230
IF(ADR 2 A + CON ST*S IG 2
GO TO 120
230 DR290 = 1
NRUN = NRUN +1
GO TO no
240 DR290=DR
NRUN = NRUN +1
GO TO 110
150 I F ( DR a GTo NR ) GO TO 210
IF(ADR 2 A*CONST*SIG 2
GO TO 120
210 DR275="1
GO TO 580
220 DR?75 = DR
580 IF(0R190oLTaQaANDaDR290aLTi0) WRITE(6,820) PI(IPI)
IF(DR175oLTa0a ANDaDR190>a0aAND:>DR290aLTa0) WRITE(6,830) PKIPI),
1 DR 190
[F(DR175oGT00,AND3DR190aGT500AND3DR2903LTa0) WRITE(6,840) PKIPI)
1 DRL75DR190
lF(DR190oLTa0,AND,DR2750LTsO0ANDaDR290aGTa0) WRITE(6,850) PI(IPI),
1 DR290
IF(DR190cLTo0sANDaDR275oGTo0&ANDaDR290aGTa0) WRITE(6,850) PKIPI),
1NR,DR275,DR290
IF(DR175oLTa0> AND,DR 190oGTa0aANDaDR275,LTa0,AND,DR290,ST,0) WRITE
1(6,870) PKIPI), DR190,DR290
I F(DR 17 5o GTa 0aAND,DRl90aGTa0aAND,DR275,LTa0AND, DR2 90aGTa0) WRITE
1(6,880) PI(IPI), DR175,DR190,DR290
IF(DR17 5oLT,0,AND,DR190aGTaOaAND,DR275,GTa0aANO,DR290a GTa 0) WRITE
1( 6,890) PKIPI), DR190DR275DR290
IF(DR175.aGTo0aANDaDR190aGTa0oANDaDR275,GTa0aAN0, 0R29 0aGTa 0)
IWRITE( 6,900) PKIPI) D R 175, OR 190, DR275 ,DR290
810 FQRMAK17X,'MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE*,/,
116X,AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,*,/*
225X,INSPECTION SAMPLE SIZE =*,15,*,*,/,
123
328X,AJDlT SAMPLE SIZE =',I4,///,
4L5X,'ALPHA ERROR = F 5=, 3, 2 X ( NORMAL ) / ,
515X,'ALPHA E K O R = F 5, 3, 2X ( C AMP ME I DE L) //
615X,OBSERVED',13X,'CASE l',19X,'CASE 2' /15X ' INSPECTION' /
715X,FRACTION',7X,'ACCURACY' 2 X ,'ACCURACY', 7X ACCURACY*, 2X,
81 ACCURACY ',/,
915X,'DEFECTIVE',8X,'=0375',5X,'=0>90',10X,'=0375',5X,'=0>93',/)
823 FORMAT!17X,F53 3,8X, 'NO TEST',3X,'NO TEST',8X,'MO TEST',3Xt
l'MO TEST',/)
833 FORMAT!L7X,F5,3,8X, 'MO TEST*,3X,I 7,8X,NO TEST',3X'NO TEST',/)
843 FORMAT!17X,F533,8X,I7,3X,I7,8X,'NO TEST',3X,'NO TEST',/)
853 FORMAT(17XF5338X, 'NO TEST',3X,'NO TEST',8X,'NO TEST 3X I 7 / )
863 FORMAT!17X,F53 3,8X, 'NO TEST',3X,'NO TEST*,8X,I 7,3X,I 7,/)
873 FORMAT!17X,F53 3,8X,'NO TEST,3X,I 7,8X,NO TEST,3X,I 7,/)
883 FORMAT!17X,F533,8X,I7,3X,17,8X,'NO TEST,3X I 7,/)
893 FORMAT!17X,F533,8X,'NO TEST,3X,I 7,8X,I 7,3X,I 7,/)
903 FORMAT!17X,F533,8X,I7,3X,I7,8X,I7,3XI7,/)
913 FORMAT (1H1,////,35X,'TABLE 7',II,/)
923 FORMAT 11H1,////,35X,'TABLE 7,',12,/i
603 CONTINJE
613 CONTINJE
STOP
END
oooooooonooooooo
THIS PROGRAM DETERMINES AUDIT SAMPLE SIZES AND THE MINIMUM NUMBER
OF DEFECTIVES NECESSARY TO REJ rC T THE NULL HYPOTHESIS FOR A
DOUBLE HYPOTHESIS TEST ON INSPECTION ACCURACY. THE NULL
HYPOTHESIS IS EQUAL TO AN ACCEPTABLE VALUE OF INSPECTION ACCURACY.
THE ALTERNATE HYPOTHESIS IS EQUAL TO AN UNACCEPTABLE VALUE OF
INSPECTION ACCURACY. THE DATA CARD INCLUDES THE FOLLOWING
INFORMATION AND CAN BE MODIFIED FOR OTHER VALUES.
NI IS THE INSPECTION SAMPLE SIZE, NNI IS AN INDEX NUMBER FOR NI IF
MORE THAN ONE VALUE IS USED PI IS THE INSPECTION FRACTION
DEFECTIVE, NPI IS ITS ASSOCIATED INDEX NUMBER.
ADRAC IS THE VALUE OF ACCEPTABLE INSPECTION ACCURACY.
ADRUN ARE THE VALUES OF UNACCEPTABLE INSPECTION ACCURACY, NUN
IS ITS INDEX NUMBER.
ALNORM IS THE ALPHA ERROR DESIRED AS9UMING A NORMAL DISTRIBUTION
AND CONST IS THE ASSOCIATED NORMALIZED Z STATISTIC.
DIMENSION NI(5),PI(19),ADRU{3)
DATA NI/lOO,500,1000,5000,lOOOO /,PI/.01,.02,.03,.04,.05,.06,
1.08,.10,.12,.14,.16,.18,.20,.25,.30,.35,.40,.45,.50/,NNI/5/,NPI/19
2/,CONST/1.645/,ALNORM/.05/,ADRU/.50,.60,.75/,ADR AC/.90/,NUN/3/
500 FORMAT!19X,F5.3,8X,16 ,5X,I6 ,5X,I6 ,5X,I6,/)
510 FORMAT!19X,F5.3,8X,'LOT SIZE TOO SMALL',4X,'LOT SIZE TOO SMALL',/)
520 FORMAT!19X,F5.3,8X,'LOT SIZE TOO SMALL',4X,16 ,5X,I6 ,/)
530 FORMAT! 19X.F5.3,8X,16 ,5X,I6 ,5X,'L0T SIZE TOO SMALL',/)
540 FORMAT! 19X,AU0IT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES
1T0*,/,22X,'REJECT ACCURACY =,F4.2, 1 X,AND ACCEPT ACCURACY =',
2F4.2,','/,30X,INSPECTION SAMPLE SIZE = I 6 ,///,17X,
3 ALPHA ERROR = BETA ERROR = ,F5.3,2X,!NORMAL) ',/, 17X,ALPHA ERROR
4 = BETA ERROR = ,F5.3,2X, {CAMn MEIDEL) ,//,
517X,'OBSERVED',12X,
6 CASE l',16X,"CASE 2, /,17X,'INSPECTION',5X,'AUDIT',6X,'AUDIT',
7 6X 'AUDIT' 7XAUDIT,/,17X,FRACTI ON,7X,'SAMPLE* ,3X,'DEFE
8CT I VES',3X,'SAMPLE',4X,* DEFECTI VES,/,17X,DEFECTIVE,6X,SIZE,18
9X 'SIZE*,/)
550 FORMAT!1H1,////,40X,'TABLE 8.',II,/)
560 FORMATl1H1,////,40X,'TABLE 8.',12,/)
125
ALCAMP=1.0/(2.25*C0NST**2)/2.0
IT AB = 0
00 610 IUN=1 NUN
ADRUN=ADRU(IUN)
DO 610 INI = 1tNNI
IT A8 = IT A8+1
IF!ITAB.LT.10) WRITE<6,550) ITAB
I F( ITAB.GE.10) WRITE(6 560) ITAB
WRI T E ( 6 540) ADRACtADRUN,NI(INT),ALNQRM,ALCAMP
DO 610 IPI = 1 NPI
NRUN = 0
100 NRUN = NRUN + 1
IF(NRUN.EQ.3) GO TO 600
NR=0
110 NR=NR+1
DEC ={ADRAC ADRUN )/CONST
IF(NRUN.EQ.2) GO TO 120
IFCNR.GT.NKINI)*<1PI(IPI)) ) r0 TO 130
PRUN = (PI( IPI) ( 1ADRUN ) )/(ADRUN *(1PI(I PI ) ) )
PRAC = (PI(IP I)*{1ADRAC ))/(ADRAC *(1P I (IPI ) ) )
SIGUN =((PRUN *PIIPI)*(1PI(IPI)))/ ((PI(IPI)+PRUN (1PKIPI)
1 ) ) **2 ) )*SQRT ( 1/ ( N I ( INI)*PI(If>I ) =M 1PI ( IPI ) ) ) + ( 1PRUN) / (NR* PRUN) )
SIGAC =( (PRAC *PI ( IPI )*(1PICIPI)>)/ ((PI(IPI)+ PRAC *( 1PI( I P I )
1) )**2) )* SORT( l/(NI(INI)*PI(II)*(lPI(IPI))) + (1PRAC)/(NR*PRAC) )
IF(SI GUN +SIGAC .GT.DEC) GO TO 110
NRDEC=NR
DR1 = NRDEC*PI(IPI)*(1.0ADRAC+CHNST*SIGAC)/( (ADRACCONST*SIGAC)*
1( l.OPK IPI ) ) ) +1.0
I DR 1 = DR 1
GO TO 100
120 IF(NR.GT.NI(INI)*(1PI(IPI)J) r0 TO 140
PRUN =(PI(IPI )*(1ADRUN ) )/( 1PI(IPI ) )
PRAC =(PI (IPI )*(1ADRAC ) )/( 1P I ( I P I ) )
SIGUN =(PRUN *(1PI(IPI))/PI(IPI))SQRT(1/(N I ( IN I)*PI(I PI)*( 1PI
1C IPI)))(1PRUN )/(NR* PRUN ))
SIGAC =(PRAC *(1PI(IPI))/PICIPI))*SQRT(l/(NIC I NI)*PI
126
11IPI)>)+(1PRAC )/
IFSIGUN +SIGAC GT.DEC) GO TO HO
NRDECB=NR
DR2=NRDECB*PI( IPI )*(1.0 + CONST#MGACADRAC)/I 1.0PI{IPIJl + 1.0
IDR2=DR2
GO TO 430
130 NRDEC=10
GO TO 100
140 NRDECB=20
430 IF(NRDEC.LT.0.0.AND.NRDECB.LT.^.0) GO TO 400
IF(NRDEC.LT.O.O.AND.NRDECB.GT.rt.O) GO TO 410
IFtNRDEC.GT.O.O.AND.NRDECB.LT.^.O) GO TO 420
WRITE(6500)PI(IPI).NRDECIDR1 NRDECB I DR2
GO TO 60C
400 WRITE(6,510) P I ( IPI )
GO TO 600
410 WRITE(6 520) PI(IPI),NRDECB,ID2
GO TO 600
420 WRITE(6,530) P I ( IPI ) NRDEC*I DR 1
600 CONTINUE
610 CONTINUE
STOP
END
127
REFERENCES
Periodicals
1. Ayers, A. W., "A Comparison of Certain Visual Factors
with the Efficiency of Textile Inspectors," Journal
of Applied Psychology, Vol. 26, No. 6, Dec. 1942,
pp. 812827.
2. Belbin, R. M., "New Fields for Quality Control," Metal
Industry, Vol. 90, No. 4, Jan. 1957, pp. 6365.
3. Blais, R. A., "True Reliability Versus Inspection Ef
ficiency," IEEE Transactions on Reliability, Vol.
R18, No. 4, Nov. 1969, pp. 201203.
4. ForsterCooper, J., "A Description of the Statistical
Quality Control System in Operation at Joseph Lucas
(Electrical) LTD," Engineering Inspection, Vol. 18,
No. 2, Feb. 1954, pp. 110121.
5. Harris, D. H., "Effect of Defect Rate on Inspection
Accuracy," Journal of Applied Psychology, Vol. 52,
No. 5, Oct. 1968, pp. 377379.
6. Harris, D. H., "Effect of Equipment Complexity on In
spection Performance," Journal of Applied Psychology,
Vol. 50, No. 3, June 1966, pp. 236237.
7. Hayes, A. S., "Control of Visual Inspection," Industrial
Quality Control, Vol. 6, No. 6, May 1950, pp. 7376.
8. Jacobson, H. J., "A Study of Inspector Accuracy," In
dustrial Quality Control, Vol. 9, No. 2, Sept. 1952,
pp. 1625.
9. Juran, J. M., "Inspectors Errors in Quality Control,"
Mechanical Engineering, Vol. 57, No. 10, Oct. 1935,
pp. 643644.
10.Lawsche, C. H., and Tiffin, J., "The Accuracy of Pre
cision Instrument Measurement in Industrial In
spection," Journal of Applied Psychology, Vol. 29,
No. 6, Dec. 1945, pp. 413419.
128
129
11. Mackworth, N. H., "Work Design and Training for Future
Industrial Skills," Institute of Production En
gineers Journal, Vol. 35, No. 4, April 1956, pp.
214228.
12. Marash, S. A., "Performing Quality Audits," Industrial
Quality Control, Vol. 22, No. 7, Jan. 1966, pp.
342347.
13. Marien, L. P., "Inspecting the Inspector," American
Machinist, Vol. 74, No. 22, May 1931, pp. 815817.
14. McKenzie, R. M., "On the Accuracy of Inspectors,"
Ergonomics, Vol. 1, No. 3, May 1958, pp. 258272.
15. Purcell, W. R., "Sampling Techniques in Quality System
Audits," Quality Progress, Vol. 1, No. 10, Oct.
1968, pp. 1315.
16. Schwartz, D. H., "Statistical Sleuthing to Detect Bias
in Visual Inspection," Industrial Quality Control,
Vol. 3, No. 6, May 1947, pp. 1417.
17. Tiffin, J., and Rogers, H. B., "The Selection and
Training of Inspectors," Personnel, Vol. 18, No.
1, July 1941, pp. 1431.
Books
18. Burr, I. W.', Engineering Statistics and Quality Con
trol McGrawHill, New York, 1953.
19. Duncan, A. J., Quality Control and Industrial Statis
tics, 3rd Edition, Richard D. Irwin, Homewood, Ill.,
1965.
20. Feigenbaum, A. V., Total Quality Control, McGrawHill,
New York, 1961.
21. Grant, E. L., Statistical Quality Control, 3rd Edition,
McGrawHill, New York, 1964.
22. Juran, J. M., Ed., Quality Control Handbook, McGraw
Hill, New York, 1951.
23. Juran, J. M., and Gryna, F. M., Quality Planning and
Analysis, McGrawHill, New York, 1970.
130
24. Kelly, M. L., A Study of Industrial Inspection by the
Method of Paired Comparisons, Psychological Mono
graphs, No. 394, American Psychological Association,
Washington, D. C., 1955.
25. Kendall, M. G., and Stuart, A., Advanced Theory of
Statistics, Vol. 1, Hafner, New York, 1963.
26. Kennedy, W., Inspection and Gaging, The Industrial
Press, New York, 1957.
27. Shewhart, W. A., Economic Control of Quality of Manu
factured Product, Van Nostrand, New York, 1 $ 31.
Transactions and Proceedings
28.Kidwell, J. L., Squeglia, N. L., and Lavender, H. J.,
"Escape Probability as a Design Consideration,"
Annual Technical Conference Transactions, American
Society for Quality Control, 1965, pp. 486499.
BIOGRAPHICAL SKETCH
Lee Allen Weaver was born April 28, 1935, at Hellertown,
Pennsylvania. In June, 1953, he was graduated from Allen
town High School, Allentown, Pennsylvania. In June, 1957,
he received the degree of Bachelor of Arts from Moravian
College. In September, 1957, he enrolled in the graduate
school of the University of Florida. He worked as a graduate
assistant in the Department of Mathematics until June, 1959,
when he received the Master of Science. From 1960 until
1967 he worked as a Product Assurance engineer for Honeywell
Inc. In 1967 he joined the faculty of the University of
South Florida. From June, 1968, until the present time he
has pursued his work toward the degree of Doctor of
Philosophy.
Lee Allen Weaver is married to the former LaRae Kathryn
Fritzinger and is the father of four children. He is a
member of the American Society for Quality Control, and the
Institute of Electronic and Electrical Engineers. He is
a statistical consultant to a Working Group of the Interna
tional Electrotechnical Commission and to the Director of
Product Assurance of Honeywell Inc.
131
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Warren Menke, gnaapman
Associate Professor of Management
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
k UJ
Ralph Blodgett ]
Professor of Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
r />
oC
Elmo Jackson
Professor of Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
V / >'
s')
jL
Ralph Kimbrough
Professor of Education
This dissertation was submitted to the Department of Manage
ment in the College of Business Administration and to the
Graduate Council, and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
December, 1972
Dean, Graduate School
24
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
= 51
ADR (new method)
51 + 5
.911
LIST OF TABLES (CONTINUED)
TABLE
Paqe
8.11
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE 100
109
8.12
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE 500
110
8.13
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE 1000
111
8.14
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 5000
112
8.15
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.75, INSPECTION SAMPLE
SIZE = 10000
113
x
40
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
NI
GI
1PI *
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 noreplacement 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:
I
5
!
Ã‚Â¡y (ADR)Ka(ADR)
I
f
!
S
3 M
i /
y
V
/
i /
\
1/
3
y (ADR)
Figure 7.1
Graphical Presentation of a Single Hypothesis
Statistical Test for Inspection Accuracy
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."
4
TABLE 4.1
SUMMARY OF EXPECTED VALUE FUNCTIONS
Expected
Value
CASE 1
Auditor Accuracy = 100%
CASE 2
Auditor Accuracy = Inspection Accuracy
PI
PI (1PI)PR
U (ADR)
PI + (1PI)PR
PI
a (ADR)
(PR) (PI) (1PI) / 1 1PR
PR(1PI) / 1 1PR
[PI+PR(1PI)]2 / (NI) (PI) (1PI) (NR) (PR)
PI / (NI) (PI) (1PI) (NR) (PR)
U(MQL)
1PR(1PI)PR
1.PT (IPI)PR
x ADR
U(IOQL)
1PR
1 PR
ADR
V(OQL)
(NI) (PR) (1PI)(NR) (PR)
(NI) (PR) (1PI)(NR) (PR) (ADR)
1 (NI) (1PI) (NR) (PR)
1 "ADR [NI (1PI)(NR) (PR)]
11
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
TABLE 5s 7
AUDIT
AUDIT
1 .
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
ST Do DEVo
MQL
100
Os 0600
Oo 847 5
Os 0520
Os 7050
100
Os 0900
Oo 7 8 74
Os 0546
Os 6825
100
Os 1500
0 o 6 3 9 7
Os 0533
Os 63 7 5
100
0,2000
Oo 6250
Os 0499
Os 6000
100
0 o 2 5 0 0
Oo 5 714
Os 0460
Os 5625
100
Os 3500
Oo 4878
Os 0386
Oo 4875
100
Os 4500
Oo 4255
Os 032 4
Os 4125
100
Os 6000
Oo 3 5 7 1
Os 0252
Os 3000
100
Os 8000
Oo 2 941
Os 0184
Op 1500
10
Os 1000
0o 7692
Os 1689
Os 6750
10
Os 2000
Oo 6250
Os 1492
Os 6000
10
Os 3000
Oo 5263
Os 1218
Os 5250
10
Os 4000
Oo 4545
Os 0977
Oo 4500
10
Os 5000
Oo 4000
Os 0779
Oo 3750
10
Os 700.0
Oo 3226
Os 0480
Os 2250
10
Os 9000
Oo 2 703
Os 0253
Oo 0750
3NTINUED
OQL
ACCURACY
STD,OEV
, MOL
OQL
Os 9476
0, 8200
0,0724
3 >
6951
0,9343
Os 9211
0 7300
0,0881
0,
6575
Os 8874
Oo 8673
Os 5500
O,1120
0,
5455
Oo 7421
Oo 8219
Os 4000
0,1277
0,
3 750
0,5137
Oo 7759
INVALID
FDR
CASE
2
Oo 6818
INVALID
FDR
CASE
2
Os 5851
INVALID
FOR
CASE
2
Oo 4348
INVALID
FOR
CASE
2
Os 2239
INVALID
FOR
CASE
2
*59012
0,7000
0,2854
0,
5429
0,B583
*s 8021
0,4000
0,3820
3 D
3750
0,5013
*b 7028
INVALID
FDR
CASE
2
*s 6032
INVALID
FOR
CASE
2
*s 5034
INVALID
FOR
CASE
2
*s 3028
INVALID
FOR
CASE
2
*s1012
INVALID
FOR
CASE
2
109
TABLE 8011
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCJRACY =0o90 AND ACCEPT ACCURACY =0>75,
INSPECTION SAMPLE SIZE = 100
ALPHA ERROR
ALPHA ERROR
BETA ERROR = 03050 (NORMAL)
BETA ERROR =0o082 (CAMP MEIDEL)
08SERVED
CASE 1
CASE 2
INSPECTION
FRACTION
DEFECTIVE
AUDIT
SAMPLE
SIZE
AUDIT
DEFECTIVES
AUDIT
SAMPLE
SI ZE
AUDIT
DEFECTIV
OcOlO
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 020
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SHALL
0o030
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 040
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 050
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 060
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 080
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 100
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 120
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 140
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 160
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0 a 18 0
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0a200
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 250
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0a300
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 350
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 400
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0 o 450
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oa 500
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
42
CUSTOMER
PI =
.200
NR = 160
PR =
.125
NI = 1000
ADR =
.667
[= Number of true good units
CD = Number of true bad units
Figure 4.1
Sample Production Flow for Case 1 with No Replacement,
Audit Accuracy = 100 Percent
79
TABLE 7,2
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE =, 100
ALPHA ERROR =0,050
ALPHA ERROR =0,082
(NORMAL)
(CAMP ME I DEL)
OBSERVED
CASE
1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
Oj 75
= 0,90
0,010
2
2
7
7
0,020
3
2
5
5
0,030
4
3
5
4
0,040
4
3
6
4
0,050
5
3
6
5
0,060
6
3
6
5
0,080
7
4
7
5
0,100
8
4
8
5
0, 120
9
5
9
6
0, 140
10
5
10
6
0,160
12
6
11
6
0, 180
13
6
12
7
0,200
14
7
13
7
0,250
18
8
15
8
0,300
21
10
18
10
0,350
26
11
21
11
0,400
31
13
25
13
0,450
37
15
30
15
0,500
43
18
35
17
31
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 reenter 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.
102
TABLE 8 4
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCJRACV = 0,90 AND ACCEPT ACCURACY =0,50,
INSPECTION SAMPLE SIZE = 5000
ALPHA ERROR = BETA ERROR =0,050 (NORMAL)
ALPHA ERROR =. BETA ERROR =0,082 (CAMP MEIDEL),
OBSERVED
CASE
1
CASE
2
INSPECTION
AUDIT
AUDIT
AUDIT
AUDIT
FRACTION
DEFECTIVE
SAMPLE
SIZE
DEFECTIVES
SAMPLE
SIZE
DEFECTIVES
0,010
474
3
2017
5
0,020
228
3
924
5
0,030
148
3
594
5
0,040
109
3
434
5
O, 050
86
3
340
4
0,060
70
3
278
4
0,080
51
2
201
4
0, 100
39
2
156
4
0,120
32
2
125
4
0, 140
26
2
104
4
0, 160
22
2
88
4
0,180
19
2
75
4
0,200
16
2
65
4
0,250
12
2
47
4
0,300
9
2
35
4
0,350
6
2
27
4
0,400
5
2
20
4
0,450
3
2
15
3
0,500
2
2
11
3
91
TABLE 7,14
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE =2000
ALPHA ERRDR =0o050 (NORMAL)
ALPHA ERROR =0o082 (CAMP ME I DEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0o 90
ACCURACY
= 0, 75
ACCURACY
=0 j 90
OoOlO
12
6
11
7
Oo 020
21
10
18
10
Oa 030
29
13
24
12
0 040
38
16
31
15
Oo 050
46
19
37
18
Oo 060
55
22
43
21
Oo 080
72
28
57
26
OolOO
90
35
70
32
Oo 120
108
41
84
38
Oo 140
127
48
98
44
Oo 160
147
55
113
50
Oo L80
168
62
129
57
Oo 200
189
69
145
64
Oo 250
248
90
190
82
Oo 300
315
113
240
103
Oo 350
391
139
298
127
Oo 400
480
170
365
154
0,450
584
205
443
186
Oo 500
709
248
538
225
116
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
outgoing 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.
108
TABLE 8910
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0o90 AND ACCEPT ACCURACY =0,60
INSPECTION SAMPLE SIZE = 10000
ALPHA ERROR
ALPHA ERROR
BETA ERROR =0,050 (NORMAL)
BETA ERROR =0,082 (CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE 1
AUDIT AUDIT
SAMPLE DEFECTIVES
SIZE
CASE 2
AUDIT AUDIT
sample defectives
SIZE
010
981
4
2909
020
474
4
1375
030
310
4
892
040
228
4
656
050
180
4
515
060
147
4
422
080
107
4
307
100
83
3
238
120
67
3
192
140
56
3
160
160
*7
3
135
180
40
3
116
200
35
3
101
250
25
3
74
300
19
3
56
350
14
3
43
400
11
3
33
450
8
3
25
500
6
3
19
6
6
6
6
6
6
6
6
6
6
6
6
6
6
5
5
5
5
5
34
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 a2 (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) =
NI
and
a2(PR) = PRdPR).
NR
The partial derivatives of ADR are given as
9ADR [PI + PR(1PI)] (1) PI(lPI)
8PI [PI + PR(1PI)]2
= PR
[PI + PR (1PI) ] 2 '
and
3 ADR [PI + PR(lPI)] (O)PI(lPI)
3 PR [PI + PR (1PI) ]
PI (1PI)
[PI + PR(1PI)]2 '
Combining the above terms we obtain
a (ADR) =
(PR) (PI) (1PI)
[PI + PR (1PI) ] 2
(NI)(PI)(1PI)
+
1PR
(NR) (PR) *
To determine the expected value of the manufacturing
quality level we apply the following modification to the
equation given in Chapter III.
17
each batch. But these were made of steel and brassplated:
thus missorts 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.
Ill
TABLE 8013
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY =0e90 AND ACCEPT ACCURACY =0,75,
INSPECTION SAMPLE SIZE = 1000
ALPHA ERROR =. BETA ERROR =0,050 (NORMAL)
ALPHA ERROR = BETA ERROR =0,082 (CAMP ME I DEL)
OBSERVED CASE l CASE 2
INSPECTION
FRACTION
DEFECTIVE
AUDIT
SAMPLE
SIZE
AUDIT
DEFECTIVES
AUDIT
SAMPLE
SI ZE
AUDIT
DEFECTIV
0,010
LOT SIZE
TOO
SMALL
L3T SIZE
TOO
SMALL
0,020
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
Os 030
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
Oo 040
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
0,050
LOT SIZE
TOO
SMALL
LOT SIZE
TOO
SMALL
0o060
795
11
LOT SIZE
TOO
SMALL
0,080
554
10
LOT SIZE
TOO
SMALL
0, LOO
419
10
851
15
0, 120
334
10
670
15
0, 140
274
9
547
14
Os 160
231
9
458
14
0,180
198
9
391
14
Os 200
171
9
338
14
Oa 250
125
9
246
13
Os 300
94
9
186
13
Os 350
73
9
144
13
0,400
57
8
112
12
0o450
44
8
88
12
0,500
34
8
69
12
122
i + rum = NnN
d0=061d0 C02
OIT 01 00
1+ NP.dN = NPbN
I=06IdQ C6T
021 01 00
002 01 00 li6coil'V I 0IS*1SI\03 + V 1 vavidi
061 oi as
011 01 00
1+ NOUN = NOUN
d(] = 511ba C91
011 01 00
1+ NPdN = NPdN
1=511bO cn
021 01 00
081 01 00 (i.leoneV 1 0IS*1SN03 + V 1 dOVIdl
on oi os nÃ‚Â¡Niscya)di coi
051 01 00 <^c035NrbN)dI
091 01 00 (Ã‚Â£cC3cNrbN)dI
001 01 00 (2 C3 eNfdN)dI
OM 01 CO (!c03cNrbN)dI
VCVd*(( Idl ) Id/VIl )=V 2 OIS
vavb*((v21*V21)/(VIi*(IdI )Id) )=V 1 OIS
((vdd*bN) / mdi )( (((i
(idi) id/vei =v 2 yav
V21 / ( Idl ) 1 <5 = V 1 dav
Vll(Idl ) ld=VÃ‚Â£i
V11+( Idl ) Id =V 2 1
Vdd*( ( Idl )IdI )=V11
dN/dO =Vdd
dN=dNV
009 01 00 ldNclle( < Idl )IdOeI )*( IM )IN)dI CS2
I*dO = da C2I
052 01 00
I = da (0o000Ie3Tc(INI )IN)dI
i = ba (crnfdO)di
82
TABLE 7s 5
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 1000,
AUOIT SAMPLE SIZE = 500
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0o082 (CAMP ME I DEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
*0o 90
ACCURACY
= 0, 75
ACCURACY
=0,90
0,010
5
3
7
6
Os 0 20
8
5
9
6
Os 0 30
11
6
10
6
Oo 040
13
7
12
7
Os 0 50
15
7
14
8
Oo 060
18
8
16
9
Os 080
23
10
19
10
OslOO
28
12
23
12
Os 120
33
14
27
14
Oo 140
38
16
31
15
Oo 160
43
18
35
17
Oo 180
49
20
39
19
0,200
55
22
43
21
Oo 250
70
28
55
26
0,300
88
34
69
31
0,350
108
41
84
38
0,400
132
49
102
45
Os 450
160
59
123
54
Os 500
193
70
148
64
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 defectives found by the inspector by the total
number of defectives in the lot. During the actual pro
duction process the only way to determine the total number
of defectives in the lot is by an audit reinspection of the
lot to determine how many defectives 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
THE MEASUREMENT OF INSPECTOR ACCURACY
By
LEE ALLEN WEAVER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1972
44
PI =
.200
NR = 160
PR =
.125
GI = 1000
ADR = .667
Number of true good units
Number of true bad units
Figure 4.3
Sample Production Flow for Case 1 with Replacement,
Audit Accuracy = 100 Percent
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 defectives 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 defectives in a reinspection
sample.
xii
TABLE 5 5 CONTIMUED
AUDIT
AUDIT
1  .
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MOL
QL
ACCURACY
STDs DEV
s MOL
OQL
200
Os 0050
Oo 9132
Os 0799
Os 9453
% 9960
Os 9050
Os 0958
0,9448
Os 9955
200
OoOlOO
Oo 8403
Os 0964
Os 9405
"s 9921
0,8100
Os 1365
0,9383
0,9897
200
Oo 0150
Oo 7782
01020
Os 93 58
''o 9881
0=7150
Os 1685
0,9301
Os 9821
200
Os 0200
Oo 7246
0S1029
Os 9310
"Ã‚Â¡s 9841
0,6200
0>1960
0,9194
Os 9718
200
Os 0300
Oa 6369
Os 0988
Oo 9215
"o 9762
Os 4300
Os 2437
Os 3B37
0,9361
200
Oo 0400
Oo 5682
Os 0921
Oo 912 0
^o 9682
Os 2400
0,2854
0,7917
Os 8404
200
0s 0500
Oo 5128
Os 0851
Os 9025
no 9601
INVALID
FOR CASE
2
200
0 07 50
Oo 4124
Os 0697
Os 8788
Os 9398
INVALID
FOR CASE
2
200
Oo1000
Oo 3448
0B 0581
Oo 8550
Os 9194
INVALID
FOR CASE
2
200
Or, 1500
Oo 2 597
Os 0427
Oo 8075
Os 8777
INVALID
FOR CASE
2
100
OoOlOO
Oo 8403
0>1349
Os 9405
0=9910
Os 8100
019L 0
Os 9 38 3
0=9337
100
0a0200
Oo 7 246
Os 1426
Os 9310
Oo 9821
0> 6200
Os 2717
0=9194
0=9698
100
Oo 0300
Oo 6369
Os 1357
Oo 9215
0 973 1
Os 4300
0, 3345
0=3837
0,9332
v
100
0a 0400
Oo 5682
Os 1254
Oo 912 0
0s 9641
0 2400
0s 38 8 3
0=7917
Os 8369
100
Oo 0500
Oo 5128
0,1148
Oo 9025
"=9550
INVALID
FOR CASE
2
100
Oo 0700
Oo 4292
Os 0961
Oo 8835
"d 9369
INVALI0
FOR CASE
2
100
Oo 0900
Oo 3690
Os 0314
Oo 8645
Oo 9187
INVALID
FOR CASE
2
100
01500
Oo 2597
Os 0536
Os 8075
Oo 8636
INVALID
FOR CASE
2
10
Oo1000
Oo 3448
0 2168
Go 8550
Os 9009
INVALID
FOR CASE
2
\
ui
130
24. Kelly, M. L., A Study of Industrial Inspection by the
Method of Paired Comparisons, Psychological Mono
graphs, No. 394, American Psychological Association,
Washington, D. C., 1955.
25. Kendall, M. G., and Stuart, A., Advanced Theory of
Statistics, Vol. 1, Hafner, New York, 1963.
26. Kennedy, W., Inspection and Gaging, The Industrial
Press, New York, 1957.
27. Shewhart, W. A., Economic Control of Quality of Manu
factured Product, Van Nostrand, New York, 1 $ 31.
Transactions and Proceedings
28.Kidwell, J. L., Squeglia, N. L., and Lavender, H. J.,
"Escape Probability as a Design Consideration,"
Annual Technical Conference Transactions, American
Society for Quality Control, 1965, pp. 486499.
22
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:
103
TABLE 80 5
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0o90 AND ACCEPT ACCURACY =0,50,
INSPECTION
SAMPLE SIZE
= 10000
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
=0,050 (NORMAL)
= 0& 082 (CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
3 010
462
3
1873
5
Oo 020
225
3
892
5
0,030
147
3
580
5
Oo 040
108
3
426
4
Oo 050
85
3
335
4
0,060
70
3
275
4
Oo 080
51
2
199
4
Oo 100
39
2
154
4
Oo 120
32
2
124
4
Oo 140
26
2
103
4
Oo 160
22
2
87
4
Oo 180
19
2
75
4
0,200
16
2
65
4
Oo 250
12
2
47
4
Oo 300
9
2
35
4
Oo 350
6
2
27
4
0o400
5
2
20
4
0 o 450
3
2
15
3
Oo 500
2
2
11
3
i/0 O')
TABLE 501
EXAMPLES
OF ACCURACY EXPECTED VALUES
STAINED
FROM DIFFERENT INSPEC
TIGN
AND
RE iMSPECT Z ON TEST
RESULTS,
INSPECTION
SAMPLE
SIZE = 100,
INSPECTION
1 FRACTION
DEFECTIVE
= 0> 05
AUDIT
AUDIT
IU ct KJ U
CASE
1
ca w* w ps
 CASE 2
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MQL
DQL
ACCURACY
STDs DEV
> MQL
OQL
95
0> 0105
Oo 3 333
Os 1521
Os 9400
IsOOOO
Os 8000
Os 2191
0,
9375
05 9973
95
Os 0211
Oc 7143
Os 1707
Oo 9300
1o 0000
O,6000
0,3347
0
9167
0,9857
95
Os 0316
Oo 6250
Os 1712
Os 9200
1o 0000
0,4000
0,4332
Os
3750
Os 9511
95
Os 0421
Oo 5556
Os 1656
Os 9100
IsOOOO
0,2000
0,5367
0,
7500
Os 8242
95
03 0526
Oo 5 000
Os 1581
Os 90 00
lo 0000
INVALID
FDR
CASE
2
95
Os 0737
Oo 4167
Os 1423
Os 88 0 0
IsOOOO
INVAL D
FDR
CASE
2
95
0s 0947
Oo 3571
Os 1281
0o 86 0 0
IsOOOO
INVALID
FOR
CASE
2
95
Os 1579
Oo 2500
0,0968
Oo 8000
1,0000
INVALID
FOR
CASE
2
50
Os 0200
Oo 7246
Os 2177
Oo 9310
Os 9904
0> 6200
O,4146
Os
9194
0,9780
50
Os 0400
Oo 5682
Os 2039
Os 9120
O0 9806
0,2400
0,6315
0,
7917
0,8513
50
Os 0600
Oo 4673
Os 1802
Oo 8930
0,9707
INVALID
FDR
CASE
2
50
Os 0800
Oo 3968
Os 1589
Os 8740
Os 9604
INVALID
FOR
CASE
2
50
Os 1000
Oo 3448
Os 1412
Os 8550
Os 9500
INVALID
FDR
CASE
2
50
Os 1400
Oc 2732
Os 1147
Oo 8170
00 92 84
INVALID
FOR
CASE
2
20
Os 0500
Oo 5128
Os 2691
Os 9025
ns 9601
INVALID
FOR
CASE
2
20
Os 1000
Oo 3 448
Os 1836
Oo 8550
*39194
INVALID
FDR
CASE
2
20
Os 1500
Oo 2597
Os 1351
Oo 8075
*s 8777
INVALID
FOR
CASE
2
10
Os 1000
Oo 3448
0 2381
Oo 8550
*s 9096
INVALID
FDR
CASE
2
6
meter was graduated every two units, and the human tendency
to read to the nearest graduated unit showed itself in the
excessive number of evennumbered 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
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
xi
112
TABLE 8o14
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY =0C90 AND ACCEPT ACCURACY =0,75,
INSPECTION SAMPLE SIZE = 8000
ALPHA ERROR
ALPHA ERROR
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
0,010
0o020
Oa 030
Oo 040
Oo 050
Oa060
Oo 080
Oo 100
Oo 120
0 a 140
Oo 160
Oo 180
0,200
0,250
0a300
0a350
Oo 400
0,450
BETA ERROR = 0o050 (NORMAL)
BETA ERROR =0,082 (CAMP ME I DEL)
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
LOT SIZE
TOO SMALL
LOT SIZE
TOO SMALL
2307
10
4634
15
1465
10
2879
14
1066
9
2074
14
834
9
1612
14
681
9
1312
14
493
9
946
13
382
9
731
13
308
9
590
13
256
9
490
13
217
9
415
13
187
9
358
13
163
9
312
13
119
9
229
13
91
8
175
12
70
8
136
12
55
8
107
12
43
8
84
12
33
7
66
11
Oo 500
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
LIST OF TABLES V
LIST OF FIGURES xi
KEY TO ABBREVIATIONS xii
ABSTRACT XV
CHAPTER
I INTRODUCTION 1
II A SURVEY OF THE LITERATURE 4
III MEASURES OF INSPECTOR ACCURACY 18
IV EXPECTED VALUES FOR THE SAMPLING DISTRI
BUTION OF THE RATIO OF DEFECTIVE PRODUCT
REJECTED 30
V CALCULATION OF EXPECTED VALUES 46
VI VERIFICATION OF DERIVED EXPECTED VALUES
BY SIMULATION 62
VII SINGLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY 73
VIII DOUBLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY 94
IX CONCLUSION 114
iii
90
TABLE 7, 13
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE =5000
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0o082 (CAMP MEIDEL)
OBSERVED CASE 1 CASE 2
INSPECT ION
FRACTION
DEFECTIVE
ACCURACY
= 0, 75
ACCURACY
= 0, 90
ACCURACY
= 0, 75
ACCURACY
=0,90
OoOlO
26
11
21
11
0 020
46
19
37
18
0 0 30
66
26
52
24
Oa 040
86
33
67
31
0050
106
40
82
37
0,060
126
47
98
44
Oa 080
168
62
129
57
Oo 100
211
77
162
70
0,120
256
92
196
84
0,140
302
108
231
99
Oo 160
351
125
267
114
Oo 180
401
142
306
130
Oo 200
455
161
346
146
0,250
599
210
455
191
Oo 300
764
266
579
241
0,350
953
330
721
299
0,400
1174
405
887
367
0,450
1434
493
1083
446
0,500
1746
598
1318
541
67
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 y(MQL) = .50, .75, and .90 with y(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.
129
11. Mackworth, N. H., "Work Design and Training for Future
Industrial Skills," Institute of Production En
gineers Journal, Vol. 35, No. 4, April 1956, pp.
214228.
12. Marash, S. A., "Performing Quality Audits," Industrial
Quality Control, Vol. 22, No. 7, Jan. 1966, pp.
342347.
13. Marien, L. P., "Inspecting the Inspector," American
Machinist, Vol. 74, No. 22, May 1931, pp. 815817.
14. McKenzie, R. M., "On the Accuracy of Inspectors,"
Ergonomics, Vol. 1, No. 3, May 1958, pp. 258272.
15. Purcell, W. R., "Sampling Techniques in Quality System
Audits," Quality Progress, Vol. 1, No. 10, Oct.
1968, pp. 1315.
16. Schwartz, D. H., "Statistical Sleuthing to Detect Bias
in Visual Inspection," Industrial Quality Control,
Vol. 3, No. 6, May 1947, pp. 1417.
17. Tiffin, J., and Rogers, H. B., "The Selection and
Training of Inspectors," Personnel, Vol. 18, No.
1, July 1941, pp. 1431.
Books
18. Burr, I. W.', Engineering Statistics and Quality Con
trol McGrawHill, New York, 1953.
19. Duncan, A. J., Quality Control and Industrial Statis
tics, 3rd Edition, Richard D. Irwin, Homewood, Ill.,
1965.
20. Feigenbaum, A. V., Total Quality Control, McGrawHill,
New York, 1961.
21. Grant, E. L., Statistical Quality Control, 3rd Edition,
McGrawHill, New York, 1964.
22. Juran, J. M., Ed., Quality Control Handbook, McGraw
Hill, New York, 1951.
23. Juran, J. M., and Gryna, F. M., Quality Planning and
Analysis, McGrawHill, New York, 1970.
12
pin points. Dayworking 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 ultraprecision 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 v/ith 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
dramatically with the large 6inch Vernier micrometer to
APPENDIX
ALCAMP=1.0/(2.25*C0NST**2)/2.0
IT AB = 0
00 610 IUN=1 NUN
ADRUN=ADRU(IUN)
DO 610 INI = 1tNNI
IT A8 = IT A8+1
IF!ITAB.LT.10) WRITE<6,550) ITAB
I F( ITAB.GE.10) WRITE(6 560) ITAB
WRI T E ( 6 540) ADRACtADRUN,NI(INT),ALNQRM,ALCAMP
DO 610 IPI = 1 NPI
NRUN = 0
100 NRUN = NRUN + 1
IF(NRUN.EQ.3) GO TO 600
NR=0
110 NR=NR+1
DEC ={ADRAC ADRUN )/CONST
IF(NRUN.EQ.2) GO TO 120
IFCNR.GT.NKINI)*<1PI(IPI)) ) r0 TO 130
PRUN = (PI( IPI) ( 1ADRUN ) )/(ADRUN *(1PI(I PI ) ) )
PRAC = (PI(IP I)*{1ADRAC ))/(ADRAC *(1P I (IPI ) ) )
SIGUN =((PRUN *PIIPI)*(1PI(IPI)))/ ((PI(IPI)+PRUN (1PKIPI)
1 ) ) **2 ) )*SQRT ( 1/ ( N I ( INI)*PI(If>I ) =M 1PI ( IPI ) ) ) + ( 1PRUN) / (NR* PRUN) )
SIGAC =( (PRAC *PI ( IPI )*(1PICIPI)>)/ ((PI(IPI)+ PRAC *( 1PI( I P I )
1) )**2) )* SORT( l/(NI(INI)*PI(II)*(lPI(IPI))) + (1PRAC)/(NR*PRAC) )
IF(SI GUN +SIGAC .GT.DEC) GO TO 110
NRDEC=NR
DR1 = NRDEC*PI(IPI)*(1.0ADRAC+CHNST*SIGAC)/( (ADRACCONST*SIGAC)*
1( l.OPK IPI ) ) ) +1.0
I DR 1 = DR 1
GO TO 100
120 IF(NR.GT.NI(INI)*(1PI(IPI)J) r0 TO 140
PRUN =(PI(IPI )*(1ADRUN ) )/( 1PI(IPI ) )
PRAC =(PI (IPI )*(1ADRAC ) )/( 1P I ( I P I ) )
SIGUN =(PRUN *(1PI(IPI))/PI(IPI))SQRT(1/(N I ( IN I)*PI(I PI)*( 1PI
1C IPI)))(1PRUN )/(NR* PRUN ))
SIGAC =(PRAC *(1PI(IPI))/PICIPI))*SQRT(l/(NIC I NI)*PI
126
8
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 nonrandom 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 onehalf hours to inspect the
unit, the average inspection time on the regular inspection
line. Jacobson reported as a result of his experiment that
107
TABLE 8,9
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY =0e90 AND ACCEPT ACCURACY =0,60t
INSPECTION SAMPLE SIZE = 5000
ALPHA ERROR = BETA ERROR = 0,050 (NORMAL)
ALPHA ERROR =
BETA ERROR
=0,082 (C
AMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
0 a 0 10
1022
4
3190
7
Oo 020
484
4
1437
6
0 a 030
314
4
918
6
Da 040
231
4
670
6
0 a 050
181
4
524
6
0 a 060
148
4
429
6
0,080
108
4
310
6
0,100
83
4
240
6
0,120
67
3
194
6
0, 140
56
3
161
6
0,160
47
3
136
6
0,180
40
3
117
6
0,200
35
3
102
6
0,250
25
3
74
6
0,300
19
3
56
5
0,350
14
3
43
5
0,400
11
3
33
5
0,450
8
3
25
5
0,500
6
3
19
5
TABLE 5.7
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
RE I NS PcCTI ON TEST
RESULTS,
INSPECTION
SAMPLE
SIZE = 1000,
INSPECTION FRACTION
1 DEFECTIVE
= 0.25
AUDIT
AUOI T
1
SAMPLE
FRACTION
SI ZE
DEFECTIVE
ACCURACY
STD.CEV.
MQL
OQL
ACCURACY
STO.OEV
. MQL
OQL
750
0.0013
0.996C
0.0040
0.7490
1.0000
0.9960
0.0040
0.7490
1.0000
750
0.0053
0.9843
0.0078
0.7460
1.0000
0.9840
0.0081
0.7459
0.9999
750
0.0093
0.9728
0.0102
0.7430
1.0000
0.9720
0.0107
0.7428
0.9997
750
0.0133
0.9615
0.0119
0.7400
1.0000
0.9600
0.0129
0.7396
0.9994
750
0.0200
0.9434
0.0142
C.7350
1.0000
0.9400
0.0159
0.7340
0.9987
750
0.0267
0.9259
0.0159
0.7300
1.0000
0.9200
0.0186
0.7283
0.9976
750
0.0333
0.9091
0.0173
0.7250
1.0000
0.9000
0.0210
0.7222
0.9962
750
0.0600
0.8475
0.0209
0.7050
1.0000
0.8200
0.0291
0.6951
0.9860
750
0.0933
0.7813
0.0231
0.6800
1.0000
0.7200
0.0379
0.6528
0.9600
750
0.1333
0.7143
0.0241
0.6500
1.0000
0.6000
0.0473
0.5833
0.8974
750
0.1733
0.6579
0.0243
0.6200
1.0000
0.4800
0.0562
0.4792
0.7728
750
0.2267
0.5952
0.0240
0.5800
1.0000
0.3200
0.0676
0.2188
0.3772
750
0.2800
0.5435
0.0232
0.5400
1.0000
INVALID
FCR CASE
2
750
0.3467
0.4902
0.0221
0.4900
1.0000
INVALID
FOR CASE
2
750
0.4267
0.4386
0.0208
0.4300
1.0000
INVALID
FCR CASE
2
750
0.5333
0.3846
0.0191
0.3500
1.0000
INVALID
FOR CASE
2
750
0.6667
0.3333
0.0172
0.2500
1.0000
INVALID
FOR CASE
2
750
0.8 533
0.2809
0.0151
0.1100
1.0000
INVALID
FUR CASE
2
500
0.0020
0.9940
0.0059
0.7485
0.9993
0.9940
0.0060
0.7485
0.9993
500
0.0060
0.9823
0.0101
0.7455
0.9980
0.9820
0.0104
0.7454
0.9979
5C0
0.0100
0.9709
0.0128
0.7425
0.9966
0.9700
0.0135
0.7423
0.9963
500
0.0140
0.9597
0.0143
0.7395
0.9953
0.9580
0.0161
0.7390
0.9947
500
0.0180
0.9488
0.0164
0.7365
0.9939
0.9460
0.0183
0.7357
0.9929
500
0.0300
0.9174
0. 0200
0.7275
0.9898
0.9100
0.0238
0.7253
0.9868
500
0.0600
0.8475
0.0248
0.7050
0.9792
0.8200
0.0345
0.6951
0.9654
Ln
VO
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 onetail 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
73
TABLE 6.2
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 2
NUMBER INSPECTED = 1000 NUMBER AUDITED = 500
MQL Initial
. 90
.90
.90
.75
.75
.75
.50
.50
.50
Initial
. 90
.75
.50
VO
o
.75
.50
.90
.75
. 50
Accuracy
Observed
.8993
.7484
.4943
.8999
.7496
.4985
.9000
.7500
. 4996
Expected
. 0461
. 0833
.1542
.0269
.0495
. 0937
. 0163
.0312
.0619
a (ADR)
Observed
. 0456
.0832
.1562
.0266
.0489
.0933
.0160
.0307
.0616
Expected
. 0900
. 0750
.0500
.2250
.1875
.1250
.4500
.3750
.2500
PI
Observed
. 0901
. 0751
. 0501
.2252
.1877
.1251
..4502
. 3752
.2501
Expected
. 0099
. 0203
. 0263
. 0290
.0577
.0714
. 0818
.1500
.1667
PR
Observed
. 0099
.0202
. 0262
.0290
.0576
.0712
. 0818
.1498
.1663
Correlation
. 0408
.0375
.0290
. 0389
. 0361
.0272
. 0387
.0351
.0242
Number Greater
than 1 a
309
300
307
306
309
318
313
311
316
Number Greater
than 2 o
40
42
43
41
42
43
35
45
43
cr\
vo
TABLE
EXAMPLES
OF ACCURACY EXPECTED VALUES
OBTAINED
FROM DIFFERENT INSPECTION
AND
REINSPECTION TEST
RESULTS,
INSPECTION
SAMPLE
SIZE 100,
INSPECTION
1 FRACTION
DEFECTIVE
* 0,10
AUDIT
AUDIT
1  
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCURACY
ST Do DEVo
MQL
OQL
ACCURACY
STD,DEV
MOL
OQL
90
0,0111
Oo 9091
0,0867
0,8900
l,0000
0,9000
0,1049
0,3889
0,9988
90
0,0222
Oo 8333
Oo1076
Oo 8800
loOOOO
0,8000
0,1549
0,8750
0,9943
90
0,0333
Oo 7692
0,1169
0o 8700
1,0000
0,7000
0,1975
0,8571
0,9852
90
0o 0444
Oo 7143
0,1207
Oo 8600
loOOOO
0,6000
0,2366
0,3333
3,9690
90
0,0667
Oo 6250
0#1210
Oo 8400
1,0000
0,4000
0,3098
3,7530
0,8929
90
0,0889
Oo 5556
0,1171
0,8200
1,0000
0,2000
0,3795
0,5000
0,6098
90
0,1111
Oo 5000
0,1118
0,8000
loOOOO
INVALID
FDR CASE
2
90
0,1667
Oo 4030
0,0980
Oo 7500
loOOOO
INVALID
FDR CASE
2
90
0,2222
Oo 3333
0,0861
0,7000
loOOOO
INVALID
FDR CASE
2
50
0,0200
Oo 8475
0,1350
Oo 8820
0,9910
0, 8200
0, 1833
0,8780
3,9866
50
0,0400
Oo 7353
0,1496
Oo 8640
0 9818
0,6400
0,2758
3,8438
0,9583
50
0,0600
0o 6494
Oo1483
0,8460
0,9724
0,4600
0,3518
3,7826
0,8996
50
0,0800
Oo 5 814
0,1421
Oo 8280
Oo 9628
0,2800
0,4205
0,6429
0,7475
50
0,1000
Oo 5263
Oo1345
0, 8100
0,9529
INVALID
FOR CASE
2
50
0,1400
Oo 4425
0,1193
Oo 7740
0,9325
INVALID
FDR CASE
2
50
0,1800
Oo 3817
0,1061
Oo 73 8 0
Oo 9111
INVALID
FDR CASE
2
50
0,3000
Oo 2703
Oo 0733
Oo 6300
", 8400
INVALID
FDR CASE
2
20
0,0500
Oo 6 397
0,2205
0,8550
0,9607
0,5500
0,4635
0,3182
0,9193
20
0,1000
Oo 5263
Oo1867
Oo 8100
0,9205
INVALID
FDR CASE
2
20
0,1500
Oo 4255
Oo1535
0,7650
0,8793
INVALID
FOR CASE
2
20
0,2000
Oo 3571
0,1281
0,7200
0,8372
INVALID
FDR CASE
2
20
0, 3000
Oo 2703
0,0941
Oo 6300
",7500
INVALID
FDR CASE
2
10
0,1000
Oo 5263
Oo 2507
0, 8100
0,9101
INVALID
FOR CASE
2
87
TABLE 7a10
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =1000
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0o082 (CAMP ME I DEL )
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0a 75
ACCURACY
= 0a 90
0 010
8
4
8
5
0a020
12
6
11
7
0,030
17
8
15
8
0a 040
21
10
18
10
0,050
26
11
21
ll
0,060
30
13
25
13
0,080
39
16
32
16
O0 100
48
20
39
19
Oa 1 20
58
23
46
22
Oa 140
68
27
53
25
Oo 160
78
30
61
28
0, 180
89
34
69
32
Oo 200
100
38
78
35
Oo 250
130
49
100
45
O0 300
164
61
126
56
Oa 350
202
74
155
68
Oo 400
247
90
189
82
Oa 450
300
108
229
98
Oo 500
363
130
277
118
85
TABLE 7,8
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = IOOO,
AUDIT SAMPLE SIZE =. 20
ALPHA ERROR
ALPHA ERROR
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
0,010
0,020
0,030
0,040
0,050
0,060
0,080
0, 100
0,120
0, 140
0, 160
0,180
0,200
0,250
0,300
0,350
0,400
0,450
0,500
= Qo 050 (NORMAL)
= 0,082 (CAMP ME I DEL)
CASE l
ACCURACY ACCURACY
= 0o 75 =0,90
l l
1 l
2 1
2 l
2 2
2 2
2 2
3 2
3 2
3 2
4 3
4 3
4 3
5 3
6 4
7 4
8 5
9 5
11 6
CASE 2
ACCURACY ACCURACY
=0,75 =0,90
4
3
3
3
3
4
4
4
4
4
4
5
5
5
6
7
7
8
9
4
3
3
3
3
3
3
3
3
3
4
4
4
4
4
5
5
5
6
This dissertation was submitted to the Department of Manage
ment in the College of Business Administration and to the
Graduate Council, and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
December, 1972
Dean, Graduate School
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.
18
TABLE 2.2
DEFECTS FOUND IN FOUR
SUCCESSIVE VISUAL
AND GAGING
INSPECTIONS OF 30,000 UNITS
Submitted
Regular I
Regular II
Regular III
Experts
Good 29900
Defective 100
v Found 68
Left 32
> Found 18
Left 14
> Found 8
Left 6
* Found 4
Left 2
Accuracy
0.6800
0.5625
0.5714
0.6667
TABLE 5,4
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST
RESULTS,
INSPECTION
SAMPLE
SIZE 1000,
INSPECTION FRACTION
1 DEFECTIVE
 0,01
AUDIT
AUDIT
L  
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCJRACY
STDoDEV,
MOL
OQL
ACCURACY
STD,DEV
', MOL
OQL
990
Oo 0010
Oo 909 1
0,0867
0,9890
1,0000
0,9000
0,1049
0,9889
0,9999
990
0,0020
0,8333
0,1076
0,9880
1,0000
0,8000
0, 1549
0,9875
05 9995
990
0,0030
Oo 7692
0,1169
0,9870
1,0000
0,7000
0,1975
0,9857
0, 9987
990
0,0040
Oo 7 14 3
0,1207
0,9860
1,0000
0,6000
0,2366
0,9333
0,9973
990
0,0061
0,6250
0,1210
0,9840
1,0000
0,4000
0,3098
0,9750
0,9909
990
0,0081
0,5556
0,1171
0,9820
1,0000
0,2000
0,3795
0,9500
0,9674
990
0,0101
Oo 5000
0,1118
0,9800
1,0000
INVALID
FOR CASE
2
990
0,0152
Oo 4000
0,0980
0, 9750
1,0000
INVALID
FOR CASE
2
990
0,0202
Oo 3333
0,0861
0,9700
1,00 00
INVALID
FOR CASE
2
500
0,0020
Oo 8347
0,1446
Oo 9880
0,9990
0,8020
0,2076
0,9875
0,9985
500
0,0040
Oo 716 3
0,1573
Oo 9860
0,9980
0 6040
0,3065
0,9334
0,9954
500
0,0060
Oo 6274
0,1537
Oo 98 41
0,9970
0,4060
0,3906
0,9754
0,9882
500
0,0080
Oo 5580
0,1457
Oo 98 2 1
0,9960
0,2080
0,4679
0,9519
0,9654
500
0,0100
0,5025
0,1367
Oo 9801
0,9950
INVALID
FOR CASE
2
500
0,0140
Oo 4191
0,1197
0,9761
0,9930
INVALID
FOR CASE
2
500
0,0180
0,3595
0,1055
0o 9722
0,9910
INVALID
FOR CASE
2
500
0,0300
Oo 2519
0,0767
Oo 9603
0,9849
INVALID
FOR CASE
2
200
0,0050
Oo 6689
0,2319
0,9851
0,9960
0,5050
0,5182
0,9302
0,9911
200
0,0100
Oo 502 5
0,1930
Oo 98 01
",9920
INVALID
FOR CASE
2
200
0,0150
Oo 4024
0,1576
0,9752
",9880
INVALID
FOR CASE
2
200
0,0200
O,3356
0,1312
0,9702
",9840
INVALID
FOR CASE
2
200
0,0300
Oo 2519
0,0966
0,9603
",9759
INVALID
FDR CASE
2
100
0,0100
Oo 5025
0,2611
Oo 9801
0,9910
INVALID
FOR CASE
2
93
TA8LE 7s16
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE ^ 200
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0o082 (CAMP MEIDEL)
OBSERVED
INSPECTION
CASE
1
CASE
2
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0,75
= 0o 90
= 0,75
= 0,90
0,010
3
2
4
4
Oo 0 20
4
3
5
4
09 0 30
5
3
6
4
Oo 0 40
6
4
7
5
0,050
7
4
7
5
Oo 060
9
5
8
5
Oo 080
11
6
10
6
0,100
13
6
11
7
Oo 120
15
7
13
8
0,140
17
8
15
8
Oo 160
19
9
16
9
Oo 180
22
10
18
10
Oo 200
24
11
20
11
0,2 50
30
13
25
13
0, 300
38
16
30
15
Oo 350
46
19
36
18
0,400
55
22
43
21
Oo 450
66
26
52
24
0,500
78
31
61
29
104
TA8LE 8o6
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0C90 AND ACCEPT ACCURACY =0,60t
INSPECTION SAMPLE SIZE = 100
ALPHA ERROR =
ALPHA ERROR =
BETA
BETA
ERROR
ERROR
n a
O O
ti II
050
082
(NORMAL)
(CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
l CASE
AUDIT AUDIT
DEFECTIVES SAMPLE
SIZE
2
AUDIT
DEFECTIV
0,010
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Do 020
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,030
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,040
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0o050
LOT
SIZE
TOO
SMALL
LOT
SI ZE
TOO
SMALL
0,060
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 080
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,100
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,120
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 140
80
4
LOT
SIZE
TOO
SMALL
Oo 160
65
4
LOT
SIZE
TOO
SMALL
Oo 180
54
4
LOT
SIZE
TOO
SMALL
0,200
46
4
LOT
SIZE
TOO
SMALL
Oo 250
32
4
LOT
SIZE
TOO
SMALL
Oo 300
23
4
LOT
SIZE
TOO
SMALL
Oo 350
17
3
LOT
SIZE
TOO
SMALL
Oo 400
13
3
49
7
0o450
10
3
38
7
Oo 500
7
3
29
6
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 y (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.
62
REFERENCES
Periodicals
1. Ayers, A. W., "A Comparison of Certain Visual Factors
with the Efficiency of Textile Inspectors," Journal
of Applied Psychology, Vol. 26, No. 6, Dec. 1942,
pp. 812827.
2. Belbin, R. M., "New Fields for Quality Control," Metal
Industry, Vol. 90, No. 4, Jan. 1957, pp. 6365.
3. Blais, R. A., "True Reliability Versus Inspection Ef
ficiency," IEEE Transactions on Reliability, Vol.
R18, No. 4, Nov. 1969, pp. 201203.
4. ForsterCooper, J., "A Description of the Statistical
Quality Control System in Operation at Joseph Lucas
(Electrical) LTD," Engineering Inspection, Vol. 18,
No. 2, Feb. 1954, pp. 110121.
5. Harris, D. H., "Effect of Defect Rate on Inspection
Accuracy," Journal of Applied Psychology, Vol. 52,
No. 5, Oct. 1968, pp. 377379.
6. Harris, D. H., "Effect of Equipment Complexity on In
spection Performance," Journal of Applied Psychology,
Vol. 50, No. 3, June 1966, pp. 236237.
7. Hayes, A. S., "Control of Visual Inspection," Industrial
Quality Control, Vol. 6, No. 6, May 1950, pp. 7376.
8. Jacobson, H. J., "A Study of Inspector Accuracy," In
dustrial Quality Control, Vol. 9, No. 2, Sept. 1952,
pp. 1625.
9. Juran, J. M., "Inspectors Errors in Quality Control,"
Mechanical Engineering, Vol. 57, No. 10, Oct. 1935,
pp. 643644.
10.Lawsche, C. H., and Tiffin, J., "The Accuracy of Pre
cision Instrument Measurement in Industrial In
spection," Journal of Applied Psychology, Vol. 29,
No. 6, Dec. 1945, pp. 413419.
128
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,
1
99
TABLE 8,1
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER OF DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,50,
INSPECTION! SAMPLE SIZE = 100
ALPHA ERROR
ALPHA ERROR
= BETA
= BETA
ERROR
ERROR
 Oo
= 3o
050 (NORMAL)
082 (CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUOIT
SAMPLE
SIZE
i
AUDIT
DEFECTIVES
CASE
AUDIT
sample
SIZE
2
AUDIT
DEFECTIVES
0,010
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,020
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
Oo 030
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,040
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,050
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
0,060
LOT
SIZE
TOO
SMALL
LOT
SIZE
TOO
SMALL
OoOBO
74
3
LOT
SIZE
TOO
SMALL
0, 100
53
3
LOT
SIZE
TOO
SMALL
0,120
41
3
LOT
SIZE
TOO
SMALL
0, 140
33
3
LOT
SIZE
TOO
SMALL
0,160
27
3
LOT
SIZE
TOO
SMALL
0, 180
23
3
LOT
SIZE
TOO
SMALL
0,200
19
3
LOT
SIZE
TOO
SMALL
0,250
13
2
72
6
0,300
10
2
51
5
0,350
7
2
38
5
0,400
5
2
28
5
0,450
4
2
21
4
0,500
2
2
16
4
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.
xv
38
Since this function is also nonlinear 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
Sadr pi(1+pr) [pipr(ipi)](l) pr
s pi ~ pi"2 pi2
Sadr _ pi(ipi) [pipr(ipi)](0) = (1pi)
5 PR PI2 PI
Combining the above terms with
obtain
o ADR = PR(1PI) / 1 !pR .
PI J (NI) (PI) (1PI) (NR) (PR) '
Using the methods previously discussed for Case 1
we obtain the following additional expected values for Case
2.
U(MQL, = 1 P! .
pdOQL) = 1 .
u(OQL) = 1 (NI) (PR) (1PI) ~(NR) (PR) (ADR)
ADR[NI (1PI) (NR) (PR) ] '
The equations derived for Cases 1 and 2 are summarized
in Table 4.1.
27
For the Jacobson data the following results are ob
tained:
MQL = 38,195 +25
.9800
39,000
i
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 defectives
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
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INGEST IEID EHDZQ299I_FK3MV0 INGEST_TIME 20141006T23:24:13Z PACKAGE AA00025720_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
113
TABLE 8015
AUDIT SAMPLE SIZE AMD MINIMUM NUMBER DF DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,75,
INSPECTION SAMPLE SIZE = 10000
ALPHA ERROR = BETA ERROR =0,050 (NORMAL)
ALPHA ERROR = BETA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDI T
sample
SIZE
2
AUDIT
DEFECTI
0,010
4666
10
9364
15
0,020
2185
9
4243
14
0,030
1414
9
2719
14
0,040
1038
9
1988
14
0,050
816
9
1558
13
0,060
669
9
1275
13
0,080
486
9
926
13
0,100
378
9
719
13
0,120
305
9
581
13
0,140
254
9
483
13
0,160
216
9
410
13
0, 180
186
9
354
13
0,200
162
9
309
13
O0 250
119
9
227
13
0,300
90
8
173
12
0,350
70
8
135
12
0,400
55
8
106
12
0,450
43
8
83
11
0,500
33
7
66
11
89
TABLE 7,12
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND 8Y THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =. 100
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED
INSPECTION
CASE
1
CASE
2
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o 75
= 0,90
= 0,75
= 0,90
0,010
2
2
4
3
0,020
3
2
4
4
0,030
3
2
5
4
0,040
4
3
5
4
0,050
5
3
5
4
0,060
5
3
6
4
0,080
7
4
7
5
o
o
O
O
8
4
8
5
0,120
9
5
8
5
0, 140
10
5
9
6
0,160
11
6
10
6
0, 180
12
6
11
7
0,200
14
7
12
7
0,250
17
8
15
8
0,300
21
10
17
10
0,350
25
11
21
11
0,400
30
13
24
13
0,450
35
15
28
14
0,500
42
18
33
17
TABLE 3.1
Accuracy Measure
ACI = Ratio of Correct
Inspections
AGA = Ratio of Good
Product Accepted
ADR = Ratio of Defective
Product Rejected
API = Accuracy of
Product Improvement
SUMMARY OF ACCURACY MEASURES OBTAINED FROM
TWO SETS OF SAMPLE DATA
Solder
Connection
Example
Television
Old Inspection
Method
Panels
New Inspection
Method
.996
.507
.897
.999
.654
. 886
.828
.286
.911
. 825
.053
.831
831
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.
30
88
TABLE 7, 1 i
MINIMUM NUMBER OF DEFECTIVE ITEMS FBUND BY. THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE =. 500
ALPHA ERROR
ALPHA ERROR
OBSERVED
=0,050 (NORMAL)
=0,082 (CAMP MEIDEL)
CASE 1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0o75
= 0,90
= 0,75
= 0,90
0,010
5
3
6
4
0 020
7
4
8
5
0,030
10
5
9
6
Oo 040
12
6
11
7
Oo 050
15
7
13
8
Os 060
17
8
15
8
0,080
22
10
18
10
0,100
27
12
22
12
0,120
32
14
26
13
0,140
37
16
30
15
0, 160
42
17
34
17
0,180
47
19
38
18
0,200
53
22
42
20
0,250
68
27
54
25
0,300
86
33
67
31
0,350
105
40
82
37
0,400
128
48
99
44
0,450
155
58
119
53
0,500
186
69
143
63
98
Table 8.14 shows that with an inspection sample size of 5000
and observed fraction defective of .010 no test is possible
for the above hypotheses. However, higher inspection ob
served fraction defective will allow for adequate audit
inspection to determine inspection accuracy.
63
1. A value for the manufacturing quality level (MQL)
was predetermined. Since the actual number of defectives in
a lot of fixed size will vary from lot to lot, the number
of defectives 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 defectives in the lot (D) was de
termined by generating a normal random deviate (Z) and solving
for D in the following equation
D = NI(lMQL) + Z /NI (MQL) (IMOL)".
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 defectives 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 defectives, ADR also follows a binomial
distribution which can be approximated by a normal distribu
tion. For each lot the number of defectives 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) (1ADR) .
3. The lot PI was calculated.
4. The absolute number of defectives 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.
36
and
GGI =
GI [lDR\,
NR
therefore
IQL = 1 gg .
or
y(IOQL) = 1 PR
For the OQL we have that
GGI + DGI = GI DR
and
GGI = GI
Solving we obtain
OQL = GI
fl 
\ NR/
:ain
ljd .
GI DR
Substituting the appropriate expected values in order to
have OQL as a function of NI, NR, PI and PR we obtain
U(OQL)
(NI) (PR) (1PI) (NR) (PR)
(NI) (1PI) (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 = QL ~ MQL
1 MQL *
The above equations will hold for all the cases and, there
fore, will not be discussed in the following paragraphs.
81
TABLE 7,4
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 500,
AUDIT SAMPLE SIZE =t 10
ALPHA ERROR =0o050 (NORMAL)
ALPHA ERROR =0,082 (CAMP ME I DEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
0,010
0,020
0,030
0,040
0,050
0,060
0,080
0, 100
0,120
0, 140
0,160
0,180
0,200
0,250
0, 300
0,350
0,400
0,450
0,500
CASE l
ACCURACY ACCURACY
= 0o 75 =0,90
l
l
I
l
1
2
2
2
2
2
3
3
3
3
4
4
5
5
6
1
l
I
I
1
L
1
2
2
2
2
2
2
2
3
3
3
3
4
CASE 2
ACCURACY ACCURACY
=0,75 =0,90
4
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
5
5
6
4
3
3
3
3
3
3
3
3
3
3
3
Y
3
3
3
3
4
4
4
33
We can define the following expected values
E Iff] PI
If DI/NI and DR/NR are independent variables we can
write the expected value of ADR as (25, p. 52)
y (ADR)
PI
PI + (1PI)PR
Intuitively the number of defectives 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.
o2 (g)
3g
37
p axay.
For our problem the above equation would be written
as
a2 (ADR)
(3 ADrY
\3 PI )
3ADRV(PI)
+
/3ADR\2
\3 PR J
O2 (PR)
3 ADR 3 ADR
3 PI 3PR
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
48
If the accuracy of the inspector is .10 this means the whole
inspection sample is defective, which is verified by the
equation for y(MQL) for Case 2
y (MQL) = 1 PI (1 PI.MPJll
ADR
= 1 .10 
(.90) (.10)
.10
= 0.
If the audit fraction defective is greater than the inspec
tion fraction defective, negative values for y(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.
GO TO 160
170 WRITE( 6 520)JNRPR ADR 1,SIG1 MQL 1 OQL l
GO TO 160
500 FORMAT2CX,'EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIF
1FERENT INSPECTION AND',/,29X,
2REINSPECT ION TEST RESULTS INSPECTION SAMPLE SIZE =,I5,*,,
3/,37X,INSPECTION FRACTION DEFECTIVE =,F5.2,/)
505 FORMAT!10X,AUDIT*,2X,'AUDIT*,7X,* CASE 1  
1 6X * CASE 2  ,/,10X, SAMPLE FRACTION',/,
210X,'SIZE DEFECTIVE ACCURACY STD.DEV MQL 0QL,6X,
3 ACCURACY STD.DEV. MQL' ^QL,/>
510 FORMAT(10X,15,F9.4 2X, 4F9.4,2X,4F9.4)
520 FORMAT(10X,15,F9.4 ,2X,4F9.413X,INVALID FOR CASE 2)
530 FORMAT!IH1,//////,50X,TABLE 5.',II,/)
540 FORMAT!1H1,//////,45X,TABLE 5.,11' CONTINUED',/)
560 FORMAT(IX, )
110 CONTINUE
100 CONTINUE
STOP
END
32
The following expected values will be determined.
y(ADR) = mean value for the sampling distribution of
the ratio of defective product rejected.
2
c (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
therefore
ADR =
(GI) (DR)
NR
DI
DI + (GI) TORT
NR
Dividing both the numerator and denominator by NI, we
have
DI
NI
ADR
110
TABLE 8o12
AUDIT SAMPLE SIZE AND MINIMUM NUMBER 3F DEFECTIVES TO
REJECT ACCURACY = 0,90 AND ACCEPT ACCURACY =0,75,
INSPECTION
SAMPLE SIZE
= 500
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
=0,050 (NORMAL)
= 0o 082 (CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
l
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SIZE
2
AUDIT
DEFECTIVi
0,010
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 020
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 030
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,040
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 050
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Oo 060
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
Do 080
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
OolOO
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,120
372
11
LOT SIZE
TOO
SMALL
O, 140
301
10
LOT SIZE
TOO
SMALL
0,160
251
10
LOT SIZE
TOO
SMALL
0,180
213
10
LOT SIZE
TOO
SMALL
0,200
183
10
378
15
0,250
132
9
270
15
Oo 300
99
9
201
14
0,350
76
9
155
14
0,400
59
9
121
13
0,450
46
8
95
13
0,500
36
8
74
12
78
TABLE 7,1
MINIMUM NUMBER OF DEFECTIVE ITEMS F^UND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 5*0,
AUDIT SAMPLE SIZE = 250
ALPHA ERROR =0oC50 (NORMAL)
ALPHA ERROR =0o0B2 (CAMP ME I DEL)
OBSERVED
CASE
1
CASE
2
INSPECTION
FRACTION
ACCURACY
ACCURACY
ACCURACY
ACCURACY
DEFECTIVE
= 0 75
= 0o 90
= 0,75
= 0,90
OoOlO
4
2
9
7
Oo 0 20
5
3
7
6
Oo 0 30
7
4
8
6
0,040
8
5
9
6
Oo 050
9
5
10
6
Oo 060
11
6
10
6
0,080
13
7
12
7
Oo 100
16
8
14
8
Oo 120
19
9
16
9
Oo 140
22
10
18
10
Oo 160
24
ll
21
11
o
o
co
o
27
12
23
12
Oo 200
30
13
25
13
Oo 2 50
39
16
31
16
Oo 300
43
20
38
19
Oo 350
59
23
46
22
Oo 4 00
71
28
56
26
Oo 450
85
33
67
31
Oo 500
103
39
80
36
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
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 100 99
8.2 AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.50, INSPECTION SAMPLE
SIZE = 500 100
viii
BIOGRAPHICAL SKETCH
Lee Allen Weaver was born April 28, 1935, at Hellertown,
Pennsylvania. In June, 1953, he was graduated from Allen
town High School, Allentown, Pennsylvania. In June, 1957,
he received the degree of Bachelor of Arts from Moravian
College. In September, 1957, he enrolled in the graduate
school of the University of Florida. He worked as a graduate
assistant in the Department of Mathematics until June, 1959,
when he received the Master of Science. From 1960 until
1967 he worked as a Product Assurance engineer for Honeywell
Inc. In 1967 he joined the faculty of the University of
South Florida. From June, 1968, until the present time he
has pursued his work toward the degree of Doctor of
Philosophy.
Lee Allen Weaver is married to the former LaRae Kathryn
Fritzinger and is the father of four children. He is a
member of the American Society for Quality Control, and the
Institute of Electronic and Electrical Engineers. He is
a statistical consultant to a Working Group of the Interna
tional Electrotechnical Commission and to the Director of
Product Assurance of Honeywell Inc.
131
20
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:
AC I = 38,195 + 646 = .9959
39,000
43
CUSTOMER
OQL = ^769)
PI = .200
PR = .125
ADR = .500
NR = 160
NI = 1000
= Number of
= Number of
true good units
true bad units
Figure 4.2
Sample Production Flow for Case 2 with No Replacements
Audit Accuracy = Inspection Accuracy
92
TABLE 715
MINIMUM NUMBER OF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE =10000,
AUDIT SAMPLE SIZE =1000
ALPHA ERRDR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP ME I DEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0,75
ACCURACY
= 0,90
0,010
7
4
8
5
0, 020
12
6
11
7
0,030
17
8
14
8
Oo 040
21
10
18
10
0,050
26
11
21
11
0,060
30
13
25
13
0,080
39
16
31
16
0,100
48
20
38
19
0, 120
58
23
46
22
0, 140
67
27
53
25
0,160
78
30
61
28
0, 180
88
34
69
32
0,200
99
38
77
35
0,250
129
48
100
45
0,300
163
60
125
55
0,350
201
74
155
68
0,400
246
89
188
82
0,450
298
108
228
98
0,500
361
129
275
118
7
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, illdefined 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 everyday rates.
McKenzie offers three solutions to the controlled
experiment problem. One way is the introduction of known
defectives, 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 defectives being detected by inspection.
2
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 defectives or using an audit
inspector. Required sample sizes and the calculation of
quantitative accuracy measures are left unansv/ered.
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
TABLE 5s 6
AUDIT
AUDIT
1 
SAMPLE
FRACTION!
SIZE
DEFECTIVE
ACCURACY
STDoDEVo
MQL
500
0o 0800
Oo 5 814
Os 0449
0, 8280
500
Os 1000
0 o 5 2 6 3
Oo 0425
08100
500
Os 1400
Oo 4425
Os 0377
Os 7740
500
0,1800
Oo 3817
0,0336
Os 7380
500
Os 2400
Oo 3165
Os 0286
Os 6840
500
Os 3200
Oo 2 577
0,0237
0,6120
200
Os 0050
Oo 9569
Oo 0413
Oo 8955
200
OsOlOO
Oo 9 174
0,0539
Os 8910
200
Os 0200
Oo 8 47 5
Os 0654
0,8820
200
Os 0300
Oo 7874
0,0696
Os 8730
200
Os 0400
Oo 7353
0,0705
0,8640
200
Os 0750
Oo 5970
0,0649
Os 8325
200
Os 1000
Oo 5263
0s 0591
0, 8100
200
Os 1250
Oo 4706
0,0535
0,7875
200
Os 1750
Oo 3883
0,0442
0,7425
200
0s 2250
Oo 3306
0,0372
Os 6975
200
Os 3000
0 2703
Oo 0298
Oo 6300
100
0,0100
Oo 9174
Os 0758
0,8910
100
Os 0200
Oo 8475
0,0915
0,8820
100
0 0300
Oo 7874
0,0968
0,8730
100
Os 0400
Oo 7353
0,0975
Oo 8640
100
0,0600
Oo 6494
0,0933
0,8460
100
Os 0800
Oo 5814
0,0864
0,8280
100
Os 1000
Oo 5263
Os 0793
Oo 8100
100
Os 1500
Oo 4255
Os 0636
Oo 7650
100
Os 2000
Oo 3571
0,0519
0, 7200
100
Oo 3000
Oo 2 703
0,0366
Oo 6300
10
Os 1000
Oo 5263
0,2380
0,8100
3MTIMUE0
CASE 2
DQL
ACCURACY
STD,DEV
, MQL
OQL
0,9628
0, 2800
Os 1330
0,5429
0,7475
Oo 9529
INVALID
FDR CASE
2
0,9325
INVALID
FOR CASE
2
0,9111
INVALID
FOR CASE
2
"a 8769
INVALID
FOR CASE
2
"a 82 70
INVALID
FOR CASE
2
0,9961
0,9550
0>0451
Os 3953
0,9959
0,9922
0,9100
0,0640
0,8901
0,9912
0,9844
0> 8200
0,0911
0,8780
0,9800
0,9765
0,7300
0,1122
0,3630
0,9653
'',9686
0> 6400
0,1304
0, 8438
Os 9459
", 9407
0,3250
0,1821
0,5923
0,7823
", 9205
INVALID
FOR CASE
2
", 9000
INVALID
FOR CASE
2
",8584
INVALID
FOR CASE
2
", 8158
INVALID
FOR CASE
2
",7500
INVALID
FOR CASE
2
",9911
0> 9100
0,0900
0,3901
0,9901
", 9822
0> 8200
0,1274
0,8780
0,9778
",9732
0,7300
0,1551
0,8530
0,9521
",9643
0,6400
0,1804
0,3438
0,9417
",9463
0,4600
0,2212
0,7826
0,8754
",9283
0,2800
0,2557
0,6429
0,7207
",9101
INVALID
FOR CASE
2
", 8644
INVALID
FOR CASE
2
",8182
INVALID
FOR CASE
2
Oa 7241
INVALID
FOR CASE
2
0,9010
INVALID
FOR CASE
2
49
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.
26
Accuracy of Product Improver,'t
As noted earlier the purpose of the ii ection 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 *
96
If the inspection sample size is 100 and the observed in
spection fraction defective is .20, the auditor should take
a sample of 19 items. If 3 or more defectives are found the
alternate hypothesis is accepted, if 2 or less are found the
null hypothesis is accepted. Situations where the audit
sample size required to reduce the standard deviation are
greater than the original inspection sample size are noted
by the remark that the "lot size is too small."
Tables in the back of the chapter are given for in
spection sample sizes of 100, 500, 1000, and 5000, and for
alternate hypotheses of y ]_ (ADR) = .50, .60, and .75.
The computer program for these sampling plans is given
in the Appendix and can be modified for different hypotheses,
alpha and beta errors, and inspection sample sizes by modify
ing the data card.
The equations used to determine the audit sample sizes
are based on the assumption of normality; however, the equi
valent value for the alpha and beta errors for the Camp Meidel
inequality were calculated and are printed in the tables.
The following procedure for Case 1 was used to deter
mine the minimum audit sample sizes and decision criteria.
For each hypothesis the following two equations are deter
mined:
ADR y (ADR)
Z
a (ADR)
TABLE 6.4
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 2
NUMBER INSPECTED = 1000 NUMBER AUDITED = 200
MQL Initial
.90
.90
.90
.75
.75
.75
.50
.50
.50
Initial
Accuracy
Observed
.90
.8978
.75
.7487
.50
.4961
.90
. 8999
.75
.7500
.50
.4996
.90
.9000
.75
.7502
.50
.5004
Expected
a (ADR)
Observed
.0716
. 0683
.1265
.1265
. 2270
.2311
.0416
.0410
. 0743
.0737
.1361
.1365
.0245
.0242
.0451
.0446
. 0871
. 0869
Expected
PI
Observed
. 0900
. 0901
.0750
.0751
.0500
. 0501
.2250
.2252
.1875
.1877
.1250
.1251
.4500
.4502
. 3750
. 3752
.2500
.2501
Expected
PR
Observed
.0099
.0101
. 0203
.0202
. 0263
. 0261
. 0290
.029 0
. 0577
.0575
.0714
.0710
.1818
.1817
. 1500
.1497
.1667
.1660
Correlation
.0361
.0404
.0307
. 0407
. 0373
.0280
.0393
.0354
.0245
Number Greater
than 1 a
331
304
301
311
308
317
309
308
319
Number Greater
than 2 o
35
36
44
46
44
46
41
43
49
21
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
GGI
NI
V (GGI) +
GDI
NI
V (GDI) +
DDI
NI
V (DDI) +
DGI
NI
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.
65
y(MQL) = population manufacturing quality level
y(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,
y(MQL) = .75
y (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
y(ADR)
.9000
.9002
PI
.2250
.2252
PR
.0323
.0323
0 (ADR)
. 0231
.0225
The table of
frequencies for
the 1000 ADR values
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.
accuracy is not being met based on the number of defectives
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 defectives 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.
xvxi
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:
y(ADR) = accuracy
a(ADR) = accuracy standard deviation
y(MQL) = manufacturing quality level
y(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
46
CHAPTER VIII
DOUBLE HYPOTHESIS SAMPLING PLANS FOR
INSPECTION ACCURACY
The sampling plans developed in this chapter are based
on a null hypothesis which states that the accuracy is equal
to some acceptable value and an alternate hypothesis which
states that the accuracy is equal to some unacceptable value.
To develop these sampling plans both the alpha and beta errors
are fixed. Figure 8.1 is a graphic presentation of the above
described classical statistical test. In tests of hypotheses
of this type two values need to be determined. The minimum
sample size necessary to reduce the standard deviation to
meet the alpha and beta errors and the actual decision value
need to be calculated.
The sampling plans developed are for a fixed inspection
sample size. Based on an observed inspection fraction de
fective, the audit sample size and the number of defects in
the audit sample to reject the null hypothesis are given.
For example in Table
8.1,
the hypotheses are:
Null hypotheses
^o
(ADR) =
.90
Alternate hypothesis
(ADR) =
.50
94
TABLE 5,7
AUDIT
AUDIT
1 
SAMPLE
FRACTIQM
SIZE
DEFECTIVE
ACCJRACY
S T Do 0 EVo
MQL
500
0,0900
Oo 7874
0,0268
0,6825
500
0,1200
Oo 7353
0,0275
0,6600
500
0,1600
Oo 6757
0,0276
0,6300
500
0,2000
Oo 6250
0,0271
0,6000
500
0,2600
Oo 5 618
0,0258
Oo 5550
500
0,3200
Oo 5102
0,0245
0,5100
500
0,4000
Oo 4545
0,0226
0,4500
500
0,5000
Oo 4000
0,0206
0,3750
500
0,6200
Oo 3497
0,0184
0,2850
500
0,7300
Oo 2 994
0,0161
0,1650
200
0, 0050
Oo 9852
0,0146
0,7463
200
0,0100
Oo 9709
0,0200
0,7425
200
0,0150
Oo 9569
0,0238
0,7388
200
0,0200
Oo 9434
0,0267
0,7350
200
0,0250
Oo 9302
0,0290
0,7313
200
0,0300
Oo 9174
0,0310
0,7275
200
0,0350
Oo 9050
0,0325
0,7238
200
0,0750
Oo 8163
0,0388
Oo 6938
200
0,1250
0r> 7273
0,0398
Oo 65 63
200
0,1750
Oo 6557
0,0384
0,6188
200
0,2250
Oo 5970
0,0361
Oo 5313
200
0,3000
Oo 5263
0,0325
0,5250
200
0,4000
Oo 4545
0,0281
0,4500
200
0,5000
Oo 4000
0,0244
0,3750
200
0,6500
Oo 3390
0,0201
O,2625
200
0,8500
Oo 2 817
0,0160
0, 1125
100
0,0100
Oo 9709
0,0282
0,7425
100
0,0200
Oo 9434
0,0376
0,7350
100
0,0300
Oo 9174
0,0434
0, 7275
OMTINUED
EASE 2
OQL
AEEURAEY
STD,DEV
, MQL
OQL
0,9681
0,7300
0,0432
0,6575
0,9327
0,9565
0,6400
0,0509
0,5094
0,8832
0,9403
0,5200
0,0604
0,5192
0,7750
0,9231
0,4000
0,0693
0,3750
0,5769
0,8952
INVALID
FDR EASE
2
0,8644
INVALID
FOR EASE
2
0,8182
INVALID
FOR EASE
2
^,7500
INVALID
FOR EASE
2
0,6477
INVALID
FOR EASE
2
^34583
INVALID
FOR EASE
2
0,9963
0,9850
0,0150
0,7462
0,9963
0,9926
0,9700
0,0212
0,7423
0,9923
0,9890
0,9550
0,0250
0,7332
0,9882
0,9353
0,9400
0,0300
0,7340
0,9840
0,9815
0,9250
0,0336
0,7297
0,9795
0,9778
0,9100
0,0358
0,7253
0,9748
n,9741
0,8950
0,0397
0,7207
0:9699
9439
0,7750
0,0582
0,6774
0,9217
0,9052
0,6250
0,0753
0,6000
0,8276
0,8654
0,4750
0,0893
0,4737
0,6625
0,8245
0,3250
0, 1014
0,2308
0,3273
9,7609
INVALID
FOR EASE
2
0,6716
INVALID
FOR EASE
2
0,5769
INVALID
FOR EASE
2
0,4234
INVALID
FOR EASE
2
0,1940
INVALID
FOR EASE
2
0,9913
0> 9700
0,0299
0,7423
0,9910
0,9826
0,9400
0,0422
0,7340
0,9813
0,9739
0,9100
0,0516
0,7253
0,9709
v
TABLE 6.1
RESULTS OF SIMULATION ANALYSIS BASED ON 1000 RUNS FOR CASE 1
NUMBER INSPECTED = 1000 NUMBER AUDITED = 500
MQL Initial
.90
.90
.90
.75
.75
.75
.50
.50
. 50
Initial
Accuracy
Observed
.90
. 9006
.75
.7516
.50
.5021
.90
.9002
.75
.7504
.50
.5006
.90
.9000
.75
.7501
.50
.5002
Expected
a (ADR)
Observed
. 0395
. 0386
. 0551
.0538
.0597
. 0580
. 0231
.0225
.0328
.0317
.0364
.0350
. 0140
.0136
.0208
.0200
.0242
.0232
Expected
PI
Observed
. 0900
.0901
.0750
. 0751
. 0500
.0501
.2250
.2252
.1875
.1877
.1250
.1251
.4500
.4502
.3750
. 3752
.2500
.2501
Expected
PR
Observed
. 0110
.0110
. 0270
.0271
.0526
. 0527
.0323
. 0323
. 0769
.0770
.1429
.1430
.0909
. 0910
.2000
.2001
.3333
.3335
Correlation
. 0378
.0389
. 0388
.0379
. 0396
.0391
.0390
.0404
. 0391
Number Greater
than 1 a
321
317
315
320
319
310
315
313
309
Number Greater
than 2 a
49
41
42
44
40
45
39
41
45
CTl
03
83
TABLE 7,6
MINIMUM NUMBER DF DEFECTIVE ITEMS FOUND BY THE
AUDITOR TO REJECT SPECIFIED INSPECTION ACCURACY,
INSPECTION SAMPLE SIZE = 1000,
AUDIT SAMPLE SIZE =. 200
ALPHA ERROR =0,050 (NORMAL)
ALPHA ERROR =0,082 (CAMP MEIDEL)
OBSERVED CASE 1 CASE 2
INSPECTION
FRACTION
DEFECTIVE
ACCURACY
= 0o 75
ACCURACY
= 0,90
ACCURACY
= 0,75
ACCURACY
=0,90
0,010
3
2
5
5
0,020
4
3
6
4
0,030
5
3
6
5
0,040
7
4
7
5
0,050
8
4
8
5
0,060
9
5
9
6
0,080
11
6
10
6
0, 100
13
7
12
7
0, 120
15
7
13
8
0,140
17
8
15
3
0,160
20
9
17
9
0, 180
22
10
19
10
0,200
25
11
20
11
0,250
31
13
25
13
0,300
38
16
31
15
0,350
47
19
37
18
0,400
56
23
45
21
0,450
67
27
53
25
0,500
81
31
63
29
9
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
Frequency
66
200
150
100
50
80 85 90 95 100
Observed Accuracy
Initial Conditions
Figure 6.1
Histogram of
Simulation Results
101
TA8LE 8a 3
AUDIT SAMPLE SIZE AND MINIMUM NUMBER 3F DEFECTIVES TO
REJECT
ACCURACY =0o
INSPECTION
90 AND
sample
ACCEPT
SIZE =
ACCURACY
1000
= 0
>50,
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
= 0o 050
= 0c 082
i (NORMAL)
(CAMP MEIDEL)
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
S IZE
l
AUDIT
DEFECTIVES
CASE
AUDIT
SAMPLE
SI ZE
2
AUDIT
DEFECTIVES
OaOlO
592
3
LOT SIZE
TOO
SMALL
Oo 020
253
3
LOT SIZE
TOO
SMALL
Oo 030
159
3
732
5
0 a 040
115
3
506
5
Oo 050
90
3
384
5
Oo 060
73
3
307
5
Oo 080
52
3
217
5
Oo 100
40
2
165
5
Oo 120
32
2
132
5
Oo 140
27
2
109
4
Oo 160
22
2
91
4
Oo 180
19
2
78
4
Oo 200
17
2
67
4
Oo 250
12
2
49
4
Oo 300
9
2
36
4
Oo 350
6
2
27
4
Oo 400
5
2
21
4
0 a 450
3
2
16
4
O 500
2
2
12
3
CHAPTER IX
CONCLUSION
Five possible measures of inspection accuracy were
investigated: the ratio of correct inspections, 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 have been reviewed.
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 was selected as the measure that should
be further examined. The ratio of defective product re
jected is determined by dividing the total number of de
fectives found by the inspector by the total number of de
fectives in the lot. During the actual production process
the only way to determine the total number of defectives in
the lot is by an audit reinspection of the lot to determine
how many defectives were missed by the original inspector.
Two types of sampling plans were derived. Single
hypothesis plans determine whether an acceptable inspection
accuracy is not being met, based on the number of defectives
114
TABLE OF CONTENTS (CONTINUED)
Page
APPENDIX 117
REFERENCES 128
BIOGRAPHICAL SKETCH 131
iv
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 Poissonlooking
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
41
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.
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 defectives necessary to
be found by the auditor.
11IPI)>)+(1PRAC )/
IFSIGUN +SIGAC GT.DEC) GO TO HO
NRDECB=NR
DR2=NRDECB*PI( IPI )*(1.0 + CONST#MGACADRAC)/I 1.0PI{IPIJl + 1.0
IDR2=DR2
GO TO 430
130 NRDEC=10
GO TO 100
140 NRDECB=20
430 IF(NRDEC.LT.0.0.AND.NRDECB.LT.^.0) GO TO 400
IF(NRDEC.LT.O.O.AND.NRDECB.GT.rt.O) GO TO 410
IFtNRDEC.GT.O.O.AND.NRDECB.LT.^.O) GO TO 420
WRITE(6500)PI(IPI).NRDECIDR1 NRDECB I DR2
GO TO 60C
400 WRITE(6,510) P I ( IPI )
GO TO 600
410 WRITE(6 520) PI(IPI),NRDECB,ID2
GO TO 600
420 WRITE(6,530) P I ( IPI ) NRDEC*I DR 1
600 CONTINUE
610 CONTINUE
STOP
END
127
115
found during the audit inspection. Doublehypothesis plans
determine whether a preselected acceptable inspection accur
acy level or a preselected unacceptable inspection accuracy
level is being attained. The double sampling plans require
that the audit sample size as well as the number of audit
defectives be stated in the sampling plan.
The sampling plans are affected by a major assumption
regarding the accuracy of the audit inspector. Two cases
are explored, one where the audit inspector is 100 percent
accurate and the other where the audit inspector accuracy is
equal to that of the initial inspector.
A study of the sampling plans reveals that large lot
sizes are generally required if an adequate measure of
inspector accuracy is to be determined. As the observed
inspection defect rate goes down, the lot size requirements
become fairly large. The audit sample size also must be
fairly large with respect to the original lot size as the
observed inspection defect rate goes down. For example, in
a doublehypothesis test with the acceptable accuracy of .90
and an unacceptable accuracy of .50, for an observed inspec
tion fraction defective of .01 the inspection sample size
should be 1000. Table 8.3 also shows that the audit sample
size should be 592 if a 100 percent accurate audit inspector
is assumed. For the case where the audit inspector has the
same accuracy as the initial inspector an even larger in
spection sample size would be required.
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,
ADR =
DR
DR*
and
ADR =
DDI
Since
DDI + DGI
DDI=DI,
and
DGI =
. (GI)(DR*) (GI)(DR)
NR (NR) (ADR) '
we obtain
DI
ADR =
nr 4. (GD'CdRT *
(ADR) (NR)
Solving for ADR and substituting the expected values
for DI/NI and DR/NR we obtain
. PI (1PI)PR
y (ADR)
PI
LIST OF TABLES (CONTINUED)
TABLE
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
Page
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 1000 101
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY = 0.90 AND
ACCEPT ACCURACY = 0.50, INSPECTION SAMPLE
SIZE 5000 102
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.50, INSPECTION
SAMPLE SIZE = 10000 103
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 100 104
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 500 105
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.60, INSPECTION
SAMPLE SIZE = 1000 106
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90 AND
ACCEPT ACCURACY = 0.60, INSPECTION SAMPLE
SIZE = 5000 107
AUDIT SAMPLE SIZE AND MINIMUM NUMBER OF
DEFECTIVES TO REJECT ACCURACY =0.90
AND ACCEPT ACCURACY = 0.60, INSPECTION
SAMPLE SIZE = 10000 108
IX
105
TABLE 8, 7
AUOIT SAMPLE SIZE AND MINIMUM NUMBER OF DEFECTIVES TO
REJECT
ACCURACY =3o
INSPECTION
90 AND ACCEPT ACCURACY.
SAMPLE SIZE = 500
=0,
60
ALPHA ERROR
ALPHA ERROR
= BETA ERROR
= BETA ERROR
= 0,050 (NORMAL)
=0,082 (CAMP MEIDEL).
OBSERVED
INSPECTION
FRACTION
DEFECTIVE
CASE
AUDIT
SAMPLE
SIZE
1
AUDIT
DEFECTIVES
CASE
AUDIT
sample
SIZE
2
AUDIT
DEFECTIVI
OoOlO
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,020
LOT SIZE
TOO SMALL
LOT SIZE
TOO
SMALL
0,030
411
4
LOT SIZE
TOO
SMALL
0 3 040
281
4
LOT SIZE
TOO
SMALL
Os 050
212
4
LOT SIZE
TOO
SMALL
Os 060
169
4
LOT SIZE
TOO
SMALL
Os 080
119
4
388
7
OsIOO
90
4
287
7
Os 120
72
4
225
7
0,140
59
4
183
6
Os 160
50
4
153
6
Oa 180
42
3
130
6
Oa 200
37
4
112
6
0,250
26
3
80
6
Os 300
20
3
60
6
0,350
15
3
46
6
Oo 400
11
3
35
5
Os 450
8
3
27
5
0,500
6
3
21
5
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Warren Menke, gnaapman
Associate Professor of Management
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
k UJ
Ralph Blodgett ]
Professor of Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
r />
oC
Elmo Jackson
Professor of Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
V / >'
s')
jL
Ralph Kimbrough
Professor of Education
14
scarcely one inspector in 10. A 6inch 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 22.
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
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 SAMPLE SIZE = 5000,
AUDIT SAMPLE SIZE = 2 500 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
vxi
35
mot. = 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
y(MQL) = 1PI (1PI)PR.
The true fraction defective in the inspection sample is
1MQL.
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
OQL
GGI
GGI + DGI
For the IOQL we have that
= GI,
GGI + DGI
DR=DR+5
GO TO 220
210 DR =DR +1 0
220 PR=DR/NR
J DR = DR
JNR = NR
IF(PR.GT.l.O) GO TO 110
T 1 = ( 1.0PI(IPI ))*PR
T2= PI( IPI J+Tl
T3= PI{ IPI JTl
RAD=SQRT((1.0/NI!INI)*PI
ADR 1 =P I( IPI )/T2
IF(DR.LE.l.O) GO TO 150
IF ISADRADR 1 LT.0.01) GO TO 160
IF(ADR 1 .GT..90) GO TO 150
IF(SADRADR i .LT.0.05JG0 TO 1*0
150 SADR =ADR 1
IF(ADR 1 .LT.0.25) GO TO 110
KOUNT =K0UNT+1
IFIKONTIN.EQ.O.AND.KOUNT.LT.271 GO TO 180
IF(KONTIN.GT.O.AND.KOUNT.LT.311 GO TO 180
WRITE!6 540) I TAB
WRITE!6,505)
K0NTIN=K0NTIN+1
KOUNT = 0
180 ADR 2 =T3/PI(IPI )
SIG1 =((PI(IPI)*T1>/
MQL1 = 10T 2
OQL1=1.0{NI(INI)*TlDR)/(NI(INi)*ci.oPI(IPI))DR)
KADR2=1000*ADR 2
KPI = 1000*PI
IFtKADR2.LE.KPI ) GO TO 170
SIG2 =(T1/PI( IPI) )*RAD
MQL2=1.OPIIIPI)Tl/ADR 2
00L2=1.0!NI(INI)*T1DR*ADR 2)/( !NI!IN I)*(l.OPIIPI))DR)*ADR 2)
WRITE!6,510)JNR,PR ,ADR 1,SIG1,MOL 1,00L1,ADR 2,SIG2,MQL2,0QL2
119
23
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
LIST OF TABLES
TABLE Page
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
/
v
TABLE S6
EXAMPLES OF ACCURACY EXPECTED VALUES OBTAINED FROM DIFFERENT INSPECTION AND
REINSPECTION TEST RESULTS, INSPECTION SAMPLE SIZE 1000,
INSPECTION FRACTION DEFECTIVE = 0,10
AUDIT
AUDIT
m n U M
CASE
1  
M M M
m *a as ma
 CASE 2 
SAMPLE
FRACTION
SIZE
DEFECTIVE
ACCJRACY
STDoDEVo
MQL
DQL
ACCURACY
STD,DEV
, MOL
OQL
900
0,0011
0o 9901
Os 0099
Oo 8990
1,0000
Os 9900
0,0100
0,3990
1,0000
900
Od 0033
0o 9709
Os 0166
Os 8970
1,0000
0,9700
0,0176
0,3969
0,9999
900
0, 0056
Oo 9524
Os 0208
Os 8950
,0000
0,9500
0,0229
0,3947
Os 9997
900
Os 0078
Oo 9346
Os 0239
Os 8930
I,0000
Os 9300
0,0274
0,3925
0,9994
900
OsOlOO
Oo 9174
Os 0264
0,8910
I,0000
0,9100
0,0313
0,3901
0,9990
900
Os 0222
Oo 8 3 3 3
Os 0340
Os 8800
,0000
0,8000
0,0490
0,3750
0,9943
900
Os 0333
Oo 7692
Os 0370
Os 8700
,0000
0,7000
0,0625
Os 8571
0,9852
900
Os 0444
Oo 7143
0> 0382
Oo 8600
IsOOOO
0,6000
0,0748
0,3333
0,9690
900
Os 0667
Oo 62 50
Os 038 3
Os 8400
,0000
0,4000
0,0980
0,7500
0,8929
900
Os 0889
Oo 5556
Os 0370
Os 8200
1,0000
0,2000
0, 1200
0,5000
0,6098
900
Os 1111
Oo 5000
0,0354
Os 8000
I,0000
INVALID
FOR CASE
2
900
Os 1444
Oo 4 348
0,0327
0,7700
l,0000
INVALID
FDR CASE
2
900
0,1778
Oo 3846
O,0302
0 7400
1,0000
INVALID
FDR CASE
2
900
Os 2222
Oo 3333
Oo 0272
Os 7000
Is oooo
INVALID
FOR CASE
2
900
Os 2889
Oo 277d
0,0236
Os 6400
1,0000
INVALID
FOR CASE
2
500
Os 0020
Oo 9823
0,0174
0,8982
0,9991
0,9820
0,0181
0,8982
0,9991
500
Os 0040
Oo 9 653
0,0239
Os 8964
0,9982
0,9640
0,0257
0, 3963
On 9981
500
Os 0060
Oo 9483
0,0284
Os 8946
0,9973
Os 9460
0,0316
0,3943
0,9970
500
Os 0080
Oo 9328
Os 0319
0,8928
0,9964
Os 9280
0,0367
0,3922
0,9958
500
OsOlOO
Oo 9174
0s 0346
Os 8910
0,9955
0, 9100
0,0412
0,3901
0,9945
500
Os 0120
Oo 9025
0,0369
O,8892
*,9946
0,8920
0,0453
0,8879
0,9932
500
Os 0200
Oo B475
0,0427
Os 8820
*,9910
Os 8200
0,0595
0,8780
0,9866
500
Os 0300
Oo 7874
0,0461
On 8730
fts9864
0,7300
0,0743
0,8630
0,9752
500
Os 0400
Oo 7353
0,0473
On 8640
0,9818
0,6400
0,0875
0,8438
0,9588
500
Os 0600
Os 6494
0,0469
Os 8460
0,9724
0,4600
0, 1113
0,7826
0,8996
ui
