a process incapability index

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International Journal of Quality & Reliability ManagementA process incapability index

Michael Greenwich Betty L. Jahr-Schaffrath

Ar tic le information:To cite this document:Michael Greenwich Betty L. Jahr-Schaffrath, (1995),"A process incapability index", International Journal of Quality &Reliability Management, Vol. 12 Iss 4 pp. 58 - 71Permanent link to this document:http://dx.doi.org/10.1108/02656719510087328

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Mats Deleryd, (1998),"On the gap between theory and practice of process capability studies", International Journal of Quality

& Reliability Management, Vol. 15 Iss 2 pp. 178-191 http://dx.doi.org/10.1108/02656719810204892

Jann-Pygn Chen, Cherng G. Ding, (2001),"A new process capability index for non-normal distributions", International Journalof Quality & Reliability Management, Vol. 18 Iss 7 pp. 762-770 http://dx.doi.org/10.1108/02656710110396076

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IJQRM12,4

58

A process incapability indexMichael Greenwich and Betty L. Jahr-Schaffrath

Depar tment of Mathemat ics, Computer Science and Stat istics, PurdueUniversity Calumet, Hammond, Indiana, USA

IntroductionThere have been a number of process capability indices proposed over the yearsfor the purpose of assessing the capability of a process to meet certainspecifications. Kane[l] investigated comprehensively theC

pandC

pkindices and

their estimators. Generally, theCpkindex takes into account the processvariation as well as the location of the process mean relative to the specificationlimits while the Cpindex reflects only the magnitude of the process variation.Chan et a l.[2] developed the Cpmindex in order to take into account thedeparture of the process mean from the target (nominal) value. TheCpmindex isdefined mathematically as

USL is the upper specification limit, LSL is the lower specification limit and Tis the target (nominal) value such that LSL < T < USL.

A general form ofCpmisC*pmwhich was defined by Chanet al.[2] as

whereD= min[(USL T)/3, (T LSL)/3].

In this article, C*pmis used for both CpmandC*pm(except in those instanceswhereCpmneeds to be specified) sinceCpmis merely a special case ofC*pmwhereUSL T=T LSL. Unfortunately, the statistical properties of the estimator,

for C*pmare analytically intractable, regardless of the choices for the variationestimator, 2, and the mean estimator,. This process capability index is also

( )

,*C D

T

pm=+ 2 2

C D

pm* ,=

C

E X T

pm=

=

USL LSL

where6

2

[( ) ],

Received July 1993Revised April 1994

International Journal of Quality& Reliability Management,Vol. 12 No. 4, 1995, pp. 58-71,MCBUniversity Press,0265-671X

The research for this article was supported partially by the School of Liberal Arts and Sciences,Purdue University Calumet. The authors also wish to thank Mr R.J. Whittaker and his IntegratedProcess Control Group, LTV Steel Co., for their support.


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A processincapability

index

59

discussed in Spirings article[3]. Boyles[4] pointed out that the index C*pmisidentical to the process capability index proposed by Taguchi[5]. A table whichgives the critical values thatCpmmust exceed in order to conclude thatCpm> 1at various significance levels was provided by Boyles[4].

A simple transformation of theC*pmindex,Cpp, will be introduced in thisarticle. This Cppindex can be regarded as a process capability index or moreprecisely as a process incapabilityindex. Since the transformation is bijective,theCppindex contains the same information as that ofC*pm; namely, inaccuracy(the departure of the process mean,, from the target value,T) and imprecision(the magnitude of the process variation, 2). Moreover, theCppindex provides

an uncontaminated separation between information concerning the processaccuracy and precision while this kind of information separation is notavailable with the C*pmindex. Often, C*pmis employed to assess processcapability because the process accuracy is of significance. Therefore, thisinformation separation is highly beneficial because it indicates to what degreethe process inaccuracy contributes to the process being incapable of meetingthe specifications.

Recently, Pearn et al.[6] introduced a new (so-called third generation)process capability index,Cpmk, in an attempt to detect a smaller departure of theprocess mean from the target value. However, the degree of this processinaccuracy (large or small) is indicated explicitly withCpp as a result of theinformation separation by the transformation. Moreover, since theCpmkindex is

sensitive to the process inaccuracy, it may produce a small value for a process,indicating that the process is not capable of meeting the specification althoughthe process has a high proportion (for instance, greater than 99.73 per cent) ofconforming output. The same is true for CpmandCpp(but to a lesser extentsince they are less sensitive to process inaccuracy). The separated informationcan be used to segregate processes (which are indicated to be less capable by aprocess capability index) between those with a high proportion of conformingoutput and those with a low proportion of conforming output.

The process incapability index, Cpp

LetCppbe defined as

That is,

and, since2= ( T)2+2, it follows that

CD

pp=

2

,

C

Cpp

pm

=

1 1

2

*. ( )


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IJQRM12,4

60

whereis the process mean and2 is the process variance. Due to the relation(1),Cppassumes a smaller value for a process more capable of meeting itsspecifications and a larger value for a less capable process. A process is mostcapable whenCpp= 0. For this, the process mean must be at the process target(=T)andthe process variance must be zero (2 = 0). Any non-zero value ofCppindicates some degree of incapability of the process. Thus,Cppis a processincapabilityindex.

A (,) plot can be constructed forCppby plotting the contours ofCpp=kforvarious kvalues. They are a set of (,) values satisfying the equation,

( T)2+2=kD2.

These contours are semicircles centred at=Twith radiuskD, which are the

same as those of Cpm=c= 1/(kD) plotted in Boyles[4, Figure 5]. The more

capable the process, the smaller the semicircle is. This geometric interpretationofCppis immediate from the transformation (1).

The first and second terms of the right-hand side of (2) exclusively andrespectively reflect the process inaccuracy (departure of the process mean fromthe process target) and the process imprecision (process variation). Thisdecomposition (non-contaminated information separation) is not available withC*pm. If the process variation is negligible (that is, the second term is close tozero), thenCppprovides a concrete measure of process centring.

For instance, ifCpp= 1, the process mean,, is located within one-third of thedistance between the target, T, and the nearest specification limit. The right-hand side of (2) also shows that if the process shifts away from its target, thenCppincreases. If the process variance increases, thenCppincreases. When theprocess mean and variance both change,Cppreflects these changes additively.

This indicates that Cpphas the necessary properties for assessing processcapability, using both the departure from the target value and the magnitude ofthe process variation.

Let ( T)2/D2 be denoted byCia(inaccuracy index) and (/D)2 by Cip

(imprecision index), then

Cpp= Cia+ Cip.

The inaccuracy index is the squared ratio of the distance between the processmean and the process target to one-third of the distance between the processtarget and the nearest specification limit. The imprecision index is the squaredratio of the process standard deviation to one-third of the distance between theprocess target and the nearest specification limit. Since the denominators areidentical, these subindices provide the relative magnitudes of the contributionsto the process in capability indicated byCpp. In fact,

C T

D Dpp=

+

2 2

2, ( )


5/17

provide the proportions of the process incapability contributed by thedeparture of the process mean from the target and by the process variation,respectively.

An example of three processes with LSL = 10,T= 13 and USL = 16 is givenin Pearnet al.[6, Figure 2] and is used to illustrate that Cpmfails to distinguishbetween on-target and off-target processes. The three processes are Adistributed asN(13.00, 1.00), B distributed asN(13.50, 0.87) and C distributed as

N(13.87, 0.50) whereN(, ) stands for a normal distribution with meanandstandard deviation. The value ofCppis one for each of A, B and C and, hence,Cppalso fails to distinguish between on-target and off-target processes.However, the values of (Cia, Cip) are (0, 1) for A, (0.25, 0.75) for B and (0.75, 0.25)for C. These processes are clearly distinguished by the separated information.

The values of the indices for the three processes are summarized in Table I.One disadvantage that Cpmkhas failed to overcome is that it cannot

distinguish processes of high conforming output proportions from those of lowconforming output proportions. For instance, the value of 0.83 for Cpmkcanoccur from B, which has high conforming output, as well as from a processdistributed as N(13, 1.20) with the same LSL, USL and T, which has lowconforming output.

All three processes given in Pearnet al.[6, Figure 2] are capable because their

conforming output proportions (COP) are higher than or equal to 99.73 per cent.The new process capability index,Cpmk, produces a value of 1.00 for A, a valueof 0.83 for B and a value of 0.71 for C although the conforming outputproportion of C is higher than that of B, which is, in turn, higher than that of A.

The same result happens toCpm, but to a lesser degree, with a larger departure(| T | > D ) of the process mean from the target value. This is illustrated byPearn et al.[6, Figure 1]. In the figure, three processes (A ofN(14.00, 4/3), B ofN(16.00, 2/3) and C of N(13.87, 1/2)) are presented with LSL = 10,T= 14 andUSL = 18. The values ofCpmkare respectively 1.00, 0.32 and 0.11 for A, B and C.

Those of Cpm(Cpp) are 1.00 (1.00), 0.63 (2.50) and 0.44 (5.13) for the threeprocesses respectively. With these values, all three indices indicate that B and Care not capable. However, they are highly capable of turning out a high

proportion of products that meet the specifications. The values of the indices forthe three processes are summarized in Table II.

100 100C

C

C

C

ia

pp

ip

pp

andA process

incapabilityindex

61

Table I.Values of various

indices for processes,A, B and C

Cpkm Cpp Cia Cip Ccop

A 1.00 1.00 0.00 1.00 1.00B 0.83 1.00 0.25 0.75 1.00C 0.71 1.00 0.75 0.25 1.00


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IJQRM12,4

62

The two subindices can be used to identify processes with high conformingoutput proportions by computing

IfCcop 1, then the process has a conforming output proportion of at least 99.73per cent (assuming normality). The values ofCcopfor the three processes inPearn et al.[6, Figure 2] are all identical to 1, indicating that these threeprocesses are indeed capable of producing at least 99.73 per cent of the time

products which meet the specifications. If another process is distributed asN(17, 1/4), thenCpm= 0.44 andCpp= 5.10 (Cia= 81/16 andCip= 9/256) for thesame process target and specification limits as in Pearn et al.[6, Figure 2].However,Ccopis 0.75, indicating a very high (far greater than 99.73 per cent,assuming normality) conforming output proportion.

Confidence intervals forCpp

based on a small sampleLetXbe the normally distributed process quality characteristic of interest, andX1,,Xnbe a random sample of nobservations ofXtaken while the process isin control. It is assumed that such a normal sample of a relatively small samplesize nis available for inference onCppin this section. Let X

and s2 denote the

usual sample mean and variance respectively.

Kushler and Hurley[7] compared the performances of a variety of confidencebounds (intervals) for Cpmusing the actual miss rates of the differentconfidence intervals. The miss rate is the probability of a confidence intervalmissing the true value ofCpm. If the method (and the approximation used in themethod) of constructing a confidence interval is good, then the miss rate isclose to the nominal rate of in the 100(1 ) confidence. According to themiss rates comparisons, the method suggested by Boyles[4] performs betteroverall than other methods such as the central2 method, the Bonferronimethod and the method suggested by Chanet al.[2].The unbiased estimator for 2,

, ( )2

2

1 3=

( )=

X T

n

i

i

n

CC

Ccop

ip

ia

=

3

3.

Table II.Values of variousindices for processes,A, B and C

Cpkm Cpm Cpp Ccop

A 1.00 1.00 1.00 1.00B 0.32 2.50 0.63 1.00C 0.11 5.13 0.44 1.00


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index

63

will be used in this article. This estimator was originally proposed byTaguchi[5] and used by Boyles[4]. The variance of the estimator is smaller thanthe mean squared error of the biased estimator proposed by Chanet al.[2].Under normality, the estimatoris also the maximum likelihood estimator(MLE) for. Using (3), an unbiased and consistent point estimator forCppcan beobtained as

Intervals with 100(1 ) per cent confidence forCppwill be obtained byapproximating a non-central2 distribution with a scaled2 distribution byequating the first two moments of the distributions. This is the sameapproximation method used by Boyles[4] in obtaining confidence intervals forCpm.The quantity2/2 is approximately distributed as2m/mwhere

This means that the pivotal quantity is approximately distributed as2m/msince

where cup(p; k) is the upper ppercentage point of the2 distribution with k

degrees of freedom, mis the MLE formgiven as

,

*

*

m

n X T

s

X T

s

=

+

+

1

1 2

2

2

2

.

( / ; ),

( / ; ), ( )

C

C

C

mC

m

mC

m

pp

pp

pp

pp pp

=

2

2

2 1 24

Then,an approximate 100(1 ) confidence interval for is given as

cup cup

m

n T

T

=

+

+

1

1 2

2

2

2

.

.CD

pp=

2

2


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IJQRM12,4

64

The upper confidence bound is likely to be of interest.An unbiased and consistent point estimator forCipcan be defined obtained as

This bias factor shows that the quantity is asymptotically unbiased but couldbe considerably biased for a small sample size, n, and a large standarddeviation, .

Table II I shows 30 computer-generated observations (rounded to threedecimal places) using a normal distribution with mean 13.50 and standarddeviation 13/15. Using a process target of 13 and upper and lower specificationlimits of 16 and 10, respectively,Cpp= 1.00 results. Using the same USL, LSL,Tand the observations in Table III, confidence intervals forCppand estimates for

the sub-indices will be determined as an example. A 95 per cent confidence levelwill be used.

.

.

( )

.

C s

D

C

C C C

D

s

D

X T

D

C

nD

ip

ia

ia pp ip

ia

=

=

=

2

2

2

2

2

2

2

2

2

2

An unbiased and consistent point estimator for can be obtained as

The quantity

can be used to estimate but it is biased by the factor

( ; ),

( ; ). ( )

mC

m

mC

m

pp pp

cupand 0,

cup

1

5

( ) / .*s n s n

Cpp

= 1 2and The lower and upper 100(1 ) confidence bounds (inter-

vals) for are respectively

Table III.Computer-generatednormal observations

12.921 13.363 14.632 15.270 12.72714.281 13.626 13.468 12.664 13.04013.588 14.656 13.556 11.920 13.11714.528 12.417 13.906 14.497 12.45311.951 14.639 12.852 13.367 11.90113.409 14.807 12.874 13.705 13.275


9/17


index

65

The point estimate of^Cppis 1.01 with the two-sided 95 per cent confidence

interval of [0.65, 1.75] by (4). This interval is equivalent to the 95 per centconfidence interval of [0.76, 1.24] forCpm. The one-sided 95 per cent confidenceintervals for Cppare [0.69,) and (0, 1.60] by (5). These are equivalentrespectively to the one-sided 95 per cent confidence intervals, (0, 1.20] and [0.79,), forCpm. The estimate, 1.01, is close to the true value, 1.00, ofCppand these95 per cent confidence intervals contain the trueCpp.The point estimate of

^Cipis 0.83, slightly higher than the trueCipof 0.75. The

point estimate^Ciafor Ciais 0.18 (1.01 0.83), determined by subtracting the

point estimate forCipfrom that ofCpp. Thus, the estimated proportion of theprocess incapability due to inaccuracy is 17.8 per cent which is determined bydividing ^Ciaby

^Cpp. The estimated proportion of the process incapability due to

the imprecision is 82.2 per cent. Both estimates are close to the true proportions.These relative proportions suggest that the imprecision contributes more to theincapability of the process and that, in order to improve the process, the processvariation should be reduced before attempting to adjust the process to thetarget.

Boyles[4] conducted simulation studies to assess the accuracy of theapproximate lower confidence bounds forCpmby the procedure based on thesame approximation. The accuracy of approximation was reported to besatisfactory. This is expected to hold for the approximate confidence intervalsfor Cppintroduced in this section since Cppand its confidence intervals are abijective transformation ofC

pmand its confidence bounds.

Confidence intervals for Cpp

based on a large sampleUsing a large sample sample (n 80), the confidence intervals both for Cppand for its subindices are developed in this section. It is assumed thatthe process quality characteristic of interestXhas the fourth moment (that is,E(X4)


10/17

IJQRM12,4

66

Due to the theorems and the corollary by Serfling mentioned above,

Using this estimator, a 100(1) per cent confidence interval for Cppcan beconstructed as

where generally zprepresents the upper 100pper cent point of the standardnormal distribution. One-sided 100(1 ) per cent confidence intervals arerespectively,

is asymptotically normally distributed with meanCiaand variance2ia/nwhere

By substitutingXands*2for and2, an estimator for2iais obtained as

( )

.*

iaX T s

D

22 2

4

4=

ia

T

D

22 2

4

4=

( ).

/ , , / . ( )

,

( ) ,

C z n C z n

C

C X T

D

pp pp pp pp

ia

ia

+

=

and

An estimator for

0 7

2

2

/ ,

/ , ( )/ /C z n C z n pp pp pp pp +

2 2 6

( )

/

( ) ( ),

[( ) [( ) ]

, ,

*

*

C

D

X T

D

s

D

C n

T

D

T

D D

E X E X

X s M

X X

pp

pp pp

pp

i

i

= =

+

=

+

+

= =

=

( )=

2

2

2

2

2

2

2

22 2

4

3

4

44

4

33

44

23

3

4 4

is asymptotically normally distributed with mean and variance where

] and .

By substituting the sample moments,

114

4

1

23 4

2

22 2

4

3

4

44

4

4 4

n

i

i

n

pp

pp

nM

X X

n

X T s

D

M X T

D

M s

D

=

( )

=

+

+

=and

for and , an estimator for is obtained as

,

, ,

( ) ( )

.* *


11/17


index

67

Using this estimator, a 100(1 ) per cent confidence interval forCiacan beconstructed as

One-sided 100(1 ) per cent confidence intervals are respectively,

Similarly, an estimator forCia,

is asymptotically normally distributed with meanCipand variance2ip /n where

By substituting s*4andM4 for 4and4, an estimator for

2ipis obtained as



According to2pp,2iaand

2ip, the two subindices,

Ciaand Cip, are uncorrelated ifthe underlying distribution is symmetric (and hence 3 = 0). If the process

distribution is normal, then these estimators are independent as expected.An estimator forCcop,

whereCiaandCipare estimators forCiaandCipdefined earlier in this section, isasymptotically normally distributed with meanCcopand variance

2cop /nwhere

,C

C

Ccop

ip

ia

=

3

3

/ , , / . ( )C z n C z n ip ip ip ip

+

and 0 11

/ , / . ( )/ /C z n C z n ip ip ip ip +[ ] 2 2 10

.*

ipM s

D

2 44

4=

ip

D

2 44

4=

.

,C s

Dip=

2

2

/ , , / . ( )C z n C z n ia ia ia ia

+

and 0 9

/ , / . ( )/ /C z n C z n ia ia ia ia +

2 2 8


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IJQRM12,4

68

By using the same substitution with the sample moments, an estimator for 2copis obtained as



In order to assess the accuracy of the approximate confidence intervals forCpp,Cia, CipandCcop, simulation studies were conducted for 95 per cent and 99 percent confidence levels. A simulation study consisted of taking 10,000 randomsamples from a process and constructing a confidence interval (based on eachof the 10,000 samples) for every one of the four indices.

The ratio of the number of confidence intervals that contain the index to10,000 (the total number of intervals) is the observed coverage rate. Theaccuracy of the approximation can be measured by how close the observedcoverage rate is to the significance level preset for the intervals. This simulationstudy was conducted for normal and non-normal processes using varioussample sizes.

The results of the simulation studies generally indicate that the observedcoverage rates tend to be lower than the preset significance level but theapproximation accuracy seems to be satisfactory for a sample size of 200 orgreater. A typical result of one such study is given in Table IV. The observedcoverage rates of 95 per cent two-sided confidence intervals for the indices ofthe same normal process used as a numerical example in the last section are

given. As expected, the accuracy is lower for a smaller sample size. Incidentally,

/ , , / . ( )C z n C z n cop cop cop cop

+

and 0 13

/ , / . ( )/ /C z n C z n cop cop ip cop +

2 2 12

( ) ( )

( )

( ).

* *

*cop

s

D X T

M

D X T

M s

s D X T

24

4

3

3

44

2 2

9

3

9

3

9

4 3=

++

++

+

cop

D T D T D T

24

4

3

3

44

2 2

9

3

9

3

9

4 3=

++

++

+( ) ( )

( )

( ).

Table IV.The observed coveragerates

n Cpp Cia Cip Ccop

100 93.57 93.60 92.51 93.35200 94.16 94.10 93.95 94.26300 94.51 94.47 95.25 94.28400 94.60 94.21 94.18 94.66500 94.59 94.64 94.13 94.45


13/17


index

69

the width of a confidence interval for a given significance level becomes wideras the sample size gets smaller.

Since the observed coverage rates are indicated to be generally lower than theconfidence level, the approximation accuracy of the confidence intervals can beimproved by using innkinstead of n as the denominator in (6), (7), (8), (9),(10), (11), (12) and (13). For instance, further simulation studies indicate that thedenominator of n 16 seems to work well for normal processes in general.

Table V shows the observed coverage rates of the same confidence intervals

Table V.The observed coverage

rates withn 16

n Cpp Cia Cip Ccop

50 95.56 95.21 94.60 95.4960 95.39 95.00 94.48 95.2570 95.15 95.19 94.82 94.8980 95.01 95.07 94.20 94.8890 95.06 94.69 94.24 94.91

100 95.24 95.26 94.51 95.25200 94.84 95.10 94.67 94.88300 95.20 95.08 94.58 95.10400 95.00 94.64 94.67 95.10500 94.99 94.99 94.50 94.77

Table VI.

Computer-generatednormal observations

13.452 13.312 13.122 12.740 15.25812.276 13.158 14.473 15.386 13.97112.197 13.973 11.938 14.971 12.44713.883 13.188 14.450 14.383 13.37513.544 13.316 14.144 12.490 14.34513.341 13.938 13.003 13.467 11.96812.505 11.487 13.982 12.702 12.05312.315 13.765 13.852 13.056 14.66414.156 14.592 14.254 12.505 13.03013.228 12.656 13.680 12.594 14.13513.933 13.895 13.321 13.020 13.36213.193 13.952 15.123 14.015 13.34614.095 12.686 14.633 12.661 15.09013.548 13.090 13.780 12.211 13.418

14.194 14.144 12.866 14.068 14.08412.925 14.928 12.074 12.369 12.765

Two-sided One-sided One-sided

Cpp [0.68, 1.28] [0.72,) (0, 1.23]Cia [0.02, 0.42] [0.05,) (0, 0.39]Cip [0.54, 0.98] [0.58,) (0, 0.94]Ccop [0.86, 1.21] [0.89,) (0, 1.18]

Table VII.95 per cent confidence

intervals


14/17

for the identical normal process as those listed in Table IV. However, this time,n 16 is used with various sample sizen.

With the correction factor k = 16, the accuracy seems to be satisfactory for asample size as small as 50. However, the confidence interval, usingn 16 forthe denominator, is wider than a confidence interval usingn.

Table VI shows 80 computer-generated observations (rounded to threedecimal places) from the same normal distribution used to generate the 30observations in Table III. Using these 80 observations with the same target andspecification limits used in the example given in Table III, confidence intervalsforCppand its subindices will be constructed as an example of the large samplemethod introduced at the beginning of this section. The confidence level used is95 per cent.The point estimate ofCppis 0.98 with the two-sided 95 per cent confidence

interval of [0.68, 1.28] by (6) while the one-sided 95 per cent confidence intervalsforCppare [0.72,) and (0, 1.23] by (7). This estimate is close to the true value,1.00, of Cppand these 95 per cent confidence intervals contain the trueCpp.The point estimate ofCiais 0.22 with the two-sided 95 per cent confidence

interval of [0.02, 0.42] by (8) while the one-sided 95 per cent confidence intervalsforCiaare [0.05,) and (0, 0.39] by (9). This estimate is close to the true value,0.25, of Ciaand these 95 per cent confidence intervals contain the trueCia.The point estimate ofCipis 0.76 with the two-sided 95 per cent confidence

interval of [0.54, 0.98] by (10) while the one-sided 95 per cent confidenceintervals for

Cipare [0.58,

)and (0, 0.94] by (11). This estimate is close to the

true value, 0.75, ofCipand these 95 per cent confidence intervals contain thetrueCip.

Similarly, the point estimate ofCcopis 1.04 with the two-sided 95 per centconfidence interval of [0.86, 1.21] by (12) while the one-sided 95 per centconfidence intervals forCcopare [0.89,) and (0, 1.18] by (13). This estimate isclose to the true value, 1.03, ofCcopand these 95 per cent confidence intervalscontain the trueCcop. These confidence intervals are summarized in Table VII.

Using the confidence intervals developed in this section, a sample size of 80provides practical confidence intervals forCppand its subindices with adequateaccuracy as demonstrated with the simulation studies and numerical examplegiven in Table VI. If a subgroup of size 5 is used for an

X-chart to monitor a

process, 80 observations constitute 16 points on the chart. The number ofobservations, 80, is not prohibitively large for many processes.

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a process incapability index

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