ON ESTIMATION OF PROPORTION OF CONFORMANCE Teresa Lam Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI 48109-2117 C. Ming Wang National Institute of Standards & Technology Boulder, CO 80303 Technical Report 93-20 August 1993

On Estimation of Proportion of Conformance C. Teresa Lam University of Michigan, Ann Arbor, MI 48109 C. Ming Wang National Institute of Standards and Technology, Boulder, CO 80303 Abstract This paper presents results on point and interval estimation problems for proportion of conformance. Proportion of conformance is defined as the proportion of products with quality characteristic inside the specification limits. Five point estimators are presented and compared with respect to their root mean squared error. Two approximate methods for constructing lower confidence limits are proposed and the performance of each is assessed by simulation. In tlie case when the nominal value of the quality characteristic is not centered in the middle of the specification range, a modified proportion of conformance is introduced. Numerical examples are also given to illustrate the procedures. Dr. Lam is an Assistant Professor in the Industrial and Operations Engineering Department. She is a member of ASQC. Dr. Wang is a Mathematical Statistician in the Statistical Engineering Division. He is a member of ASQC. 1

Introduction The quality of any manufacturing product is ultimately determined by customer satisfaction. The purpose of any quality improvement process should aim to improve customer satisfaction by reducing and eliminating defects, and to continuously improve processes throughout the organization, thereby reducing sources of variation and improving quality and productivity. The quality of a product can usually be quantified by observable characteristics of the product or the manufacturing process which produces the product. The performance of a quality characteristic can often be specified by a nominal (target) value, and a tolerance (acceptable) region. These specifications are set by engineering requirements or by customers. Ideally, the quality characteristic value should be at the nominal value with no variation. In reality, variation in a manufacturing process is unavoidable and hence there is a statistical distribution associated with the quality characteristic. Let X' be the quality characteristic of interest with a nominal value T, a lower specification limit L and an upper specification limit U. Also, let PL = P[X < L] and pu = P[X > U] be the proportion of products with its quality characteristic X below L and above U, respectively. If pc represents the proportion of conformance, that is, the proportion of products with its quality characteristic X. within the specification limits L and U, then it is given by pc = P[L < X < U] = 1 -PL -PU (1) Proportion of conformance is a measure of how well the output of a process meets the specification limits. In recent years, process capability indices have been widely used for assessing the performance of manufacturing processes. However, as pointed out by Pearn, IKotz, and Johnson (1992), the underlying motives for the introduction of process capability indices are clearly related to monitoring the I)roplortion of conforming products. Proportion of conformance is, thus, a more direct and easily understood measure. Furthermore, under proper distributional assumptions, proportion of conformance can usually be estimated not only for univariate but also multivariate quality characteristics over fairly general shapes of tolerance regions. Since there is no consensus on how to extend the capability indices developed for univariate processes to multivariate processes, proportion of conformance is 2

perhaps a preferable measure of process performance. The minimum variance unbiased estimator (MVUE) of PL (and pu) has been derived by several authors. In particular, Folks, Pierce, and Steward (1965) derived the MVUE of PL under various distributional assumptions of X. When X is normally distributed, Wheeler (1970) obtained the variance of the MVUE, and Owen and Hua (1977) derived confidence limits on PL and pu. Chou and Owen (1984) constructed one-sided simultaneous confidence limits for PL and pu when X has a normal distribution. However, there is no known general procedure for constructing confidence intervals for pc. In this paper, we consider point and interval estimation of pc when X is normally distributed with unknown mean y/ and variance a2. Specifically, we compare five different estimators of pc with respect to root mean squared error of the estimate. We also present two methods for constructing approximate lower confidence limits for pc. Only lower confidence limits are considered here since, in practice, we would like pc to be greater than some threshold in order for us to conclude that a process is capable. Simulations are used to compare the five different estimators and to assess the performance of the proposed confidence limits. When the nominal value T is centered between L and U, i.e., T = M = (L + U)/2, the tolerance region for the quality characteristic is symmetric. If T is off-center, the tolerance region is not symmetric. Also, if T = Al, pc is maximized when,u = T (for fixed a). However, in many assembly-fit processes, the nominal value of the quality characteristic is off-center, indicating that deviation in one direction is less acceptable than deviation in the other. In such cases, we would like to maxinize pc with the additional constraint that the majority of products be produced near the nominal value. We introduce a modified pc which is maximized when the process mean /c is located at the nominal value T for both symmetric and nonsymmetric tolerance regions. A lower confidence limit is also given for the modified proportion of conformance. 3

Point Estimation In this section, we consider five different estimators for pc defined in equation (1). In the discussion below, let XI,X2,...,X be a random sample from a normal distribution with unknown mean. and variance a2. Let X = Enl Xi/n and S2 = Eni(Xi-X)2/(n-1) be the sample mean and sample variance respectively. Define rci = (y - L)/u and /C2 = (U - 1)/c7. Then, pc can be written equivalently as PC = ((DC2)- -(-1) where <4(.) is the cumulative distribution function of the standard normal. Let K1 and K2 be the estimators of sr and rc2, respectively, obtained by substituting X for pa and S for a. The five estimators of pc we consider are: 1. Uniformly minimum variance unbiased estimator (UMVUE): From Wheeler (1970), the UMVUE of pu = 1 - pu is given by 0 if KI2 < -(n - 1)//n( Pu = < 1 if IK2 > (n- l)/l/n (2) 7n-2 (wu) otherwise where T- 2(-) is the cumulative distribution function of a Student's t distribution with, 2- 2 degrees of freedom and wU = [n(n - 2)]1/2K2 [(n - 1)2 - nK2]1/2. Similarly, the U'lVU\ E /L for PL is given by equation (2) with K2 replaced by K1. Since 1 if L<X <U I(.X,)= { 0 otherwise is an unbiased estimator for pc, it follows from the Blackwell-Rao Theorem (e.g., see Mood, Graybill and Boes (1974) page:321) that the UMVUE of pc is PC(iUMA'lE) = E[I(X1) I X, S]. lUsing the arguments given by Johnison and IKotz (1970, page 73) and Wheeler (1970), we find that PC(UMvVUE) = PU - PL 4

2. Modified maximum likelihood estimator I (MMLE1): Since X and S(n - 1)/n are maximum likelihood estimators for I and a respectively, it follows that PC(MMLE1) f ( K2) -4 (-/ KI1) is the maximum likelihood estimator for pc. 3. Modified maximum likelihood estimator 2 (MMLE2): If we estimate # by X and a by the biased estimator S, then pc can be estimated by PC(MMLE2) = ~ (KI2) - (-K1). 4. Modified maximum likelihood estimator 3 (MMLE3): Alternatively, we can estimate a by its unbiased estimator S/C4 and pc by PC(MMLE3) = ( (C4K2) - ( (-C4Ki) where v'_r(n/2) C4 = c:4 = 'n-Tr((n - 1)/2) and F(-) is the gamma function. 5. Modified maximum likelihood estimator 4 (NMMLE4): It is readily verified that \/iTK2 follows a noncentral t distribution with n - 1 degrees of freedom and noncentrality parameter V/h2. From Johnson and Kotz (1970, page 203), we know that cK2 is an unbiased estimator for K2 where ~r((, - 1)/2) Cx/ - F((n - 2)/2) Similarly, cKl is an unbiased estimator for s1. This suggests that we can estimate pc by PC(AMMILEI) = - (c'K2) - i (-cKA'). 5

We compare the five estimators listed above using the root mean squared error criterion. The root mean squared error of an estimator pjc of pc is defined as the square root of EK, 2 [(C -pC)2] which is a function of Kci and K2.- We want to choose an estimator with the smallest root mean squared error for all practical values of *1 and KC2. The root mean squared errors of the five estimators can be computed either by numerical integrations or by simulations. Simulations are conducted in this paper to evaluate the root mean squared errors. Given n, a standard normal random deviate Z and an independent chisquared random deviate Y with n - 1 degrees of freedom were generated using IMSL (1987) routines RANNOR and RNCHI. For each pair of values ci and K2, the resulting values r,1 + z/,/ K2 - z/I~ A, = and K2 = 2 (3) /Y/(n-1) /Y/(n-1) were used to calculate each of the five estimates of pc. Monte Carlo estimates of the root mean squared error for each pc were obtained by repeating the procedure above 10000 times. Simulations were carried out for sample size n = 5(1)15(5)30 and a wide range of values of K1 and K2. Figure 1 shows the root mean squared errors of the five estimates when n = 5 and K1 = 3.0. Tlhe root mean squared errors of all estimates attain their maximum at K2 = 0.0, corresponding to the situation where the process mean it is located at the upper specification limit U. This is nlot surprising since a nornlal density function has the most mass concentrated around nwan op and with it = U, a small discrepanci in sample value of K2 would result in a large difference in pc. As K2 deviates from zero, the root mean squared errors decrease rapidly for all five estimators. Similar conclusions apply to other values of n, Kc1 and K2. In practice, we expect both sK and K2 to be positive and greater than 3.0, meaning that tihe process mean lies within the specification limits and the process standard deviation conlsumes no more than one-sixth of the specification range (see for examples, Kane (1986) and MIcFadden (1993)). In Figure 1, UMhVUE has the lowest root mean squared error whereas MMILE4 has the highest for all K2 > 3. Keeping K1 = 3.0, the maximum differences of the 6

root mean squared errors among the five estimates over the range of K2 > 2.0 are computed for different sample sizes and are plotted in Figure 2. As the sample size increases, this maximum difference decreases rapidly. When n > 15, the root mean squared errors of the five different estimation methods become essentially indistinguishable from each other. The performance of UMVUE and MMLE1 are very close for all sample size and all practical value of cl and C2. Both can be recommended in practice. Example As an illustration, consider n = 30, K1 = 2.4 and K2 = 3.0. In this case, PC(UMVUE) = 0.99351, PC(MMLE1) = 0.99154, PC(MMLE2) = 0.99045, PC(MMLE3) = 0.98986 and PC(MMLE4) = 0.98855. Note that PC(MMLE1) > PC(MMLE2) > PC(MMLE3) > PC(MMLE4). This is as expected since V~2/(n- 1) > 1 > 4 > c. Confidence Limits for Proportion of Conformance We are interested in constructing lower confidence limits on pc; i.e., with a pre-specified confidence coefficient 1 - a, we seek p(a, AK, K2) such that PAT1 AT2 [IC > p(a, i, )] 1 - a. Since PC = 1 - Pu we first obtain confidence limits for PL aind pU and then attempt to find a confidence limit for PC based on these limits. A 1 - I upper confidence limit, pi = pi(a, I1), for pi, satisfies P[IL < Pi] = 1 - a. Owen and Hua (1977) showed that p) can be obtained by solving the equation PT,_, [Tn_- (-V/'- (P1)) ~< VnKi] =1 - a (4) where 7',(6) is distributed as a noncentral t distribution with v degrees of freedom and noncentrality parameter 6, and <>-l(.) is the inverse distribution function of the standard 7

normal. A 1 - a upper confidence limit, P2 = P(a, K2), for pu is similarly obtained by solving the equation PT.n- [Tn-1 (-V -1 (P2)) < X/nK2] =1-a. (5) Now, P[pL + PU < Pi + P2] > P[PL < Pl, PU < P2] > 1 - P[pL > Pl]- P[PU > p2] = - 2a. The last inequality is justified by the Bonferroni inequality (e.g. see Graybill (1976), page 360). Thus, a conservative lower-limit for a 100(1 - 2a)% one-sided confidence interval for pc is given by PC > 1- - P2. (6) Values of Pi and p2 can be found in Odeh and Owen (1980, Table 7). They provide tables of pi and P2 for 1 - a = 0.5, 0.75, 0.9, 0.95, 0.975, 0.99, 0.995 and for n = 2(1)18(3)30, 40(20)120, 240, 600, 1000, 1200. The Bonferroni inequality has been used by other authors in many similar problems, especially, in interval estimation for variance components, e.g., see Williams (1962), Wang (1991, 1992). In variance components problems, it was shown, analytically or numerically, that the confidence coefficient 1 - 2a can be replaced by 1 - a. This also seems to be the case for the interval in (6) as indicated by the simulation studies described below. To evaluate the true confidence coefficient of the interval (6), simulation studies described in the previous section were again carried out. For each value of cl and n2, Kl and K2 were conmputed as given in equation (3). Also. pi and p2 were obtained by solving equations (4) and (5). respectively. The procedure was repeated 10000 times and the percentage of times that 1 - pi - P2 < PC = ((n2)- )(-lI) was recorded. Table 1 reports the results for 1 - a = 0.95, n = 10,30,50, and for some selected values of K, and K2. Since the interval in (6) is symmetric about il and 22, only cases with 2 > I1 are shown in the table. 8

The results indicate that the proposed interval is successful in maintaining the stated confidence level. The sample size n seems not to play an important role on the converge of the interval except when either Ic or K2 are negative, a rare situation in practice. In Table 1, the proposed confidence interval is conservative when ri. is near K2, and as the difference between Kc1 and K2 increases (or pu becomes smaller) the confidence coefficient approaches to 1 - a (0.95 in this example). The latter can be verified by observing that P[pc > 1 - P1 - P2] = P[PL + PU < P1 + P2] = P[PL < P1] + P[PL + PU < Pi + P2, PL > P] - P[PL + PU > P1 + P2, PL < Pi < PL + PU] and the second and third terms vanish as pu approaches to zero. A similar argument can also be used to verify the findings reported by Kushler and Hurley (1992) regarding the coverage of several confidence intervals for the process capability index Cpk = min(Ki/3, '2/3). That is, the intervals are conservative when T is near T and give nominal coverage when, moves away from T. In the case when,1 is near K2, an alternative lower confidence limit p* = p*( K, IK2, K2) is derived in the appendix and is given by p \ + max(l) Xe — ) ( min (K,2) X ):n-1 The simulated confidence coefficient corresponding to confidence limit p* is given in Table 2. Again, the interval is successful in maintaining the nominal confidence level and the sample sizes considered have little effect on the performance of the interval. For the purpose of comparing with confidence limit 1 - pi - p2, Table'2 also includes the simulated value of E[p*]/E[1 - P1 - P2]. From Tables 1 and 2, we can conclude that confidence limit 1 - P1 - P2 is less conservative and produces a tighter bound for most cases. On the other hand, the confidence limit p* is easier to compute and performs better when both 1i and 2 are small and are close to each other. However, p* is derived based on the assumption that the probabilities of K1 < 0 or KA2 < 0 are negligible, meaning that l lies between L and U. Although in practice, we 9

expect the process mean to lie within the specification limits, nevertheless, it is a limitation for p*. Examples For illustration, again consider n = 30. The 95% lower confidence limits 1 - pi - P2 and p* for pc are tabulated in Table 3 for a few combinations of K1 and K2. For other values of c, n, K1 and K2, a computer program to compute these lower confidence limits is described in Lam and Wang (1993). Table 3 confirms that the confidence limit 1 - pi - P2 provides a tighter bound for pc. Modified Proportion of Conformance It is common that the nominal value of a quality characteristic is not centered between the upper and lower specifications. This generally occurs in an assembly-fit process where deviation of the quality characteristic in one direction is less acceptable than deviation in the other. The primary goal for design engineers in this case is not to simply maximize the proportion of conformance. Instead, the engineers want to maximize proportion of conformance under the additional constraint that the majority of products be produced near to the nominal value. Consider the two processes A and B in Figure 3, both processes have the same standard deviation. Hlowever, the mean of process A is located at the center of the specification range A1! andl the mean of process B is located at the nominal value T. Obviously, the proportion of confornmance pc of process A will be greater than that of process B even though the mean of process B is at the nominal value. Hlowever, process B is preferable to process A if the intent of the design engineers is to penalize deviation toward the lower specification less than deviation toward the upper specification. This important fact is ignored in computing the proportion of conformance of processes A and B. This problem can be overcome by defining a modified proportion of conformance as discussed in Littig and Lam (1993). This modified proportion of conformance is maximized 10

when the process mean is located at the nominal value. Since deviation in one direction is penalized more than deviation in the other, it is reasonable to use different distance scales in measuring deviation from the nominal value. In particular, we choose constants dl and d2 such that T-L U -T dl d2 The modified proportion of conformance, pM, is then given by ( U- T T )- (( - L if T > p d2a dl - dia pm = <. (7) U - 2 TdL ) 2c - if T < ( d2 Or doa d2 O In this paper, we choose d2 = 1 whenever T - L > U - T and dl = 1 whenever T - L < U - T. This choice of d1 and d2 ensures that processes C and D in Figure 4 has the same modified proportion of conformance. This is a desirable property since processes C and D and their corresponding design specifications are mirror reflection of each other and should be considered as performing equally well in meeting design specifications. Furthermore, it is readily verified that pm defined in equation (7) is indeed maximized when the process mean pt is located at the nominal value T. Also, if the mean of process E is T - r(T - L) and the mean of process F is T + r(U - T) where 0 < r < 1 (Figure 5), then both processes have the same modified proportion of conformance. This is reasonable since both processes have the identical proportion of allowable process mean deviation below and above the nominal value. The maximum likelihood estimator of pa;n can be readily obtained by replacing X for a and S(n -- 1)/n for a in equation (7). Ju(lgilg by the results on pc in the previous section, this should be an adequate estimator for p I. Let p = (U - T)/(T - L) be the relative location of T to the specification limits L and U. In particular, p = 1 means that the nominal value is centered between the specification limits and pr = pc. It is readily verified that equation (7) is equivalent to = 1 - -p - 11

where "if2- Pri > o 1+p { max(l, 1/p))I M 13L (K (1 + K2) + (K 1 - N21P) if K2 -PKI (I + I1p)max(l, p) i +p and ~((K1 + K2) +(rt,2- pr,,) ifK2 -PK1. (I + p) max(l, 11p) i M PU Kf K2 -PI1> Tlax(lp)} if p - In the equation above, (r12 - prj)!(1 + p) = (T - ji)/o. Hence, we would expect I - Ircl to increase as u deviates from the nominal value T. The same technique used to derive the confidence limit for pc can be extended to obtain the lower confidence bound for the modified proportion of conformance pm. In particular, this confidence limit can be obtained by solving for pm = p7(a, K1, K2) and pm = pm(a, Ki, K2) from the following equations. If (K2 - pK1)/l1 + p) ~ 0, the equations are PT.1 [Tn-1 (-#VPmax (1, l/p) -1 (pml) < #nK1] = 1 - a, T(l) rtKi + K2) + (K2 - (8) TnI T-1(- Jf max (1, l/p) <D-1(pm )) < -a Otherwise, we solve for pm and pm in the following equations. \iY[(Ki + A2) + (K1 - K2/p)l PT,,_1 7TT (-#n max (1, p) (D-1(pm)) ~ 1+ 1j 1 (9) P-r,, [Tn-1 (-M /7max(lp) 1 (p~)) ~ fnK2] = 1 - a. A 1 - (TI lower-limit one-sided confidence interval for the modified proportion of conformance is then given by 1 - pm - pg. Again, We carried out simulation studies described earlier to evaluate tie true confidence coefficient of this lower confidence limit. Simulations were carried out for I - a = 0.95, n = 10, 30, 50, p = 0.25, 0.5, 0.75 and a variety of combinations of K1 and K2. Note that the lower confidence bound is no longer symmetric about 'ci and 12

K2 and we have to consider both i1 > K2 and r1 < K2. Since pC = pc when p = 1 and the definition of pC ensures that processes C and D in Figure 4 have the same modified proportion of conformance, it is only necessary to consider p < 1 in our simulation. Table 4 reports the results for p = 0.75. The results for p = 0.25,0.5 are very similar and hence omitted here. From Table 4, it is clear that the proposed confidence interval is again conservative. Also, as K2 - pli deviates from zero (,i deviates from T) and keeping K1 or K2 fixed (pL or pm become smaller), the confidence coefficient approaches 1 - a. This fact can also be verified using a similar argument as given for pc. Examples The 95% lower confidence limit 1 - p - p for pm is tabulated in Table 5 for n = 30, p = 0.75 and some common combinations of K1 and IK2. For other values of a, n, p, K1 and KI2, a computer program in Lam and Wang (1993) can be used. Note that the lower confidence limit is not symmetric in K1 and K'2. For example, when K1, = 2.4 and K12 = 3.0 (with 2 - pK1 = 1.2), the 95% lower confidence limit is 0.8954, while K1 = 3.0 and K2 = 2.4 (with KI2 - pKi = 0.15), the 95% lower confidence limit is 0.9208. This is not surprising since K2 - pK1 is smaller in the later case indicating that even though both cases have the same sample standard deviation, the sample mean of the later case is closer to the nominal value. Conclusions Wle have considered both point and interval estimation for the proportion of conformance and a modified proportion of conformance. Proportion of conformance measures bow well the output of a process meets the specification limits. If the objective is to meet design specifications and at the same time to requlire that the majority of products be produced near the nominal value, then the modified proportion of conformance can be used. The computer program used to obtain point estimates and confidence limits for pc and p1 is described in Lam and Wang (1993). 13

All statistical procedures developed in this paper are based on the assumption that the quality characteristic of interest can be modeled well by a normal distribution. As discussed in Littig and Lam (1993), many quality characteristics such as flatness and goodness of surface finish are modeled better by a skewed distribution such as a three-parameter gamma distribution. Also, in many applications quality characteristics and tolerance regions are multi-dimensional such as the hole location problem in a gear carrier with a circular tolerance region (see, for example, Littig, Lam and Pollock (1993)). These topics as well as the statistical tolerance interval problems for the distribution of Pc will be the subject of future research. Acknowledgment Dr. Lam was visiting the National Institute of Standards and Technology under a joint ASA/NSF/NIST fellowship program while this research was carried out. This work is a contribution of the National Institute of Standards and Technology and is not subject to copyright in the United States. Appendix We seek p* = p*(ac, K1, K2) such that 1 -a = P[P[L< < ] > p'] = PXS [P[X - KlS < < X + K2S] > p ] 'pX - At IVS X - <X- + K2S] _= S p< < - + >P = (PzI r ) (+ lK2 - _ P ] ng 71t - 1 ) en(- n 1 j where Z is a standard normal random variable, Y is a chi-squared random variable with n-l1 degrees of freedom and is independent of Z. If the probabilities that KI < 0 and '2 < 0 14

are negligible, or if u lies between L and U, a result described in Wald and Wolfowitz (1946) can be used to approximate the above equation. Specifically, for u > 0, Ez [o (Z/n+ — u) - $ (Z/nu - u)] is closely approximated by (1/Vn + u) - 4 (1/ - u) the difference being of the order 1/n2. Making use of this approximation, an approximate lower confidence limit p* can be obtained from the following equation. Py ~ =+ K2 n- -~ v /< _Kjn — _ >P* =1-a. (10) I 1 / Y ) ( -1 ] Let q be the solution of I) 7 — + K2q -I (x/ Klq p(\ ) n lq) p* then by equation (10) PY [/Y/(n-l) > ] = 1 -or q= \/X:n-l/(n - 1) where \2 is the a percentile of a chi-squared distribution with v degrees of freedom. Taking into account of symmetry, the result follows. References [1] Chou, Y.; and Owen, D. B. (1984). "One-Sided Confidence Regions on the Upper and Lower Tail Areas of Normal Distribution". Journal of Quality Technology 16, pp. 150 -158. [2] Folks, J. L.; Pierce, D. A.; and Stewart, C. (1965). "Estimating the Fraction of Acceptable Product". Technometrics 7, pp. 43-50. 15

[3] Graybill, F. A. (1976). Theory and Applications of the Linear Model. Duxbury Press, North Scituate, MA. [4] IMSL Inc. (1987). The IMSL Math/Library. Houston, Texas. [5] Johnson, N. L.; and Kotz. S. (1970). Continuous Univariate Distributions-1. John Wiley & Sons, New York. [6] Kane, V. E. (1986). "Process Capability Indices". Journal of Quality Technology 18, pp. 41-52. [7] Kushler, R. H.; and Hurley, H. (1992). "Confidence Bounds for Capability Indices". Journal of Quality Technology 24, pp. 188-195. [8] Lam, C. T.; and Wang, C. M. (1993). "Lower Confidence Limits for Proportion of Conformance". submitted to Journal of Quality Technology. [9] Littig, S. J.; and Lam, C. T. (1993). "Case Studies in Process Capability Measurement". ASQC: 47th Annual Quality Congress pp. 569-575. [10] Littig, S. J.; Lam, C. T.; and Pollock, S. M. (1993). "Capability Measurements for liulltivariate Processes: Definitions and and Example for a Gear Carrier". Technical rep:)ort #142, Industrial and Operations Engineering Department, University of Michigan. [11] Mlood, A. N.; GraSbill, F. A.; and Boes, D. C. (1974). Introduction to the Theory of Statistics. MNcGraw-Hill, New York. [121] l cFadden, F. R. (1993). "Six-Sigmna Quality Programs". Quality Progress 26, 6, pp. 37-1'2. [13] Odeh, R. E.; and Owen, D. B. (1980). Tables for Normal Tolerance Limits, Sampling Plans, and Screening. Marcel Dekker, Inc., New York. [1-4] Owen, D. B.; and Hua, T. A. (1977). "Tables of Confidence Limits on the Tail Area of the Normal Distribution". Com1rnunications in Statistics - Simulation and Computation B6, pp. 285-311. 16

[15] Pearn, W. L.; Kotz, S.; and Johnson, N. L. (1992). "Distributional and Inferential Properties of Process Capability Indices". Journal of Quality Technology 24, pp. 216 -231. [16] Wald, A.; and Wolfowitz, J. (1946). "Tolerance Limits for a Normal Distribution". Annals of Mathematical Statistics 14, pp. 45-55. [17] Wang, C. M. (1991). "Approximate Confidence Intervals on Positive Linear Combinations of Expected Mean Squares". Communications in Statistics - Simulation and Computation B20, pp. 81-96. [18] Wang, C. M. (1992). "Approximate Confidence Intervals on Linear Combinations of Expected Mean Squares". Journal of Statistical Computation and Simulation 43, pp. 229-241. [19] Wheeler, D. J. (1970). "The Variance of an Estimator in Variables Sampling". Technometrics 12, pp. 751-755. [20] Williams, J. S. (1962). "A Confidence Interval for Variance Components". Biometrika 49, pp. 278-281. Key Words: Capability Indices, Confidence Interval, Modified Proportion of Conformance, Point Estimation. 17

Table 1. Simulated confidence coefficients for confidence limit 1 - pi - P2 of PC 1 - a = 0.95 n i K1 2 10 30 50 1.0 1.0 0.9889 0.9937 0.9941 1.0 2.0 0.9760 0.9772 0.9783 1.0 3.0 0.9603 0.9570 0.9563 1.0 4.0 0.9496 0.9481 0.9496 1.0 5.0 0.9479 0.9471 0.9490 1.0 6.0 0.9474 0.9471 0.9490 2.0 2.0 0.9728 0.9738 0.9742 2.0 3.0 0.9604 0.9586 0.9583 2.0 4.0 0.9491 0.9486 0.9491 2.0 5.0 0.9470 0.9475 0.9486 2.0 6.0 0.9469 0.9474 0.9486 3.0 3.0 0.9657 0.9661 0.9654 3.0 4.0 0.9542 0.9521 0.9507 3.0 5.0 0.9472 0.9471 0.9475 3.0 6.0 0.9468 0.9470 0.9474 4.0 4.0 0.9621 0.9611 0.9614 4.0 5.0 0.9503 0.9498 0.9493 4.0 6.0 0.9459 0.9478 0.9480 5.0 5.0 0.9597 0.9587 0.9588 5.0 6.0 0.9196 0.9479 0.9493 6.0 6.0 0.9577 0.9574 0.9573 4.0 -1.0 0.9823 0.9596 0.9538 5.0 -2.0 0.9948 0.9558 0.9517 6.0 -3.0 0.9999 0.9561 0.9518 7.0 -1.0 0.9514 0.9491 0.9491 8.0 -2.0 0.9506 0.9500 0.9497 9.0 -3.0 0.9518 0.9517 0.9512 18

Table 2. Simulated confidence coefficients for confidence limit p* of pc 1 - a = 0.95 n Kl K2 l 10 30 50 1.0 1.0 0.9815a 1.470b 0.9750a 1.169b 0.9707a 1.116b 1.0 2.0 0.9786 1.121 0.9737 1.036 0.9747 1.021 1.0 3.0 0.9738 1.020 0.9701 0.999 0.9691 0.997 1.0 4.0 0.9726 0.986 0.9688 0.992 0.9663 0.993 1.0 5.0 0.9719 0.977 0.9682 0.989 0.9658 0.992 1.0 6.0 0.9718 0.974 0.9681 0.991 0.9658 0.992 2.0 2.0 0.9795 1.043 0.9737 1.014 0.9701 1.010 2.0 3.0 0.9742 0.998 0.9718 0.995 0.9717 0.995 2.0 4.0 0.9714 0.975 0.9707 0.988 0.9709 0.993 2.0 5.0 0.9712 0.967 0.9705 0.988 0.9707 0.992 2.0 6.0 0.9712 0.955 0.9705 0.987 0.9707 0.992 3.0 3.0 0.9765 1.000 0.9720 1.000 0.9694 1.000 3.0 4.0 0.9694 0.989 0.9676 0.997 0.9692 0.998 3.0 5.0 0.9668 0.983 0.9676 0.996 0.9686 0.997 3.0 6.0 0.9668 0.981 0.9676 0.996 0.9686 0.998 4.0 4.0 0.9735 0.99 0.9710 1.000 0.9681 1.000 4.0 5.0 0.9665 0.994 0.9651 0.999 0.9668 1.000 4.0 6.0 0.9643 0.992 0.96-19 0.999 0.9667 1.000 5.0 5.0 0.9697 0.998 0.9696 1.000 0.9674 1.000 5.0 6.0 0.9643 0.998 0.9629 1.000 0.9643 1.000 6.0 6.0 0.9677 0.999 0.9)6712 1.000 0.9661 1.000 asimulated confidence coefficient bsimulated value of E[p*]/E[1 - p, - p2] 19

Table 3. 1 Lower confidence limits for pc - a = 0.95 and n = 30 K K2 1 - pl - p2 p* 2.4 3.0 0.9519 0.9490 3.0 3.0 0.9771 0.9789 3.0 4.0 0.9875 0.9842 4.0 4.0 0.9979 0.9979 4.0 6.0 0.9989 0.9984 20

Table 4. Simulated confidence coefficients for confidence limit of pC 1 - a = 0.95 and p = 0.75 n Kl K2 - PK1 10 30 50 1.0 0.0 0.9899 0.9957 0.9958 1.0 1.0 0.9767 0.9792 0.9800 1.0 2.0 0.9648 0.9639 0.9633 2.0 0.0 0.9732 0.9777 0.9786 2.0 1.0 0.9625 0.9611 0.9625 2.0 2.0 0.9515 0.9512 0.9513 3.0 0.0 0.9670 0.9684 0.9675 3.0 1.0 0.9559 0.9539 0.9541 3.0 2.0 0.9480 0.9479 0.9481 4.0 0.0 0.9624 0.9626 0.9625 4.0 1.0 0.9518 0.9506 0.9515 4.0 2.0 0.9464 0.9179 0.9482 5.0 0.0 0.9597 0.9591 0.9598 5.0 1.0 0.9506 0.9486 0.9505 5.0 2.0 0.9469 0.9470 0.9486 6.0 0.0 0.9581 0.9579 0.9577 6.0 1.0 0.9494 0.9484 0.9504 6.0 2.0 0.9462 0.9470 0.9491 n K2 2 1 - K2/P 10 30 50 1.0 0.0 0.9838 0.9893 0.9904 1.0 1.0 0.9750 0.9823 0.9819 1.0 2.0 0.9620 0.9662 0.9648 2.0 0.0 0.9683 0.9704 0.9699 2.0 1.0 0.9615 0.9659 0.9645 2.0 2.0 0.9516 0.9559 0.9538 3.0 0.0 0.9624 0.9626 0.9625 3.0 1.0 0.9559 0.9570 0.9562 3.0 2.0 0.9487 0.9518 0.9517 4.0 0.0 0.9591 0.9586 0.9595 4.0 1.0 0.9523 0.9526 0.9528 4.0 2.0 0.9475 0.9499 0.9508 5.0 0.0 0.9572 0.9570 0.9570 5.0 1.0 0.9508 0.9511 0.9510 5.0 2.0 0.9481 0.9492 0.9503 6.0 0.0 0.9560 0.9560 0.9561 6.0 1.0 0.9503 0.9501 0.9506 6.0 2.0 0.9485 0.9487 0.9501 21

Table 5. Lower confidence limits for pc 1 - a = 0.95, p = 0.75 and n = 30 |K l Kz 1 -pi - P2 2.4 3.0 0.8954 2.4 4.0 0.9082 2.4 6.0 0.9104 3.0 2.4 0.9208 3.0 3.0 0.9428 3.0 4.0 0.9542 3.0 6.0 0.9560 4.0 2.4 0.9538 4.0 3.0 0.9788 4.0 4.0 0.9880 4.0 6.0 0.9894 6.0 2.4 0.9633 6.0 3.0 0.9884 6.0 4.0 0.9987 6.0 6.0 0.9998 22

o 0 *1/..'. ',,....... UMVUE.,//... \ -..-., *MMLE1 MMLE2 /./'/ '\ \ -- MMLE3 o^~" ".... * ' —MMLE4 -F-. —..:.I -2 0 2 4 6 KC2 Figure 1: Root mean squared error of five different estimators for pc when n = 5 and KI = 3.0 23

CY) 0 - 0 \ c5 0 0 - 10 15 20 25 30 Sample size, n Figure 2: Maximum difference in root mean squared errors among the five estimators over the range of K2 > 2.0 and K1 = 3.0 24

A B L M T U Figure 3: Process B is preferable to process A since it is centered at the nominal value T 25

L T U L T U Figure 4: Processes C and D are mirror reflection of each other 26

E L T-r(T-L) T T+r(U-T) U Figure 5: Processes E and F have the same modified proportion of conformance for any 0 < r < 27