COMPUTER PROGRAM FOR CALCULATING THE p-VALUE IN
TESTING PROCESS CAPABILITY INDEX C
pmkW. L. PEARN1∗AND P. C. LIN2
1Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, Republic of China 2Center of General Education, National Chin-Yi Institute of Technology, Taiwan, Republic of China
SUMMARY
Many process capability indices, including Cp, Cpk, and Cpm, have been proposed to provide numerical measures
on the process potential and performance. Combining the advantages of these indices, Pearn et al. (1992) introduced a new capability index called Cpmk, which has been shown to be a useful capability index for processes
with two-sided specification limits. In this paper, we implement the theory of a testing hypothesis using the natural estimator of Cpmk, and provide an efficient Maple computer program to calculate the p-values. We also provide tables of the critical values for some commonly used capability requirements. Based on the test we develop a simple step-by-step procedure for in-plant applications. The practitioners can use the proposed procedure to determine whether their process meets the preset capability requirement, and make reliable decisions. Copyright
2002 John Wiley & Sons, Ltd.
KEY WORDS: process capability index; testing hypothesis; critical value; p-value
1. INTRODUCTION
Process capability indices, including Cp, Cpk, and
Cpm, have been proposed in the manufacturing
industry to provide numerical measures on whether a process is capable of reproducing items meeting the quality requirement preset in the factory. Combining the advantages of these indices, Pearn et al. [1] introduced a new capability index called Cpmk, which
has been shown to be a useful capability index for processes with two-sided specification limits. V¨annman [2] constructed a unified superstructure for the four basic indices, Cp, Cpk, Cpm, and Cpmk.
The superstructure has been referred to as Cp(u, v),
which can be defined as
Cp(u, v) = d − u|µ − m|
3σ2+ v(µ − T )2,
where µ is the process mean, σ is the process standard deviation, T is the target value preset by the product designer, d = (USL − LSL)/2 is half of the length of the specification interval, m = (USL + LSL)/2 is the mid-point between the lower and the upper specification limits (LSL and USL), and u, v 0. It is easy to verify that Cp(0, 0) = Cp, Cp(1, 0) = Cpk, ∗Correspondence to: W. L. Pearn, Department of Industrial Engineering and Management, National Chiao Tung University, 1001 TA Hsueh Road, Hsinchu, Taiwan 30050, Republic of China. Email: roller@cc.nctu.edu.tw
Cp(0, 1) = Cpm, and Cp(1, 1) = Cpmk, which have
been defined explicitly as Cp= USL− LSL 6σ , Cpk= min USL− µ 3σ , µ − LSL 3σ , Cpm= USL− LSL 6σ2+ (µ − T )2, Cpmk = min USL− µ 3σ2+ (µ − T )2, µ − LSL 3σ2+ (µ − T )2
The index Cpmkis constructed [1] by combining the
yield-based index Cpkand the loss-based index Cpm,
taking into account the process yield (meeting the manufacturing specifications) as well as the process loss (variation from the target). When the process mean µ departs from the target value T , the reduced value of Cpmk is more significant than those of Cp,
Cpk, and Cpm. Hence, the index Cpmkresponds to the
departure of the process mean µ from the target value T faster than the other three basic indices Cp, Cpk,
and Cpm, while it remains sensitive to the changes
of process variation (see [1]). We note that a process meeting the capability requirement ‘Cpk C’ may
Received 2 April 2001
not be meeting the capability requirement ‘Cpm
C’. On the other hand, a process meeting the capability requirement ‘Cpm C’ may not be
meeting the capability requirement ‘Cpk C’ either.
The discrepancy between the two indices may be contributed to the fact that the Cpk index primarily
measures the process yield, but the Cpmindex focuses
mainly on the process loss.
However, if the process meets the capability requirement ‘Cpmk C’, then the process must meet
both capability requirements ‘Cpk C’ and ‘Cpm
C’ since Cpmk Cpkand Cpmk Cpm. According to
today’s modern quality improvement theory, reduction of the process loss is as important as increasing the process yield. While Cpk remains the more popular
and widely used index, Cpmk is considered to be an
advanced and useful index for processes with two-sided specification limits.
2. DISTRIBUTION OF THE ESTIMATED Cpmk
For a normally distributed process that is demonstra-bly stable (under statistical control), Pearn et al. [1] considered the maximum likelihood estimator (MLE) of Cpmk: ˆCpmk = min USL− X 3 S2 n+ (X − T )2 , X − LSL 3 S2 n+ (X − T )2 , where X = n i=1 Xi/n and Sn2= n i=1 (Xi − ¯X)2/n
are the MLEs of µ and σ2, respectively. We note that
Sn2+ (X − T )2=
n
i=1
(Xi− T )2/n,
which is the major part of the denominator of ˆCpmk, is
the uniformly minimum variance unbiased estimator (UMVUE) of
σ2+ (µ − T )2= E[(X − T )2] in the denominator of Cpmk.
Under the assumption of normality, Pearn et al. [1] obtained the rth moment and the first two moments, as well as the mean and the variance of ˆCpmk
for the common cases with T = m. Chen and
Hsu [3] showed that the estimator ˆCpmk is consistent,
and asymptotically unbiased. V¨annman and Kotz [4] obtained the distribution of the estimated Cp(u, v) for
cases with T = m. V¨annman [5] further provided a simplified form for the obtained distribution. The cumulative distribution function of ˆCpmk can,
therefore, be expressed (using our notation) as FˆCpmk(x) = 1 − b√ n/(1+3x) 0 G (b√n − t)2 9x2 − t 2 × [φ(t + ξ√n) + φ(t − ξ√n )] dt (1) for x > 0, where b = d/σ , ξ = (µ − T )/σ , G(·) is the cumulative distribution function of the chi-squared distribution χn−12 , and φ(·) is the probability density function of the standard normal distribution N(0, 1).
In practice, sample data must be collected in order to calculate the index value; therefore, a great degree of uncertainty may be introduced into capability assessments due to sampling errors. The approach of simply looking at the index value calculated from the given sample and then making a conclusion on whether the given process is capable or not is intuitively reasonable but not reliable because sampling errors are ignored. Taking into account the sampling errors, we implement the theory of a testing hypothesis using the natural estimator of Cpmk, and provide an efficient Maple computer
program to calculate the p-values for making reliable decisions. This approach is similar to the one proposed by Cheng [6] for testing process capability Cpm.
We also provide the tables of the critical values for some commonly used capability requirements. Using these tables, the practitioners may choose not to run the computer program. Based on the test we develop a simple step-by-step procedure for in-plant applications. The practitioners can use the proposed procedure to judge whether or not their process meets the preset capability requirement (capable) and runs under the desired quality condition.
3. TESTING THE PROCESS CAPABILITY To test whether a given process is capable using the index Cpmk, we consider the following statistical
testing hypotheses:
H0: Cpmk C (process is not capable),
H1: Cpmk > C (process is capable).
Based on a given α(c0) = α, the chance
of incorrectly concluding an incapable process as capable, the decision rule is to reject H0if ˆCpmk > c0
> # Input parameter values LSL, USL, T, C, n, X_bar, Sn.
LSL:=2.40; USL:=3.40; T:=2.90; C:=1.00;
n:=100; X_bar:=2.865; Sn:=0.125;
d:=(USL - LSL)/2; ξ =(X_bar - T)/Sn;
c1:=(d - abs(X_bar - T))/(3*(Snˆ2 + (X_bar - T)ˆ2)ˆ0.5): # Note that c1 = c* = Cpmk_hat.
b:=(3*C*(1 + ξ ˆ2)ˆ0.5 + abs(ξ )): G:=(c1,t)->stats[statevalf,cdf,chisquare[n-1]] ((b*nˆ0.5 - t)ˆ2/(9*c1ˆ2) - tˆ2): h:=t->stats[statevalf,pdf,normald](t + ξ *nˆ0.5) +stats[statevalf,pdf,normald](t - ξ *nˆ0.5): pV:=c1->int(G(c1,t)*h(t),t=0..(b*nˆ0.5/(1 + 3*c1))): Estimated_Cpmk:=c1; p_Value:=evalf(pV(c1)); The output is:
LSL:= 2.40 USL:= 3.40 T:= 2.90 C:= 1.00 n:= 100 X bar:= 2.865 Sn:= 0.125 d:= 0.500000000 ξ := −0.2800000000 Estimated Cpmk:= 1.194075384 p Value:= 0.02529584382. Figure 1.
Given values of α and C, the critical value c0can
be obtained by solving the equation P ( ˆCpmk c0 |
Cpmk = C) = α using available numerical methods.
For processes with a target value set to the mid-point of the specification limits (T = m), the index may be rewritten as Cpmk = d − |µ − T | 3σ2+ (µ − T )2 = d/σ − |ξ | 31+ ξ2, where ξ = (µ − T )/σ .
Given Cpmk = C, b = d/σ can be expressed as b =
3C1+ ξ2+ |ξ|. Given a value of C (the capability
requirement), the p-value corresponding to c∗, a specific value of ˆCpmk calculated from the sample
data, is (by equation (1))
P { ˆCpmk c∗| Cpmk = C} = b√ n/(1+3c∗) 0 G (b√n − t)2 9(c∗)2 − t 2 × [φ(t + ξ√n) + φ(t − ξ√n )] dt (2) Hence, given values of the capability requirement C, the parameter ξ , the sample size n, and risk α,
the critical value c0 can be obtained by solving the
following equation: b√ n/(1+3c0) 0 G (b√n − t)2 9c20 − t2 × [φ(t + ξ√n) + φ(t − ξ√n)] dt = α (3) Given values of C, n, and α, the critical value c0for
ξ = ξ0and ξ = −ξ0is the same because equation (3)
is an even function of ξ.
4. COMPUTER PROGRAM
An efficient Maple computer program is developed to calculate equation (2), to obtain the p-value for given c∗. We note that similar programs can also be written using ‘Mathematica’ or ‘MatLab’ software. The program is listed in Figure1, with input parameters set to LSL= 2.40, USL = 3.40, T = 2.90,
C = 1.00, n = 100, X = 2.865, and Sn = 0.125.
Here, we set ξ = ˆξ = (X − T )/Sn, since generally
Figure 2. Plots of c0versus|ξ| for Cpmk = 1.00, α = 0.05, and
n = 30, 50, 70, 100 200, 300 (top to bottom in plot)
Figure 3. Plots of c0versus|ξ| for Cpmk = 1.33, α = 0.05, and
n = 30, 50, 70, 100 200, 300 (top to bottom in plot)
Figure 4. Plots of c0versus|ξ| for Cpmk = 1.50, α = 0.05, and
n = 30, 50, 70, 100 200, 300 (top to bottom in plot)
the one proposed by Cheng [5] for testing the process capability index Cpm. On the other hand,
c∗= ˆCpmk = (d − |X − T |)/{3[Sn2+ (X − T ) 2]1/2}
can be calculated from the sample data. The program gives ˆξ = −0.28 and ˆCpmk = 1.194, and the
corresponding p-value as 0.0253.
Figure 5. Plots of c0versus|ξ| for Cpmk = 1.67, α = 0.05, and
n = 30, 50, 70, 100 200, 300 (top to bottom in plot)
Figure 6. Plots of c0versus|ξ| for Cpmk = 2.00, α = 0.05, and
n = 30, 50, 70, 100 200, 300 (top to bottom in plot)
5. CRITICAL VALUES c0AND ξ
Since the process parameters µ and σ are unknown, then the parameter ξ = (µ − T )/σ is also unknown, which has to be estimated in real applications, naturally by substituting µ and σ by its sample mean and sample standard deviation. Such an approach certainly would make our approach less reliable. To eliminate the need for estimating the parameter ξ, we examine the behavior of the critical values c0as a
function of ξ. We calculate the critical values c0 for
ξ = 0(0.05)3.00, n = 30, 50, 70, 100, 200, 300, C = 1.00, 1.33, 1.50, 1.67, 2.00, and α = 0.01, 0.025, 0.05. Noting that ξ = 0(0.05)3.00 covers a wide range of applications with process capability Cpmk 0.
We find that the critical value c0obtains its maximum
either at ξ = 0.50 (for most cases), or at 0.45 (in a few cases), and the difference between the two critical values is less than 10−3. Hence, for practical purposes we may solve equation (3) for ξ = 0.50 to obtain the required critical value for the given C, n, and α, without having to estimate the parameter ξ . We note the above result is almost impossible to prove theoretically.
Table 1. Critical values c0for C = 1.00, n = 10(5)405, and α = 0.01, 0.025, 0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 2.148 1.892 1.704 210 1.160 1.133 1.111 15 1.831 1.660 1.530 215 1.158 1.132 1.110 20 1.675 1.542 1.439 220 1.156 1.130 1.108 25 1.578 1.468 1.380 225 1.154 1.129 1.107 30 1.512 1.416 1.339 230 1.152 1.127 1.106 35 1.463 1.377 1.309 235 1.151 1.126 1.105 40 1.425 1.347 1.285 240 1.149 1.124 1.103 45 1.394 1.323 1.265 245 1.147 1.123 1.102 50 1.369 1.303 1.249 250 1.146 1.121 1.101 55 1.348 1.286 1.235 255 1.144 1.120 1.100 60 1.330 1.271 1.224 260 1.143 1.119 1.099 65 1.314 1.259 1.213 265 1.141 1.118 1.098 70 1.301 1.248 1.205 270 1.140 1.116 1.097 75 1.289 1.238 1.197 275 1.138 1.115 1.096 80 1.278 1.229 1.189 280 1.137 1.114 1.095 85 1.268 1.221 1.183 285 1.136 1.113 1.094 90 1.259 1.214 1.177 290 1.134 1.112 1.093 95 1.251 1.208 1.172 295 1.133 1.111 1.093 100 1.244 1.202 1.167 300 1.132 1.110 1.092 105 1.237 1.196 1.162 305 1.131 1.109 1.091 110 1.231 1.191 1.158 310 1.130 1.108 1.090 115 1.225 1.186 1.154 315 1.128 1.107 1.089 120 1.219 1.182 1.151 320 1.127 1.106 1.089 125 1.214 1.178 1.147 325 1.126 1.105 1.088 130 1.210 1.174 1.144 330 1.125 1.105 1.087 135 1.205 1.170 1.141 335 1.124 1.104 1.087 140 1.201 1.167 1.138 340 1.123 1.103 1.086 145 1.197 1.164 1.136 345 1.122 1.102 1.085 150 1.193 1.161 1.133 350 1.121 1.101 1.085 155 1.190 1.158 1.131 355 1.120 1.101 1.084 160 1.186 1.155 1.129 360 1.119 1.100 1.083 165 1.183 1.152 1.127 365 1.119 1.099 1.083 170 1.180 1.150 1.125 370 1.118 1.098 1.082 175 1.177 1.147 1.123 375 1.117 1.098 1.082 180 1.175 1.145 1.121 380 1.116 1.097 1.081 185 1.172 1.143 1.119 385 1.115 1.096 1.080 190 1.169 1.141 1.117 390 1.114 1.095 1.080 195 1.167 1.139 1.116 395 1.114 1.095 1.079 200 1.165 1.137 1.114 400 1.113 1.094 1.079 205 1.162 1.135 1.112 405 1.112 1.094 1.078
Table 2. Critical values c0for C = 1.33, n = 10(5)405, and α = 0.01, 0.025, 0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 2.792 2.464 2.224 210 1.530 1.497 1.469 15 2.383 2.165 1.999 215 1.528 1.495 1.467 20 2.182 2.013 1.882 220 1.525 1.493 1.465 25 2.059 1.918 1.808 225 1.523 1.491 1.464 30 1.974 1.852 1.756 230 1.520 1.489 1.462 35 1.912 1.804 1.717 235 1.518 1.487 1.460 40 1.864 1.765 1.687 240 1.516 1.485 1.459 45 1.825 1.735 1.662 245 1.514 1.483 1.458 50 1.793 1.710 1.642 250 1.512 1.482 1.456 55 1.767 1.688 1.625 255 1.510 1.480 1.455 60 1.744 1.670 1.610 260 1.508 1.478 1.454 65 1.724 1.654 1.597 265 1.506 1.477 1.452 70 1.707 1.640 1.586 270 1.504 1.475 1.451 75 1.692 1.628 1.576 275 1.503 1.474 1.450 80 1.678 1.617 1.567 280 1.501 1.473 1.449 85 1.666 1.607 1.559 285 1.499 1.471 1.448 90 1.654 1.598 1.552 290 1.498 1.470 1.447 95 1.644 1.590 1.545 295 1.496 1.469 1.445 100 1.635 1.582 1.539 300 1.495 1.467 1.444 105 1.627 1.575 1.533 305 1.493 1.466 1.443 110 1.619 1.569 1.528 310 1.492 1.465 1.442 115 1.611 1.563 1.523 315 1.490 1.464 1.442 120 1.605 1.557 1.518 320 1.489 1.463 1.441 125 1.598 1.552 1.514 325 1.488 1.462 1.440 130 1.592 1.547 1.510 330 1.486 1.461 1.439 135 1.587 1.543 1.507 335 1.485 1.459 1.438 140 1.581 1.539 1.503 340 1.484 1.458 1.437 145 1.576 1.535 1.500 345 1.483 1.457 1.436 150 1.572 1.531 1.497 350 1.481 1.456 1.435 155 1.567 1.527 1.494 355 1.480 1.456 1.145 160 1.563 1.524 1.491 360 1.479 1.455 1.434 165 1.559 1.520 1.488 365 1.478 1.454 1.433 170 1.555 1.517 1.486 370 1.477 1.453 1.432 175 1.552 1.514 1.483 375 1.476 1.452 1.432 180 1.548 1.511 1.481 380 1.475 1.451 1.431 185 1.545 1.509 1.479 385 1.474 1.450 1.430 190 1.542 1.506 1.476 390 1.473 1.449 1.430 195 1.539 1.504 1.474 395 1.472 1.449 1.429 200 1.536 1.501 1.472 400 1.471 1.448 1.428 205 1.533 1.499 1.470 405 1.470 1.447 1.428
Table 3. Critical values c0for C = 1.50, n = 10(5)405, and α = 0.01, 0.025, 0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 3.124 2.759 2.492 210 1.721 1.684 1.653 15 2.668 2.425 2.241 215 1.718 1.682 1.651 20 2.444 2.256 2.111 220 1.715 1.679 1.649 25 2.306 2.151 2.028 225 1.713 1.677 1.647 30 2.213 2.078 1.971 230 1.710 1.675 1.646 35 2.143 2.023 1.927 235 1.708 1.673 1.644 40 2.090 1.981 1.894 240 1.705 1.671 1.642 45 2.047 1.947 1.867 245 1.703 1.669 1.641 50 2.012 1.919 1.844 250 1.701 1.667 1.639 55 1.982 1.896 1.825 255 1.699 1.666 1.638 60 1.957 1.876 1.809 260 1.696 1.664 1.636 65 1.935 1.858 1.795 265 1.694 1.662 1.635 70 1.916 1.843 1.783 270 1.692 1.660 1.634 75 1.899 1.829 1.771 275 1.691 1.659 1.632 80 1.884 1.817 1.762 280 1.689 1.657 1.631 85 1.871 1.806 1.753 285 1.687 1.656 1.630 90 1.858 1.796 1.745 290 1.685 1.654 1.629 95 1.847 1.787 1.737 295 1.683 1.653 1.627 100 1.837 1.779 1.730 300 1.682 1.652 1.626 105 1.827 1.771 1.724 305 1.680 1.650 1.625 110 1.819 1.764 1.718 310 1.678 1.649 1.624 115 1.811 1.757 1.713 315 1.677 1.648 1.623 120 1.803 1.751 1.708 320 1.675 1.646 1.622 125 1.796 1.745 1.703 325 1.674 1.645 1.621 130 1.790 1.740 1.699 330 1.673 1.644 1.620 135 1.783 1.735 1.695 335 1.671 1.643 1.619 140 1.778 1.730 1.691 340 1.670 1.642 1.618 145 1.772 1.726 1.687 345 1.668 1.641 1.617 150 1.767 1.722 1.684 350 1.667 1.640 1.616 155 1.762 1.718 1.681 355 1.666 1.638 1.615 160 1.757 1.714 1.677 360 1.665 1.637 1.615 165 1.753 1.710 1.674 365 1.663 1.636 1.614 170 1.749 1.707 1.672 370 1.662 1.635 1.613 175 1.745 1.703 1.669 375 1.661 1.634 1.612 180 1.741 1.700 1.666 380 1.660 1.634 1.611 185 1.737 1.697 1.664 385 1.659 1.633 1.611 190 1.734 1.694 1.662 390 1.658 1.632 1.610 195 1.730 1.692 1.659 395 1.657 1.631 1.609 200 1.727 1.689 1.657 400 1.656 1.630 1.608 205 1.724 1.686 1.655 405 1.654 1.629 1.608
Table 4. Critical values c0for C = 1.67, n = 10(5)405, and α = 0.01, 0.025, 0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 3.457 3.054 2.760 210 1.912 1.871 1.837 15 2.953 2.686 2.483 215 1.909 1.869 1.835 20 2.705 2.499 2.339 220 1.906 1.866 1.833 25 2.554 2.383 2.249 225 1.903 1.864 1.831 30 2.451 2.303 2.185 230 1.900 1.862 1.829 35 2.375 2.243 2.138 235 1.897 1.859 1.827 40 2.316 2.197 2.101 240 1.895 1.857 1.826 45 2.269 2.160 2.072 245 1.892 1.855 1.824 50 2.231 2.129 2.047 250 1.890 1.853 1.822 55 2.198 2.103 2.026 255 1.887 1.851 1.821 60 2.171 2.081 2.008 260 1.885 1.849 1.819 65 2.147 2.062 1.993 265 1.883 1.847 1.818 70 2.126 2.045 1.979 270 1.881 1.846 1.816 75 2.107 2.030 1.967 275 1.878 1.844 1.815 80 2.091 2.017 1.956 280 1.876 1.842 1.813 85 2.076 2.005 1.947 285 1.874 1.840 1.812 90 2.062 1.994 1.938 290 1.873 1.839 1.811 95 2.050 1.984 1.930 295 1.871 1.837 1.809 100 2.039 1.975 1.922 300 1.869 1.836 1.808 105 2.029 1.967 1.915 305 1.867 1.834 1.807 110 2.019 1.959 1.909 310 1.865 1.833 1.806 115 2.010 1.952 1.903 315 1.864 1.832 1.805 120 2.002 1.945 1.898 320 1.862 1.830 1.803 125 1.994 1.939 1.893 325 1.860 1.829 1.802 130 1.987 1.933 1.888 330 1.859 1.828 1.801 135 1.980 1.927 1.883 335 1.857 1.826 1.800 140 1.974 1.922 1.879 340 1.856 1.825 1.799 145 1.968 1.917 1.875 345 1.854 1.824 1.798 150 1.962 1.912 1.871 350 1.853 1.823 1.797 155 1.957 1.908 1.868 355 1.851 1.821 1.796 160 1.952 1.904 1.864 360 1.850 1.820 1.795 165 1.947 1.900 1.861 365 1.849 1.819 1.794 170 1.942 1.896 1.858 370 1.847 1.818 1.794 175 1.938 1.893 1.855 375 1.846 1.817 1.793 180 1.934 1.889 1.852 380 1.845 1.816 1.792 185 1.930 1.886 1.849 385 1.844 1.815 1.791 190 1.926 1.883 1.847 390 1.842 1.814 1.790 195 1.922 1.880 1.844 395 1.841 1.813 1.789 200 1.919 1.877 1.842 400 1.840 1.812 1.789 205 1.915 1.874 1.840 405 1.839 1.811 1.788
Table 5. Critical values c0for C = 2.00, n = 10(5)405, and α = 0.01, 0.025, 0.05 n α = 0.01 α = 0.025 α = 0.05 n α = 0.01 α = 0.025 α = 0.05 10 4.102 3.627 3.280 210 2.282 2.235 2.195 15 3.506 3.192 2.953 215 2.279 2.232 2.193 20 3.214 2.972 2.784 220 2.275 2.229 2.190 25 3.036 2.835 2.677 225 2.272 2.226 2.188 30 2.915 2.741 2.603 230 2.269 2.224 2.186 35 2.825 2.671 2.548 235 2.265 2.221 2.184 40 2.756 2.616 2.504 240 2.262 2.218 2.182 45 2.701 2.573 2.469 245 2.259 2.216 2.180 50 2.656 2.537 2.441 250 2.257 2.214 2.178 55 2.618 2.507 2.416 255 2.254 2.211 2.176 60 2.586 2.481 2.395 260 2.251 2.209 2.174 65 2.558 2.458 2.377 265 2.248 2.207 2.172 70 2.533 2.439 2.361 270 2.246 2.205 2.171 75 2.511 2.421 2.347 275 2.243 2.203 2.169 80 2.492 2.405 2.335 280 2.241 2.201 2.167 85 2.474 2.391 2.323 285 2.239 2.199 2.166 90 2.459 2.379 2.313 290 2.236 2.197 2.164 95 2.444 2.367 2.303 295 2.234 2.195 2.163 100 2.431 2.356 2.295 300 2.232 2.194 2.161 105 2.419 2.346 2.287 305 2.230 2.192 2.160 110 2.408 2.337 2.279 310 2.228 2.190 2.158 115 2.397 2.329 2.272 315 2.226 2.189 2.157 120 2.388 2.321 2.266 320 2.224 2.187 2.156 125 2.379 2.314 2.260 325 2.222 2.185 2.154 130 2.370 2.307 2.254 330 2.220 2.184 2.153 135 2.362 2.300 2.249 335 2.219 2.182 2.152 140 2.355 2.294 2.244 340 2.217 2.181 2.151 145 2.348 2.289 2.239 345 2.215 2.180 2.150 150 2.341 2.283 2.235 350 2.213 2.178 2.148 155 2.335 2.278 2.231 355 2.212 2.177 2.147 160 2.329 2.273 2.227 360 2.210 2.176 2.146 165 2.323 2.269 2.223 365 2.209 2.174 2.145 170 2.318 2.264 2.219 370 2.207 2.173 2.144 175 2.313 2.260 2.216 375 2.206 2.172 2.143 180 2.308 2.256 2.212 380 2.204 2.171 2.142 185 2.303 2.252 2.209 385 2.203 2.169 2.141 190 2.299 2.248 2.206 390 2.201 2.168 2.140 195 2.294 2.245 2.203 395 2.200 2.167 2.139 200 2.290 2.241 2.201 400 2.199 2.166 2.138 205 2.286 2.238 2.198 405 2.197 2.165 2.137
Similarly, without having to estimate the parameter ξ we can set ξ = 0.50 and calculate the p-value using the program provided in Section 4. The p-value for ξ = 0.50 is greater than the p-values for other choice of ξ in all cases. Based on the conservative p-value, the decision making is more reliable. In the example displayed in Section4, if we set ξ = 0.50, then the program gives ˆCpmk = 1.194 and the corresponding
p-value as 0.0290 for the same input parameters: LSL = 2.40, USL = 3.40, T = 2.90, C = 1.00, n = 100, X = 2.865, and Sn= 0.125.
Figures 2–6 plot the curves of c0 versus the
parameter ξ for sample size n = 30 (top curve 1), 50 (top curve 2), 70 (top curve 3), 100 (top curve 4), 200 (top curve 5), 300 (bottom curve), and C = 1.00, 1.33, 1.50, 1.67, 2.00, with α = 0.05.
6. TESTING PROCEDURE
Tables 1–5 display the critical values c0 for C =
1.00, 1.33, 1.50, 1.67, and 2.00, with sample sizes n = 10(5)405, and α-risk = 0.01, 0.025, 0.05. To judge if a given process meets the capability requirement, we first determine the value of C, the capability requirement, and the α-risk. Checking the appropriate table from Tables1–5, we may obtain the critical value c0 based on the given values of α-risk,
C, and the sample size n. If the estimated value ˆCpmk
is greater than the critical value c0 ( ˆCpmk > c0),
then we conclude that the process meets the capability requirement (Cpmk > C). Otherwise, we do not have
sufficient information to conclude that the process meets the present capability requirement. In this case, we would believe that Cpmk C.
Step 1. Decide the definition of ‘capable’ (C, nor-mally set to 1.00, or 1.33), and the α-risk (nor-mally set to 0.01, 0.025, or 0.05), the chance of wrongly concluding an incapable process as capable.
Step 2. Calculate the values of ˆCpmkfrom the sample.
Step 3. Check the appropriate table and find the critical value c0 based on α-risk, C, and the
sample size n.
Step 4. Conclude that the process is capable (Cpmk > C) if the ˆCpmk value is greater than the
critical value c0( ˆCpmk > c0). Otherwise, we do
not have enough information to conclude that the process is capable.
REFERENCES
1. Pearn WL, Kotz S, Johnson NL. Distributional and inferential properties of process capability indices. Journal of Quality Technology 1992; 24:216–231.
2. V¨annman K. A unified approach to capability indices. Statistica Sinica 1995; 5:805–820.
3. Chen SM, Hsu NF. The asymptotic distribution of the process capability index: Cpmk. Communications in Statistics—Theory
and Methods 1995; 24:1279–1291.
4. V¨annman K, Kotz S. A superstructure of capability indices: Distributional properties and implications. Scandinavian Jour-nal of Statistics 1995; 22:477–491.
5. V¨annman K. Distribution and moments in simplified form for a general class of capability indices. Communications in Statistics—Theory and Methods 1997; 26(1):159–179. 6. Cheng SW. Practical implementation of the process capability
indices. Quality Engineering 1994; 7(2):239–259.
Authors’ biographies:
W. L. Pearn is a professor of operations research and quality
management at the Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan, Republic of China. He received his MS degree in statistics and his PhD degree in operations research from the University of Maryland, College Park, MD, USA. He worked for AT&T Bell Laboratories at the Switch Network Control and Process Quality Centers before he joined National Chiao Tung University.
P. C. Lin received his MS degree in statistics from National
Chung Hsing University and his PhD degree in quality management from National Chiao Tung University, Taiwan, Republic of China. Currently, he is an associate professor at the General Education Center, National Chin-Yi Institute of Technology, Taiwan, Republic of China.
