Testing process capability for one-sided specification limit with application to the voltage level translator

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Testing process capability for one-sided speciﬁcation limit

with application to the voltage level translator

P.C. Lin

a,*

, W.L. Pearn

b

a

Center of General Education, National Chin-Yi Institute of Technology, Taipin, Taichung, Taiwan 411, ROC

b

Department of Industrial Engineering and Management, National Chiao Tung University, Taiwan 30050, ROC Received 8April 2002; received in revised form 30 May 2002

Abstract

Process capability indices CPU and CPL have been widely used in the microelectronics manufacturing industry as

capability measures for processes with one-sided speciﬁcation limits. In this paper, the theory of statistical hypothesis testing is implemented for normal processes, using the uniformly minimum variance unbiased estimators of CPU and

CPL. Eﬃcient SAS computer programs are provided to calculate the critical values and the p-values required for making

decisions. Useful critical values for some commonly used capability requirements are tabulated. Based on the test a simple but practical step-by-step procedure is developed for in-plant applications. An example on the voltage level translator manufacturing process is given to illustrate how the proposed procedure may be applied to test whether the process meets the preset capability requirement.

1. Introduction

Several capability indices including Cp, CPU, CPL, and

Cpkhave been widely used in the manufacturing industry

as well as the service industry providing common quantitative measures on process potential and perfor-mance (see Kane [1], Pearn et al. [2], and Pearn and Chen [3,4]). Those indices are deﬁned in the following:

Cp¼ USL LSL 6r ; CPU¼ USL l 3r ; CPL¼ l LSL 3r ; Cpk¼ min USL l 3r ; l LSL 3r ;

where USL is the upper speciﬁcation limit, LSL is the lower speciﬁcation limit, l is the process mean, and r is the process standard deviation (overall process varia-tion). While Cp and Cpk are appropriate measures for

processes with two-sided speciﬁcations (which require both USL and LSL), CPU and CPL have been designed

speciﬁcally for processes with one-sided speciﬁcation limit (which require only USL or LSL). The index CPU

measures the capability of a smaller-the-better process with an upper speciﬁcation limit USL, whereas the index CPL measures the capability of a larger-the-better

pro-cess with a lower speciﬁcation limit LSL.

For normally distributed processes with one-sided speciﬁcation limit USL, the process yield is

PðX < USLÞ ¼ P X l 3r <USL l 3r ¼ P 1 3Z < CPU ¼ P ðZ < 3CPUÞ ¼ Uð3CPUÞ;

where Z follows the standard normal distribution Nð0; 1Þ with the cumulated distribution function UðÞ. Similarly, for normally distributed processes with one-sided speciﬁcation limit LSL, the process yield is

www.elsevier.com/locate/microrel

*

Corresponding author.

E-mail address:linpc@ncit.edu.tw(P.C. Lin).

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PðX > LSLÞ ¼ P l X 3r <l LSL 3r ¼ P 1 3Z < CPL ¼ P ðZ > 3CPLÞ ¼ 1 Uð3CPLÞ ¼ Uð3CPLÞ:

Table 1 displays the corresponding non-conforming units in parts per million (NCPPM) for a well controlled normally distributed process.

Montgomery [5] recommended some minimum quality requirements of CPU and CPLfor processes runs

under some designated capable conditions. In particular, 1.25 for existing processes, and 1.45 for new processes; 1.45 also for existing processes on safety, strength, or critical parameter, and 1.60 for new processes on safety, strength, or critical parameter.

The formulae for these indices are easy to understand and straightforward to apply. In practice, however, sample data must be collected in order to calculate these indices. Therefore, a great degree of uncertainty may be introduced into capability assessments due to sampling errors. To ensure the capability assessment reliable, the uniformly minimum variance unbiased estimators (UMVUEs) of CPU and CPL obtained by Pearn and

Chen [6], is considered under normality assumption. The theory of testing hypothesis using the UMVUEs of CPU

and CPL is implemented, and eﬃcient SAS computer

programs are provided to calculate the critical values and the p-values required for making decisions.

Based on the test, two practical step-by-step proce-dures are developed for in-plant applications. The practitioners or the engineers can apply the proposed procedure to judge whether or not their processes meet the preset capability requirement and run under the desired quality conditions. An example taken from the factory involving the voltage level translator manufac-turing is investigated. The proposed procedures are ap-plied to test whether the process meets the preset capability requirement. If the process meets the capa-bility requirement, the translators made are considered to be reliable.

2. Estimations of CPUand CPL

To estimate the indices CPUand CPL, Chou and Owen

[7] considered bCCPU and bCCPL, the natural estimators of

CPUand CPL, which are deﬁned as the following:

b C CPU¼ USL X 3S ; CCbPL¼ X LSL 3S ; where X ¼Pn

i¼1Xi=n is the sample mean, and S2¼

ðn 1Þ1Pn_i¼1ðXi X Þ 2

is the sample variance, which may be obtained from a process that is demonstrably stable (under statistical control).

Chou and Owen [7] showed that under normality as-sumption, the estimator bCCPU is distributed as Cntn1ðdÞ,

where Cn¼ ð3 ffiffiffin

p

Þ1, and t_n1ðdÞ is a non-central t dis-tribution with n 1 degrees of freedom and non-centrality parameter d¼ 3pffiffiffinCPU. The estimator bCCPLhas

the same sampling distribution as bCCPU but with d ¼

3pffiffiffinCPL.

Both estimators bCCPU and bCCPL are biased. But, Pearn

and Chen [6] showed that by adding the correction factor bn1¼ ½2=ðn 1Þ

1=2

C½ðn 1Þ=2=C½ðn 2Þ=2 to b

C

CPU and bCCPL, we may obtain unbiased estimators

bn1CCbPU and bn1CCbPL which have been denoted as eCCPU

and eCCPL. Thus, we have Eð eCCPUÞ ¼ CPU, and Eð eCCPLÞ ¼

CPL. Since bn1<1, then varð eCCPUÞ < varð bCCPUÞ and

varð eCCPLÞ < varð bCCPLÞ.

Pearn and Chen [6] further showed that eCCPUand eCCPL

are the UMVUEs of CPU and CPL, respectively.

There-fore, in practice using the UMVUEs eCCPU and eCCPL to

calculate the capability measures would be desirable. The probability density function (PDF) of eCCPU(or eCCPL)

may be easily obtained as

fðxÞ ¼ 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n=ðn 1Þ p 2n=2 bn1 ffiffiffip p C½ðn 1Þ=2 Z 1 0 yðn2Þ=2 exp ( 1 2 y " þ 3x ffiffiffiffiffiny p bn1 ffiffiffiffiffiffiffiffiffiffiffi n 1 p d 2#) dy; where bn1¼ ½2=ðn 1Þ1=2C½ðn 1Þ=2=C½ðn 2Þ=2 and d¼ 3pffiffiffinCPU (or d¼ 3 ffiffiffin p

CPL). Figs. 1–4 display the

PDF plots of eCCPU (or eCCPL) for CPU ðor CPLÞ ¼ 1:00,

1.25, 1.45, and 1.60, with various sample sizes of n¼ 10, 20, 30, 40, and 50 (from bottom to top in plot).

3. Testing process capability

To test whether a given process with an upper spec-iﬁcation limit USL, meets the preset capability require-ment and runs under the desired quality condition, the hypothesis testing: H0: CPU6C versus H1: CPU> C, is

considered, where C is a known constant preset by the product designer or process engineer. A process meets

Table 1

CPUand the corresponding non-conforming units (in ppm)

Values of CPU NCPPM 0.80 8198 1.00 1350 1.25 88 1.45 7 1.60 1 1.80 0.33 2.00 0.00099

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the capability requirement if CPU> C, and fails to meet

the capability requirement if CPU6C.

Based on a given aðc0Þ ¼ a, the chance of incorrectly

judging an incapable process as capable, the decision rule is to reject H0 if eCCPU> c0 and do not reject H0

otherwise, for some critical value c0> C. It is noted that

the procedures for testing the hypothesis: ÔH0: CPL6C

versus H1: CPL> CÕ and ÔH0: CPU6C versus H1:

CPU> CÕare exactly the same.

The p-value, which is the actual probability of in-correctly concluding an incapable process as a capable one, corresponding to a speciﬁc value of eCCPU¼ w, can

be calculated from the sample as

Pf eCCPUP wjCPU6Cg ¼ P 3p bCCffiffiffin PU P3 ffiffiffi n p w bn1 jCPU6C ¼ P tn1ðdÞ P3 ffiffiffi n p w bn1 ;

where d¼ 3pffiffiffinC. An efficient SAS computer program is developed to calculate the p-value given a specific value of eCCPU¼ w obtained from the sample data. The

pro-gram is listed in Appendix A, with input parameters set to: C¼ 1:25, sample size n ¼ 120, and the calculated

e C

CPU¼ w ¼ 1:433. The program gives the p-value as

0.025.

On the other hand, the critical value, c0, with ﬁxed a

risk, may be determined by solving the following equa-tions. An eﬃcient SAS computer program (see Appen-dix B) is developed to calculate the critical values c0for a

specific value of C. P eCCPU n P c0jCPU¼ C o ¼ a; P 3p bCCffiffiffin PU P3 ffiffiffi n p c0 bn1 jCPU¼ C ¼ a;

Fig. 1. PDF plot of eCCPUfor CPU¼ 1:00 and n ¼ 10, 20, 30, 40,

50 (from bottom to top in plot).

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P tn1ðdÞ P3 ffiffiffi n p c0 bn1 ¼ a;

where d¼ 3pffiffiffinC. Hence, 3pffiffiffinc0=bn1¼ tn1;aðdÞ, the

upper ath percentile of tn1ðdÞ, the non-central t

distri-bution. Thus, the critical value c0 may be obtained as

c0¼ bn1tn1;aðdÞ=ð3 ffiffiffin

p

Þ , where d ¼ 3pffiffiffinC.

Critical values for those capability requirements CPU¼ 1:25, 1.45, 1.60, as recommended by

Montgo-mery [5], are summarized in Tables 2–4 for sample sizes

Table 2

Critical values c0for C¼ 1:25, n ¼ 10(5)505 and a ¼ 0:01, 0.025, 0.05

n a¼ 0:01 a¼ 0:025 a¼ 0:05 n a¼ 0:01 a¼ 0:025 a¼ 0:05 10 2.420 2.124 1.910 260 1.396 1.371 1.350 15 2.087 1.895 1.750 265 1.395 1.370 1.349 20 1.927 1.780 1.667 270 1.393 1.369 1.348 25 1.830 1.708 1.614 275 1.392 1.368 1.347 30 1.763 1.659 1.576 280 1.391 1.367 1.346 35 1.715 1.622 1.548 285 1.389 1.366 1.346 40 1.677 1.593 1.526 290 1.388 1.365 1.345 45 1.647 1.570 1.508295 1.387 1.363 1.344 50 1.622 1.551 1.493 300 1.386 1.362 1.343 55 1.602 1.535 1.481 305 1.384 1.362 1.342 60 1.584 1.521 1.470 310 1.383 1.361 1.342 65 1.5681.509 1.460 315 1.382 1.360 1.341 70 1.555 1.4981.452 320 1.381 1.359 1.340 75 1.543 1.489 1.444 325 1.380 1.358 1.339 80 1.532 1.480 1.438 330 1.379 1.357 1.339 85 1.522 1.472 1.432 335 1.378 1.356 1.338 90 1.513 1.465 1.426 340 1.377 1.355 1.337 95 1.505 1.459 1.421 345 1.376 1.355 1.337 100 1.4981.453 1.416 350 1.375 1.354 1.336 105 1.491 1.4481.412 355 1.374 1.353 1.335 110 1.485 1.443 1.408 360 1.373 1.352 1.335 115 1.479 1.4381.404 365 1.372 1.351 1.334 120 1.474 1.434 1.401 370 1.371 1.351 1.333 125 1.469 1.430 1.398375 1.370 1.350 1.333 130 1.464 1.426 1.394 380 1.370 1.349 1.332 135 1.460 1.422 1.392 385 1.369 1.349 1.332 140 1.455 1.419 1.389 390 1.368 1.348 1.331 145 1.451 1.416 1.386 395 1.367 1.347 1.331 150 1.4481.413 1.384 400 1.366 1.347 1.330 155 1.444 1.410 1.382 405 1.366 1.346 1.330 160 1.441 1.407 1.379 410 1.365 1.345 1.329 165 1.4381.405 1.377 415 1.364 1.345 1.329 170 1.434 1.402 1.375 420 1.363 1.344 1.328 175 1.432 1.400 1.373 425 1.363 1.344 1.328 180 1.429 1.398 1.372 430 1.362 1.343 1.327 185 1.426 1.395 1.370 435 1.361 1.343 1.327 190 1.424 1.393 1.368440 1.361 1.342 1.326 195 1.421 1.391 1.367 445 1.360 1.341 1.326 200 1.419 1.389 1.365 450 1.359 1.341 1.325 205 1.417 1.388 1.364 455 1.359 1.340 1.325 210 1.414 1.386 1.362 460 1.358 1.340 1.325 215 1.412 1.384 1.361 465 1.357 1.339 1.324 220 1.410 1.383 1.359 470 1.357 1.339 1.324 225 1.4081.381 1.358475 1.356 1.3381.323 230 1.406 1.379 1.357 480 1.356 1.338 1.323 235 1.405 1.3781.356 485 1.355 1.337 1.323 240 1.403 1.377 1.355 490 1.354 1.337 1.322 245 1.401 1.375 1.353 495 1.354 1.337 1.322 250 1.400 1.374 1.352 500 1.353 1.336 1.321 255 1.3981.373 1.351 505 1.353 1.336 1.321

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n¼ 10(5)505, a-risk ¼ 0:05, 0.025, and 0.01. Using those tables, the practitioners may perform the capability testing without having to run the SAS computer pro-grams. In practice, the sample data taken from a stable

normal process is ﬁrst collected. The sample mean and the sample standard deviation, X and S, are calculated. The estimator eCCPU (or eCCPL) is then calculated, and

compared with the critical value c0found from the table.

Table 3

n a¼ 0:01 a¼ 0:025 a¼ 0:05 n a¼ 0:01 a¼ 0:025 a¼ 0:05 10 2.793 2.452 2.206 260 1.617 1.589 1.565 15 2.410 2.189 2.023 265 1.616 1.587 1.563 20 2.225 2.057 1.927 270 1.614 1.586 1.562 25 2.114 1.975 1.866 275 1.612 1.584 1.561 30 2.038 1.918 1.823 280 1.611 1.583 1.560 35 1.982 1.876 1.791 285 1.609 1.582 1.559 40 1.939 1.843 1.766 290 1.608 1.581 1.558 45 1.904 1.816 1.745 295 1.606 1.580 1.557 50 1.876 1.794 1.728300 1.605 1.5781.556 55 1.852 1.776 1.714 305 1.603 1.577 1.555 60 1.832 1.760 1.701 310 1.602 1.576 1.554 65 1.814 1.746 1.690 315 1.601 1.575 1.554 70 1.7981.734 1.681 320 1.600 1.574 1.553 75 1.785 1.723 1.672 325 1.598 1.573 1.552 80 1.772 1.713 1.664 330 1.597 1.572 1.551 85 1.761 1.704 1.657 335 1.596 1.571 1.550 90 1.751 1.696 1.651 340 1.595 1.570 1.550 95 1.742 1.689 1.645 345 1.594 1.569 1.549 100 1.734 1.682 1.640 350 1.593 1.568 1.548 105 1.726 1.676 1.635 355 1.592 1.5681.547 110 1.719 1.670 1.630 360 1.590 1.567 1.547 115 1.712 1.665 1.626 365 1.589 1.566 1.546 120 1.706 1.660 1.622 370 1.588 1.565 1.545 125 1.700 1.655 1.619 375 1.587 1.564 1.545 130 1.695 1.651 1.615 380 1.587 1.563 1.544 135 1.689 1.647 1.612 385 1.586 1.563 1.543 140 1.685 1.643 1.609 390 1.585 1.562 1.543 145 1.680 1.639 1.606 395 1.584 1.561 1.542 150 1.676 1.636 1.603 400 1.583 1.560 1.542 155 1.672 1.633 1.600 405 1.582 1.560 1.541 160 1.6681.630 1.598410 1.581 1.559 1.540 165 1.664 1.627 1.595 415 1.580 1.558 1.540 170 1.661 1.624 1.593 420 1.579 1.5581.539 175 1.657 1.621 1.591 425 1.579 1.557 1.539 180 1.654 1.619 1.589 430 1.578 1.556 1.538 185 1.651 1.616 1.587 435 1.577 1.556 1.538 190 1.6481.614 1.585 440 1.576 1.555 1.537 195 1.645 1.612 1.583 445 1.576 1.554 1.537 200 1.643 1.609 1.581 450 1.575 1.554 1.536 205 1.640 1.607 1.580 455 1.574 1.553 1.536 210 1.6381.605 1.578460 1.573 1.553 1.535 215 1.635 1.603 1.576 465 1.573 1.552 1.535 220 1.633 1.601 1.575 470 1.572 1.552 1.534 225 1.631 1.600 1.573 475 1.571 1.551 1.534 230 1.629 1.5981.572 480 1.571 1.550 1.533 235 1.627 1.596 1.571 485 1.570 1.550 1.533 240 1.625 1.595 1.569 490 1.569 1.549 1.532 245 1.623 1.593 1.568495 1.569 1.549 1.532 250 1.621 1.591 1.567 500 1.5681.5481.532 255 1.619 1.590 1.566 505 1.567 1.5481.531

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A simple step-by-step procedure (Test Procedure I) based on the critical value, is presented bellow for in-plant applications. The practitioners can use the pro-posed procedure to determine whether or not their

processes meet the preset capability requirements and run under the desired quality conditions. If the SAS computer program is used, then Test Procedure II based on the p-value may be applied.

Table 4

n a¼ 0:01 a¼ 0:025 a¼ 0:05 n a¼ 0:01 a¼ 0:025 a¼ 0:05 10 3.074 2.700 2.430 260 1.783 1.752 1.725 15 2.652 2.410 2.228265 1.781 1.750 1.724 20 2.450 2.265 2.122 270 1.779 1.749 1.723 25 2.327 2.175 2.056 275 1.7781.747 1.722 30 2.244 2.112 2.009 280 1.776 1.746 1.721 35 2.183 2.066 1.973 285 1.774 1.744 1.719 40 2.135 2.030 1.946 290 1.773 1.743 1.718 45 2.0982.001 1.923 295 1.771 1.742 1.717 50 2.066 1.977 1.904 300 1.769 1.741 1.716 55 2.040 1.956 1.889 305 1.768 1.739 1.715 60 2.0181.939 1.875 310 1.767 1.7381.714 65 1.9981.924 1.863 315 1.765 1.737 1.713 70 1.981 1.910 1.852 320 1.764 1.736 1.712 75 1.966 1.899 1.843 325 1.762 1.735 1.712 80 1.953 1.888 1.835 330 1.761 1.734 1.711 85 1.941 1.878 1.827 335 1.760 1.733 1.710 90 1.930 1.870 1.820 340 1.758 1.732 1.709 95 1.920 1.862 1.814 345 1.757 1.731 1.708 100 1.910 1.854 1.808 350 1.756 1.730 1.707 105 1.902 1.847 1.803 355 1.755 1.729 1.707 110 1.894 1.841 1.798 360 1.754 1.728 1.706 115 1.887 1.835 1.793 365 1.753 1.727 1.705 120 1.880 1.830 1.789 370 1.752 1.726 1.704 125 1.874 1.825 1.784 375 1.750 1.725 1.704 130 1.868 1.820 1.781 380 1.749 1.724 1.703 135 1.862 1.816 1.777 385 1.748 1.723 1.702 140 1.857 1.811 1.774 390 1.747 1.722 1.701 145 1.852 1.807 1.770 395 1.746 1.722 1.701 150 1.847 1.804 1.767 400 1.745 1.721 1.700 155 1.843 1.800 1.764 405 1.744 1.720 1.700 160 1.839 1.797 1.762 410 1.743 1.719 1.699 165 1.834 1.793 1.759 415 1.743 1.719 1.698 170 1.831 1.790 1.757 420 1.742 1.718 1.698 175 1.827 1.787 1.754 425 1.741 1.717 1.697 180 1.824 1.785 1.752 430 1.740 1.716 1.696 185 1.820 1.782 1.750 435 1.739 1.716 1.696 190 1.817 1.779 1.748 440 1.738 1.715 1.695 195 1.814 1.777 1.746 445 1.737 1.714 1.695 200 1.811 1.774 1.744 450 1.737 1.714 1.694 205 1.808 1.772 1.742 455 1.736 1.713 1.694 210 1.805 1.770 1.740 460 1.735 1.712 1.693 215 1.803 1.768 1.738 465 1.734 1.712 1.693 220 1.800 1.766 1.737 470 1.733 1.711 1.692 225 1.7981.764 1.735 475 1.733 1.710 1.692 230 1.796 1.762 1.734 480 1.732 1.710 1.691 235 1.793 1.760 1.732 485 1.731 1.709 1.691 240 1.791 1.7581.731 490 1.731 1.709 1.690 245 1.789 1.756 1.729 495 1.730 1.708 1.690 250 1.787 1.755 1.728500 1.729 1.7081.689 255 1.785 1.753 1.727 505 1.729 1.707 1.689

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3.1. Test Procedure I (based on critical value c0)

Step 1: Determine the value of C, the desired quality condition, and the a-risk (normally set to 0.01, 0.025, or 0.05), the chance of incorrectly concluding a bad process (quality does not meet the capability requirement) as a good process (quality meets the preset capability re-quirement).

Step 2: Calculate the value of the estimator, eCCPU(or

e C

CPL), from the sample.

Step 3: Check the appropriate Tables 2–4 and ﬁnd the corresponding c0 based on C, a, and n.

Step 4: Conclude that the process meets the capability requirement if eCCPU (or eCCPL) is greater than c0.

Other-wise, we do not have suﬃcient information to conclude that the given process satisﬁes the capability require-ment. In this case, we tend to believe that the process is incapable.

3.2. Test Procedure II (based on the p-value)

Step 1: Determine the value of C, the desired quality condition, and the a-risk (normally set to 0.01, 0.025, or 0.05), the chance of incorrectly concluding a bad process (quality does not meet the capability requirement) as a good process (quality meets the preset capability re-quirement).

Step 2: Calculate the value of the estimator, eCCPU(or

e C

CPL), from the sample.

Step 3: Input C, n, and w¼ eCCPU(or eCCPL), and execute

the provided SAS program listed in Appendix A to ﬁnd the corresponding p-value.

Step 4: Conclude that the process meets the capability requirement if the p-value is less than the chosen risk a. Otherwise, we do not have enough information to con-clude that the given process satisﬁes the capability re-quirement. In this case, we tend to believe that the process is incapable.

4. The voltage level translator

The transistor–transistor logic (TTL) family has been the major family of bipolar digital ICs for over 30 years. TTL had been the leading IC family in the small-scale integration (SSI, with fewer than 12 gates per chip) and median-scale integration (MSI, with 12–99 gates per chip) categories up until the last 10 or so years. Since then the leading position has been challenged by the CMOS, the complementary metal–oxide semiconductor family, which has gradually displaced TTL from that position. CMOS belongs to the class of unipolar digital ICs, which uses fewer components in many high per-formance applications, one of the main advantages of CMOS over TTL. CMOS and TTL ICs dominate the ﬁeld of SSL and MSI devices. Both the TTL and CMOS

circuits have a dc power supply voltage connected to one of their pins, and ground to another. The power supply pin is labeled VCCfor the TTL circuit, and VDD for the

CMOS circuit. Many of the newer CMOS ICs that are designed to be compatible with TTL ICs also use the VCC

designation as their power pin. For TTL devices, VCCis

nominally 5 V. For CMOS ICs, VDDcan range from 3 to

18V, although 5 V is most often used when CMOS ICs are used in the same circuit with TTL ICs. When TTL ICs are used with the high-voltage CMOS operating at VDD¼ 15 V, then a voltage-level translator is necessary

if TTL is to be driven directly from the high-voltage CMOS since the speciﬁcations of most TTL types are unable to handle an input voltage for more than 7 V before becoming damaged. The voltage-level translator converts the high-voltage input to a suitable voltage output 5 V that can be connected to the TTL.

The example investigated is taken from a microelec-tronics factory, located on the Science-Based Industrial Park in Taiwan, manufacturing the voltage level trans-lators. This factory manufactures various types of the voltage-level translator. For a particular model of the voltage-level translator investigated, the upper speciﬁ-cation limit, USL, for the out voltage level was set to 6.8 V. The capability requirement for this particular model of voltage-level translator was set to CPU¼ 1:25. The

collected sample data (a total of 120 observations) are displayed in Table 5.

Fig. 5 displays the histogram of the 120 observations. Fig. 6 displays the normal probability plot of the sample data. The sample data appears to be normal. Shapiro– Wilk test for normality check is also performed,

Table 5

Sample data of 120 voltage level translators

4.57 4.72 5.00 5.17 5.14 5.23 4.91 5.95 4.70 4.085.19 5.37 4.42 5.16 5.56 5.485.07 5.34 5.62 4.63 4.585.55 5.67 5.46 4.57 4.94 4.94 5.16 5.35 5.35 4.64 4.55 5.16 4.85 5.08 5.70 4.54 4.33 5.64 4.52 5.35 4.74 5.49 4.90 5.484.49 5.06 4.85 4.984.24 4.87 4.17 5.07 5.03 4.50 4.47 4.66 4.50 4.51 4.19 4.63 5.46 4.47 4.67 3.95 4.90 5.34 5.32 4.55 4.84 4.85 4.73 5.27 5.17 5.07 4.15 4.74 4.70 4.56 5.01 4.29 5.41 4.35 4.70 5.47 4.99 5.09 4.90 4.34 5.47 5.03 4.14 5.24 5.36 4.69 5.19 5.07 4.67 5.33 5.50 4.76 4.82 5.13 5.12 4.84 4.89 5.64 5.10 4.64 4.63 5.12 5.06 5.284.46 5.14 4.95 5.96 5.35 3.87 4.65

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obtaining W ¼ 0:9922. Thus, the sample data can be regarded as taken from a normal process.

The sample mean X¼ 4:94 and sample standard de-viation S¼ 0:43 are ﬁrst calculated. For n ¼ 120, we obtain the correction factor bn1¼ 0:9937, and calculate

the value of the estimator eCCPU¼ bn1ðUSL X Þ=ð3SÞ ¼

1:433. Assume the a-risk is 0.05, the critical value is found to be c0¼ 1:401 from Table 2 based on C ¼ 1:25,

a¼ 0:05, and n ¼ 120. Since eCCPU¼ 1:433 is greater than

the critical value c0¼ 1:401 in this case, it is therefore

concluded with 95% conﬁdence (a¼ 0:05) that the voltage level translator manufacturing process satisﬁes the requirement ÔCPU>1:25Õ. It is noted that

corre-sponding to the value of the estimator w¼ eCCPU¼ 1:433,

sample size n¼ 120, and the capability requirement C¼ 1:25, the p-value is found to be 0.025 by executing the provided SAS program. Thus, the actual probability of incorrectly concluding an incapable process as a ca-pable one is 2.5%.

Appendix A

The output is: The SAS System

Appendix B

/******************************/; /* SAS PROGRAM forp-Value */; /******************************/; OPTIONS REPLACE PAGESIZE¼ 78; OPTIONS LINESIZE¼ 78 NODATE; DATA VoLeTranlator;

/******************************/; /*Input quality requirement C */ /*Input Sample size n */ /*Input w¼ Estimated CPU */ /******************************/; C¼ 1.25; n ¼ 120; w ¼ 1.433; F¼ n-1; ND ¼ 3*SQRT(n) *C; /******************************/; /*Calculate bf */ /*Find the P-value */ /******************************/; DN¼ SQRTððn 2Þ=2Þ ð1 1=ð4 ðn 2ÞÞþ 1=ð32 ðn 2Þ 2Þ þ 5=ð128 ðn 2Þ 3ÞÞ; BF¼ SQRTð2=ðn 1ÞÞ DN; X¼ 3 SQRTðnÞ w=BF; PV¼ 1 PROBTðX; F; NDÞ; OUTPUT; FORMAT C 4.2 PV 5.3;

PROC PRINT DATA¼ VoLeTranlator; VAR C n w PV;

RUN;

Fig. 5. Histogram of the sample data.

Fig. 6. The normal probability plot.

******************************/; /* SAS PROGRAM for the */; /* Critical Value co */;

/******************************/; OPTIONS REPLACE PAGESIZE¼ 78; OPTIONS LINESIZE¼ 78 NODATE; DATA VoLeTranlator;

/******************************/; /*Input capability Requirement C */ /*Input Risk Alpha */ /*Input Sample Size n */ /*******************************/; C¼ 1:25; Alpha¼ 0:05; n¼ 120; Obs C n w PV 1 1.25 120 1.433 0.025

(9)

The output is: The SAS System

References

[1] Kane VE. Process capability indices. J Qual Technol 1986; 18(1):41–52.

[2] Pearn WL, Kotz S, Johnson NL. Distributional and inferential properties of process capability indices. J Qual Technol 1992;24(4):216–33.

[3] Pearn WL, Chen KS. A practical implementation of the process capability index C0

pk. Qual Eng 1997;9(4):721–

37.

[4] Pearn WL, Chen KS. Making decisions in assessing process capability index Cpk. Qual Reliab Eng Int 1999;15:

321–6.

[5] Montgomery DC. Introduction to statistical quality control. 4th ed. New York, NY: John Wiley & Sons, Inc., 2001. [6] Pearn WL, Chen KS. One-sided capability indices CPUand

CPL: decision making with sample information. Working

Paper, National Chiao Tung University, Taiwan, ROC, 2001.

[7] Chou YM, Owen DB. On the distributions of the estimated process capability indices. Commun Stat: Theory Meth 1989;18(2):4549–60. Obs C Alpha n C0 1 1.25 0.05 120 1.401 F¼ n 1; ND¼ 3 SQRTðnÞ C; /******************************/; /*Calculate bf */ /*Find Critical value c0 */

/******************************/; DN¼ SQRTððn 2Þ=2Þ ð1 1=ð4 ðn 2ÞÞ þ 1=ð32 ðn 2Þ 2Þ þ 5=ð128 ðn 2Þ 3ÞÞ; BF¼ SQRTð2=ðn 1ÞÞ DN; C0¼ BF=ð3 SQRTðnÞ TINVð1 Alpha; F; NDÞ; FORMAT C0 5.3;

PROC PRINT DATA¼ VoLeTranlator; VAR C Alpha n C0;