Document O3xJbqD0EqDVgZqb0ornmJZqv

engineering statistics AND QUALITY CONTROL IRVING W. BURR Professor of Mathematics and Research Associate in the Statistical Laboratory, Purdue University Ti 1 SB 61 New York Toronto London McGRAW-HILL BOOK COMPANY, INC. 1953 ft 5 ucc 051519 CHAPTER 15 SINGLE SAMPLING OF MEASUREMENTS 16.1. One-way Protection on the Mean, o' Known. Single Sampling. In this chapter we shall present some sampling plans which require rather strong assumptions about the distribution from which we draw our sample and concerning which we are to make a decision. Throughout the chapter, unless the contrary is specifically indicated, we shall assume that the distribution is normal. Furthermore in the present section we assume that the population standard deviation o' is known. Then we seek a sampling plan which protects against, say, too low a population mean X' but which is not concerned with how high the mean may be. Let us consider an example. Suppose that the distribution of tensile strength of lots of castings is known to be normal with a* = 2,500 psi. A casting with a tensile strength less than 65,000 psi is likely to fail in service. Hence the lot mean should be far enough above 65,000 psi so that very few castings will have a tensile below this figure. Would 70,000 psi be adequate for X'? Let us see. We find min. specification -- X' 65,000 -- 70,000 t----------------------------- ------------------------------------------------------------ ----------------- -,,2.M00 Using Table A (page 404), we learn that we can expect to have about 2.3 per cent of the castings below 65,000 psi. If the casting is at all critical, such a percentage would probably be considered unsatisfactory. Would X' - 72,500 psi be safe? 65,000 - 72,500 2,500 3.00 so that only about .13 per cent of the castings from such a lot will lie below 65,000 psi. For the purposes of this example we shall consider such a percentage to be clearly acceptable. Hence we may take X\ -- 70,000 psi - rejectable quality X't = 72,500 psi -- acceptable quality Now we must go further. How much risk shall we take of accepting a lot having if, * 70,000 psi if such a lot should be offered us? And how 361 i Ucc 0$1$2q 362 engineering statistics and quality control [Sec. 15.1 much risk shall we take of rejecting a lot having X\ = 72,500 psl whenever one is offered us? Just as with the choice of what constitutes acceptable and rejectable quality, the setting of the risks is a practical engineering matter. We shall use the following notation: a = risk of rejecting acceptable quality, if offered d = risk of accepting rejectable quality, if offered We shall arbitrarily set a * .02 and $ = .05 in our example. Although there are many possible kinds of plans, we shall try to find a single sampling plan of the following type to meet our desired requirements: Choose n pieces at random from the lot. Test these. If their mean measurement is not less than K accept; otherwise reject. It is our job therefore to find n and K in order to determine the plan. Flo. 15.1. Graphs showing the distributions of X tor acceptable and rejectable quality levels and the corresponding risks. One-way protection. Tensile strengths of castings. Since this plan accepts X > K, we must work with the distribution of J?'s. We know that the distribution is normal with some mean X' and standard deviation a'/y/n -- 2,500/y/n. Figure 15.1 shows the two X distributions and a and 0 corresponding. To use the condition on a, we look in Table A for that value of t for which the area below is .02. It w --2.054. Hence we have / Kt - -2.054 a'/y/n K - 72,500 2,500/y/n ucc 051521 t Sec. 15.1] i SINGLE SAMPLING OF MEASUREMENTS 363 Thus we have one equation containing K and n. Next we derive an equation from 0. Using Table A, the value of t for which the area below is .95 (area above = .05 = 0) ia +1.645. Hence K-X[ K- 70,000 + 1-640 = ------t=- = ---------;-- a jyjn 2,500/\/n Subtracting the preceding equation from this one yields & j-1-1.645 + 2:054 2,500 n 2,500/Vn frorq which y/n = 3.699, or n = 13.68. But for a sample size, n must be a whole number; so in order to avoid having either risk exceed the desired value we take n = 14. t Now in finding K, if we substitute n => 14 into the equation arising from (8, we shall preserve 0 at .05, while a will be slightly below .02. On the other hand, if we substitute 14 into the equation from a, then a will remain at .02 while 0 will be a bit below .05. We shall do the latter. -2.054 -1,372 K K - 72,500 2,500/VU K - 72,500 71,128 Thus our sampling plan reads: Choose 14 pieces at random, and obtain the tensile strength of each. Find the mean tensile. If it ia at least 71,128 psi, accept the lot; otherwise reject. Now let us check back and find out what happened to the risk 0 of accepting material at X'y, if offered. For the X distribution with mean X', we have K-X[ 71,128 - 70,000 ` " a'/y/n " 2,500/VTi The risk of erroneous acceptance is therefore .0457, rather than .05. What else would we like to know about the plan? Since, in practice, lots may contain a wide variety of values of X', rather than just 70,000 or 72,500 psi, we would like to know how our sampling plan behaves on lots of other quality levels. Thus we must construct the operating characteristic curve. Table 15.1 shows the calculations for such a curve. The table is largely self-explanatory. Column (3) gives the standard score value of K relative to the lot mean. The area of the X distribution below K ~ 71,128 is given in column (4). Since this is the probability of t We always round such an approximate n upward to the next whole number. , ucc 051522 : 364 engineering statistics and quality control [Sec. 15.1 Table 15.1. Calculation por OC Curve for the Sampling Plan: n - 14; Accept ip X > 71,128, When ' - 2,500 (r'x - 2,500/vTi -- 668.2) X' (1) 68,000 68,500 69,000 69,500 70,000 70,500 71,000 71,500 72,000 72.500 73,000 73,a00 71,128 - X' (2) +3,128 +2,628 +2,128 + 1,628 +1,128 + 628 + 128 - 372 - 872 -1,372 -1,872 -2,372 t- (2) 668.2 (3) +4.681 +3.933 +3.185 +2.436 +1.688 + .940 + .192 - .557 -1.305 -2.053 -2.802 -3 550 Area below t in (3) (4) 1.0000 1.0000 .9993 .9926 .9543 .8264 .5762 .2888 .0960 .0200 .0026 .0002 P. 1 - (4) (5) t65,000 - X' 2,500 (6) Area below t in (6) (7) .0000 .0000 .0007 .0074 .0457 .1736 .4238 .7112 .9041 .9800 .9974 .9998 -1.20 -1.40 -1.60 -1.80 -2.00 -2.20 -2.40 -2.60 -2.80 -3.00 -3.20 -3.40 .1151 .0808 .0548 .0359 .0227 .0139 .0082 .0047 .0026 .0013 .0007 .0003 rejecting the lot, we subtract from 1.0000 to get P*, the probability of acceptance. The last two columns are for determining what pc centage of pieces may be expected to have tensile strengths below 65,000 psi. Fin. 15.2. OC curve for the single sampling plan for tensile strengths, n 14, accept if X > 71,128 psi (' -- 2500 psi). Column (6) gives the standard score of this minimum specification in the distribution for individual strengths, X, while column (7) lists the cor responding proportion of pieces below the minimum. Figure 15.2 shows the operating characteristic curve for our sampling plan [column (5) vs. column (1)]. There are several features we should *A Occ 0s'S?3 Se<". 15.1] single sampling of measurements 3G5 (1) As the average tensile strength increases, the probability of acceptance always increases. (2) The risk of acceptance of a lot of average tensile below 70,000 psi is less than .0457 and rapidly approaches zero as X' decreases. (3) For lots of average tensile above 72,500 psi the risk of rejection is less than .02 and rapidly approaches zero as X' increases. (4) There is an intermediate band of tensile strength of indifferent quality where both the probabilities of acceptance and rejec tion are rather high (about 70,500 to 72,000 psi). Fig. 15.3. OC curve for the single sampling plan for tensile strengths, n -- 14. accept if X > 71,138 psi (S -- 2500 psi). The horizontal scale is in terms of the proportion of pieces below the minimum specification. In Fig. 15.3 we have plotted columns (5) and (7) of Table 15.1, which gives the probability of acceptance of the lot vs, the fraction defective (pieces below 65,000 psi). This curve is thus just like the OC curves we saw in Chaps. 12 and 13. It is interesting to note that, in order to obtain an OC curve comparable with that in Fig. 15.3 by means of an attributes plan, a sample size of about 300 must be used (see Prob. 15.2 at the end of the chapter). Hence one can easily see the amount of money to be saved through using the measurement sampling plan. (The attribute test here would call for subjecting each casting to a 65,000 psi tensile and noting whether or not the casting breaks.) Let us now summarize the method of calculation of a single sampling plan when we can assume normality and when o' is known. Protection against a Minimum Specification 1. Choose an acceptable value for a lot mean, say It should be enough S'* above the minimum specification to give an acceptably small percentage of individual pieces below that minimum specification. I 366 engineering statistics and quality control [Sec. 15.1 2. Similarly chooae a rejectable value for a lot mean, say X',, which is so close to the i"i"i"n specification that we want to be quite sure of rejecting lots with mean rt. 3. Corresponding to the acceptable mean X\, choose a risk a of erroneously rejecting a lot having a mean Xt, if offered. 4. Likewise choose a risk 0 that a lot of mean Xl shall be erroneously accepted, if offered. 5. Now seek K and n to determine the plan from the following equations: K ~ X" t (area below -- ) " _ 7VS JC - Jt* t (area below -- 1 -- 0) ----------7= 7V' (15.1) (15.2) where the parentheses following the t's tell what areas in Table A (page 404) are used in determining that value of L By subtraction determine the approximate \fn and n, rounding the latter upward to the next whole number. Substitute the last into which* ever of Eqs. (15.1) and (15.2) we want to have exactly fulfilled. (Thus either <* or 0 will be exact.) This substitution determine* K. Next substitute the values of K and n into the right-hand side of the equation which is to be approximately satisfied. This give* a value of ( which we use in Table A to find the risk which is only to be approximate. It will, however, always be smaller than that risk 0 or a originally specified. 6. If the sample eise determined in (5) is uneconomically large, then we must either increase one or both of the risks a and 0 or we must widen the distance between X'l and X',. Then we recompute a new plan until we seem to have a good balance between the sample sise, cost, and the power of the plan to discriminate. Thus a large sample sise a will give quite a square-shouldered OC curve which discriminates sharply between lots, while a small n will give a gently sloping OC curve with little discriminating power, 7. If desired, a complete OC curve may be computed aa in Table 15.1. 8. The sampling plan is then: "C. Take n measurements, and accept if X K, or reject if Jt < K. y `JBa. Protection against a Maximum Specification ^ --y: 1. Step* are analogous to the foregoing. 2. We still retain X1, > X\, but now represents rejectable quality and -M acceptable quality. As before a is the risk of rejection of acceptable quality (now [)* and 0 is the risk of acceptance of rejectable quality (now X1,), if either is offered for inspection. 3. Solve as before for , K. t (area below TM 1 -- a) t (area below -- 0) K-X'i 7V* K --P, *7V (15.3) (15.4) 4. By trial obtain an economic balance between n and the risks of wrong decisions. 5. The plan is: Take n measurements, and accept if X ^ K. ucc 051525 404 ENGINEERING STATISTICS and quality control Table A. Abbas under ths Normal Curve and below ( -- (X -- X')/rx, Teat Is ( *.00 .01 .03 .03 .04 .05 .06. .07 .08 .09 -3.5 -3.4 -3.3 -3.2 -3.1 .0002 .0003 .0005 .0007 .0010 .0002 .0003 .0005 .0007 .0009 .0003 .0003 .0005 .0006 .0009 .0002 .0003 .0004 .0006 .0009 .0002 .0003 .0004 .0006 .0008 .0002 .0003 .0004 .0006 .0008 .0002 .0003 .0004 .0006 .0008 .0003 .0003 .0004 .0005 .0008 .0002 .0003 .0004 .0005 .0007 .0002 .0002 .0003 .0005 .0007 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 .0013 .0019 .0026 .0035 .0047 .0013 .0018 .0025 .0034 .0045 .0013 .0018 .0024 .0033 .0044 .0012 .0017 .0023 .0032 .0043 .0013 .0018 .0023 .0031 .0041 .0011 .0016 .0022 .0030 .0040 .0011 .0015 .0031 .0039 .0039 .0011 .0015 .0021 .0028 .0038 .0010 .0014 .0020 .0027 .0037 .0010 .0014 .0019 .0026 .0036 .0062 .0082 .0107 .0139 .0179 .0060 .0080 .0104 .0136 .0174 .0059 .0078 .0102 .0132 .0170 .0057 .0075 .0099 .0129 .0166 .0055 .0073 .0096 .0125 .0162 .0054 .0071 .0094 .0122 .0158 .0052 .0069 .0091 .0119 .0154 .0051 .0068 .0089 .0116 .0150 .0049 .0065 .0087 .0113 .0146 .0048 .0064 .0084 .0110 .0143 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.8 -0.4 -0.3 -0.2 -0.1 -0.0 .0228 .0287 .0359 .0446 .0548 .0222 .0281 .0351 .0438 .0537 .0217 .0274 .0344 .0427 .0526 .0212 .0268 .0336 .0418 .0516 .0307 .0202 .0262 .0256 .0329 .0332 .0409 .0401 .0505 , 0495 tl .0197 0S3& .0314 .0392 .0485 .0192 .0244 .0307 .0384 .0475 .0188 .0239 .0301 .0375 .0465 .0183 .0233 .0294 .0367 .0455 .0668 .0808 .0968 .1151 .1357 .0655 .0793 .0951 .1131 .1335 .0643 .0778 .0934 .1113 .1314 .0630 .0764 .0918 .1093 .1202 .0618 .0749 .0901 .1075 .1271 .0606 .0735 .0888 .1056 .1251 .0594 .0721 .0869 .1038 .1230 .0582 .0708 .0853 .1020 .1210 .0571 .0694 .0838 .1190 .0559 .0681 .0823 .0985 .1170 .1387 .1841 .2119 .2420 .2743 .1563 .1814 .2090 .2389 .2709 .1539 .1788 .2061 .2358 .2676 .1515 .1763 .2033 .2327 .2643 .1493 .1736 .3005 .2296 .2811 .1469 .1711 .1977 .2266 .2578 .1446 .1685 .1949 .2236 .2546 .1423 .1660 .1923 .3306 .2514 .1401 .1635 .1894 .2177 .2483 .1379 .1611 .1867 .2148 .2451 ,3085 .3446 .3831 .4307 .4603 .5000 .3050 .3409 .3783 .4168 .4562 .4960 .3015 .3372 .3745 :4l29 .4522 .4920 .2981 .3336 .3707 .4090 .4483 .4880 .2946 .3300 .3669 4052 .4443 .4840 .2913 .3264 .3633 .4013 .4404 .4801 .2877 .3328 .3594 .3974 .4364 .4761 .3843 .3193 .3557 .3936 .4325 .4721 .2810 .2775 .3156 .3121 .3520 .3483 .3897 .3359 .4286 .4217 .4681 .4641 Occ APPENDIX 405 Tabus JL Areas omra the Nohsaal Curve and below t * (X -- ')/<tx. That Ii J dL (Continued) t .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 +0.0 +0.1 +0.2 +0.3 +0.4 +0.0 .5000 .5398 -.5793 .6179 .6554 .6915 .5040 .5438 .5832 .6217 .6591 .6950 .3080 .5478 .5871 .6255 .6628 .6985 .5120 .5517 .5910 .6293 .6664 .7019 .5160 .5557 .5948 .6331 .6700 .7054 .5199 .5596 .5987 .6368 .6736 .7088 .5239 .5636 .6026 .6406 .6772 .7123 .3279 .5319 .5675 .5714 .6064 .3123 .6443 r.64S0 .6808 .6844 .7157 .7190 .5359 .5753 .6141 .6517 6879 .7224 +0.6 +0.7 +0 8 +0.9 +1.0 .7257 .7580 .7881 .8159 .8413 .7291, .7611 .7910 .8186 .8438 .7324 .7357 .7642 .7673 .7939 .7967 .8212 .8238 .8461" .8485 .7389 .7704 .7995 .8264 .8508 .7422 .7454 .7734 .7764 .8023 .8051 .8239 .8315 .8531. 8554 .7486 .7794 /sois .8340 .8577 .7517 .7823 .8106 .8365 .8599 .7549 .7852 .8133 .8389 8621 +1.1 +1.2 +1.3 +1.4 +1.5 .8643 .8849 .9032 .9192 .9332 .8665 .8869 :9049 .9207 .9345 .8686 .8888 .9066 .9222 .9357 .8708 .8907 .9082 .9236 .9370 .8729 .8925 .9099 .9251 .9382 .8749 .8944 .9115 .9265 .9394 .8770 .8962 .9131 .9279 .9406 .8790 .8810 .8980 .8997 .9147 .9162 .9292 .9306 .9418 .9429 .8830 .9015 .9177 .9319 .9441 +1.6 +1.7 +1.8 +1.# +2.0 .9452 .9554 .9641 .9713 .9772 .9463 .9564 .9649 .9719 .9778 .9474 .9573 .9656 .9726 .9783 .9484 .9582 .9664 .9732 .9738 .9495 ,9505 .9591 .9599 .9671 .9678 .9738 .9744 .9793 .9798 .9515 .9608 .9686 .9750 .9803 .9525 .9535 .9616" 9525 .9693 9699 .9756 .9761 .9808 .9812 .9545 .9633 .9706 .9767 .9817 +2.1 +2.2 +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +3.9 +3.0 .9821 .9861 .9893 .9918 .9938 .9826 .9864 .9896 .9920 .9940 .9830 .9868 .9898 .9922 .0941 .9834 .9871 .9901 .9925 .9943 .9838 .9875 .9904 .9927 .9945 .9842 .9878 .9906 .9929 .9946 .9846 .9850 .9881 .9884 .9909- .9911 .9931 .9932 .9948 .9949 .9854 .9887 .9913 .9934 .9951 .9857 .9890 .9916 .9936 .9952 .9953 .9965 .9974 .9981 .9987 .9955 .9966 .9975 .9982 .9987 .9956 .9967 .9976 .9982 .9987 .9957 .9968 .9977 .9983 .9988 ..9Mw9o5Mw9 .9977 .9984 .9988 .9960 .9961 .9970 .9971 .9978 .9979 .9984 '.9985 .9989 m nnnVan* .9962 .9972 .9979 .9985 .9989 .9963 .9973 .9980 .9986 .9990 . AVUAU*1i .9974 .9981 .9986 .9990 +3.1 +3.2 +3.3 +3.4 +3.5 .9990 .9993 .9995 .9997 .9998 .9991 .9993 .9995 .9997 .9998 .9991 .9994 .9995 .9997 .9998 .9991 .9994 .9996 .9997 .9998 .9992 .9994 ,vAyMvv* .9997 .9998 .9992 .9994 .9996 .9997 1 AwflvAOo .9992 VAAV4Un .9996 .9997 .9998 .9992 .9995 .9996 .9997 .9998 .9993 .9995 .nAVAAQ* .9997 .9998 .9993 .9995 .9997 .9998 .9998 J' ?p\ V -- 7 - - /-?/> i ) UCC 051527