2. Suppose that we are making an auto part and that the mean length of the part is 5 inches with a standard deviation of 0.25 inches, and the length is normally distributed. The upper specification limit is 5.5 inches and the lower specification limit is 4.2 inches. The z-score table is provided on the next page. a. What is the process capability index, Cpk? b. Under this process, what is the rate of defects per unit (DPU) and the defects per million units (DPMU)? c. If you have five auto parts made under this process, what is the probability that all five of them are not defective? Number in the table represents P(Z≤2) z O Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 -3.6 .0002 .0002 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 -3.5 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 -3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002 -3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003 -3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005 -3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007 -3.0.0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010 -2.9 .0019 .0018 -2.8 .0026 .0025 .0024 -2.7 .0035 .0034 .0033 .0032 -2.6 .0047 .0045 .0044 .0043 -2.5 .0062 .0060 .0059 .0057 -2.4 .0082 .0080 .0078 .0075 -2.3 .0107 .0104 .0102 .0099 -2.2 .0139 .0136 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014 .0023 .0023 .0022 .0021 .0021 .0020 .0019 .0030 .0132 .0129 -2.1 .0179 .0174 0170 .0166 -2.0 .0228 -1.9 .0287 .0281 .0222 .0217 .0212 .0031 .0041 .0055 .0054 .0052 .0051 .0073 .0071 .0069 .0068 .0096 .0094 .0091 .0089 .0125 .0122 .0119 .0116 0162 .0158 .0154 .0150 .0146 .0207 .0202 .0197 .0029 .0028 .0027 .0026 .0040 .0039 .0038 .0037 .0036 .0049 .0048 .0066 .0064 .0087 .0113 .0110 .0084 .0143 .0192 .0188 .0183 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233 -1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294 -1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367 -1.6 .0548 .0537 .0526 .0516 .0495 -1.5 .0668 .0655 -0.9 1841 -0.8 .2119 .2090 -0.7 .2420 2389 -1.4 .0808 .0793 -1.3 .0968 .0951 -1.2 .1151 .1131 -1.1 .1357 .1335 -1.0 .1587 .1562 .1814 .0643 .0630 .0778 .0505 .0618 .0606 .0594 .0582 .0571 .0764 .0749 .0735 .0721 .0708 .0694 .0934 .0918 .0901 .0885 1112.1093 .1075 .1056 .1038 .1020 .1003 .0485 .0475 .0465 .0455 .0559 .0681 .0869 .0853 .0838 .0823 .0985 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170 -0.6 .2743 2709 2676 2643 .2611 .2578 .1539 .1515 .1492 .1469 .1446 .1788 .1762 .1736 .1711 .1685 .2061 .2033 .2005 .1977 .1949 .2358 .2327 .2296 .2266 .2236 2546 -0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 -0.4 .3446 3409 .3372 .3336 .3300 .3264 .3228 -0.3 .3821 .3783 .3745 .3707 .3669 .3632 3594 -0.2 .4207 4168 .4129 .4090 .4052 -0.1 .4602 .4562 .4522 .4483 .4443 -0.0 .5000 .4960 .4920 .4840 .1423 .1401 .1379 .1660 .1635 .1611 .1922 .1894 .1867 2206 2177 .2148 2514 .2483 .2451 2843 2810 .2776 .3192 3156 .3121 .4880 .4364 .4325 .4286 .4761 .4721 .4681 .3557 .3520 .3483 .4013 .3974 3936 3897 .3859 .4404 .4801 .4247 .4641