Question

1. For constructing a X-bar chart, subgroup sample average (x bar) and range (r) are computed for each of 24 preliminary samples using a sample size n=5. We have the following data summary: 24 24 Ex;=816 and Er;= 113 i = 1 i = 1 (a) Find the following for X-bar chart: 06 pts) Upper control limit, Center line, and Lower control limit (b) Find the following for R chart: (6 pts) Upper control limit, Center line, and Lower control limit: (c) Calculate the fraction of defective when the specification is 34 plus and minus 2. Assume the process is normally distributed. (8 pts)

Answer 1

918 |l8 2.4 - : Eñi = 84 - 24 29 เE | 려 R = 2น E Ri . 13 e 61) 29 E Ri isl AM 29

Important measures giren Az = 0.577 for n=5 from the giren problem, are: table, for the =0 3 DA - 2:115 d₂=2.326

s Chart 0 Table X Factors for Constructing Variables Control Charts Factor for Control Limits X Chart R Chart 4 43 dy D D. 2 3.760 1.880 1.128 0 3.267 3 2.394 1.023 1,693 2.575 4 1.880 .729 2.059 0 2.282 5 1.596 .577 2.326 0 2.115 6 1.410 .483 2.534 0 2.004 7 1.277 .419 2.704 ..076 1.924 8 1.175 .373 2.847 .136 1.864 9 1.094 .337 2.970 .184 1.816 10 1.028 .308 3.078 .223 1.777 11 973 285 3.173 .256 1.744 12. .925 266 3.258 284 1.716 13 .884 .249 3.336 .308 1.692 14 .848 .235 3.407 .329 1.671 15 .816 .223 3.472 .348 1.652 16 .788 .212 3.532 .364 1.636 17 .762 -203 3.588 .379 1.621 18 .738 .194 3.640 .392 1.608 19 .717 .187 3.689 .404 1.596 20 .697 .180 3.735 .414 1.586 21 .679 .173 3.778 .425 1.575 22 .662 .167 3.819 .434 1.566 23 .647 .162 3.858 443 1.557 24 .632 .157 3.895 .452 1.548 25 .619 .153 3.931 .459 1.541 * > 25:4, = 3/Vn where n-number of observations in sample. CA 0.7979 0.8862 0.9213 0.9400 0.9515 0.9994 0.9650 0.9693 0.9727 0.9754 0.9776 0.9794 0.9810 0.9823 0.9835 0.9845 0.9854 0.9862 0.9869 0.9876 0.9882 0.9887 0.9892 0.9896 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16 17 18 19 20 21 22 23 24 25

(a) x-chart ! Lc 1 = .A 211 xli 211 CL - UCL 2 G + A, R. LCL 34 - 0.577 X 4.7) ~ 31.28 CL 34 VOL = 34 +0.577 X 4.71 36.72 R- chart O x 4.71 = 0 12 L - D₂R CL = 4.71 UCL DA 2.115x4.71 a 9.96 (4) fraction defective, = 9 (4)+1-16) Here, we will use ü=ă= 34 R 4.71 22:02 d. 2.326 L = 34-2 = 32 U = 34 +2 = 36

D() is caf of N(0:1) distribution and the values here are taken froom standard normal distribution table. So, 32-34 2.02 p=0 +1-7(33.3%) = ~ (-0.99)+1 € (0.99) - 1-9 (0.99) +1- (0.99) [: Standard normal distribution is symmetric about mean of = 2 - 27(0.44) = 2:- 2x0.8389 = 0.3222

Standard normal table :

P(X) - -L Odt * ~ N0,1) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5060 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0,5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7519 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.00.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9481 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9564 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990