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662 APPENDIX A STATISTICAL TABLES AND CHARTS 0.21 234 0.4 0.6 0.8 5 0 0 1.0 d Probability of accepting H 0 n = 1 2 4 100 56710 1520 30405075 8 3 (b)O.C. curves for different values of nfor the two-sided normal test for a level of signif icance  0.01. Source:Charts VIa, e, f, k, m,and qare reproduced with permission from “Operating Characteristics for the Common Statistical Tests of Signif icance,’’ by C. L. Ferris, F. E. Grubbs, and C. L. Weaver, Annals of Mathematical Statistics,June 1946. Charts VIb, c, d, g, h, i, j, l, n, o, p,and rare reproduced with permission from Engineering Statistics, 2nd Edition, by A. H. Bowker and G. J. Lieberman, Prentice-Hall, 1972.( a)O.C. curves for different values of nfor the two-sided normal test for a level of signif icance  0.05. 0 0 1234 0.20 0.40 0.60 0.80 1.00 d Probability of accepting H 0 n = 1 3 4 100 5 67 10 1520 30405075 89 2 Chart VI Operating Characteristic Curves PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:31 PM Page 662 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : APPENDIX A 663 –1.000 n = 1 3 100 5 –0.50 0.0 2.00 3.00 0.20 0.40 0.60 0.80 1.00 Probability of accepting H 0 0.50 1.00 1.50 2.50 d 67 10 1520 30405075 89 4 2 –1.00 0 –0.50 0.0 0.50 1.00 1.50 2.00 2.50 3.00 0.20 0.40 0.60 0.80 1.00 n = 1 4 100 567 10 1520 30405075 89 Probability of accepting H 0 d 3 2 (c)O.C. curves for different values of nfor the one-sided normal test for a level of signif icance  0.05. (d)O.C. curves for different values of nfor the one-sided normal test for a level of signif icance  0.01. Chart VI Operating Characteristic Curves ( continued) PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:31 PM Page 663 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : 664 APPENDIX A STATISTICAL TABLES AND CHARTS n = 2 100 4 0 0 12 0.2 0.4 0.6 0.8 1.0 7 15 3 d Probability of accepting H 0 20 30 4050 75 3 5 10 0.2 0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 0 Probability of accepting H 0 d n = 4 n = 5 n = 7 n = 10 n = 15 n = 20 n = 30 n = 40 n = 100 n = 75 n = 50 n = 3 (e)O.C. curves for different values of nfor the two-sided t-test for a level of signif icance  0.05. (f)O.C. curves for different values of nfor the two-sided t-test for a level of signif icance  0.01. Chart VI Operating Characteristic Curves ( continued) PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:31 PM Page 664 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : APPENDIX A 665 Chart VI Operating Characteristic Curves (continued) 0 –0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 – 0.4 –0.6 –0.8 Probability of accepting H 0 d n = 40 n = 50 n = 75 n = 3 n = 4 n = 5 n = 7 n = 10 n = 15 n = 20n = 30 n = 100 0 –0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 –0.4 –0.6 –0.8 Probability of accepting H 0 d n = 5 n = 40 n = 50 n = 75 n = 100 n = 3 n = 4 n = 7 n = 10 n = 15 n = 20 n = 30 (g)O.C. curves for different values of nfor the one-sided t-test for a level of signif icance  0.05. (h)O.C. curves for different values of nfor the one-sided t-test for a level of signif icance  0.01. PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:32 PM Page 665 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : 666 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves (continued) 0 0.40 0.80 1.20 1.60 2.00 2.40 2.80 03.20 3.60 4.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.00 Probability of accepting H 0 65 5 4 3 λ 4 n = 2 3 6 n = 2 10075 50 40 30 20 15 10 9 8 7 0 0.40 0.80 1.20 1.60 2.00 2.40 2.80 03.20 3.60 4.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.00 Probability of accepting H 0 5 3 6 5 4 λ 6 n = 2 4 10075 50 40 30 20 15 10 9 8 7 n = 3 ( i)O.C. curves for different values of nfor the two-sided chi-square test for a level of signif icance  0.05. (j)O.C. curves for different values of nfor the two-sided chi-square test for a level of signif icance  0.01. PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:32 PM Page 666 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : APPENDIX A 667 Chart VI Operating Characteristic Curves (continued) 0 0 1.0 2.0 3.0 4.0 0.20 0.40 0.60 0.80 1.00 Probability of accepting H 0 n = 2 4 567 100 10 15 20 3040 5075 λ 3 8 0 0 1.0 0.20 n = 3 56 100 10 15 20 30 40 50 75 7 98 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0.40 0.60 0.80 1.00 λ Probability of accepting H 0 4 (k)O.C. curves for different values of nfor the one-sided (upper tail) chi-square test for a level of signif icance  0.05. (l)O.C. curves for different values of nfor the one-sided (upper tail) chi-square test for a level of signif icance  0.01. PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:32 PM Page 667 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : 668 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves (continued) 0 0 0.40 0.80 1.20 1.40 0.20 0.40 0.60 0.80 1.00 Probability of accepting H 0 0.20 0.60 1.00 n = 2 3 4 5 6 10 15 20 30405075 789 λ 100 (m)O.C. curves for different values of nfor the one-sided (lower tail) chi-square test for a level of signif icance  0.05. (n)O.C. curves for different values of nfor the one-sided (lower tail) chi-square test for a level of signif icance  0.01. 0 0 n = 2 3 45 100 0.5 01. 1.5 2.0 0.2 0.4 0.6 0.8 1.0 Probability of accepting H 0 10 1520 30 40 50 75 78 λ 6 PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:32 PM Page 668 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : APPENDIX A 669 Chart VI Operating Characteristic Curves (continued) 0 0 0.40 0.80 1.40 1.80 2.20 2.60 3.00 3.40 3.80 4.00 0.20 0.40 0.60 0.80 1.00 λ 1.00 Probability of accepting H 0 6 7 9 16 101 4 5 4 3 8 10 21 31 51 5 n1 = n 2 = 3 6 7 8 9 10 16 21 31 51 101 0 0 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 0.20 0.40 0.60 0.80 1.00 λ 1.00 Probability of accepting H 0 6 7 9 51 101 5 4 8 10 16 21 31 n1 = n 2 = 3 6 7 8 9 10 16 21 31 51 1013 4 5 ( o)O.C. curves for different values of nfor the two-sided F-test for a level of signif icance  0.05. (p) O.C. curves for different values of nfor the two-sided F-test for a level of signif icance  0.01. PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:33 PM Page 669 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : 670 APPENDIX A STATISTICAL TABLES AND CHARTS Chart VI Operating Characteristic Curves (continued) 1 0 23 4 0.20 0.40 0.60 0.80 1.00 3 4 5 100 10 1520 30 78 Probability of accepting H 0 4050 75 λ n1 = n 2 = 2 6 0 0 2.00 4.00 5.00 8.00 10.00 12.00 14.00 0.20 0.40 0.60 0.80 1.00 16.00 n1 = n 2 = 2 3 4 5 6 7 8 109 162141 5131 1.00 λ Probability of accepting H 0 121 (q)O.C. curves for different values of nfor the one-sided F-test for a level of signif icance  0.05. (r)O.C. curves for different values of nfor the one-sided F-test for a level of signif icance  0.01. PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:33 PM Page 670 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : APPENDIX A 671 Table VIICritical Values for the Sign Test r* 0.10 0.05 0.01 Two-sided tests0.10 0.05 0.01 Two-sided tests n0.05 0.025 0.005 One-sided testsn0.05 0.025 0.005 One-sided tests 5 0 23 7 6 4 600 24765 700 25775 8 1 0 0 26 8 7 6 9 1 1 0 27 8 7 6 10 1 1 0 28 9 8 6 11 2 1 0 29 9 8 7 12 2 2 1 30 10 9 7 13 3 2 1 31 10 9 7 14 3 2 1 32 10 9 8 15 3 3 2 33 11 10 8 16 4 3 2 34 11 10 9 17 4 4 2 35 12 11 9 18 5 4 3 36 12 11 9 19 5 4 3 37 13 12 10 20 5 5 3 38 13 12 10 21 6 5 4 39 13 12 11 22 6 5 4 40 14 13 11 Table VIIICritical Values for the Wilcoxon Signed-Rank Test w* 0.10 0.05 0.02 0.01 Two-sided tests n*0.05 0.025 0.01 0.005 One-sided tests 4 50 620 7320 85310 98531 10 10 8 5 3 11 13 10 7 5 12 17 13 9 7 13 21 17 12 9 14 25 21 15 12 15 30 25 19 15 16 35 29 23 19 17 41 34 27 23 18 47 40 32 27 19 53 46 37 32 20 60 52 43 37 21 67 58 49 42 22 75 65 55 48 23 83 73 62 54 24 91 81 69 61 25 100 89 76 68 * If n25, W (or W ) is approximately normally distributed with mean n(n 1)4 and variance n(n 1)(2n 1)24. PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:33 PM Page 671 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : 672 APPENDIX A STATISTICAL TABLES AND CHARTS Table IXCritical Values for the Wilcoxon Rank-Sum Test w0.05 n1* n2 45678910 1112 1314 15 410 51117 6121826 713202736 81421293849 9 152231405163 10 15 23 32 42 53 65 78 11 16 24 34 44 55 68 81 96 12 17 26 35 46 58 71 85 99 115 13 18 27 37 48 60 73 88 103 119 137 14 19 28 38 50 63 76 91 106 123 141 160 15 20 29 40 52 65 79 94 110 127 145 164 185 16 21 31 42 54 67 82 97 114 131 150 169 17 21 32 43 56 70 84 100 117 135 154 18 22 33 45 58 72 87 103 121 139 19 23 34 46 60 74 90 107 124 20 24 35 48 62 77 93 110 21 25 37 50 64 79 95 22 26 38 51 66 82 23 27 39 53 68 24 28 40 55 25 28 42 26 29 27 28 *For n 1and n 28, W 1is approximately normally distributed with mean and variance n 1n2(n1 n 2 1)12. 12n11 n1 n 2 12 Table IXCritical Values for the Wilcoxon Rank-Sum Test (continued) w0.01 n1 n2 45678910 1112 1314 15 515 6101623 710172432 81117253443 9 111826354556 10 12 19 27 37 47 58 71 11 12 20 28 38 49 61 74 87 12 13 21 30 40 51 63 76 90 106 13 14 22 31 41 53 65 79 93 109 125 14 14 22 32 43 54 67 81 96 112 129 147 15 15 23 33 44 56 70 84 99 115 133 151 171 16 15 24 34 46 58 72 86 102 119 137 155 17 16 25 36 47 60 74 89 105 122 140 18 16 26 37 49 62 76 92 108 125 19 17 27 38 50 64 78 94 111 20 18 28 39 52 66 81 97 21 18 29 40 53 68 83 22 19 29 42 55 70 23 19 30 43 57 24 20 31 44 25 20 32 26 21 27 28 PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:33 PM Page 672 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : APPENDIX A 673 Table XFactors for Constructing Variables Control Charts Factor for Control Limits ChartRChartSChart n*A 1 A2 d2 D3 D4 c4 n 2 3.760 1.880 1.128 0 3.267 0.7979 2 3 2.394 1.023 1.693 0 2.575 0.8862 3 4 1.880 .729 2.059 0 2.282 0.9213 4 5 1.596 .577 2.326 0 2.115 0.9400 5 6 1.410 .483 2.534 0 2.004 0.9515 6 7 1.277 .419 2.704 .076 1.924 0.9594 7 8 1.175 .373 2.847 .136 1.864 0.9650 8 9 1.094 .337 2.970 .184 1.816 0.9693 9 10 1.028 .308 3.078 .223 1.777 0.9727 10 11 .973 .285 3.173 .256 1.744 0.9754 11 12 .925 .266 3.258 .284 1.716 0.9776 12 13 .884 .249 3.336 .308 1.692 0.9794 13 14 .848 .235 3.407 .329 1.671 0.9810 14 15 .816 .223 3.472 .348 1.652 0.9823 15 16 .788 .212 3.532 .364 1.636 0.9835 16 17 .762 .203 3.588 .379 1.621 0.9845 17 18 .738 .194 3.640 .392 1.608 0.9854 18 19 .717 .187 3.689 .404 1.596 0.9862 19 20 .697 .180 3.735 .414 1.586 0.9869 20 21 .679 .173 3.778 .425 1.575 0.9876 21 22 .662 .167 3.819 .434 1.566 0.9882 22 23 .647 .162 3.858 .443 1.557 0.9887 23 24 .632 .157 3.895 .452 1.548 0.9892 24 25 .619 .153 3.931 .459 1.541 0.9896 25 * where nnumber of observations in sample. n25: A 13 1n X PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:33 PM Page 673 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : 674 APPENDIX A STATISTICAL TABLES AND CHARTS Table XIFactors for Tolerance Intervals Values of kfor Two-Sided Intervals Confidence Level 0.90 0.95 0.99 Percent Coverage 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.99 2 15.978 18.800 24.167 32.019 37.674 48.430 160.193 188.491 242.300 3 5.847 6.919 8.974 8.380 9.916 12.861 18.930 22.401 29.055 4 4.166 4.943 6.440 5.369 6.370 8.299 9.398 11.150 14.527 5 3.949 4.152 5.423 4.275 5.079 6.634 6.612 7.855 10.260 6 3.131 3.723 4.870 3.712 4.414 5.775 5.337 6.345 8.301 7 2.902 3.452 4.521 3.369 4.007 5.248 4.613 5.488 7.187 8 2.743 3.264 4.278 3.136 3.732 4.891 4.147 4.936 6.468 9 2.626 3.125 4.098 2.967 3.532 4.631 3.822 4.550 5.966 10 2.535 3.018 3.959 2.839 3.379 4.433 3.582 4.265 5.594 11 2.463 2.933 3.849 2.737 3.259 4.277 3.397 4.045 5.308 12 2.404 2.863 3.758 2.655 3.162 4.150 3.250 3.870 5.079 13 2.355 2.805 3.682 2.587 3.081 4.044 3.130 3.727 4.893 14 2.314 2.756 3.618 2.529 3.012 3.955 3.029 3.608 4.737 15 2.278 2.713 3.562 2.480 2.954 3.878 2.945 3.507 4.605 16 2.246 2.676 3.514 2.437 2.903 3.812 2.872 3.421 4.492 17 2.219 2.643 3.471 2.400 2.858 3.754 2.808 3.345 4.393 18 2.194 2.614 3.433 2.366 2.819 3.702 2.753 3.279 4.307 19 2.172 2.588 3.399 2.337 2.784 3.656 2.703 3.221 4.230 20 2.152 2.564 3.368 2.310 2.752 3.615 2.659 3.168 4.161 21 2.135 2.543 3.340 2.286 2.723 3.577 2.620 3.121 4.100 22 2.118 2.524 3.315 2.264 2.697 3.543 2.584 3.078 4.044 23 2.103 2.506 3.292 2.244 2.673 3.512 2.551 3.040 3.993 24 2.089 2.489 3.270 2.225 2.651 3.483 2.522 3.004 3.947 25 2.077 2.474 3.251 2.208 2.631 3.457 2.494 2.972 3.904 30 2.025 2.413 3.170 2.140 2.529 3.350 2.385 2.841 3.733 40 1.959 2.334 3.066 2.052 2.445 3.213 2.247 2.677 3.518 50 1.916 2.284 3.001 1.996 2.379 3.126 2.162 2.576 3.385 60 1.887 2.248 2.955 1.958 2.333 3.066 2.103 2.506 3.293 70 1.865 2.222 2.920 1.929 2.299 3.021 2.060 2.454 3.225 80 1.848 2.202 2.894 1.907 2.272 2.986 2.026 2.414 3.173 90 1.834 2.185 2.872 1.889 2.251 2.958 1.999 2.382 3.130 100 1.822 2.172 2.854 1.874 2.233 2.934 1.977 2.355 3.096 PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:33 PM Page 674 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX : APPENDIX A 675 Table XIFactors for Tolerance Intervals (continued) Values of kfor One-Sided Intervals Confidence Level 0.90 0.95 0.99 Percent Coverage 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.99 2 10.253 13.090 18.500 20.581 26.260 37.094 103.029 131.426 185.617 3 4.258 5.311 7.340 6.155 7.656 10.553 13.995 17.370 23.896 4 3.188 3.957 5.438 4.162 5.144 7.042 7.380 9.083 12.387 5 2.742 3.400 4.666 3.407 4.203 5.741 5.362 6.578 8.939 6 2.494 3.092 4.243 3.006 3.708 5.062 4.411 5.406 7.335 7 2.333 2.894 3.972 2.755 3.399 4.642 3.859 4.728 6.412 8 2.219 2.754 3.783 2.582 3.187 4.354 3.497 4.285 5.812 9 2.133 2.650 3.641 2.454 3.031 4.143 3.240 3.972 5.389 10 2.066 2.568 3.532 2.355 2.911 3.981 3.048 3.738 5.074 11 2.011 2.503 3.443 2.275 2.815 3.852 2.898 3.556 4.829 12 1.966 2.448 3.371 2.210 2.736 3.747 2.777 3.410 4.633 13 1.928 2.402 3.309 2.155 2.671 3.659 2.677 3.290 4.472 14 1.895 2.363 3.257 2.109 2.614 3.585 2.593 3.189 4.337 15 1.867 2.329 3.212 2.068 2.566 3.520 2.521 3.102 4.222 16 1.842 2.299 3.172 2.033 2.524 3.464 2.459 3.028 4.123 17 1.819 2.272 3.137 2.002 2.486 3.414 2.405 2.963 4.037 18 1.800 2.249 3.105 1.974 2.453 3.370 2.357 2.905 3.960 19 1.782 2.227 3.077 1.949 2.423 3.331 2.314 2.854 3.892 20 1.765 2.028 3.052 1.926 2.396 3.295 2.276 2.808 3.832 21 1.750 2.190 3.028 1.905 2.371 3.263 2.241 2.766 3.777 22 1.737 2.174 3.007 1.886 2.349 3.233 2.209 2.729 3.727 23 1.724 2.159 2.987 1.869 2.328 3.206 2.180 2.694 3.681 24 1.712 2.145 2.969 1.853 2.309 3.181 2.154 2.662 3.640 25 1.702 2.132 2.952 1.838 2.292 3.158 2.129 2.633 3.601 30 1.657 2.080 2.884 1.777 2.220 3.064 2.030 2.515 3.447 40 1.598 2.010 2.793 1.697 2.125 2.941 1.902 2.364 3.249 50 1.559 1.965 2.735 1.646 2.065 2.862 1.821 2.269 3.125 60 1.532 1.933 2.694 1.609 2.022 2.807 1.764 2.202 3.038 70 1.511 1.909 2.662 1.581 1.990 2.765 1.722 2.153 2.974 80 1.495 1.890 2.638 1.559 1.964 2.733 1.688 2.114 2.924 90 1.481 1.874 2.618 1.542 1.944 2.706 1.661 2.082 2.883 100 1.470 1.861 2.601 1.527 1.927 2.684 1.639 2.056 2.850 PQ220 6234F .pp [ 1 26] .qxd 5/8/02 8:33 PM Page 675 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX :