appendix a statistical tables and proofs
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7/30/2019 Appendix a Statistical Tables and Proofs
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7/30/2019 Appendix a Statistical Tables and Proofs
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428 Appendix A Statistical Tables and Proofs
Table A.1 Binomial Probability Sumsr
x=0
b(x;n, p)
p
n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
1 0 0.9000 0.8000 0.7500 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000
1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 0 0.8100 0.6400 0.5625 0.4900 0.3600 0.2500 0.1600 0.0900 0.0400 0.01001 0.9900 0.9600 0.9375 0.9100 0.8400 0.7500 0.6400 0.5100 0.3600 0.19002 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 0 0.7290 0.5120 0.4219 0.3430 0.2160 0.1250 0.0640 0.0270 0.0080 0.00101 0.9720 0.8960 0.8438 0.7840 0.6480 0.5000 0.3520 0.2160 0.1040 0.02802 0.9990 0.9920 0.9844 0.9730 0.9360 0.8750 0.7840 0.6570 0.4880 0.27103 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 0 0.6561 0.4096 0.3164 0.2401 0.1296 0.0625 0.0256 0.0081 0.0016 0.00011 0.9477 0.8192 0.7383 0.6517 0.4752 0.3125 0.1792 0.0837 0.0272 0.0037
2 0.9963 0.9728 0.9492 0.9163 0.8208 0.6875 0.5248 0.3483 0.1808 0.05233 0.9999 0.9984 0.9961 0.9919 0.9744 0.9375 0.8704 0.7599 0.5904 0.34394 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5 0 0.5905 0.3277 0.2373 0.1681 0.0778 0.0313 0.0102 0.0024 0.0003 0.00001 0.9185 0.7373 0.6328 0.5282 0.3370 0.1875 0.0870 0.0308 0.0067 0.00052 0.9914 0.9421 0.8965 0.8369 0.6826 0.5000 0.3174 0.1631 0.0579 0.00863 0.9995 0.9933 0.9844 0.9692 0.9130 0.8125 0.6630 0.4718 0.2627 0.08154 1.0000 0.9997 0.9990 0.9976 0.9898 0.9688 0.9222 0.8319 0.6723 0.40955 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
6 0 0.5314 0.2621 0.1780 0.1176 0.0467 0.0156 0.0041 0.0007 0.0001 0.0000
1 0.8857 0.6554 0.5339 0.4202 0.2333 0.1094 0.0410 0.0109 0.0016 0.00012 0.9842 0.9011 0.8306 0.7443 0.5443 0.3438 0.1792 0.0705 0.0170 0.00133 0.9987 0.9830 0.9624 0.9295 0.8208 0.6563 0.4557 0.2557 0.0989 0.01594 0.9999 0.9984 0.9954 0.9891 0.9590 0.8906 0.7667 0.5798 0.3446 0.11435 1.0000 0.9999 0.9998 0.9993 0.9959 0.9844 0.9533 0.8824 0.7379 0.46866 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 0 0.4783 0.2097 0.1335 0.0824 0.0280 0.0078 0.0016 0.0002 0.00001 0.8503 0.5767 0.4449 0.3294 0.1586 0.0625 0.0188 0.0038 0.0004 0.00002 0.9743 0.8520 0.7564 0.6471 0.4199 0.2266 0.0963 0.0288 0.0047 0.00023 0.9973 0.9667 0.9294 0.8740 0.7102 0.5000 0.2898 0.1260 0.0333 0.00274 0.9998 0.9953 0.9871 0.9712 0.9037 0.7734 0.5801 0.3529 0.1480 0.0257
5 1.0000 0.9996 0.9987 0.9962 0.9812 0.9375 0.8414 0.6706 0.4233 0.14976 1.0000 0.9999 0.9998 0.9984 0.9922 0.9720 0.9176 0.7903 0.52177 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
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Table A.1 Binomial Probability Table 429
Table A.1 (continued) Binomial Probability Sumsr
x=0
b(x;n, p)
p
n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
8 0 0.4305 0.1678 0.1001 0.0576 0.0168 0.0039 0.0007 0.0001 0.0000
1 0.8131 0.5033 0.3671 0.2553 0.1064 0.0352 0.0085 0.0013 0.00012 0.9619 0.7969 0.6785 0.5518 0.3154 0.1445 0.0498 0.0113 0.0012 0.00003 0.9950 0.9437 0.8862 0.8059 0.5941 0.3633 0.1737 0.0580 0.0104 0.00044 0.9996 0.9896 0.9727 0.9420 0.8263 0.6367 0.4059 0.1941 0.0563 0.00505 1.0000 0.9988 0.9958 0.9887 0.9502 0.8555 0.6846 0.4482 0.2031 0.03816 0.9999 0.9996 0.9987 0.9915 0.9648 0.8936 0.7447 0.4967 0.18697 1.0000 1.0000 0.9999 0.9993 0.9961 0.9832 0.9424 0.8322 0.56958 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
9 0 0.3874 0.1342 0.0751 0.0404 0.0101 0.0020 0.0003 0.00001 0.7748 0.4362 0.3003 0.1960 0.0705 0.0195 0.0038 0.0004 0.00002 0.9470 0.7382 0.6007 0.4628 0.2318 0.0898 0.0250 0.0043 0.0003 0.00003 0.9917 0.9144 0.8343 0.7297 0.4826 0.2539 0.0994 0.0253 0.0031 0.00014 0.9991 0.9804 0.9511 0.9012 0.7334 0.5000 0.2666 0.0988 0.0196 0.00095 0.9999 0.9969 0.9900 0.9747 0.9006 0.7461 0.5174 0.2703 0.0856 0.00836 1.0000 0.9997 0.9987 0.9957 0.9750 0.9102 0.7682 0.5372 0.2618 0.05307 1.0000 0.9999 0.9996 0.9962 0.9805 0.9295 0.8040 0.5638 0.22528 1.0000 1.0000 0.9997 0.9980 0.9899 0.9596 0.8658 0.61269 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
10 0 0.3487 0.1074 0.0563 0.0282 0.0060 0.0010 0.0001 0.00001 0.7361 0.3758 0.2440 0.1493 0.0464 0.0107 0.0017 0.0001 0.00002 0.9298 0.6778 0.5256 0.3828 0.1673 0.0547 0.0123 0.0016 0.00013
0.9872 0.8791 0.7759 0.6496 0.3823 0.1719 0.0548 0.0106 0.0009 0.00004 0.9984 0.9672 0.9219 0.8497 0.6331 0.3770 0.1662 0.0473 0.0064 0.00015 0.9999 0.9936 0.9803 0.9527 0.8338 0.6230 0.3669 0.1503 0.0328 0.00166 1.0000 0.9991 0.9965 0.9894 0.9452 0.8281 0.6177 0.3504 0.1209 0.01287 0.9999 0.9996 0.9984 0.9877 0.9453 0.8327 0.6172 0.3222 0.07028 1.0000 1.0000 0.9999 0.9983 0.9893 0.9536 0.8507 0.6242 0.26399 1.0000 0.9999 0.9990 0.9940 0.9718 0.8926 0.6513
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11 0 0.3138 0.0859 0.0422 0.0198 0.0036 0.0005 0.00001 0.6974 0.3221 0.1971 0.1130 0.0302 0.0059 0.0007 0.00002 0.9104 0.6174 0.4552 0.3127 0.1189 0.0327 0.0059 0.0006 0.0000
3 0.9815 0.8389 0.7133 0.5696 0.2963 0.1133 0.0293 0.0043 0.00024 0.9972 0.9496 0.8854 0.7897 0.5328 0.2744 0.0994 0.0216 0.0020 0.00005 0.9997 0.9883 0.9657 0.9218 0.7535 0.5000 0.2465 0.0782 0.0117 0.00036 1.0000 0.9980 0.9924 0.9784 0.9006 0.7256 0.4672 0.2103 0.0504 0.00287 0.9998 0.9988 0.9957 0.9707 0.8867 0.7037 0.4304 0.1611 0.01858 1.0000 0.9999 0.9994 0.9941 0.9673 0.8811 0.6873 0.3826 0.08969 1.0000 1.0000 0.9993 0.9941 0.9698 0.8870 0.6779 0.3026
10 1.0000 0.9995 0.9964 0.9802 0.9141 0.686211 1.0000 1.0000 1.0000 1.0000 1.0000
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430 Appendix A Statistical Tables and Proofs
Table A.1 (continued) Binomial Probability Sumsr
x=0
b(x; n, p)
p
n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
12 0 0.2824 0.0687 0.0317 0.0138 0.0022 0.0002 0.0000
1 0.6590 0.2749 0.1584 0.0850 0.0196 0.0032 0.0003 0.00002 0.8891 0.5583 0.3907 0.2528 0.0834 0.0193 0.0028 0.0002 0.00003 0.9744 0.7946 0.6488 0.4925 0.2253 0.0730 0.0153 0.0017 0.00014 0.9957 0.9274 0.8424 0.7237 0.4382 0.1938 0.0573 0.0095 0.0006 0.00005 0.9995 0.9806 0.9456 0.8822 0.6652 0.3872 0.1582 0.0386 0.0039 0.00016 0.9999 0.9961 0.9857 0.9614 0.8418 0.6128 0.3348 0.1178 0.0194 0.00057 1.0000 0.9994 0.9972 0.9905 0.9427 0.8062 0.5618 0.2763 0.0726 0.00438 0.9999 0.9996 0.9983 0.9847 0.9270 0.7747 0.5075 0.2054 0.02569 1.0000 1.0000 0.9998 0.9972 0.9807 0.9166 0.7472 0.4417 0.1109
10 1.0000 0.9997 0.9968 0.9804 0.9150 0.7251 0.341011 1.0000 0.9998 0.9978 0.9862 0.9313 0.7176
12 1.0000 1.0000 1.0000 1.0000 1.000013 0 0.2542 0.0550 0.0238 0.0097 0.0013 0.0001 0.0000
1 0.6213 0.2336 0.1267 0.0637 0.0126 0.0017 0.0001 0.00002 0.8661 0.5017 0.3326 0.2025 0.0579 0.0112 0.0013 0.00013 0.9658 0.7473 0.5843 0.4206 0.1686 0.0461 0.0078 0.0007 0.00004 0.9935 0.9009 0.7940 0.6543 0.3530 0.1334 0.0321 0.0040 0.00025 0.9991 0.9700 0.9198 0.8346 0.5744 0.2905 0.0977 0.0182 0.0012 0.00006 0.9999 0.9930 0.9757 0.9376 0.7712 0.5000 0.2288 0.0624 0.0070 0.00017 1.0000 0.9988 0.9944 0.9818 0.9023 0.7095 0.4256 0.1654 0.0300 0.00098 0.9998 0.9990 0.9960 0.9679 0.8666 0.6470 0.3457 0.0991 0.00659 1.0000 0.9999 0.9993 0.9922 0.9539 0.8314 0.5794 0.2527 0.0342
10 1.0000 0.9999 0.9987 0.9888 0.9421 0.7975 0.4983 0.133911 1.0000 0.9999 0.9983 0.9874 0.9363 0.7664 0.378712 1.0000 0.9999 0.9987 0.9903 0.9450 0.745813 1.0000 1.0000 1.0000 1.0000 1.0000
14 0 0.2288 0.0440 0.0178 0.0068 0.0008 0.0001 0.00001 0.5846 0.1979 0.1010 0.0475 0.0081 0.0009 0.00012 0.8416 0.4481 0.2811 0.1608 0.0398 0.0065 0.0006 0.00003 0.9559 0.6982 0.5213 0.3552 0.1243 0.0287 0.0039 0.00024 0.9908 0.8702 0.7415 0.5842 0.2793 0.0898 0.0175 0.0017 0.00005 0.9985 0.9561 0.8883 0.7805 0.4859 0.2120 0.0583 0.0083 0.00046 0.9998 0.9884 0.9617 0.9067 0.6925 0.3953 0.1501 0.0315 0.0024 0.00007 1.0000 0.9976 0.9897 0.9685 0.8499 0.6047 0.3075 0.0933 0.0116 0.00028 0.9996 0.9978 0.9917 0.9417 0.7880 0.5141 0.2195 0.0439 0.00159 1.0000 0.9997 0.9983 0.9825 0.9102 0.7207 0.4158 0.1298 0.0092
10 1.0000 0.9998 0.9961 0.9713 0.8757 0.6448 0.3018 0.044111 1.0000 0.9994 0.9935 0.9602 0.8392 0.5519 0.158412 0.9999 0.9991 0.9919 0.9525 0.8021 0.415413 1.0000 0.9999 0.9992 0.9932 0.9560 0.771214 1.0000 1.0000 1.0000 1.0000 1.0000
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Table A.1 Binomial Probability Table 431
Table A.1 (continued) Binomial Probability Sumsr
x=0
b(x;n, p)
p
n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
15 0 0.2059 0.0352 0.0134 0.0047 0.0005 0.0000
1 0.5490 0.1671 0.0802 0.0353 0.0052 0.0005 0.00002 0.8159 0.3980 0.2361 0.1268 0.0271 0.0037 0.0003 0.00003 0.9444 0.6482 0.4613 0.2969 0.0905 0.0176 0.0019 0.00014 0.9873 0.8358 0.6865 0.5155 0.2173 0.0592 0.0093 0.0007 0.00005 0.9978 0.9389 0.8516 0.7216 0.4032 0.1509 0.0338 0.0037 0.00016 0.9997 0.9819 0.9434 0.8689 0.6098 0.3036 0.0950 0.0152 0.00087 1.0000 0.9958 0.9827 0.9500 0.7869 0.5000 0.2131 0.0500 0.0042 0.00008 0.9992 0.9958 0.9848 0.9050 0.6964 0.3902 0.1311 0.0181 0.00039 0.9999 0.9992 0.9963 0.9662 0.8491 0.5968 0.2784 0.0611 0.0022
10 1.0000 0.9999 0.9993 0.9907 0.9408 0.7827 0.4845 0.1642 0.012711 1.0000 0.9999 0.9981 0.9824 0.9095 0.7031 0.3518 0.0556
12 1.0000 0.9997 0.9963 0.9729 0.8732 0.6020 0.184113 1.0000 0.9995 0.9948 0.9647 0.8329 0.451014 1.0000 0.9995 0.9953 0.9648 0.794115 1.0000 1.0000 1.0000 1.0000
16 0 0.1853 0.0281 0.0100 0.0033 0.0003 0.00001 0.5147 0.1407 0.0635 0.0261 0.0033 0.0003 0.00002 0.7892 0.3518 0.1971 0.0994 0.0183 0.0021 0.00013 0.9316 0.5981 0.4050 0.2459 0.0651 0.0106 0.0009 0.00004 0.9830 0.7982 0.6302 0.4499 0.1666 0.0384 0.0049 0.00035 0.9967 0.9183 0.8103 0.6598 0.3288 0.1051 0.0191 0.0016 0.00006 0.9995 0.9733 0.9204 0.8247 0.5272 0.2272 0.0583 0.0071 0.0002
7 0.9999 0.9930 0.9729 0.9256 0.7161 0.4018 0.1423 0.0257 0.0015 0.00008 1.0000 0.9985 0.9925 0.9743 0.8577 0.5982 0.2839 0.0744 0.0070 0.00019 0.9998 0.9984 0.9929 0.9417 0.7728 0.4728 0.1753 0.0267 0.0005
10 1.0000 0.9997 0.9984 0.9809 0.8949 0.6712 0.3402 0.0817 0.003311 1.0000 0.9997 0.9951 0.9616 0.8334 0.5501 0.2018 0.017012 1.0000 0.9991 0.9894 0.9349 0.7541 0.4019 0.068413 0.9999 0.9979 0.9817 0.9006 0.6482 0.210814 1.0000 0.9997 0.9967 0.9739 0.8593 0.485315 1.0000 0.9997 0.9967 0.9719 0.814716 1.0000 1.0000 1.0000 1.0000
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432 Appendix A Statistical Tables and Proofs
Table A.1 (continued) Binomial Probability Sumsr
x=0
b(x; n, p)
p
n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
17 0 0.1668 0.0225 0.0075 0.0023 0.0002 0.0000
1 0.4818 0.1182 0.0501 0.0193 0.0021 0.0001 0.00002 0.7618 0.3096 0.1637 0.0774 0.0123 0.0012 0.00013 0.9174 0.5489 0.3530 0.2019 0.0464 0.0064 0.0005 0.00004 0.9779 0.7582 0.5739 0.3887 0.1260 0.0245 0.0025 0.00015 0.9953 0.8943 0.7653 0.5968 0.2639 0.0717 0.0106 0.0007 0.00006 0.9992 0.9623 0.8929 0.7752 0.4478 0.1662 0.0348 0.0032 0.00017 0.9999 0.9891 0.9598 0.8954 0.6405 0.3145 0.0919 0.0127 0.00058 1.0000 0.9974 0.9876 0.9597 0.8011 0.5000 0.1989 0.0403 0.0026 0.00009 0.9995 0.9969 0.9873 0.9081 0.6855 0.3595 0.1046 0.0109 0.0001
10 0.9999 0.9994 0.9968 0.9652 0.8338 0.5522 0.2248 0.0377 0.000811 1.0000 0.9999 0.9993 0.9894 0.9283 0.7361 0.4032 0.1057 0.0047
12 1.0000 0.9999 0.9975 0.9755 0.8740 0.6113 0.2418 0.022113 1.0000 0.9995 0.9936 0.9536 0.7981 0.4511 0.082614 0.9999 0.9988 0.9877 0.9226 0.6904 0.238215 1.0000 0.9999 0.9979 0.9807 0.8818 0.518216 1.0000 0.9998 0.9977 0.9775 0.833217 1.0000 1.0000 1.0000 1.0000
18 0 0.1501 0.0180 0.0056 0.0016 0.0001 0.00001 0.4503 0.0991 0.0395 0.0142 0.0013 0.00012 0.7338 0.2713 0.1353 0.0600 0.0082 0.0007 0.00003 0.9018 0.5010 0.3057 0.1646 0.0328 0.0038 0.00024 0.9718 0.7164 0.5187 0.3327 0.0942 0.0154 0.0013 0.0000
5 0.9936 0.8671 0.7175 0.5344 0.2088 0.0481 0.0058 0.00036 0.9988 0.9487 0.8610 0.7217 0.3743 0.1189 0.0203 0.0014 0.00007 0.9998 0.9837 0.9431 0.8593 0.5634 0.2403 0.0576 0.0061 0.00028 1.0000 0.9957 0.9807 0.9404 0.7368 0.4073 0.1347 0.0210 0.00099 0.9991 0.9946 0.9790 0.8653 0.5927 0.2632 0.0596 0.0043 0.0000
10 0.9998 0.9988 0.9939 0.9424 0.7597 0.4366 0.1407 0.0163 0.000211 1.0000 0.9998 0.9986 0.9797 0.8811 0.6257 0.2783 0.0513 0.001212 1.0000 0.9997 0.9942 0.9519 0.7912 0.4656 0.1329 0.006413 1.0000 0.9987 0.9846 0.9058 0.6673 0.2836 0.028214 0.9998 0.9962 0.9672 0.8354 0.4990 0.098215 1.0000 0.9993 0.9918 0.9400 0.7287 0.2662
16 0.9999 0.9987 0.9858 0.9009 0.549717 1.0000 0.9999 0.9984 0.9820 0.849918 1.0000 1.0000 1.0000 1.0000
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434 Appendix A Statistical Tables and Proofs
Table A.2 Poisson Probability Sumsr
x=0
p(x;)
r 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066
1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.77252 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.93713 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.98654 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.99775 1.0000 1.0000 1.0000 0.9999 0.9998 0.99976 1.0000 1.0000 1.0000
r 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.00671 0.7358 0.5578 0.4060 0.2873 0.1991 0.1359 0.0916 0.0611 0.04042 0.9197 0.8088 0.6767 0.5438 0.4232 0.3208 0.2381 0.1736 0.12473 0.9810 0.9344 0.8571 0.7576 0.6472 0.5366 0.4335 0.3423 0.26504 0.9963 0.9814 0.9473 0.8912 0.8153 0.7254 0.6288 0.5321 0.44055 0.9994 0.9955 0.9834 0.9580 0.9161 0.8576 0.7851 0.7029 0.6160
6 0.9999 0.9991 0.9955 0.9858 0.9665 0.9347 0.8893 0.8311 0.76227 1.0000 0.9998 0.9989 0.9958 0.9881 0.9733 0.9489 0.9134 0.86668 1.0000 0.9998 0.9989 0.9962 0.9901 0.9786 0.9597 0.93199 1.0000 0.9997 0.9989 0.9967 0.9919 0.9829 0.9682
10 0.9999 0.9997 0.9990 0.9972 0.9933 0.986311 1.0000 0.9999 0.9997 0.9991 0.9976 0.994512 1.0000 0.9999 0.9997 0.9992 0.998013 1.0000 0.9999 0.9997 0.999314 1.0000 0.9999 0.999815 1.0000 0.999916 1.0000
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Table A.2 Poisson Probability Table 435
Table A.2 (continued) Poisson Probability Sumsr
x=0
p(x; )
r 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5
0 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.0001
1 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.00082 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.00423 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.01494 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.04035 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.0885
6 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.16497 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.26878 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.39189 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218
10 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.6453
11 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.752012 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.836413 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.898114 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.940015 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.9665
16 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.982317 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.991118 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.995719 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.998020 0.9999 0.9998 0.9996 0.9991
21 1.0000 0.9999 0.9998 0.999622 1.0000 0.9999 0.999923 1.0000 0.999924 1.0000
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Table A.2 (continued) Poisson Probability Sumsr
x=0
p(x;)
r 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0
0 0.0000 0.0000 0.0000
1 0.0005 0.0002 0.0001 0.0000 0.00002 0.0028 0.0012 0.0005 0.0002 0.0001 0.0000 0.00003 0.0103 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.00004 0.0293 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.00015 0.0671 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003
6 0.1301 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.00107 0.2202 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.00298 0.3328 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.00719 0.4579 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154
10 0.5830 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304
11 0.6968 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.054912 0.7916 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.091713 0.8645 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.142614 0.9165 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.208115 0.9513 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867
16 0.9730 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.375117 0.9857 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.468618 0.9928 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.562219 0.9965 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.650920 0.9984 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307
21 0.9993 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.799122 0.9997 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.855123 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.898924 1.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.931725 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554
26 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.971827 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.982728 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.989729 1.0000 0.9999 0.9996 0.9989 0.9973 0.994130 0.9999 0.9998 0.9994 0.9986 0.9967
31 1.0000 0.9999 0.9997 0.9993 0.9982
32 1.0000 0.9999 0.9996 0.999033 0.9999 0.9998 0.999534 1.0000 0.9999 0.999835 1.0000 0.9999
36 0.999937 1.0000
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Table A.4 Student t-Distribution Probability Table 439
0 t
Table A.4 Critical Values of the t-Distribution
v 0.40 0.30 0.20 0.15 0.10 0.05 0.025
1 0.325 0.727 1.376 1.963 3.078 6.314 12.706
2 0.289 0.617 1.061 1.386 1.886 2.920 4.303
3 0.277 0.584 0.978 1.250 1.638 2.353 3.182
4 0.271 0.569 0.941 1.190 1.533 2.132 2.776
5 0.267 0.559 0.920 1.156 1.476 2.015 2.571
6 0.265 0.553 0.906 1.134 1.440 1.943 2.447
7 0.263 0.549 0.896 1.119 1.415 1.895 2.365
8 0.262 0.546 0.889 1.108 1.397 1.860 2.306
9 0.261 0.543 0.883 1.100 1.383 1.833 2.262
10 0.260 0.542 0.879 1.093 1.372 1.812 2.228
11 0.260 0.540 0.876 1.088 1.363 1.796 2.201
12 0.259 0.539 0.873 1.083 1.356 1.782 2.179
13 0.259 0.538 0.870 1.079 1.350 1.771 2.160
14 0.258 0.537 0.868 1.076 1.345 1.761 2.145
15 0.258 0.536 0.866 1.074 1.341 1.753 2.131
16 0.258 0.535 0.865 1.071 1.337 1.746 2.120
17 0.257 0.534 0.863 1.069 1.333 1.740 2.110
18 0.257 0.534 0.862 1.067 1.330 1.734 2.101
19 0.257 0.533 0.861 1.066 1.328 1.729 2.093
20 0.257 0.533 0.860 1.064 1.325 1.725 2.086
21 0.257 0.532 0.859 1.063 1.323 1.721 2.080
22 0.256 0.532 0.858 1.061 1.321 1.717 2.074
23 0.256 0.532 0.858 1.060 1.319 1.714 2.069
24 0.256 0.531 0.857 1.059 1.318 1.711 2.064
25 0.256 0.531 0.856 1.058 1.316 1.708 2.060
26 0.256 0.531 0.856 1.058 1.315 1.706 2.056
27 0.256 0.531 0.855 1.057 1.314 1.703 2.052
28 0.256 0.530 0.855 1.056 1.313 1.701 2.048
29 0.256 0.530 0.854 1.055 1.311 1.699 2.045
30 0.256 0.530 0.854 1.055 1.310 1.697 2.042
40 0.255 0.529 0.851 1.050 1.303 1.684 2.02160 0.254 0.527 0.848 1.045 1.296 1.671 2.000
120 0.254 0.526 0.845 1.041 1.289 1.658 1.980
0.253 0.524 0.842 1.036 1.282 1.645 1.960
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Table A.4 (continued) Critical Values of the t-Distribution
v 0.02 0.015 0.01 0.0075 0.005 0.0025 0.0005
1 15.894 21.205 31.821 42.433 63.656 127.321 636.5782 4.849 5.643 6.965 8.073 9.925 14.089 31.600
3 3.482 3.896 4.541 5.047 5.841 7.453 12.9244 2.999 3.298 3.747 4.088 4.604 5.598 8.6105 2.757 3.003 3.365 3.634 4.032 4.773 6.869
6 2.612 2.829 3.143 3.372 3.707 4.317 5.9597 2.517 2.715 2.998 3.203 3.499 4.029 5.4088 2.449 2.634 2.896 3.085 3.355 3.833 5.0419 2.398 2.574 2.821 2.998 3.250 3.690 4.781
10 2.359 2.527 2.764 2.932 3.169 3.581 4.587
11 2.328 2.491 2.718 2.879 3.106 3.497 4.43712 2.303 2.461 2.681 2.836 3.055 3.428 4.31813
2.282 2.436 2.650 2.801 3.012 3.372 4.22114 2.264 2.415 2.624 2.771 2.977 3.326 4.14015 2.249 2.397 2.602 2.746 2.947 3.286 4.073
16 2.235 2.382 2.583 2.724 2.921 3.252 4.01517 2.224 2.368 2.567 2.706 2.898 3.222 3.96518 2.214 2.356 2.552 2.689 2.878 3.197 3.92219 2.205 2.346 2.539 2.674 2.861 3.174 3.88320 2.197 2.336 2.528 2.661 2.845 3.153 3.850
21 2.189 2.328 2.518 2.649 2.831 3.135 3.81922 2.183 2.320 2.508 2.639 2.819 3.119 3.79223 2.177 2.313 2.500 2.629 2.807 3.104 3.768
24 2.172 2.307 2.492 2.620 2.797 3.091 3.74525 2.167 2.301 2.485 2.612 2.787 3.078 3.725
26 2.162 2.296 2.479 2.605 2.779 3.067 3.70727 2.158 2.291 2.473 2.598 2.771 3.057 3.68928 2.154 2.286 2.467 2.592 2.763 3.047 3.67429 2.150 2.282 2.462 2.586 2.756 3.038 3.66030 2.147 2.278 2.457 2.581 2.750 3.030 3.646
40 2.123 2.250 2.423 2.542 2.704 2.971 3.55160 2.099 2.223 2.390 2.504 2.660 2.915 3.460
120 2.076 2.196 2.358 2.468 2.617 2.860 3.373 2.054 2.170 2.326 2.432 2.576 2.807 3.290
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442 Appendix A Statistical Tables and Proofs
Table A.5 (continued) Critical Values of the Chi-Squared Distribution
v 0.30 0.25 0.20 0.10 0.05 0.025 0.02 0.01 0.005 0.001
1 1.074 1.323 1.642 2.706 3.841 5.024 5.412 6.635 7.879 10.8272 2.408 2.773 3.219 4.605 5.991 7.378 7.824 9.210 10.597 13.815
3 3.665 4.108 4.642 6.251 7.815 9.348 9.837 11.345 12.838 16.2664 4.878 5.385 5.989 7.779 9.488 11.143 11.668 13.277 14.860 18.4665 6.064 6.626 7.289 9.236 11.070 12.832 13.388 15.086 16.750 20.515
6 7.231 7.841 8.558 10.645 12.592 14.449 15.033 16.812 18.548 22.4577 8.383 9.037 9.803 12.017 14.067 16.013 16.622 18.475 20.278 24.3218 9.524 10.219 11.030 13.362 15.507 17.535 18.168 20.090 21.955 26.1249 10.656 11.389 12.242 14.684 16.919 19.023 19.679 21.666 23.589 27.877
10 11.781 12.549 13.442 15.987 18.307 20.483 21.161 23.209 25.188 29.588
11 12.899 13.701 14.631 17.275 19.675 21.920 22.618 24.725 26.757 31.26412 14.011 14.845 15.812 18.549 21.026 23.337 24.054 26.217 28.300 32.90913
15.119 15.984 16.985 19.812 22.362 24.736 25.471 27.688 29.819 34.52714 16.222 17.117 18.151 21.064 23.685 26.119 26.873 29.141 31.319 36.12415 17.322 18.245 19.311 22.307 24.996 27.488 28.259 30.578 32.801 37.698
16 18.418 19.369 20.465 23.542 26.296 28.845 29.633 32.000 34.267 39.25217 19.511 20.489 21.615 24.769 27.587 30.191 30.995 33.409 35.718 40.79118 20.601 21.605 22.760 25.989 28.869 31.526 32.346 34.805 37.156 42.31219 21.689 22.718 23.900 27.204 30.144 32.852 33.687 36.191 38.582 43.81920 22.775 23.828 25.038 28.412 31.410 34.170 35.020 37.566 39.997 45.314
21 23.858 24.935 26.171 29.615 32.671 35.479 36.343 38.932 41.401 46.79622 24.939 26.039 27.301 30.813 33.924 36.781 37.659 40.289 42.796 48.26823 26.018 27.141 28.429 32.007 35.172 38.076 38.968 41.638 44.181 49.728
24 27.096 28.241 29.553 33.196 36.415 39.364 40.270 42.980 45.558 51.17925 28.172 29.339 30.675 34.382 37.652 40.646 41.566 44.314 46.928 52.619
26 29.246 30.435 31.795 35.563 38.885 41.923 42.856 45.642 48.290 54.05127 30.319 31.528 32.912 36.741 40.113 43.195 44.140 46.963 49.645 55.47528 31.391 32.620 34.027 37.916 41.337 44.461 45.419 48.278 50.994 56.89229 32.461 33.711 35.139 39.087 42.557 45.722 46.693 49.588 52.335 58.30130 33.530 34.800 36.250 40.256 43.773 46.979 47.962 50.892 53.672 59.702
40 44.165 45.616 47.269 51.805 55.758 59.342 60.436 63.691 66.766 73.40350 54.723 56.334 58.164 63.167 67.505 71.420 72.613 76.154 79.490 86.66060 65.226 66.981 68.972 74.397 79.082 83.298 84.58 88.379 91.952 99.608
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Table A.6 F-Distribution Probability Table 443
0 f
Table A.6 Critical Values of the F-Distribution
f0.05(v1, v2)
v1
v2 1 2 3 4 5 6 7 8 91 161.448 199.500 215.707 224.583 230.162 233.986 236.768 238.883 240.54332 18.513 19.000 19.164 19.247 19.296 19.330 19.353 19.371 19.3853 10.128 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.8124 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.9995 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772
6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.0997 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.725 3.6868 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.3889 5.117 4.256 3.863 3.633 3.482 3.373 3.293 3.230 3.179
10 4.964 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020
11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.89612 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.79613 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.71414 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.64615 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588
16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.53817 4.451 3.592 3.197 2.965 2.810 2.699 2.614 2.548 2.49418 4.414 3.555 3.160 2.928 2.773 2.661 2.578 2.510 2.45619 4.381 3.522 3.127 2.895 2.740 2.628 2.544 2.477 2.42320 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.392
21 4.325 3.467 3.072 2.840 2.685 2.573 2.488 2.420 2.36622 4.301 3.443 3.049 2.817 2.661 2.549 2.464 2.397 2.34223 4.279 3.422 3.028 2.795 2.640 2.528 2.442 2.375 2.32024 4.260 3.403 3.009 2.776 2.621 2.508 2.423 2.355 2.30025 4.242 3.385 2.991 2.759 2.603 2.490 2.405 2.337 2.282
26 4.225 3.369 2.975 2.743 2.587 2.474 2.388 2.321 2.26527 4.210 3.354 2.960 2.728 2.572 2.459 2.373 2.305 2.25028 4.196 3.340 2.947 2.714 2.558 2.445 2.359 2.291 2.23629 4.183 3.328 2.934 2.701 2.545 2.432 2.356 2.278 2.22330 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.226 2.211
40 4.085 3.232 2.839 2.606 2.449 2.339 2.249 2.180 2.12460 4.001 3.150 2.758 2.525 2.368 2.254 2.167 2.097 2.040
120 3.920 3.072 2.680 2.447 2.290 2.175 2.087 2.016 1.959 3.841 2.996 2.605 2.372 2.214 2.099 2.010 1.938 1.880
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Table A.6 (continued) Critical Values of the F-Distribution
f0.05(v1, v2)
v1
v2 10 12 15 20 24 30 40 60 120
1 241.882 243.906 245.950 248.013 249.052 250.095 251.143 252.196 253.253 254.3142 19.396 19.413 19.429 19.446 19.454 19.462 19.471 19.479 19.487 19.4963 8.786 8.745 8.703 8.660 8.639 8.617 8.594 8.572 8.549 8.5264 5.964 5.912 5.858 5.803 5.774 5.746 5.717 5.688 5.658 5.6285 4.735 4.678 4.619 4.558 4.527 4.496 4.464 4.431 4.398 4.365
6 4.060 4.000 3.938 3.874 3.841 3.808 3.774 3.740 3.705 3.6697 3.637 3.575 3.511 3.445 3.410 3.376 3.340 3.304 3.267 3.2308 3.347 3.284 3.218 3.150 3.115 3.079 3.043 3.005 2.967 2.9289 3.137 3.073 3.006 2.936 2.900 2.864 2.826 2.787 2.748 2.707
10 2.978 2.913 2.845 2.774 2.737 2.700 2.661 2.621 2.580 2.538
11 2.854 2.788 2.719 2.646 2.609 2.570 2.531 2.490 2.448 2.404
12 2.753 2.687 2.617 2.544 2.505 2.466 2.429 2.384 2.341 2.29613 2.671 2.602 2.533 2.459 2.420 2.380 2.339 2.297 2.252 2.20614 2.602 2.534 2.463 2.388 2.349 2.308 2.266 2.223 2.178 2.13115 2.544 2.475 2.403 2.328 2.288 2.247 2.204 2.160 2.114 2.066
16 2.494 2.425 2.352 2.276 2.235 2.194 2.151 2.106 2.059 2.01017 2.450 2.381 2.308 2.230 2.190 2.148 2.104 2.058 2.011 1.96018 2.412 2.342 2.269 2.191 2.150 2.107 2.063 2.017 1.968 1.91719 2.378 2.308 2.234 2.155 2.114 2.071 2.026 1.980 1.930 1.87820 2.348 2.278 2.203 2.124 2.082 2.039 1.994 1.946 1.896 1.843
21 2.321 2.250 2.176 2.096 2.054 2.010 1.965 1.916 1.866 1.81222 2.297 2.229 2.151 2.071 2.028 1.984 1.938 1.889 1.838 1.783
23 2.275 2.204 2.128 2.048 2.005 1.961 1.914 1.865 1.813 1.75724 2.255 2.183 2.108 2.027 1.984 1.939 1.892 1.842 1.790 1.73325 2.236 2.165 2.089 2.007 1.964 1.919 1.872 1.822 1.768 1.711
26 2.220 2.148 2.072 1.990 1.946 1.901 1.853 1.803 1.749 1.69127 2.204 2.132 2.058 1.974 1.930 1.884 1.836 1.785 1.731 1.67228 2.190 2.118 2.041 1.959 1.915 1.869 1.820 1.769 1.714 1.65429 2.177 2.104 2.027 1.945 1.901 1.854 1.806 1.754 1.698 1.63830 2.165 2.092 2.015 1.932 1.887 1.841 1.792 1.740 1.683 1.622
40 2.077 2.003 1.924 1.839 1.793 1.744 1.693 1.637 1.577 1.50960 1.993 1.917 1.836 1.748 1.700 1.649 1.594 1.534 1.467 1.389
120 1.910 1.834 1.750 1.659 1.608 1.554 1.495 1.429 1.352 1.254 1.831 1.752 1.666 1.570 1.517 1.459 1.394 1.318 1.221 1.000
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Table A.6 F-Distribution Probability Table 445
Table A.6 (continued) Critical Values of the F-Distribution
f0.01(v1, v2)
v1
v2 1 2 3 4 5 6 7 8 9
1 4052.181 4999.500 5403.352 5624.583 5763.650 5858.986 5928.356 5981.070 6022.4732 98.503 99.000 99.166 99.249 99.299 99.333 99.356 99.374 99.3883 34.116 30.817 29.457 28.710 28.237 27.911 27.672 27.489 27.3454 21.198 18.000 16.694 15.977 15.522 15.207 14.976 14.799 14.6595 16.258 13.274 12.060 11.392 10.967 10.672 10.455 10.289 10.158
6 13.745 10.925 9.780 9.148 8.746 8.466 8.260 8.102 7.9767 12.246 9.547 8.451 7.847 7.460 7.191 6.993 6.840 6.7198 11.259 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.9119 10.561 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.351
10 10.044 7.559 6.552 5.994 5.636 5.386 5.200 5.057 4.942
11 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.632
12 9.330 6.927 5.953 5.412 5.064 4.821 4.640 4.499 4.38813 9.074 6.701 5.739 5.205 4.862 4.620 4.441 4.302 4.19114 8.862 6.515 5.564 5.035 4.694 4.456 4.278 4.140 4.03015 8.683 6.359 5.417 4.893 4.556 4.318 4.142 4.004 3.895
16 8.531 6.226 5.292 4.773 4.437 4.202 4.026 3.890 3.78017 8.400 6.112 5.185 4.669 4.336 4.102 3.927 3.791 3.68218 8.285 6.013 5.092 4.579 4.248 4.015 3.841 3.705 3.59719 8.185 5.926 5.010 4.500 4.171 3.939 3.765 3.631 3.52320 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.457
21 8.017 5.780 4.874 4.369 4.042 3.812 3.640 3.506 3.39822 7.945 5.719 4.817 4.313 3.988 3.758 3.587 3.453 3.346
23 7.881 5.664 4.765 4.264 3.939 3.710 3.539 3.406 3.29924 7.823 5.614 4.718 4.218 3.895 3.667 3.496 3.363 3.25625 7.770 5.568 4.675 4.177 3.855 3.627 3.458 3.324 3.217
26 7.721 5.527 4.637 4.140 3.818 3.591 3.421 3.288 3.18227 7.677 5.488 4.601 4.106 3.785 3.558 3.388 3.256 3.14928 7.636 5.453 4.568 4.074 3.754 3.528 3.358 3.226 3.12029 7.598 5.420 4.538 4.045 3.725 3.499 3.330 3.198 3.09230 7.562 5.390 4.510 4.018 3.699 3.473 3.304 3.173 3.067
40 7.314 5.179 4.313 3.828 3.514 3.291 3.124 2.993 2.88860 7.077 4.977 4.126 3.649 3.339 3.119 2.953 2.823 2.718
120 6.851 4.787 3.949 3.480 3.174 2.956 2.792 2.663 2.559 6.635 4.605 3.782 3.319 3.017 2.802 2.639 2.511 2.407
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Table A.6 (continued) Critical Values of the F-Distribution
f0.01(v1, v2)
v1
v2 10 12 15 20 24 30 40 60 120
1 6055.847 6106.321 6157.285 6208.730 6234.631 6260.649 6286.782 6313.030 6339.391 6365.8642 99.399 99.416 99.433 99.449 99.458 99.466 99.474 99.483 99.491 99.4993 27.229 27.052 26.872 26.690 26.598 26.505 26.411 26.316 26.221 26.1254 14.546 14.374 14.198 14.020 13.929 13.838 13.745 13.652 13.558 13.4635 10.051 9.888 9.722 9.553 9.466 9.379 9.291 9.202 9.112 9.020
6 7.874 7.718 7.559 7.396 7.313 7.229 7.143 7.057 6.969 6.8807 6.620 6.469 6.314 6.155 6.074 5.992 5.908 5.824 5.737 5.6508 5.814 5.667 5.515 5.359 5.279 5.198 5.116 5.032 4.946 4.8599 5.256 5.111 4.962 4.808 4.729 4.649 4.567 4.483 4.398 4.311
10 4.849 4.706 4.558 4.405 4.327 4.247 4.165 4.082 3.996 3.909
11 4.539 4.397 4.251 4.099 4.021 3.941 3.860 3.776 3.690 3.602
12 4.296 4.155 4.010 3.858 3.780 3.701 3.619 3.535 3.449 3.36113 4.100 3.960 3.815 3.664 3.587 3.507 3.425 3.341 3.255 3.16514 3.939 3.800 3.656 3.505 3.427 3.348 3.266 3.181 3.094 3.00415 3.805 3.666 3.522 3.372 3.294 3.214 3.132 3.047 2.959 2.868
16 3.691 3.553 3.409 3.259 3.181 3.101 3.018 2.933 2.845 2.75317 3.593 3.455 3.312 3.162 3.084 3.003 2.920 2.835 2.746 2.65318 3.508 3.371 3.227 3.077 2.999 2.919 2.835 2.749 2.660 2.56619 3.434 3.297 3.153 3.003 2.925 2.844 2.761 2.674 2.584 2.48920 3.368 3.231 3.088 2.938 2.859 2.778 2.695 2.608 2.517 2.421
21 3.310 3.173 3.030 2.880 2.801 2.720 2.636 2.548 2.457 2.36022 3.258 3.121 2.978 2.827 2.749 2.667 2.583 2.495 2.403 2.305
23 3.211 3.074 2.931 2.781 2.702 2.620 2.535 2.447 2.354 2.25624 3.168 3.031 2.889 2.738 2.659 2.577 2.492 2.403 2.310 2.21125 3.129 2.993 2.850 2.699 2.620 2.538 2.453 2.363 2.270 2.169
26 3.094 2.958 2.815 2.664 2.585 2.503 2.417 2.327 2.233 2.13127 3.062 2.926 2.782 2.632 2.552 2.470 2.384 2.294 2.198 2.09728 3.032 2.896 2.753 2.602 2.522 2.440 2.354 2.263 2.167 2.06429 3.005 2.868 2.726 2.574 2.495 2.412 2.335 2.234 2.138 2.03430 2.979 2.843 2.726 2.549 2.469 2.386 2.299 2.208 2.111 2.006
40 2.801 2.665 2.522 2.369 2.288 2.203 2.114 2.019 1.917 1.80460 2.632 2.496 2.352 2.198 2.115 2.028 1.936 1.836 1.726 1.601
120 2.472 2.336 2.192 2.035 1.950 1.860 1.763 1.656 1.533 1.381 2.321 2.185 2.039 1.878 1.791 1.696 1.592 1.473 1.325 1.000
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448 Appendix A Statistical Tables and Proofs
Table A.8 Critical Values for Bartletts Test
bk(0.01;n)
Number of Populations, k
n 2 3 4 5 6 7 8 9 10
3 0.1411 0.16724 0.2843 0.3165 0.3475 0.3729 0.3937 0.41105 0.3984 0.4304 0.4607 0.4850 0.5046 0.5207 0.5343 0.5458 0.5558
6 0.4850 0.5149 0.5430 0.5653 0.5832 O.5978 0.6100 0.6204 0.62937 0.5512 0.5787 0.6045 0.6248 0.6410 0.6542 0.6652 0.6744 0.68248 0.6031 0.6282 0.6518 0.6704 0.6851 0.6970 0.7069 0.7153 0.72259 0.6445 0.6676 0.6892 0.7062 0.7197 0.7305 0.7395 0.7471 0.7536
10 0.6783 0.6996 0.7195 0.7352 0.7475 0.7575 0.7657 0.7726 0.7786
11 0.7063 0.7260 0.7445 0.7590 0.7703 0.7795 0.7871 0.7935 0.799012 0.7299 0.7483 0.7654 0.7789 0.7894 0.7980 0.8050 0.8109 0.8160
13 0.7501 0.7672 0.7832 0.7958 0.8056 0.8135 0.8201 0.8256 0.830314 0.7674 0.7835 0.7985 0.8103 0.8195 0.8269 0.8330 0.8382 0.842615 0.7825 0.7977 0.8118 0.8229 0.8315 0.8385 0.8443 0.8491 0.8532
16 0.7958 0.8101 0.8235 0.8339 0.8421 0.8486 0.8541 0.8586 0.862517 0.8076 0.8211 0.8338 0.8436 0.8514 0.8576 0.8627 0.8670 0.870718 0.8181 0.8309 0.8429 0.8523 0.8596 0.8655 0.8704 0.8745 0.878019 0.8275 0.8397 0.8512 0.8601 0.8670 0.8727 0.8773 0.8811 0.884520 0.8360 0.8476 0.8586 0.8671 0.8737 0.8791 0.8835 0.8871 0.8903
21 0.8437 0.8548 0.8653 0.8734 0.8797 0.8848 0.8890 0.8926 0.895622 0.8507 0.8614 0.8714 0.8791 0.8852 0.8901 0.8941 0.8975 0.900423 0.8571 0.8673 0.8769 0.8844 0.8902 0.8949 0.8988 0.9020 0.9047
24 0.8630 0.8728 0.8820 0.8892 0.8948 0.8993 0.9030 0.9061 0.908725 0.8684 0.8779 0.8867 0.8936 0.8990 0.9034 0.9069 0.9099 0.9124
26 0.8734 0.8825 0.8911 0.8977 0.9029 0.9071 0.9105 0.9134 0.915827 0.8781 0.8869 0.8951 0.9015 0.9065 0.9105 0.9138 0.9166 0.919028 0.8824 0.8909 0.8988 0.9050 0.9099 0.9138 0.9169 0.9196 0.921929 0.8864 0.8946 0.9023 0.9083 0.9130 0.9167 0.9198 0.9224 0.924630 0.8902 0.8981 0.9056 0.9114 0.9159 0.9195 0.9225 0.9250 0.9271
40 0.9175 0.9235 0.9291 0.9335 0.9370 0.9397 0.9420 0.9439 0.945550 0.9339 0.9387 0.9433 0.9468 0.9496 0.9518 0.9536 0.9551 0.956460 0.9449 0.9489 0.9527 0.9557 0.9580 0.9599 0.9614 0.9626 0.963780 0.9586 0.9617 0.9646 0.9668 0.9685 0.9699 0.9711 0.9720 0.9728
100 0.9669 0.9693 0.9716 0.9734 0.9748 0.9759 0.9769 0.9776 0.9783
Table 1 from On the Determination of Critical Values for Bartletts Test, by Danny D. Dyerand Jerome P. Keating, Vol 75, No. 370 (Jun., 1980), pp.313319. Reprinted with permissionfrom The Journal of the American Statistical Association. Copyright 1980 by the AmericanStatistical Association. All rights reserved.
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Table A.8 Table for Bartletts Test 449
Table A.8 (continued) Critical Values for Bartletts Test
bk(0.05;n)
Number of Populations, k
n 2 3 4 5 6 7 8 9 10
3 0.3123 0.3058 0.3173 0.32994 0.4780 0.4699 0.4803 0.4921 0.5028 0.5122 0.5204 0.5277 0.53415 0.5845 0.5762 0.5850 0.5952 0.6045 0.6126 0.6197 0.6260 0.6315
6 0.6563 0.6483 0.6559 0.6646 0.6727 0.6798 0.6860 0.6914 0.69617 0.7075 0.7000 0.7065 0.7142 0.7213 0.7275 0.7329 0.7376 0.74188 0.7456 0.7387 0.7444 0.7512 0.7574 0.7629 0.7677 0.7719 0.77579 0.7751 0.7686 0.7737 0.7798 0.7854 0.7903 0.7946 0.7984 0.8017
10 0.7984 0.7924 0.7970 0.8025 0.8076 0.8121 0.8160 0.8194 0.8224
11 0.8175 0.8118 0.8160 0.8210 0.8257 0.8298 0.8333 0.8365 0.839212 0.8332 0.8280 0.8317 0.8364 0.8407 0.8444 0.8477 0.8506 0.853113 0.8465 0.8415 0.8450 0.8493 0.8533 0.8568 0.8598 0.8625 0.8648
14 0.8578 0.8532 0.8564 0.8604 0.8641 0.8673 0.8701 0.8726 0.874815 0.8676 0.8632 0.8662 0.8699 0.8734 0.8764 0.8790 0.8814 0.8834
16 0.8761 0.8719 0.8747 0.8782 0.8815 0.8843 0.8868 0.8890 0.890917 0.8836 0.8796 0.8823 0.8856 0.8886 0.8913 0.8936 0.8957 0.897518 0.8902 0.8865 0.8890 0.8921 0.8949 0.8975 0.8997 0.9016 0.903319 0.8961 0.8926 0.8949 0.8979 0.9006 0.9030 0.9051 0.9069 0.908620 0.9015 0.8980 0.9003 0.9031 0.9057 0.9080 0.9100 0.9117 0.9132
21 0.9063 0.9030 0.9051 0.9078 0.9103 0.9124 0.9143 0.9160 0.917522 0.9106 0.9075 0.9095 0.9120 0.9144 0.9165 0.9183 0.9199 0.921323 0.9146 0.9116 0.9135 0.9159 0.9182 0.9202 0.9219 0.9235 0.924824 0.9182 0.9153 0.9172 0.9195 0.9217 0.9236 0.9253 0.9267 0.9280
25 0.9216 0.9187 0.9205 0.9228 0.9249 0.9267 0.9283 0.9297 0.9309
26 0.9246 0.9219 0.9236 0.9258 0.9278 0.9296 0.9311 0.9325 0.933627 0.9275 0.9249 0.9265 0.9286 0.9305 0.9322 0.9337 0.9350 0.936128 0.9301 0.9276 0.9292 0.9312 0.9330 0.9347 0.9361 0.9374 0.938529 0.9326 0.9301 0.9316 0.9336 0.9354 0.9370 0.9383 0.9396 0.940630 0.9348 0.9325 0.9340 0.9358 0.9376 0.9391 0.9404 0.9416 0.9426
40 0.9513 0.9495 0.9506 0.9520 0.9533 0.9545 0.9555 0.9564 0.957250 0.9612 0.9597 0.9606 0.9617 0.9628 0.9637 0.9645 0.9652 0.965860 0.9677 0.9665 0.9672 0.9681 0.9690 0.9698 0.9705 0.9710 0.971680 0.9758 0.9749 0.9754 0.9761 0.9768 0.9774 0.9779 0.9783 0.9787
100 0.9807 0.9799 0.9804 0.9809 0.9815 0.9819 0.9823 0.9827 0.9830
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Table A.9 Critical Values for Cochrans Test
= 0.01
n
k 2 3 4 5 6 7 8 9 10 11 17
2 0.9999 0.9950 0.9794 0.9586 0.9373 0.9172 0.8988 0.8823 0.8674 0.8539 0.7949
3 0.9933 0.9423 0.8831 0.8335 0.7933 0.7606 0.7335 0.7107 0.6912 0.6743 0.6059
40.9676 0.8643 0.7814 0.7212 0.6761 0.6410 0.6129 0.5897 0.5702 0.5536 0.4884
5 0.9279 0.7885 0.6957 0.6329 0.5875 0.5531 0.5259 0.5037 0.4854 0.4697 0.4094
6 0.8828 0.7218 0.6258 0.5635 0.5195 0.4866 0.4608 0.4401 0.4229 0.4084 0.3529
7 0.8376 0.6644 0.5685 0.5080 0.4659 0.4347 0.4105 0.3911 0.3751 0.3616 0.3105
8 0.7945 0.6152 0.5209 0.4627 0.4226 0.3932 0.3704 0.3522 0.3373 0.3248 0.2779
9 0.7544 0.5727 0.4810 0.4251 0.3870 0.3592 0.3378 0.3207 0.3067 0.2950 0.2514
10 0.7175 0.5358 0.4469 0.3934 0.3572 0.3308 0.3106 0.2945 0.2813 0.2704 0.2297
12 0.6528 0.4751 0.3919 0.3428 0.3099 0.2861 0.2680 0.2535 0.2419 0.2320 0.196
15 0.5747 0.4069 0.3317 0.2882 0.2593 0.2386 0.2228 0.2104 0.2002 0.1918 0.1612
20 0.4799 0.3297 0.2654 0.2288 0.2048 0.1877 0.1748 0.1646 0.1567 0.1501 0.1248
24 0.4247 0.2871 0.2295 0.1970 0.1759 0.1608 0.1495 0.1406 0.1338 0.1283 0.1060
30 0.3632 0.2412 0.1913 0.1635 0.1454 0.1327 0.1232 0.1157 0.1100 0.1054 0.0867
40 0.2940 0.1915 0.1508 0.1281 0.1135 0.1033 0.0957 0.0898 0.0853 0.0816 0.0668
60 0.2151 0.1371 0.1069 0.0902 0.0796 0.0722 0.0668 0.0625 0.0594 0.0567 0.046
120 0.1225 0.0759 0.0585 0.0489 0.0429 0.0387 0.0357 0.0334 0.0316 0.0302 0.0242
0 0 0 0 0 0 0 0 0 0 0
Reproduced from C. Eisenhart, M. W. Hastay, and W. A. Wallis, Techniques of Statistical Ana
Hill Book Company, New, York, 1947.
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Table A.9 (continued) Critical Values for Cochrans Test
= 0.05
n
k 2 3 4 5 6 7 8 9 10 11 17
2 0.9985 0.9750 0.9392 0.9057 0.8772 0.8534 0.8332 0.8159 0.8010 0.7880 0.733 0.9669 0.8709 0.7977 0.7457 0.7071 0.6771 0.6530 0.6333 0.6167 0.6025 0.54
4 0.9065 0.7679 0.6841 0.6287 0.5895 0.5598 0.5365 0.5175 0.5017 0.4884 0.435 0.8412 0.6838 0.5981 0.5441 0.5065 0.4783 0.4564 0.4387 0.4241 0.4118 0.366 0.7808 0.6161 0.5321 0.4803 0.4447 0.4184 0.3980 0.3817 0.3682 0.3568 0.317 0.7271 0.5612 0.4800 0.4307 0.3974 0.3726 0.3535 0.3384 0.3259 0.3154 0.27
8 0.6798 0.5157 0.4377 0.3910 0.3595 0.3362 0.3185 0.3043 0.2926 0.2829 0.249 0.6385 0.4775 0.4027 0.3584 0.3286 0.3067 0.2901 0.2768 0.2659 0.2568 0.22
10 6.6020 0.4450 0.3733 0.3311 0.3029 0.2823 0.2666 0.2541 0.2439 0.2353 0.20
12 0.5410 0.3924 0.3264 0.2880 0.2624 0.2439 0.2299 0.2187 0.2098 0.2020 0.1715 0.4709 0.3346 0.2758 0.2419 0.2195 0.2034 0.1911 0.1815 0.1736 0.1671 0.1420 0.3894 0.2705 0.2205 0.1921 0.1735 0.1602 0.1501 0.1422 0.1357 0.1303 0.11
240.3434 0.2354 0.1907 0.1656 0.1493 0.1374 0.1286 0.1216 0.1160 0.1113 0.0930 0.2929 0.1980 0.1593 0.1377 0.1237 0.1137 0.1061 0.1002 0.0958 0.0921 0.07
40 0.2370 0.1576 0.1259 0.1082 0.0968 0.0887 0.0827 0.0780 0.0745 0.0713 0.05
60 0.1737 0.1131 0.0895 0.0765 0.0682 0.0623 0.0583 0.0552 0.0520 0.0497 0.04120 0.0998 0.0632 0.0495 0.0419 0.0371 0.0337 0.0312 0.0292 0.0279 0.0266 0.02 0 0 0 0 0 0 0 0 0 0 0
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452 Appendix A Statistical Tables and Proofs
Table A.10 Upper Percentage Points of the Studentized Range Distribution: q(0.05; k, v)
Degrees of Number of Treatments, k
Freedom, v 2 3 4 5 6 7 8 9 10
1 18.0 27.0 32.8 37.2 40.5 43.1 15.1 47.1 49.12 6.09 8.33 9.80 10.89 11.73 12.43 13.03 13.54 13.99
3 4.50 5.91 6.83 7.51 8.04 8.47 8.85 9.18 9.464 3.93 5.04 5.76 6.29 6.71 7.06 7.35 7.60 7.835 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99
6 3.46 4.34 4.90 5.31 5.63 5.89 6.12 6.32 6.497 3.34 4.16 4.68 5.06 5.35 5.59 5.80 5.99 6.158 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.929 3.20 3.95 4.42 4.76 5.02 5.24 5.43 5.60 5.74
10 3.15 3.88 4.33 4.66 4.91 5.12 5.30 5.46 5.60
11 3.11 3.82 4.26 4.58 4.82 5.03 5.20 5.35 5.4912 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.4013
3.06 3.73 4.15 4.46 4.69 4.88 5.05 5.19 5.3214 3.03 3.70 4.11 4.41 4.65 4.83 4.99 5.13 5.2515 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20
16 3.00 3.65 4.05 4.34 4.56 4.74 4.90 5.03 5.1517 2.98 3.62 4.02 4.31 4.52 4.70 4.86 4.99 5.1118 2.97 3.61 4.00 4.28 4.49 4.67 4.83 4.96 5.0719 2.96 3.59 3.98 4.26 4.47 4.64 4.79 4.92 5.0420 2.95 3.58 3.96 4.24 4.45 4.62 4.77 4.90 5.01
24 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.9230 2.89 3.48 3.84 4.11 4.30 4.46 4.60 4.72 4.8340 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.74
60 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65120 2.80 3.36 3.69 3.92 4.10 4.24 4.36 4.47 4.56 2.77 3.32 3.63 3.86 4.03 4.17 4.29 4.39 4.47
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Section A.12 Proof of Mean of the Hypergeometric Distribution 453
Table A.11 The Incomplete Gamma Function: F(x;) =x0
1()y
1ey dy
x 1 2 3 4 5 6 7 8 9 10
1 0.6320 0.2640 0.0800 0.0190 0.0040 0.0010 0.0000 0.0000 0.0000 0.00002 0.8650 0.5940 0.3230 0.1430 0.0530 0.0170 0.0050 0.0010 0.0000 0.0000
3 0.9500 0.8010 0.5770 0.3530 0.1850 0.0840 0.0340 0.0120 0.0040 0.00104 0.9820 0.9080 0.7620 0.5670 0.3710 0.2150 0.1110 0.0510 0.0210 0.00805 0.9930 0.9600 0.8750 0.7350 0.5600 0.3840 0.2380 0.1330 0.0680 0.0320
6 0.9980 0.9830 0.9380 0.8490 0.7150 0.5540 0.3940 0.2560 0.1530 0.08407 0.9990 0.9930 0.9700 0.9180 0.8270 0.6990 0.5500 0.4010 0.2710 0.17008 1.0000 0.9970 0.9860 0.9580 0.9000 0.8090 0.6870 0.5470 0.4070 0.28309 0.9990 0.9940 0.9790 0.9450 0.8840 0.7930 0.6760 0.5440 0.4130
10 1.0000 0.9970 0.9900 0.9710 0.9330 0.8700 0.7800 0.6670 0.5420
11 0.9990 0.9950 0.9850 0.9620 0.9210 0.8570 0.7680 0.659012 1.0000 0.9980 0.9920 0.9800 0.9540 0.9110 0.8450 0.758013 0.9990 0.9960 0.9890 0.9740 0.9460 0.9000 0.834014 1.0000 0.9980 0.9940 0.9860 0.9680 0.9380 0.891015 0.9990 0.9970 0.9920 0.9820 0.9630 0.9300
A.12 Proof of Mean of the Hypergeometric Distribution
To find the mean of the hypergeometric distribution, we write
E(X) =n
x=0
x
kx
Nknx
Nn = k
n
x=1
(k 1)!
(x
1)!(k
x)!
Nknx
Nn
= kn
x=1
k1x1
Nk
nx
N
n
.
Since N k
n 1 y
=
(N 1) (k 1)
n 1 y
and
N
n
=
N!
n!(N n)!=
N
n
N 1
n 1
,
letting y = x 1, we obtain
E(X) = k
n1y=0
k1y Nk
n1y
Nn
=nk
N
n1y=0
k1y
(N1)(k1)
n1y
N1n1
= nkN
,
since the summation represents the total of all probabilities in a hypergeometric experiment when N1items are selected at random from N 1, of which k 1 are labeled success.
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