Appendix a Statistical Tables and Proofs

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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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


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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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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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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12/28


13/28

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


14/28


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


15/28


16/28


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


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


20/28



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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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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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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Documents

Appendix a Statistical Tables and Proofs