appendix 3 tables

APPENDIX 3

Tables

Table 1 Binomial Probabilities

Tabulated values are P(Y ≤ a) =a∑

y=0p(y). (Computations are rounded at third decimal place.)

(a) n = 5

p

a 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 a

0 .951 .774 .590 .328 .168 .078 .031 .010 .002 .000 .000 .000 .000 01 .999 .977 .919 .737 .528 .337 .188 .087 .031 .007 .000 .000 .000 12 1.000 .999 .991 .942 .837 .683 .500 .317 .163 .058 .009 .001 .000 23 1.000 1.000 1.000 .993 .969 .913 .812 .663 .472 .263 .081 .023 .001 34 1.000 1.000 1.000 1.000 .998 .990 .969 .922 .832 .672 .410 .226 .049 4

(b) n = 10

p

a 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 a

0 .904 .599 .349 .107 .028 .006 .001 .000 .000 .000 .000 .000 .000 01 .996 .914 .736 .376 .149 .046 .011 .002 .000 .000 .000 .000 .000 12 1.000 .988 .930 .678 .383 .167 .055 .012 .002 .000 .000 .000 .000 23 1.000 .999 .987 .879 .650 .382 .172 .055 .011 .001 .000 .000 .000 34 1.000 1.000 .998 .967 .850 .633 .377 .166 .047 .006 .000 .000 .000 45 1.000 1.000 1.000 .994 .953 .834 .623 .367 .150 .033 .002 .000 .000 56 1.000 1.000 1.000 .999 .989 .945 .828 .618 .350 .121 .013 .001 .000 67 1.000 1.000 1.000 1.000 .998 .988 .945 .833 .617 .322 .070 .012 .000 78 1.000 1.000 1.000 1.000 1.000 .998 .989 .954 .851 .624 .264 .086 .004 89 1.000 1.000 1.000 1.000 1.000 1.000 .999 .994 .972 .893 .651 .401 .096 9

839

840 Appendix 3 Tables

Table 1 (Continued )

(c) n = 15

p

a 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 a

0 .860 .463 .206 .035 .005 .000 .000 .000 .000 .000 .000 .000 .000 01 .990 .829 .549 .167 .035 .005 .000 .000 .000 .000 .000 .000 .000 12 1.000 .964 .816 .398 .127 .027 .004 .000 .000 .000 .000 .000 .000 23 1.000 .995 .944 .648 .297 .091 .018 .002 .000 .000 .000 .000 .000 34 1.000 .999 .987 .836 .515 .217 .059 .009 .001 .000 .000 .000 .000 45 1.000 1.000 .998 .939 .722 .403 .151 .034 .004 .000 .000 .000 .000 56 1.000 1.000 1.000 .982 .869 .610 .304 .095 .015 .001 .000 .000 .000 67 1.000 1.000 1.000 .996 .950 .787 .500 .213 .050 .004 .000 .000 .000 78 1.000 1.000 1.000 .999 .985 .905 .696 .390 .131 .018 .000 .000 .000 89 1.000 1.000 1.000 1.000 .996 .966 .849 .597 .278 .061 .002 .000 .000 9

10 1.000 1.000 1.000 1.000 .999 .991 .941 .783 .485 .164 .013 .001 .000 1011 1.000 1.000 1.000 1.000 1.000 .998 .982 .909 .703 .352 .056 .005 .000 1112 1.000 1.000 1.000 1.000 1.000 1.000 .996 .973 .873 .602 .184 .036 .000 1213 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .995 .965 .833 .451 .171 .010 1314 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .995 .965 .794 .537 .140 14

(d) n = 20

p

a 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 a

0 .818 .358 .122 .012 .001 .000 .000 .000 .000 .000 .000 .000 .000 01 .983 .736 .392 .069 .008 .001 .000 .000 .000 .000 .000 .000 .000 12 .999 .925 .677 .206 .035 .004 .000 .000 .000 .000 .000 .000 .000 23 1.000 .984 .867 .411 .107 .016 .001 .000 .000 .000 .000 .000 .000 34 1.000 .997 .957 .630 .238 .051 .006 .000 .000 .000 .000 .000 .000 45 1.000 1.000 .989 .804 .416 .126 .021 .002 .000 .000 .000 .000 .000 56 1.000 1.000 .998 .913 .608 .250 .058 .006 .000 .000 .000 .000 .000 67 1.000 1.000 1.000 .968 .772 .416 .132 .021 .001 .000 .000 .000 .000 78 1.000 1.000 1.000 .990 .887 .596 .252 .057 .005 .000 .000 .000 .000 89 1.000 1.000 1.000 .997 .952 .755 .412 .128 .017 .001 .000 .000 .000 9

10 1.000 1.000 1.000 .999 .983 .872 .588 .245 .048 .003 .000 .000 .000 1011 1.000 1.000 1.000 1.000 .995 .943 .748 .404 .113 .010 .000 .000 .000 1112 1.000 1.000 1.000 1.000 .999 .979 .868 .584 .228 .032 .000 .000 .000 1213 1.000 1.000 1.000 1.000 1.000 .994 .942 .750 .392 .087 .002 .000 .000 1314 1.000 1.000 1.000 1.000 1.000 .998 .979 .874 .584 .196 .011 .000 .000 1415 1.000 1.000 1.000 1.000 1.000 1.000 .994 .949 .762 .370 .043 .003 .000 1516 1.000 1.000 1.000 1.000 1.000 1.000 .999 .984 .893 .589 .133 .016 .000 1617 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .996 .965 .794 .323 .075 .001 1718 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .992 .931 .608 .264 .017 1819 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .999 .988 .878 .642 .182 19

Tables 841


(e) n = 25

p

a 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 a

0 .778 .277 .072 .004 .000 .000 .000 .000 .000 .000 .000 .000 .000 01 .974 .642 .271 .027 .002 .000 .000 .000 .000 .000 .000 .000 .000 12 .998 .873 .537 .098 .009 .000 .000 .000 .000 .000 .000 .000 .000 23 1.000 .966 .764 .234 .033 .002 .000 .000 .000 .000 .000 .000 .000 34 1.000 .993 .902 .421 .090 .009 .000 .000 .000 .000 .000 .000 .000 45 1.000 .999 .967 .617 .193 .029 .002 .000 .000 .000 .000 .000 .000 56 1.000 1.000 .991 .780 .341 .074 .007 .000 .000 .000 .000 .000 .000 67 1.000 1.000 .998 .891 .512 .154 .022 .001 .000 .000 .000 .000 .000 78 1.000 1.000 1.000 .953 .677 .274 .054 .004 .000 .000 .000 .000 .000 89 1.000 1.000 1.000 .983 .811 .425 .115 .013 .000 .000 .000 .000 .000 9

10 1.000 1.000 1.000 .994 .902 .586 .212 .034 .002 .000 .000 .000 .000 1011 1.000 1.000 1.000 .998 .956 .732 .345 .078 .006 .000 .000 .000 .000 1112 1.000 1.000 1.000 1.000 .983 .846 .500 .154 .017 .000 .000 .000 .000 1213 1.000 1.000 1.000 1.000 .994 .922 .655 .268 .044 .002 .000 .000 .000 1314 1.000 1.000 1.000 1.000 .998 .966 .788 .414 .098 .006 .000 .000 .000 1415 1.000 1.000 1.000 1.000 1.000 .987 .885 .575 .189 .017 .000 .000 .000 1516 1.000 1.000 1.000 1.000 1.000 .996 .946 .726 .323 .047 .000 .000 .000 1617 1.000 1.000 1.000 1.000 1.000 .999 .978 .846 .488 .109 .002 .000 .000 1718 1.000 1.000 1.000 1.000 1.000 1.000 .993 .926 .659 .220 .009 .000 .000 1819 1.000 1.000 1.000 1.000 1.000 1.000 .998 .971 .807 .383 .033 .001 .000 1920 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .991 .910 .579 .098 .007 .000 2021 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .967 .766 .236 .034 .000 2122 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .991 .902 .463 .127 .002 2223 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .998 .973 .729 .358 .026 2324 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .996 .928 .723 .222 24


Table 2 Table of e−x

x e−x x e−x x e−x x e−x

0.00 1.000000 2.60 .074274 5.10 .006097 7.60 .0005010.10 .904837 2.70 .067206 5.20 .005517 7.70 .0004530.20 .818731 2.80 .060810 5.30 .004992 7.80 .0004100.30 .740818 2.90 .055023 5.40 .004517 7.90 .0003710.40 .670320 3.00 .049787 5.50 .004087 8.00 .0003360.50 .606531 3.10 .045049 5.60 .003698 8.10 .0003040.60 .548812 3.20 .040762 5.70 .003346 8.20 .0002750.70 .496585 3.30 .036883 5.80 .003028 8.30 .0002490.80 .449329 3.40 .033373 5.90 .002739 8.40 .0002250.90 .406570 3.50 .030197 6.00 .002479 8.50 .0002041.00 .367879 3.60 .027324 6.10 .002243 8.60 .0001841.10 .332871 3.70 .024724 6.20 .002029 8.70 .0001671.20 .301194 3.80 .022371 6.30 .001836 8.80 .0001511.30 .272532 3.90 .020242 6.40 .001661 8.90 .0001361.40 .246597 4.00 .018316 6.50 .001503 9.00 .0001231.50 .223130 4.10 .016573 6.60 .001360 9.10 .0001121.60 .201897 4.20 .014996 6.70 .001231 9.20 .0001011.70 .182684 4.30 .013569 6.80 .001114 9.30 .0000911.80 .165299 4.40 .012277 6.90 .001008 9.40 .0000831.90 .149569 4.50 .011109 7.00 .000912 9.50 .0000752.00 .135335 4.60 .010052 7.10 .000825 9.60 .0000682.10 .122456 4.70 .009095 7.20 .000747 9.70 .0000612.20 .110803 4.80 .008230 7.30 .000676 9.80 .0000562.30 .100259 4.90 .007447 7.40 .000611 9.90 .0000502.40 .090718 5.00 .006738 7.50 .000553 10.00 .0000452.50 .082085

Tables 843

Table 3 Poisson Probabilities

P(Y ≤ a) =a∑

y=0

e−λ λy

y!

aλ 0 1 2 3 4 5 6 7 8 9

0.02 0.980 1.0000.04 0.961 0.999 1.0000.06 0.942 0.998 1.0000.08 0.923 0.997 1.0000.10 0.905 0.995 1.000

0.15 0.861 0.990 0.999 1.0000.20 0.819 0.982 0.999 1.0000.25 0.779 0.974 0.998 1.0000.30 0.741 0.963 0.996 1.000

0.35 0.705 0.951 0.994 1.0000.40 0.670 0.938 0.992 0.999 1.0000.45 0.638 0.925 0.989 0.999 1.0000.50 0.607 0.910 0.986 0.998 1.000

0.55 0.577 0.894 0.982 0.988 1.0000.60 0.549 0.878 0.977 0.997 1.0000.65 0.522 0.861 0.972 0.996 0.999 1.0000.70 0.497 0.844 0.966 0.994 0.999 1.0000.75 0.472 0.827 0.959 0.993 0.999 1.000

0.80 0.449 0.809 0.953 0.991 0.999 1.0000.85 0.427 0.791 0.945 0.989 0.998 1.0000.90 0.407 0.772 0.937 0.987 0.998 1.0000.95 0.387 0.754 0.929 0.981 0.997 1.0001.00 0.368 0.736 0.920 0.981 0.996 0.999 1.000

1.1 0.333 0.699 0.900 0.974 0.995 0.999 1.0001.2 0.301 0.663 0.879 0.966 0.992 0.998 1.0001.3 0.273 0.627 0.857 0.957 0.989 0.998 1.0001.4 0.247 0.592 0.833 0.946 0.986 0.997 0.999 1.0001.5 0.223 0.558 0.809 0.934 0.981 0.996 0.999 1.000

1.6 0.202 0.525 0.783 0.921 0.976 0.994 0.999 1.0001.7 0.183 0.493 0.757 0.907 0.970 0.992 0.998 1.0001.8 0.165 0.463 0.731 0.891 0.964 0.990 0.997 0.999 1.0001.9 0.150 0.434 0.704 0.875 0.956 0.987 0.997 0.999 1.0002.0 0.135 0.406 0.677 0.857 0.947 0.983 0.995 0.999 1.000



aλ 0 1 2 3 4 5 6 7 8 9

2.2 0.111 0.355 0.623 0.819 0.928 0.975 0.993 0.998 1.0002.4 0.091 0.308 0.570 0.779 0.904 0.964 0.988 0.997 0.999 1.0002.6 0.074 0.267 0.518 0.736 0.877 0.951 0.983 0.995 0.999 1.0002.8 0.061 0.231 0.469 0.692 0.848 0.935 0.976 0.992 0.998 0.9993.0 0.050 0.199 0.423 0.647 0.815 0.916 0.966 0.988 0.996 0.999

3.2 0.041 0.171 0.380 0.603 0.781 0.895 0.955 0.983 0.994 0.9983.4 0.033 0.147 0.340 0.558 0.744 0.871 0.942 0.977 0.992 0.9973.6 0.027 0.126 0.303 0.515 0.706 0.844 0.927 0.969 0.988 0.9963.8 0.022 0.107 0.269 0.473 0.668 0.816 0.909 0.960 0.984 0.9944.0 0.018 0.092 0.238 0.433 0.629 0.785 0.889 0.949 0.979 0.992

4.2 0.015 0.078 0.210 0.395 0.590 0.753 0.867 0.936 0.972 0.9894.4 0.012 0.066 0.185 0.359 0.551 0.720 0.844 0.921 0.964 0.9854.6 0.010 0.056 0.163 0.326 0.513 0.686 0.818 0.905 0.955 0.9804.8 0.008 0.048 0.143 0.294 0.476 0.651 0.791 0.887 0.944 0.9755.0 0.007 0.040 0.125 0.265 0.440 0.616 0.762 0.867 0.932 0.968

5.2 0.006 0.034 0.109 0.238 0.406 0.581 0.732 0.845 0.918 0.9605.4 0.005 0.029 0.095 0.213 0.373 0.546 0.702 0.822 0.903 0.9515.6 0.004 0.024 0.082 0.191 0.342 0.512 0.670 0.797 0.886 0.9415.8 0.003 0.021 0.072 0.170 0.313 0.478 0.638 0.771 0.867 0.9296.0 0.002 0.017 0.062 0.151 0.285 0.446 0.606 0.744 0.847 0.916

10 11 12 13 14 15 16

2.8 1.0003.0 1.0003.2 1.0003.4 0.999 1.0003.6 0.999 1.0003.8 0.998 0.999 1.0004.0 0.997 0.999 1.000

4.2 0.996 0.999 1.0004.4 0.994 0.998 0.999 1.0004.6 0.992 0.997 0.999 1.0004.8 0.990 0.996 0.999 1.0005.0 0.986 0.995 0.998 0.999 1.000

5.2 0.982 0.993 0.997 0.999 1.0005.4 0.977 0.990 0.996 0.999 1.0005.6 0.972 0.988 0.995 0.998 0.999 1.0005.8 0.965 0.984 0.993 0.997 0.999 1.0006.0 0.957 0.980 0.991 0.996 0.999 0.999 1.000

Tables 845


aλ 0 1 2 3 4 5 6 7 8 9

6.2 0.002 0.015 0.054 0.134 0.259 0.414 0.574 0.716 0.826 0.9026.4 0.002 0.012 0.046 0.119 0.235 0.384 0.542 0.687 0.803 0.8866.6 0.001 0.010 0.040 0.105 0.213 0.355 0.511 0.658 0.780 0.8696.8 0.001 0.009 0.034 0.093 0.192 0.327 0.480 0.628 0.755 0.8507.0 0.001 0.007 0.030 0.082 0.173 0.301 0.450 0.599 0.729 0.830

7.2 0.001 0.006 0.025 0.072 0.156 0.276 0.420 0.569 0.703 0.8107.4 0.001 0.005 0.022 0.063 0.140 0.253 0.392 0.539 0.676 0.7887.6 0.001 0.004 0.019 0.055 0.125 0.231 0.365 0.510 0.648 0.7657.8 0.000 0.004 0.016 0.048 0.112 0.210 0.338 0.481 0.620 0.741

8.0 0.000 0.003 0.014 0.042 0.100 0.191 0.313 0.453 0.593 0.7178.5 0.000 0.002 0.009 0.030 0.074 0.150 0.256 0.386 0.523 0.6539.0 0.000 0.001 0.006 0.021 0.055 0.116 0.207 0.324 0.456 0.5879.5 0.000 0.001 0.004 0.015 0.040 0.089 0.165 0.269 0.392 0.522

10.0 0.000 0.000 0.003 0.010 0.029 0.067 0.130 0.220 0.333 0.458

10 11 12 13 14 15 16 17 18 19

6.2 0.949 0.975 0.989 0.995 0.998 0.999 1.0006.4 0.939 0.969 0.986 0.994 0.997 0.999 1.0006.6 0.927 0.963 0.982 0.992 0.997 0.999 0.999 1.0006.8 0.915 0.955 0.978 0.990 0.996 0.998 0.999 1.0007.0 0.901 0.947 0.973 0.987 0.994 0.998 0.999 1.000

7.2 0.887 0.937 0.967 0.984 0.993 0.997 0.999 0.999 1.0007.4 0.871 0.926 0.961 0.980 0.991 0.996 0.998 0.999 1.0007.6 0.854 0.915 0.954 0.976 0.989 0.995 0.998 0.999 1.0007.8 0.835 0.902 0.945 0.971 0.986 0.993 0.997 0.999 1.000

8.0 0.816 0.888 0.936 0.966 0.983 0.992 0.996 0.998 0.999 1.0008.5 0.763 0.849 0.909 0.949 0.973 0.986 0.993 0.997 0.999 0.9999.0 0.706 0.803 0.876 0.926 0.959 0.978 0.989 0.995 0.998 0.9999.5 0.645 0.752 0.836 0.898 0.940 0.967 0.982 0.991 0.996 0.998

10.0 0.583 0.697 0.792 0.864 0.917 0.951 0.973 0.986 0.993 0.997

20 21 22

8.5 1.0009.0 1.0009.5 0.999 1.000

10.0 0.998 0.999 1.000



aλ 0 1 2 3 4 5 6 7 8 9

10.5 0.000 0.000 0.002 0.007 0.021 0.050 0.102 0.179 0.279 0.39711.0 0.000 0.000 0.001 0.005 0.015 0.038 0.079 0.143 0.232 0.34111.5 0.000 0.000 0.001 0.003 0.011 0.028 0.060 0.114 0.191 0.28912.0 0.000 0.000 0.001 0.002 0.008 0.020 0.046 0.090 0.155 0.24212.5 0.000 0.000 0.000 0.002 0.005 0.015 0.035 0.070 0.125 0.201

13.0 0.000 0.000 0.000 0.001 0.004 0.011 0.026 0.054 0.100 0.16613.5 0.000 0.000 0.000 0.001 0.003 0.008 0.019 0.041 0.079 0.13514.0 0.000 0.000 0.000 0.000 0.002 0.006 0.014 0.032 0.062 0.10914.5 0.000 0.000 0.000 0.000 0.001 0.004 0.010 0.024 0.048 0.08815.0 0.000 0.000 0.000 0.000 0.001 0.003 0.008 0.018 0.037 0.070

10 11 12 13 14 15 16 17 18 19

10.5 0.521 0.639 0.742 0.825 0.888 0.932 0.960 0.978 0.988 0.99411.0 0.460 0.579 0.689 0.781 0.854 0.907 0.944 0.968 0.982 0.99111.5 0.402 0.520 0.633 0.733 0.815 0.878 0.924 0.954 0.974 0.98612.0 0.347 0.462 0.576 0.682 0.772 0.844 0.899 0.937 0.963 0.97912.5 0.297 0.406 0.519 0.628 0.725 0.806 0.869 0.916 0.948 0.969

13.0 0.252 0.353 0.463 0.573 0.675 0.764 0.835 0.890 0.930 0.95713.5 0.211 0.304 0.409 0.518 0.623 0.718 0.798 0.861 0.908 0.94214.0 0.176 0.260 0.358 0.464 0.570 0.669 0.756 0.827 0.883 0.92314.5 0.145 0.220 0.311 0.413 0.518 0.619 0.711 0.790 0.853 0.90115.0 0.118 0.185 0.268 0.363 0.466 0.568 0.664 0.749 0.819 0.875

20 21 22 23 24 25 26 27 28 29

10.5 0.997 0.999 0.999 1.00011.0 0.995 0.998 0.999 1.00011.5 0.992 0.996 0.998 0.999 1.00012.0 0.988 0.994 0.997 0.999 0.999 1.00012.5 0.983 0.991 0.995 0.998 0.999 0.999 1.000

13.0 0.975 0.986 0.992 0.996 0.998 0.999 1.00013.5 0.965 0.980 0.989 0.994 0.997 0.998 0.999 1.00014.0 0.952 0.971 0.983 0.991 0.995 0.997 0.999 0.999 1.00014.5 0.936 0.960 0.976 0.986 0.992 0.996 0.998 0.999 0.999 1.00015.0 0.917 0.947 0.967 0.981 0.989 0.994 0.997 0.998 0.999 1.000

Tables 847


aλ 4 5 6 7 8 9 10 11 12 13

16 0.000 0.001 0.004 0.010 0.022 0.043 0.077 0.127 0.193 0.27517 0.000 0.001 0.002 0.005 0.013 0.026 0.049 0.085 0.135 0.20118 0.000 0.000 0.001 0.003 0.007 0.015 0.030 0.055 0.092 0.14319 0.000 0.000 0.001 0.002 0.004 0.009 0.018 0.035 0.061 0.09820 0.000 0.000 0.000 0.001 0.002 0.005 0.011 0.021 0.039 0.06621 0.000 0.000 0.000 0.000 0.001 0.003 0.006 0.013 0.025 0.04322 0.000 0.000 0.000 0.000 0.001 0.002 0.004 0.008 0.015 0.02823 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.004 0.009 0.01724 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.005 0.01125 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.003 0.006

14 15 16 17 18 19 20 21 22 23

16 0.368 0.467 0.566 0.659 0.742 0.812 0.868 0.911 0.942 0.96317 0.281 0.371 0.468 0.564 0.655 0.736 0.805 0.861 0.905 0.93718 0.208 0.287 0.375 0.469 0.562 0.651 0.731 0.799 0.855 0.89919 0.150 0.215 0.292 0.378 0.469 0.561 0.647 0.725 0.793 0.84920 0.105 0.157 0.221 0.297 0.381 0.470 0.559 0.644 0.721 0.78721 0.072 0.111 0.163 0.227 0.302 0.384 0.471 0.558 0.640 0.71622 0.048 0.077 0.117 0.169 0.232 0.306 0.387 0.472 0.556 0.63723 0.031 0.052 0.082 0.123 0.175 0.238 0.310 0.389 0.472 0.55524 0.020 0.034 0.056 0.087 0.128 0.180 0.243 0.314 0.392 0.47325 0.012 0.022 0.038 0.060 0.092 0.134 0.185 0.247 0.318 0.394

24 25 26 27 28 29 30 31 32 33

16 0.978 0.987 0.993 0.996 0.998 0.999 0.999 1.00017 0.959 0.975 0.985 0.991 0.995 0.997 0.999 0.999 1.00018 0.932 0.955 0.972 0.983 0.990 0.994 0.997 0.998 0.999 1.00019 0.893 0.927 0.951 0.969 0.980 0.988 0.993 0.996 0.998 0.99920 0.843 0.888 0.922 0.948 0.966 0.978 0.987 0.992 0.995 0.99721 0.782 0.838 0.883 0.917 0.944 0.963 0.976 0.985 0.991 0.99422 0.712 0.777 0.832 0.877 0.913 0.940 0.959 0.973 0.983 0.98923 0.635 0.708 0.772 0.827 0.873 0.908 0.936 0.956 0.971 0.98124 0.554 0.632 0.704 0.768 0.823 0.868 0.904 0.932 0.953 0.96925 0.473 0.553 0.629 0.700 0.763 0.818 0.863 0.900 0.929 0.950

34 35 36 37 38 39 40 41 42 43

19 0.999 1.00020 0.999 0.999 1.00021 0.997 0.998 0.999 0.999 1.00022 0.994 0.996 0.998 0.999 0.999 1.00023 0.988 0.993 0.996 0.997 0.999 0.999 1.00024 0.979 0.987 0.992 0.995 0.997 0.998 0.999 0.999 1.00025 0.966 0.978 0.985 0.991 0.991 0.997 0.998 0.999 0.999 1.000


Table 4 Normal Curve AreasStandard normal probability in right-hand tail(for negative values of z, areas are found by symmetry)

0 z

Area

Second decimal place of z

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .46410.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .42470.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .38590.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .34830.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121

0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .27760.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .24510.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .21480.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .18670.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611

1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .13791.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .11701.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .09851.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .08231.4 .0808 .0793 .0778 .0764 .0749 .0735 .0722 .0708 .0694 .0681

1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571 .05591.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .04551.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .03671.8 .0359 .0352 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .02941.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233

2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .01832.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .01432.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .01102.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .00842.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064

2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .00482.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .00362.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .00262.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .00192.9 .0019 .0018 .0017 .0017 .0016 .0016 .0015 .0015 .0014 .0014

3.0 .001353.5 .000 2334.0 .000 031 74.5 .000 003 405.0 .000 000 287

From R. E. Walpole, Introduction to Statistics (New York: Macmillan, 1968).

Tables 849

Table 5 Percentage Points of the t Distributions

taa

t.100 t.050 t.025 t.010 t.005 df

3.078 6.314 12.706 31.821 63.657 11.886 2.920 4.303 6.965 9.925 21.638 2.353 3.182 4.541 5.841 31.533 2.132 2.776 3.747 4.604 4

1.476 2.015 2.571 3.365 4.032 51.440 1.943 2.447 3.143 3.707 61.415 1.895 2.365 2.998 3.499 71.397 1.860 2.306 2.896 3.355 81.383 1.833 2.262 2.821 3.250 9

1.372 1.812 2.228 2.764 3.169 101.363 1.796 2.201 2.718 3.106 111.356 1.782 2.179 2.681 3.055 121.350 1.771 2.160 2.650 3.012 131.345 1.761 2.145 2.624 2.977 141.341 1.753 2.131 2.602 2.947 15

1.337 1.746 2.120 2.583 2.921 161.333 1.740 2.110 2.567 2.898 171.330 1.734 2.101 2.552 2.878 181.328 1.729 2.093 2.539 2.861 191.325 1.725 2.086 2.528 2.845 20

1.323 1.721 2.080 2.518 2.831 211.321 1.717 2.074 2.508 2.819 221.319 1.714 2.069 2.500 2.807 231.318 1.711 2.064 2.492 2.797 241.316 1.708 2.060 2.485 2.787 25

1.315 1.706 2.056 2.479 2.779 261.314 1.703 2.052 2.473 2.771 271.313 1.701 2.048 2.467 2.763 281.311 1.699 2.045 2.462 2.756 291.282 1.645 1.960 2.326 2.576 inf.

From “Table of Percentage Points of the t-Distribution.” Computed byMaxine Merrington, Biometrika, Vol. 32 (1941), p. 300.


Table 6 Percentage Points of the χ2 Distributions

0 c2a

a

df χ 20.995 χ2

0.990 χ20.975 χ 2

0.950 χ20.900

1 0.0000393 0.0001571 0.0009821 0.0039321 0.01579082 0.0100251 0.0201007 0.0506356 0.102587 0.2107203 0.0717212 0.114832 0.215795 0.351846 0.5843754 0.206990 0.297110 0.484419 0.710721 1.063623

5 0.411740 0.554300 0.831211 1.145476 1.610316 0.675727 0.872085 1.237347 1.63539 2.204137 0.989265 1.239043 1.68987 2.16735 2.833118 1.344419 1.646482 2.17973 2.73264 3.489549 1.734926 2.087912 2.70039 3.32511 4.16816

10 2.15585 2.55821 3.24697 3.94030 4.8651811 2.60321 3.05347 3.81575 4.57481 5.5777912 3.07382 3.57056 4.40379 5.22603 6.3038013 3.56503 4.10691 5.00874 5.89186 7.0415014 4.07468 4.66043 5.62872 6.57063 7.78953

15 4.60094 5.22935 6.26214 7.26094 8.5467516 5.14224 5.81221 6.90766 7.96164 9.3122317 5.69724 6.40776 7.56418 8.67176 10.085218 6.26481 7.01491 8.23075 9.39046 10.864919 6.84398 7.63273 8.90655 10.1170 11.6509

20 7.43386 8.26040 9.59083 10.8508 12.442621 8.03366 8.89720 10.28293 11.5913 13.239622 8.64272 9.54249 10.9823 12.3380 14.041523 9.26042 10.19567 11.6885 13.0905 14.847924 9.88623 10.8564 12.4011 13.8484 15.6587

25 10.5197 11.5240 13.1197 14.6114 16.473426 11.1603 12.1981 13.8439 15.3791 17.291927 11.8076 12.8786 14.5733 16.1513 18.113828 12.4613 13.5648 15.3079 16.9279 18.939229 13.1211 14.2565 16.0471 17.7083 19.7677

30 13.7867 14.9535 16.7908 18.4926 20.599240 20.7065 22.1643 24.4331 26.5093 29.050550 27.9907 29.7067 32.3574 34.7642 37.688660 35.5346 37.4848 40.4817 43.1879 46.4589

70 43.2752 45.4418 48.7576 51.7393 55.329080 51.1720 53.5400 57.1532 60.3915 64.277890 59.1963 61.7541 65.6466 69.1260 73.2912

100 67.3276 70.0648 74.2219 77.9295 82.3581

Tables 851


χ20.100 χ 2

0.050 χ 20.025 χ2

0.010 χ20.005 df

2.70554 3.84146 5.02389 6.63490 7.87944 14.60517 5.99147 7.37776 9.21034 10.5966 26.25139 7.81473 9.34840 11.3449 12.8381 37.77944 9.48773 11.1433 13.2767 14.8602 4

9.23635 11.0705 12.8325 15.0863 16.7496 510.6446 12.5916 14.4494 16.8119 18.5476 612.0170 14.0671 16.0128 18.4753 20.2777 713.3616 15.5073 17.5346 20.0902 21.9550 814.6837 16.9190 19.0228 21.6660 23.5893 9

15.9871 18.3070 20.4831 23.2093 25.1882 1017.2750 19.6751 21.9200 24.7250 26.7569 1118.5494 21.0261 23.3367 26.2170 28.2995 1219.8119 22.3621 24.7356 27.6883 29.8194 1321.0642 23.6848 26.1190 29.1413 31.3193 14

22.3072 24.9958 27.4884 30.5779 32.8013 1523.5418 26.2962 28.8454 31.9999 34.2672 1624.7690 27.5871 30.1910 33.4087 35.7185 1725.9894 28.8693 31.5264 34.8053 37.1564 1827.2036 30.1435 32.8523 36.1908 38.5822 19

28.4120 31.4104 34.1696 37.5662 39.9968 2029.6151 32.6705 35.4789 38.9321 41.4010 2130.8133 33.9244 36.7807 40.2894 42.7956 2232.0069 35.1725 38.0757 41.6384 44.1813 2333.1963 36.4151 39.3641 42.9798 45.5585 24

34.3816 37.6525 40.6465 44.3141 46.9278 2535.5631 38.8852 41.9232 45.6417 48.2899 2636.7412 40.1133 43.1944 46.9630 49.6449 2737.9159 41.3372 44.4607 48.2782 50.9933 2839.0875 42.5569 45.7222 49.5879 52.3356 29

40.2560 43.7729 46.9792 50.8922 53.6720 3051.8050 55.7585 59.3417 63.6907 66.7659 4063.1671 67.5048 71.4202 76.1539 79.4900 5074.3970 79.0819 83.2976 88.3794 91.9517 60

85.5271 90.5312 95.0231 100.425 104.215 7096.5782 101.879 106.629 112.329 116.321 80

107.565 113.145 118.136 124.116 128.299 90118.498 124.342 129.561 135.807 140.169 100

From “Tables of the Percentage Points of the χ2-Distribution.” Biometrika, Vol. 32(1941), pp. 188–189, by Catherine M. Thompson.


Table 7 Percentage Points of the F Distributions

0 Fa

a

Numerator dfDenominator

df α 1 2 3 4 5 6 7 8 9

1 .100 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86.050 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5.025 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3.010 4052 4999.5 5403 5625 5764 5859 5928 5982 6022.005 16211 20000 21615 22500 23056 23437 23715 23925 24091

2 .100 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38.050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38.025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39.010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39.005 198.5 199.0 199.2 199.2 199.3 199.3 199.4 199.4 199.4

3 .100 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24.050 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81.025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47.010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35.005 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88

4 .100 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94.050 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00.025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90.010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66.005 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14

5 .100 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32.050 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77.025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68.010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16.005 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77

6 .100 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96.050 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10.025 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52.010 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98.005 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39

7 .100 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72.050 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68.025 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82.010 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72.005 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.51

Tables 853

Table 7 (Continued )Fα

Numerator df

10 12 15 20 24 30 40 60 120 ∞ α df

60.19 60.71 61.22 61.74 62.00 62.26 62.53 62.79 63.06 63.33 .100 1241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3 .050968.6 976.7 984.9 993.1 997.2 1001 1006 1010 1014 1018 .025

6056 6106 6157 6209 6235 6261 6287 6313 6339 6366 .01024224 24426 24630 24836 24940 25044 25148 25253 25359 25465 .005

9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.48 9.49 .100 219.40 19.41 19.43 19.45 19.45 19.45 19.47 19.48 19.49 19.50 .05039.40 39.41 39.43 39.45 39.46 39.46 39.47 39.48 39.49 39.50 .02599.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50 .010

199.4 199.4 199.4 199.4 199.5 199.5 199.5 199.5 199.5 199.5 .005

5.23 5.22 5.20 5.18 5.18 5.17 5.16 5.15 5.14 5.13 .100 38.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53 .050

14.42 14.34 14.25 14.17 14.12 14.08 14.04 13.99 13.95 13.90 .02527.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.13 .01043.69 43.39 43.08 42.78 42.62 42.47 42.31 42.15 41.99 41.83 .005

3.92 3.90 3.87 3.84 3.83 3.82 3.80 3.79 3.78 3.76 .100 45.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63 .0508.84 8.75 8.66 8.56 8.51 8.46 8.41 8.36 8.31 8.26 .025

14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.46 .01020.97 20.70 20.44 20.17 20.03 19.89 19.75 19.61 19.47 19.32 .005

3.30 3.27 3.24 3.21 3.19 3.17 3.16 3.14 3.12 3.10 .100 54.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36 .0506.62 6.52 6.43 6.33 6.28 6.23 6.18 6.12 6.07 6.02 .025

10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02 .01013.62 13.38 13.15 12.90 12.78 12.66 12.53 12.40 12.27 12.14 .005

2.94 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.72 .100 64.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67 .0505.46 5.37 5.27 5.17 5.12 5.07 5.01 4.96 4.90 4.85 .0257.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88 .010

10.25 10.03 9.81 9.59 9.47 9.36 9.24 9.12 9.00 8.88 .005

2.70 2.67 2.63 2.59 2.58 2.56 2.54 2.51 2.49 2.47 .100 73.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23 .0504.76 4.67 4.57 4.47 4.42 4.36 4.31 4.25 4.20 4.14 .0256.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65 .0108.38 8.18 7.97 7.75 7.65 7.53 7.42 7.31 7.19 7.08 .005




df α 1 2 3 4 5 6 7 8 9

8 .100 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56.050 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39.025 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36.010 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91.005 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.34

9 .100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44.050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18.025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03.010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35.005 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.54

10 .100 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35.050 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02.025 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78.010 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94.005 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.97

11 .100 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27.050 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90.025 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59.010 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63.005 12.23 8.91 7.60 6.88 6.42 6.10 5.86 5.68 5.54

12 .100 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21.050 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80.025 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44.010 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39.005 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 5.20

13 .100 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16.050 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71.025 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31.010 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19.005 11.37 8.19 6.93 6.23 5.79 5.48 5.25 5.08 4.94

14 .100 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12.050 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65.025 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21.010 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03.005 11.06 7.92 6.68 6.00 5.56 5.26 5.03 4.86 4.72

Tables 855


Numerator df

10 12 15 20 24 30 40 60 120 ∞ α df

2.54 2.50 2.46 2.42 2.40 2.38 2.36 2.34 2.32 2.29 .100 83.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93 .0504.30 4.20 4.10 4.00 3.95 3.89 3.84 3.78 3.73 3.67 .0255.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86 .0107.21 7.01 6.81 6.61 6.50 6.40 6.29 6.18 6.06 5.95 .005

2.42 2.38 2.34 2.30 2.28 2.25 2.23 2.21 2.18 2.16 .100 93.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71 .0503.96 3.87 3.77 3.67 3.61 3.56 3.51 3.45 3.39 3.33 .0255.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31 .0106.42 6.23 6.03 5.83 5.73 5.62 5.52 5.41 5.30 5.19 .005

2.32 2.28 2.24 2.20 2.18 2.16 2.13 2.11 2.08 2.06 .100 102.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54 .0503.72 3.62 3.52 3.42 3.37 3.31 3.26 3.20 3.14 3.08 .0254.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91 .0105.85 5.66 5.47 5.27 5.17 5.07 4.97 4.86 4.75 4.64 .005

2.25 2.21 2.17 2.12 2.10 2.08 2.05 2.03 2.00 1.97 .100 112.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40 .0503.53 3.43 3.33 3.23 3.17 3.12 3.06 3.00 2.94 2.88 .0254.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60 .0105.42 5.24 5.05 4.86 4.76 4.65 4.55 4.44 4.34 4.23 .005

2.19 2.15 2.10 2.06 2.04 2.01 1.99 1.96 1.93 1.90 .100 122.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30 .0503.37 3.28 3.18 3.07 3.02 2.96 2.91 2.85 2.79 2.72 .0254.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36 .0105.09 4.91 4.72 4.53 4.43 4.33 4.23 4.12 4.01 3.90 .005

2.14 2.10 2.05 2.01 1.98 1.96 1.93 1.90 1.88 1.85 .100 132.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21 .0503.25 3.15 3.05 2.95 2.89 2.84 2.78 2.72 2.66 2.60 .0254.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17 .0104.82 4.64 4.46 4.27 4.17 4.07 3.97 3.87 3.76 3.65 .005

2.10 2.05 2.01 1.96 1.94 1.91 1.89 1.86 1.83 1.80 .100 142.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13 .0503.15 3.05 2.95 2.84 2.79 2.73 2.67 2.61 2.55 2.49 .0253.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00 .0104.60 4.43 4.25 4.06 3.96 3.86 3.76 3.66 3.55 3.44 .005




df α 1 2 3 4 5 6 7 8 9

15 .100 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09.050 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59.025 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12.010 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89.005 10.80 7.70 6.48 5.80 5.37 5.07 4.85 4.67 4.54

16 .100 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06.050 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54.025 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05.010 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78.005 10.58 7.51 6.30 5.64 5.21 4.91 4.69 4.52 4.38

17 .100 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03.050 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49.025 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98.010 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68.005 10.38 7.35 6.16 5.50 5.07 4.78 4.56 4.39 4.25

18 .100 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00.050 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46.025 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93.010 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60.005 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 4.14

19 .100 2.99 2.61 2.40 2.27 2.18 2.11 2.06 2.02 1.98.050 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42.025 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88.010 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52.005 10.07 7.09 5.92 5.27 4.85 4.56 4.34 4.18 4.04

20 .100 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96.050 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39.025 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84.010 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46.005 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.96

21 .100 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95.050 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37.025 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80.010 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40.005 9.83 6.89 5.73 5.09 4.68 4.39 4.18 4.01 3.88

Tables 857


Numerator df

10 12 15 20 24 30 40 60 120 ∞ α df

2.06 2.02 1.97 1.92 1.90 1.87 1.85 1.82 1.79 1.76 .100 152.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07 .0503.06 2.96 2.86 2.76 2.70 2.64 2.59 2.52 2.46 2.40 .0253.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87 .0104.42 4.25 4.07 3.88 3.79 3.69 3.58 3.48 3.37 3.26 .005

2.03 1.99 1.94 1.89 1.87 1.84 1.81 1.78 1.75 1.72 .100 162.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01 .0502.99 2.89 2.79 2.68 2.63 2.57 2.51 2.45 2.38 2.32 .0253.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75 .0104.27 4.10 3.92 3.73 3.64 3.54 3.44 3.33 3.22 3.11 .005

2.00 1.96 1.91 1.86 1.84 1.81 1.78 1.75 1.72 1.69 .100 172.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96 .0502.92 2.82 2.72 2.62 2.56 2.50 2.44 2.38 2.32 2.25 .0253.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65 .0104.14 3.97 3.79 3.61 3.51 3.41 3.31 3.21 3.10 2.98 .005

1.98 1.93 1.89 1.84 1.81 1.78 1.75 1.72 1.69 1.66 .100 182.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92 .0502.87 2.77 2.67 2.56 2.50 2.44 2.38 2.32 2.26 2.19 .0253.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57 .0104.03 3.86 3.68 3.50 3.40 3.30 3.20 3.10 2.99 2.87 .005

1.96 1.91 1.86 1.81 1.79 1.76 1.73 1.70 1.67 1.63 .100 192.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88 .0502.82 2.72 2.62 2.51 2.45 2.39 2.33 2.27 2.20 2.13 .0253.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49 .0103.93 3.76 3.59 3.40 3.31 3.21 3.11 3.00 2.89 2.78 .005

1.94 1.89 1.84 1.79 1.77 1.74 1.71 1.68 1.64 1.61 .100 202.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84 .0502.77 2.68 2.57 2.46 2.41 2.35 2.29 2.22 2.16 2.09 .0253.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42 .0103.85 3.68 3.50 3.32 3.22 3.12 3.02 2.92 2.81 2.69 .005

1.92 1.87 1.83 1.78 1.75 1.72 1.69 1.66 1.62 1.59 .100 212.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81 .0502.73 2.64 2.53 2.42 2.37 2.31 2.25 2.18 2.11 2.04 .0253.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36 .0103.77 3.60 3.43 3.24 3.15 3.05 2.95 2.84 2.73 2.61 .005



Numerator dfDenominatordf α 1 2 3 4 5 6 7 8 9

22 .100 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93.050 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34.025 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76.010 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35.005 9.73 6.81 5.65 5.02 4.61 4.32 4.11 3.94 3.81

23 .100 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92.050 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32.025 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73.010 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30.005 9.63 6.73 5.58 4.95 4.54 4.26 4.05 3.88 3.75

24 .100 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91.050 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30.025 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70.010 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26.005 9.55 6.66 5.52 4.89 4.49 4.20 3.99 3.83 3.69

25 .100 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89.050 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28.025 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68.010 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22.005 9.48 6.60 5.46 4.84 4.43 4.15 3.94 3.78 3.64

26 .100 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88.050 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27.025 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65.010 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18.005 9.41 6.54 5.41 4.79 4.38 4.10 3.89 3.73 3.60

27 .100 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87.050 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25.025 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63.010 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15.005 9.34 6.49 5.36 4.74 4.34 4.06 3.85 3.69 3.56

28 .100 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87.050 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24.025 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61.010 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12.005 9.28 6.44 5.32 4.70 4.30 4.02 3.81 3.65 3.52

Tables 859


Numerator df

10 12 15 20 24 30 40 60 120 ∞ α df

1.90 1.86 1.81 1.76 1.73 1.70 1.67 1.64 1.60 1.57 .100 222.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78 .0502.70 2.60 2.50 2.39 2.33 2.27 2.21 2.14 2.08 2.00 .0253.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31 .0103.70 3.54 3.36 3.18 3.08 2.98 2.88 2.77 2.66 2.55 .005

1.89 1.84 1.80 1.74 1.72 1.69 1.66 1.62 1.59 1.55 .100 232.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76 .0502.67 2.57 2.47 2.36 2.30 2.24 2.18 2.11 2.04 1.97 .0253.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26 .0103.64 3.47 3.30 3.12 3.02 2.92 2.82 2.71 2.60 2.48 .005

1.88 1.83 1.78 1.73 1.70 1.67 1.64 1.61 1.57 1.53 .100 242.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73 .0502.64 2.54 2.44 2.33 2.27 2.21 2.15 2.08 2.01 1.94 .0253.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21 .0103.59 3.42 3.25 3.06 2.97 2.87 2.77 2.66 2.55 2.43 .005

1.87 1.82 1.77 1.72 1.69 1.66 1.63 1.59 1.56 1.52 .100 252.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71 .0502.61 2.51 2.41 2.30 2.24 2.18 2.12 2.05 1.98 1.91 .0253.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17 .0103.54 3.37 3.20 3.01 2.92 2.82 2.72 2.61 2.50 2.38 .005

1.86 1.81 1.76 1.71 1.68 1.65 1.61 1.58 1.54 1.50 .100 262.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69 .0502.59 2.49 2.39 2.28 2.22 2.16 2.09 2.03 1.95 1.88 .0253.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.13 .0103.49 3.33 3.15 2.97 2.87 2.77 2.67 2.56 2.45 2.33 .005

1.85 1.80 1.75 1.70 1.67 1.64 1.60 1.57 1.53 1.49 .100 272.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67 .0502.57 2.47 2.36 2.25 2.19 2.13 2.07 2.00 1.93 1.85 .0253.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10 .0103.45 3.28 3.11 2.93 2.83 2.73 2.63 2.52 2.41 2.29 .005

1.84 1.79 1.74 1.69 1.66 1.63 1.59 1.56 1.52 1.48 .100 282.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65 .0502.55 2.45 2.34 2.23 2.17 2.11 2.05 1.98 1.91 1.83 .0253.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.06 .0103.41 3.25 3.07 2.89 2.79 2.69 2.59 2.48 2.37 2.25 .005



Numerator dfDenominatordf α 1 2 3 4 5 6 7 8 9

29 .100 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86.050 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22.025 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59.010 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09.005 9.23 6.40 5.28 4.66 4.26 3.98 3.77 3.61 3.48

30 .100 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85.050 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21.025 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57.010 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07.005 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.45

40 .100 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79.050 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12.025 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45.010 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89.005 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 3.22

60 .100 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74.050 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04.025 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33.010 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72.005 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 3.01

120 .100 2.75 2.35 2.13 1.99 1.90 1.82 1.77 1.72 1.68.050 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96.025 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22.010 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56.005 8.18 5.54 4.50 3.92 3.55 3.28 3.09 2.93 2.81

∞ .100 2.71 2.30 2.08 1.94 1.85 1.77 1.72 1.67 1.63.050 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88.025 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11.010 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41.005 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.74 2.62

From “Tables of percentage points of the inverted beta (F) distribution.” Biometrika, Vol. 33 (1943) by M. Merrington and C. M.Thompson and from Table 18 of Biometrika Tables for Statisticians, Vol. 1, Cambridge University Press, 1954, edited by E. S.Pearson and H. O. Hartley.

Tables 861


Numerator df

10 12 15 20 24 30 40 60 120 ∞ α df

1.83 1.78 1.73 1.68 1.65 1.62 1.58 1.55 1.51 1.47 .100 292.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64 .0502.53 2.43 2.32 2.21 2.15 2.09 2.03 1.96 1.89 1.81 .0253.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03 .0103.38 3.21 3.04 2.86 2.76 2.66 2.56 2.45 2.33 2.21 .005

1.82 1.77 1.72 1.67 1.64 1.61 1.57 1.54 1.50 1.46 .100 302.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62 .0502.51 2.41 2.31 2.20 2.14 2.07 2.01 1.94 1.87 1.79 .0252.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01 .0103.34 3.18 3.01 2.82 2.73 2.63 2.52 2.42 2.30 2.18 .005

1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.47 1.42 1.38 .100 402.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51 .0502.39 2.29 2.18 2.07 2.01 1.94 1.88 1.80 1.72 1.64 .0252.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80 .0103.12 2.95 2.78 2.60 2.50 2.40 2.30 2.18 2.06 1.93 .005

1.71 1.66 1.60 1.54 1.51 1.48 1.44 1.40 1.35 1.29 .100 601.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39 .0502.27 2.17 2.06 1.94 1.88 1.82 1.74 1.67 1.58 1.48 .0252.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60 .0102.90 2.74 2.57 2.39 2.29 2.19 2.08 1.96 1.83 1.69 .005

1.65 1.60 1.55 1.48 1.45 1.41 1.37 1.32 1.26 1.19 .100 1201.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25 .0502.16 2.05 1.94 1.82 1.76 1.69 1.61 1.53 1.43 1.31 .0252.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38 .0102.71 2.54 2.37 2.19 2.09 1.98 1.87 1.75 1.61 1.43 .005

1.60 1.55 1.49 1.42 1.38 1.34 1.30 1.24 1.17 1.00 .100 ∞1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00 .0502.05 1.94 1.83 1.71 1.64 1.57 1.48 1.39 1.27 1.00 .0252.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00 .0102.52 2.36 2.19 2.00 1.90 1.79 1.67 1.53 1.36 1.00 .005


Table 8 Distribution Function of U

P(U ≤ U0); U0 is theargument; n1 ≤ n2;3 ≤ n2 ≤ 10.

n2 = 3

n1

U0 1 2 3

0 .25 .10 .051 .50 .20 .102 .40 .203 .60 .354 .50

n2 = 4

n1

U0 1 2 3 4

0 .2000 .0667 .0286 .01431 .4000 .1333 .0571 .02862 .6000 .2667 .1143 .05713 .4000 .2000 .10004 .6000 .3143 .17145 .4286 .24296 .5714 .34297 .44298 .5571

Tables 863


n2 = 5

n1

U0 1 2 3 4 5

0 .1667 .0476 .0179 .0079 .00401 .3333 .0952 .0357 .0159 .00792 .5000 .1905 .0714 .0317 .01593 .2857 .1250 .0556 .02784 .4286 .1964 .0952 .04765 .5714 .2857 .1429 .07546 .3929 .2063 .11117 .5000 .2778 .15488 .3651 .21039 .4524 .2738

10 .5476 .345211 .420612 .5000

n2 = 6

n1

U0 1 2 3 4 5 6

0 .1429 .0357 .0119 .0048 .0022 .00111 .2857 .0714 .0238 .0095 .0043 .00222 .4286 .1429 .0476 .0190 .0087 .00433 .5714 .2143 .0833 .0333 .0152 .00764 .3214 .1310 .0571 .0260 .01305 .4286 .1905 .0857 .0411 .02066 .5714 .2738 .1286 .0628 .03257 .3571 .1762 .0887 .04658 .4524 .2381 .1234 .06609 .5476 .3048 .1645 .0898

10 .3810 .2143 .120111 .4571 .2684 .154812 .5429 .3312 .197013 .3961 .242414 .4654 .294415 .5346 .349616 .409117 .468618 .5314



n2 = 7

n1

U0 1 2 3 4 5 6 7

0 .1250 .0278 .0083 .0030 .0013 .0006 .00031 .2500 .0556 .0167 .0061 .0025 .0012 .00062 .3750 .1111 .0333 .0121 .0051 .0023 .00123 .5000 .1667 .0583 .0212 .0088 .0041 .00204 .2500 .0917 .0364 .0152 .0070 .00355 .3333 .1333 .0545 .0240 .0111 .00556 .4444 .1917 .0818 .0366 .0175 .00877 .5556 .2583 .1152 .0530 .0256 .01318 .3333 .1576 .0745 .0367 .01899 .4167 .2061 .1010 .0507 .0265

10 .5000 .2636 .1338 .0688 .036411 .3242 .1717 .0903 .048712 .3939 .2159 .1171 .064113 .4636 .2652 .1474 .082514 .5364 .3194 .1830 .104315 .3775 .2226 .129716 .4381 .2669 .158817 .5000 .3141 .191418 .3654 .227919 .4178 .267520 .4726 .310021 .5274 .355222 .402423 .450824 .5000

Tables 865


n2 = 8

n1

U0 1 2 3 4 5 6 7 8

0 .1111 .0222 .0061 .0020 .0008 .0003 .0002 .00011 .2222 .0444 .0121 .0040 .0016 .0007 .0003 .00022 .3333 .0889 .0242 .0081 .0031 .0013 .0006 .00033 .4444 .1333 .0424 .0141 .0054 .0023 .0011 .00054 .5556 .2000 .0667 .0242 .0093 .0040 .0019 .00095 .2667 .0970 .0364 .0148 .0063 .0030 .00156 .3556 .1394 .0545 .0225 .0100 .0047 .00237 .4444 .1879 .0768 .0326 .0147 .0070 .00358 .5556 .2485 .1071 .0466 .0213 .0103 .00529 .3152 .1414 .0637 .0296 .0145 .0074

10 .3879 .1838 .0855 .0406 .0200 .010311 .4606 .2303 .1111 .0539 .0270 .014112 .5394 .2848 .1422 .0709 .0361 .019013 .3414 .1772 .0906 .0469 .024914 .4040 .2176 .1142 .0603 .032515 .4667 .2618 .1412 .0760 .041516 .5333 .3108 .1725 .0946 .052417 .3621 .2068 .1159 .065218 .4165 .2454 .1405 .080319 .4716 .2864 .1678 .097420 .5284 .3310 .1984 .117221 .3773 .2317 .139322 .4259 .2679 .164123 .4749 .3063 .191124 .5251 .3472 .220925 .3894 .252726 .4333 .286927 .4775 .322728 .5225 .360529 .399230 .439231 .479632 .5204



n2 = 9

n1

U0 1 2 3 4 5 6 7 8 9

0 .1000 .0182 .0045 .0014 .0005 .0002 .0001 .0000 .00001 .2000 .0364 .0091 .0028 .0010 .0004 .0002 .0001 .00002 .3000 .0727 .0182 .0056 .0020 .0008 .0003 .0002 .00013 .4000 .1091 .0318 .0098 .0035 .0014 .0006 .0003 .00014 .5000 .1636 .0500 .0168 .0060 .0024 .0010 .0005 .00025 .2182 .0727 .0252 .0095 .0038 .0017 .0008 .00046 .2909 .1045 .0378 .0145 .0060 .0026 .0012 .00067 .3636 .1409 .0531 .0210 .0088 .0039 .0019 .00098 .4545 .1864 .0741 .0300 .0128 .0058 .0028 .00149 .5455 .2409 .0993 .0415 .0180 .0082 .0039 .0020

10 .3000 .1301 .0559 .0248 .0115 .0056 .002811 .3636 .1650 .0734 .0332 .0156 .0076 .003912 .4318 .2070 .0949 .0440 .0209 .0103 .005313 .5000 .2517 .1199 .0567 .0274 .0137 .007114 .3021 .1489 .0723 .0356 .0180 .009415 .3552 .1818 .0905 .0454 .0232 .012216 .4126 .2188 .1119 .0571 .0296 .015717 .4699 .2592 .1361 .0708 .0372 .020018 .5301 .3032 .1638 .0869 .0464 .025219 .3497 .1942 .1052 .0570 .031320 .3986 .2280 .1261 .0694 .038521 .4491 .2643 .1496 .0836 .047022 .5000 .3035 .1755 .0998 .056723 .3445 .2039 .1179 .068024 .3878 .2349 .1383 .080725 .4320 .2680 .1606 .095126 .4773 .3032 .1852 .111227 .5227 .3403 .2117 .129028 .3788 .2404 .148729 .4185 .2707 .170130 .4591 .3029 .193331 .5000 .3365 .218132 .3715 .244733 .4074 .272934 .4442 .302435 .4813 .333236 .5187 .365237 .398138 .431739 .465740 .5000

Tables 867


n2 = 10

n1

U0 1 2 3 4 5 6 7 8 9 10

0 .0909 .0152 .0035 .0010 .0003 .0001 .0001 .0000 .0000 .00001 .1818 .0303 .0070 .0020 .0007 .0002 .0001 .0000 .0000 .00002 .2727 .0606 .0140 .0040 .0013 .0005 .0002 .0001 .0000 .00003 .3636 .0909 .0245 .0070 .0023 .0009 .0004 .0002 .0001 .00004 .4545 .1364 .0385 .0120 .0040 .0015 .0006 .0003 .0001 .00015 .5455 .1818 .0559 .0180 .0063 .0024 .0010 .0004 .0002 .00016 .2424 .0804 .0270 .0097 .0037 .0015 .0007 .0003 .00027 .3030 .1084 .0380 .0140 .0055 .0023 .0010 .0005 .00028 .3788 .1434 .0529 .0200 .0080 .0034 .0015 .0007 .00049 .4545 .1853 .0709 .0276 .0112 .0048 .0022 .0011 .0005

10 .5455 .2343 .0939 .0376 .0156 .0068 .0031 .0015 .000811 .2867 .1199 .0496 .0210 .0093 .0043 .0021 .001012 .3462 .1518 .0646 .0280 .0125 .0058 .0028 .001413 .4056 .1868 .0823 .0363 .0165 .0078 .0038 .001914 .4685 .2268 .1032 .0467 .0215 .0103 .0051 .002615 .5315 .2697 .1272 .0589 .0277 .0133 .0066 .003416 .3177 .1548 .0736 .0351 .0171 .0086 .004517 .3666 .1855 .0903 .0439 .0217 .0110 .005718 .4196 .2198 .1099 .0544 .0273 .0140 .007319 .4725 .2567 .1317 .0665 .0338 .0175 .009320 .5275 .2970 .1566 .0806 .0416 .0217 .011621 .3393 .1838 .0966 .0506 .0267 .014422 .3839 .2139 .1148 .0610 .0326 .017723 .4296 .2461 .1349 .0729 .0394 .021624 .4765 .2811 .1574 .0864 .0474 .026225 .5235 .3177 .1819 .1015 .0564 .031526 .3564 .2087 .1185 .0667 .037627 .3962 .2374 .1371 .0782 .044628 .4374 .2681 .1577 .0912 .052629 .4789 .3004 .1800 .1055 .061530 .5211 .3345 .2041 .1214 .071631 .3698 .2299 .1388 .082732 .4063 .2574 .1577 .095233 .4434 .2863 .1781 .108834 .4811 .3167 .2001 .123735 .5189 .3482 .2235 .139936 .3809 .2483 .157537 .4143 .2745 .176338 .4484 .3019 .196539 .4827 .3304 .2179



n2 = 10

n1

U0 1 2 3 4 5 6 7 8 9 10

40 .5173 .3598 .240641 .3901 .264442 .4211 .289443 .4524 .315344 .4841 .342145 .5159 .369746 .398047 .426748 .455949 .485350 .5147

Computed by M. Pagano, Department of Statistics, University of Florida.

Table 9 Critical Values of T in the Wilcoxon Matched-Pairs, Signed-Ranks Test; n = 5(1)50

One-sided Two-sided n = 5 n = 6 n = 7 n = 8 n = 9 n = 10

P = .05 P = .10 1 2 4 6 8 11P = .025 P = .05 1 2 4 6 8P = .01 P = .02 0 2 3 5P = .005 P = .01 0 2 3


P = .05 P = .10 14 17 21 26 30 36P = .025 P = .05 11 14 17 21 25 30P = .01 P = .02 7 10 13 16 20 24P = .005 P = .01 5 7 10 13 16 19


P = .05 P = .10 41 47 54 60 68 75P = .025 P = .05 35 40 46 52 59 66P = .01 P = .02 28 33 38 43 49 56P = .005 P = .01 23 28 32 37 43 49


P = .05 P = .10 83 92 101 110 120 130P = .025 P = .05 73 81 90 98 107 117P = .01 P = .02 62 69 77 85 93 102P = .005 P = .01 55 68 68 76 84 92

Tables 869



P = .05 P = .10 141 152 163 175 188 201P = .025 P = .05 127 137 148 159 171 183P = .01 P = .02 111 120 130 141 151 162P = .005 P = .01 100 109 118 128 138 149

One-sided Two-sided n = 35 n = 36 n = 37 n = 38 n = 39

P = .05 P = .10 214 228 242 256 271P = .025 P = .05 195 208 222 235 250P = .01 P = .02 174 186 198 211 224P = .005 P = .01 160 171 183 195 208


P = .05 P = .10 287 303 319 336 353 371P = .025 P = .05 264 279 295 311 327 344P = .01 P = .02 238 252 267 281 297 313P = .005 P = .01 221 234 248 262 277 292

One-sided Two-sided n = 46 n = 47 n = 48 n = 49 n = 50

P = .05 P = .10 389 408 427 446 466P = .025 P = .05 361 379 397 415 434P = .01 P = .02 329 345 362 380 398P = .005 P = .01 307 323 339 356 373

From “Some Rapid Approximate Statistical Procedures” (1964), 28, F. Wilcoxon and R. A. Wilcox.


Table 10 Distribution of the Total Number of Runs R in Samples of Size (n1, n2); P(R ≤ a)

a

(n1, n2) 2 3 4 5 6 7 8 9 10

(2, 3) .200 .500 .900 1.000(2, 4) .133 .400 .800 1.000(2, 5) .095 .333 .714 1.000(2, 6) .071 .286 .643 1.000(2, 7) .056 .250 .583 1.000(2, 8) .044 .222 .533 1.000(2, 9) .036 .200 .491 1.000(2, 10) .030 .182 .455 1.000

(3, 3) .100 .300 .700 .900 1.000(3, 4) .057 .200 .543 .800 .971 1.000(3, 5) .036 .143 .429 .714 .929 1.000(3, 6) .024 .107 .345 .643 .881 1.000(3, 7) .017 .083 .283 .583 .833 1.000(3, 8) .012 .067 .236 .533 .788 1.000(3, 9) .009 .055 .200 .491 .745 1.000(3, 10) .007 .045 .171 .455 .706 1.000

(4, 4) .029 .114 .371 .629 .886 .971 1.000(4, 5) .016 .071 .262 .500 .786 .929 .992 1.000(4, 6) .010 .048 .190 .405 .690 .881 .976 1.000(4, 7) .006 .033 .142 .333 .606 .833 .954 1.000(4, 8) .004 .024 .109 .279 .533 .788 .929 1.000(4, 9) .003 .018 .085 .236 .471 .745 .902 1.000(4, 10) .002 .014 .068 .203 .419 .706 .874 1.000

(5, 5) .008 .040 .167 .357 .643 .833 .960 .992 1.000(5, 6) .004 .024 .110 .262 .522 .738 .911 .976 .998(5, 7) .003 .015 .076 .197 .424 .652 .854 .955 .992(5, 8) .002 .010 .054 .152 .347 .576 .793 .929 .984(5, 9) .001 .007 .039 .119 .287 .510 .734 .902 .972(5, 10) .001 .005 .029 .095 .239 .455 .678 .874 .958

(6, 6) .002 .013 .067 .175 .392 .608 .825 .933 .987(6, 7) .001 .008 .043 .121 .296 .500 .733 .879 .966(6, 8) .001 .005 .028 .086 .226 .413 .646 .821 .937(6, 9) .000 .003 .019 .063 .175 .343 .566 .762 .902(6, 10) .000 .002 .013 .047 .137 .288 .497 .706 .864

(7, 7) .001 .004 .025 .078 .209 .383 .617 .791 .922(7, 8) .000 .002 .015 .051 .149 .296 .514 .704 .867(7, 9) .000 .001 .010 .035 .108 .231 .427 .622 .806(7, 10) .000 .001 .006 .024 .080 .182 .355 .549 .743

(8, 8) .000 .001 .009 .032 .100 .214 .405 .595 .786(8, 9) .000 .001 .005 .020 .069 .157 .319 .500 .702(8, 10) .000 .000 .003 .013 .048 .117 .251 .419 .621

(9, 9) .000 .000 .003 .012 .044 .109 .238 .399 .601(9, 10) .000 .000 .002 .008 .029 .077 .179 .319 .510(10, 10) .000 .000 .001 .004 .019 .051 .128 .242 .414

Tables 871


a

(n1, n2) 11 12 13 14 15 16 17 18 19 20

(2, 3)(2, 4)(2, 5)(2, 6)(2, 7)(2, 8)(2, 9)(2, 10)

(3, 3)(3, 4)(3, 5)(3, 6)(3, 7)(3, 8)(3, 9)(3, 10)

(4, 4)(4, 5)(4, 6)(4, 7)(4, 8)(4, 9)(4, 10)

(5, 5)(5, 6) 1.000(5, 7) 1.000(5, 8) 1.000(5, 9) 1.000(5, 10) 1.000

(6, 6) .998 1.000(6, 7) .992 .999 1.000(6, 8) .984 .998 1.000(6, 9) .972 .994 1.000(6, 10) .958 .990 1.000

(7, 7) .975 .996 .999 1.000(7, 8) .949 .988 .998 1.000 1.000(7, 9) .916 .975 .994 .999 1.000(7, 10) .879 .957 .990 .998 1.000

(8, 8) .900 .968 .991 .999 1.000 1.000(8, 9) .843 .939 .980 .996 .999 1.000 1.000(8, 10) .782 .903 .964 .990 .998 1.000 1.000

(9, 9) .762 .891 .956 .988 .997 1.000 1.000 1.000(9, 10) .681 .834 .923 .974 .992 .999 1.000 1.000 1.000(10, 10) .586 .758 .872 .949 .981 .996 .999 1.000 1.000 1.000

From “Tables for Testing Randomness of Grouping in a Sequence of Alternatives,” C. Eisenhart andF. Swed, Annals of Mathematical Statistics, Volume 14 (1943).


Table 11 Critical Values of Spearman’s Rank CorrelationCoefficient

n α = .05 α = .025 α = .01 α = .005

5 0.900 — — —6 0.829 0.886 0.943 —7 0.714 0.786 0.893 —8 0.643 0.738 0.833 0.8819 0.600 0.683 0.783 0.833

10 0.564 0.648 0.745 0.794

11 0.523 0.623 0.736 0.81812 0.497 0.591 0.703 0.78013 0.475 0.566 0.673 0.74514 0.457 0.545 0.646 0.71615 0.441 0.525 0.623 0.689

16 0.425 0.507 0.601 0.66617 0.412 0.490 0.582 0.64518 0.399 0.476 0.564 0.62519 0.388 0.462 0.549 0.60820 0.377 0.450 0.534 0.591

21 0.368 0.438 0.521 0.57622 0.359 0.428 0.508 0.56223 0.351 0.418 0.496 0.54924 0.343 0.409 0.485 0.53725 0.336 0.400 0.475 0.526

26 0.329 0.392 0.465 0.51527 0.323 0.385 0.456 0.50528 0.317 0.377 0.448 0.49629 0.311 0.370 0.440 0.48730 0.305 0.364 0.432 0.478

From “Distribution of Sums of Squares of Rank Differ-ences for Small Samples,” E. G. Olds, Annals of Mathe-matical Statistics, Volume 9 (1938).

Tables 873

Tabl

e12

Ran

dom

Num

bers

Lin

e/C

ol.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

110

480

1501

101

536

0201

181

647

9164

669

179

1419

462

590

3620

720

969

9957

091

291

9070

02

2236

846

573

2559

585

393

3099

589

198

2798

253

402

9396

534

095

5266

619

174

3961

599

505

324

130

4836

022

527

9726

576

393

6480

915

179

2483

049

340

3208

130

680

1965

563

348

5862

94

4216

793

003

0624

361

680

0785

616

376

3944

053

537

7134

157

004

0084

974

917

9775

816

379

537

570

3997

581

837

1665

606

121

9178

260

468

8130

549

684

6067

214

110

0692

701

263

5461

3

677

921

0690

711

008

4275

127

756

5349

818

602

7065

990

655

1505

321

916

8182

544

394

4288

07

9956

272

095

5642

069

994

9887

231

016

7119

418

738

4401

348

840

6321

321

069

1063

412

952

896

301

9197

705

463

0797

218

876

2092

294

595

5686

969

014

6004

518

425

8490

342

508

3230

79

8957

914

342

6366

110

281

1745

318

103

5774

084

378

2533

112

566

5867

844

947

0558

556

941

1085

475

3685

753

342

5398

853

060

5953

338

867

6230

008

158

1798

316

439

1145

818

593

6495

2

1128

918

6957

888

231

3327

670

997

7993

656

865

0585

990

106

3159

501

547

8559

091

610

7818

812

6355

340

961

4823

503

427

4962

669

445

1866

372

695

5218

020

847

1223

490

511

3370

390

322

1309

429

9396

952

636

9273

788

974

3348

836

320

1761

730

015

0827

284

115

2715

630

613

7495

214

1036

561

129

8752

985

689

4823

752

267

6768

993

394

0151

126

358

8510

420

285

2997

589

868

1507

119

9733

671

048

0817

877

233

1391

647

564

8105

697

735

8597

729

372

7446

128

551

9070

7

1651

085

1276

551

821

5125

977

452

1630

860

756

9214

449

442

5390

070

960

6399

075

601

4071

917

0236

821

382

5240

460

268

8936

819

885

5532

244

819

0118

865

255

6483

544

919

0594

455

157

1801

011

5409

233

362

9490

431

273

0414

618

594

2985

271

585

8503

051

132

0191

592

747

6495

119

5216

253

916

4636

958

586

2321

614

513

8314

998

736

2349

564

350

9473

817

752

3515

635

749

2007

056

9762

833

787

0999

842

698

0669

176

988

1360

251

851

4610

488

916

1950

925

625

5810

4

2148

663

9124

585

828

1434

609

172

3016

890

229

0473

459

193

2217

830

421

6166

699

904

3281

222

5416

458

492

2242

174

103

4707

025

306

7646

826

384

5815

106

646

2152

415

227

9690

944

592

2332

639

3236

305

597

2420

013

363

3800

594

342

2872

835

806

0691

217

012

6416

118

296

2285

124

2933

427

001

8763

787

308

5873

100

256

4583

415

398

4655

741

135

1036

707

684

3618

818

510

2502

488

3306

228

834

0735

119

731

9242

060

952

6128

050

001

6765

832

586

8667

950

720

9495

3


Tabl

e12

(Con

tinu

ed)

Lin

e/C

ol.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

2681

525

7229

504

839

9642

324

878

8265

166

566

1477

876

797

1478

013

300

8707

479

666

9572

527

2967

620

591

6808

626

432

4690

120

849

8976

881

536

8664

512

659

9225

957

102

8042

825

280

2800

742

5739

239

064

6643

284

673

4002

732

832

6136

298

947

9606

764

760

6458

496

096

9825

329

0536

604

213

2566

926

422

4440

744

048

3793

763

904

4576

666

134

7547

066

520

3469

390

449

3091

921

2641

864

117

9430

526

766

2594

039

972

2220

971

500

6456

891

402

4241

607

844

6961

831

0058

204

711

8791

777

341

4220

635

126

7408

799

547

8181

742

607

4380

876

655

6202

876

630

3200

725

6988

462

797

5617

086

324

8807

276

222

3608

684

637

9316

176

038

6585

577

919

8800

633

6901

165

795

9587

655

293

1898

827

354

2657

508

625

4080

159

920

2984

180

150

1277

748

501

3425

976

5794

829

888

8860

467

917

4870

818

912

8227

165

424

6977

433

611

5426

285

963

0354

735

0976

383

473

7357

712

908

3088

318

317

2829

035

797

0599

841

688

3495

237

888

3891

788

050

3691

567

4259

527

958

3013

404

024

8638

529

880

9973

055

536

8485

529

080

0925

079

656

7321

137

1795

556

349

9099

949

127

2004

459

931

0611

520

542

1805

902

008

7370

883

517

3610

342

791

3846

503

1858

418

845

4961

802

304

5103

820

655

5872

728

168

1547

556

942

5338

920

562

8733

839

9215

789

634

9482

478

171

8461

082

834

0992

225

417

4413

748

413

2555

521

246

3550

920

468

4014

577

6276

535

605

8126

339

667

4735

856

873

5630

761

607

4951

889

656

2010

377

490

1806

241

9842

707

523

3336

264

270

0163

892

477

6696

998

420

0488

045

585

4656

504

102

4688

045

709

4234

914

6397

688

720

8276

534

476

1703

287

589

4083

632

427

7000

270

663

8886

377

775

6934

843

7006

028

277

3947

546

473

2321

953

416

9497

025

832

6997

594

884

1966

172

828

0010

266

794

4453

976

5491

406

990

6724

568

350

8294

811

398

4287

880

287

8826

747

363

4663

406

541

9780

945

7607

229

515

4098

007

391

5874

525

774

2298

780

059

3991

196

189

4115

114

222

6069

759

583

4690

725

5221

083

974

2999

265

831

3885

750

490

8376

555

657

1436

131

720

5737

556

228

4154

647

6436

467

412

3333

931

926

1488

324

413

5974

492

351

9747

389

286

3593

104

110

2372

651

900

4808

962

0035

831

662

2538

861

642

3407

281

249

3564

856

891

6935

248

373

4557

878

547

8178

849

9501

268

379

9352

670

765

1059

204

542

7646

354

328

0234

917

247

2886

514

777

6273

092

277

5015

664

1049

320

492

3839

191

132

2199

959

516

8165

227

195

4822

346

751

2292

332

261

8565

3

Tables 875

Tabl

e12

(Con

tinu

ed)

Lin

e/C

ol.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

5116

408

8189

904

153

5338

179

401

2143

883

035

9235

036

693

3123

859

649

9175

472

772

0233

852

1862

981

953

0552

091

962

0473

913

092

9766

224

822

9473

006

496

3509

004

822

8677

498

289

5373

115

3510

147

498

8763

799

016

7106

088

824

7101

318

735

2028

623

153

7292

435

165

4304

054

5749

116

703

2316

749

323

4502

133

132

1254

441

035

8078

045

393

4481

212

515

9893

191

202

5530

405

8394

623

792

1442

215

059

4579

922

716

1979

209

983

7435

368

668

3042

970

735

2549

9

5616

631

3500

685

900

9827

532

388

5239

016

815

6929

882

732

3848

073

817

3252

341

961

4443

757

9677

320

206

4255

978

985

0530

022

164

2436

954

224

3508

319

687

1105

291

491

6038

319

746

5838

935

6420

214

349

8267

466

523

4413

300

697

3555

235

970

1912

463

318

2968

603

387

5984

659

3162

476

384

1740

353

363

4416

764

486

6475

875

366

7655

431

601

1261

433

072

6033

292

325

6078

919

1947

423

632

2788

947

914

0258

437

680

2080

172

152

3933

934

806

0893

085

001

8782

0

6103

931

3330

957

047

7421

163

445

1736

162

825

3990

805

607

9128

468

833

2557

038

818

4692

062

7442

633

278

4397

210

119

8991

715

665

5287

273

823

7314

488

662

8897

074

492

5180

599

378

6309

066

0090

320

795

9545

292

648

4545

409

552

8881

516

553

5112

579

375

9759

616

296

6609

264

4223

812

426

8702

514

267

2097

904

508

6453

531

355

8606

429

472

4768

905

974

5246

816

834

6516

153

0800

226

504

4174

481

959

6564

274

240

5630

200

033

6710

777

510

7062

528

725

3419

1

6621

457

4074

229

820

9678

329

400

2184

015

035

3453

733

310

0611

695

240

1595

716

572

0600

467

2158

157

802

0205

089

728

1793

737

621

4707

542

080

9740

348

626

6899

543

805

3338

621

597

6855

612

7809

583

197

3373

205

810

2481

386

902

6039

716

489

0326

488

525

4278

605

269

9253

269

4465

766

999

9932

451

281

8446

360

563

7931

293

454

6887

625

471

9391

125

650

1268

273

572

7091

340

8497

946

949

8197

337

949

6102

343

997

1526

380

644

4394

289

203

7179

599

533

5050

1

7191

227

2119

931

935

2702

284

067

0546

235

216

1448

629

891

6860

741

867

1495

191

696

8506

572

5000

138

140

6632

119

924

7216

309

538

1215

106

878

9190

318

749

3440

556

087

8279

070

925

7365

390

0522

472

958

2860

981

406

3914

725

549

4854

242

627

4523

357

202

9461

723

772

0789

674

2750

496

131

8394

441

575

1057

308

619

6448

273

923

3615

205

184

9414

225

299

8438

734

925

7537

169

9485

139

117

8963

200

959

1648

765

536

4907

139

782

1709

502

330

7430

100

275

4828

0


Tabl

e12

(Con

tinu

ed)

Lin

e/C

ol.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

7611

508

7022

551

111

3835

119

444

6649

971

945

0542

213

442

7867

584

081

6693

893

654

5989

477

3744

930

362

0669

454

690

0405

253

115

6275

795

348

7866

211

163

8165

150

245

3497

152

924

7846

515

7033

185

922

3832

957

015

1576

597

161

1786

945

349

6179

666

345

8107

349

106

7986

079

3098

681

223

4241

658

353

2153

230

502

3230

586

482

0517

407

901

5433

958

861

7481

846

942

8063

798

6499

546

583

0978

544

160

7812

883

991

4288

592

520

8353

180

377

3590

981

250

5423

8

8182

486

8484

699

254

6763

243

218

5007

621

361

6481

651

202

8812

441

870

5268

951

275

8355

682

2188

532

906

9243

109

060

6429

751

674

6412

662

570

2612

305

155

5919

452

799

2822

585

762

8360

336

9878

207

408

5345

813

564

5908

926

445

2978

985

205

4100

112

535

1213

314

645

2354

184

4393

746

891

2401

025

560

8635

533

941

2578

654

990

7189

915

475

9543

498

227

2182

419

585

8597

656

6317

589

303

1627

507

100

9206

321

942

1861

147

348

2020

318

534

0386

278

095

5013

6

8603

299

0122

105

418

3898

255

758

9223

726

759

8636

721

216

9844

208

303

5661

391

511

7592

887

7962

606

486

0357

417

668

0778

576

020

7992

425

651

8332

588

428

8507

672

811

2271

750

585

8885

636

6833

547

539

0312

965

651

1197

702

510

2611

399

447

6864

534

327

1515

255

230

9344

889

1803

914

367

6133

706

177

1214

346

609

3298

974

014

6470

800

533

3539

858

408

1326

147

908

9008

362

1565

660

627

3647

865

648

1676

453

412

0901

307

832

4157

417

639

8216

360

859

7556

7

9179

556

2906

804

142

1626

815

387

1285

666

227

3835

822

478

7337

388

732

0944

382

558

0525

092

9260

882

674

2707

232

534

1707

527

698

9820

463

863

1195

134

648

8802

256

148

3492

557

031

9323

982

2583

540

055

6700

612

293

0275

314

827

2323

535

071

9970

437

543

1160

135

503

8517

194

0991

596

306

0590

897

901

2839

514

186

0082

180

703

7042

675

647

7631

088

717

3789

040

129

9559

037

3330

026

695

6224

769

927

7612

350

842

4383

486

654

7095

979

725

9387

228

117

1923

3

9642

488

7807

769

882

6165

734

136

7918

097

526

4309

204

098

7357

180

799

7653

671

255

6423

997

4676

486

273

6300

393

017

3120

436

692

4020

235

275

5730

655

543

5320

318

098

4762

588

684

9803

237

4543

055

417

6328

290

816

1734

988

298

9018

336

600

7840

606

216

9578

742

579

9073

099

8659

181

482

5266

761

582

1497

290

053

8953

476

036

4919

943

716

9754

804

379

4637

028

672

100

3853

401

715

9496

487

288

6568

043

772

3956

012

918

8653

762

738

1963

651

132

2573

956

947

Abr

idge

dfr

omH

andb

ook

ofTa

bles

for

Pro

babi

lity

and

Stat

isti

cs,2

nded

ition

,edi

ted

byW

illia

mH

.B

eyer

(Cle

vela

nd:T

heC

hem

ical

Rub

ber

Com

pany

,196

8).

appendix 3 tables

Documents

continued c n

decimal place