Appendix: Statistical Tables This appendix contains statistical tables of the common sampling distributions used in statistical inference. See Chapter 4 for examples of their use. CUMULATIVE DISTRIBUTION FUNCTION OF THE STANDARD NORMAL DISTRIBUTION Table A.1 shows the probability, F(z) that a standard normal random variable is less than the value z. For example, the probability is 0.9986 that a standard normal random variable is less than 3. CHI-SQUARE DISTRIBUTION For a given probabilities α, Table A.2 shows the values of the chi-square distribution. For example, the probability is 0.05 that a chi-square random variable with 10 degrees of freedom is greater than 18.31. STUDENT T-DISTRIBUTION For a given probability α, Table A.3 shows the values of the student t-distribution. For example, the probability is 0.05 that a student t random variable with 10 degrees of freedom is greater than 1.812. 227 228 Table A.1 Cumulative distribution function of the standard normal distribution zF(z) zF(z) ZF(z) zF(z) 0 0.5 0.32 0.625516 0.64 0.738914 0.96 0.831472 0.01 0.503989 0.33 0.6293 0.65 0.742154 0.97 0.833977 0.02 0.507978 0.34 0.633072 0.66 0.745373 0.98 0.836457 0.03 0.511967 0.35 0.636831 0.67 0.748571 0.99 0.838913 0.04 0.515953 0.36 0.640576 0.68 0.751748 1 0.841345 0.05 0.519939 0.37 0.644309 0.69 0.754903 1.01 0.843752 0.06 0.523922 0.38 0.648027 0.7 0.758036 1.02 0.846136 0.07 0.527903 0.39 0.651732 0.71 0.761148 1.03 0.848495 0.08 0.531881 0.4 0.655422 0.72 0.764238 1.04 0.85083 0.09 0.535856 0.41 0.659097 0.73 0.767305 1.05 0.85314 0.1 0.539828 0.42 0.662757 0.74 0.77035 1.06 0.85542 0.11 0.543795 0.43 0.666402 0.75 0.773373 1.07 0.85769 0.12 0.547758 0.44 0.670031 0.76 0.776373 1.08 0.85992 0.13 0.551717 0.45 0.673645 0.77 0.77935 1.09 0.86214 0.14 0.55567 0.46 0.677242 0.78 0.782305 1.1 0.86433 0.15 0.559618 0.47 0.680822 0.79 0.785236 1.11 0.8665 0.16 0.563559 0.48 0.684386 0.8 0.788145 1.12 0.86864 0.17 0.567495 0.49 0.687933 0.81 0.79103 1.13 0.87076 0.18 0.571424 0.5 0.691462 0.82 0.793892 1.14 0.87285 0.19 0.575345 0.51 0.694974 0.83 0.796731 1.15 0.87492 0.2 0.57926 0.52 0.698468 0.84 0.799546 1.16 0.87697 0.21 0.583166 0.53 0.701944 0.85 0.802338 1.17 0.87899 0.22 0.587064 0.54 0.705402 0.86 0.805106 1.18 0.881 0.23 0.590954 0.55 0.70884 0.87 0.80785 1.19 0.88297 0.24 0.594835 0.56 0.71226 0.88 0.81057 1.2 0.88493 0.25 0.598706 0.57 0.715661 0.89 0.813267 1.21 0.88686 0.26 0.602568 0.58 0.719043 0.9 0.81594 1.22 0.88876 0.27 0.60642 0.59 0.722405 0.91 0.818589 1.23 0.89065 0.28 0.610261 0.6 0.725747 0.92 0.821214 1.24 0.89251 0.29 0.614092 0.61 0.729069 0.93 0.823814 1.25 0.89435 0.3 0.617911 0.62 0.732371 0.94 0.826391 1.26 0.89616 0.31 0.621719 0.63 0.735653 0.95 0.828944 1.27 0.89795 Continued 229 Table A.1 Continued zF(z) zF(z) ZF(z) zF(z) 1.28 0.89972 1.63 0.9484493 1.98 0.976148 2.33 0.990097 1.29 0.90147 1.64 0.9494972 1.99 0.976705 2.34 0.990358 1.3 0.90319 1.65 0.9505297 2 0.97725 2.35 0.990613 1.31 0.90490 1.66 0.9515438 2.01 0.977784 2.36 0.990863 1.32 0.90658 1.67 0.952546 2.02 0.978308 2.37 0.991106 1.33 0.90824 1.68 0.9535219 2.03 0.978822 2.38 0.991344 1.34 0.90987 1.69 0.954486 2.04 0.979325 2.39 0.991576 1.35 0.911492 1.7 0.9554357 2.05 0.979818 2.4 0.991802 1.36 0.913085 1.71 0.956367 2.06 0.980301 2.41 0.992024 1.37 0.914656 1.72 0.957284 2.07 0.980774 2.42 0.99224 1.38 0.916207 1.73 0.9581857 2.08 0.981237 2.43 0.992451 1.39 0.917736 1.74 0.9590711 2.09 0.981691 2.44 0.992656 1.4 0.919243 1.75 0.9599412 2.1 0.982136 2.45 0.992857 1.41 0.92073 1.76 0.960796 2.11 0.982571 2.46 0.993053 1.42 0.922196 1.77 0.9616365 2.12 0.982997 2.47 0.993244 1.43 0.923641 1.78 0.9624628 2.13 0.983414 2.48 0.993431 1.44 0.925066 1.79 0.9632737 2.14 0.983823 2.49 0.993613 1.45 0.926471 1.8 0.96407s5 2.15 0.984222 2.5 0.99379 1.46 0.927855 1.81 0.9648529 2.16 0.984614 2.51 0.993963 1.47 0.929219 1.82 0.9656212 2.17 0.984997 2.52 0.994132 1.48 0.930563 1.83 0.9663752 2.18 0.985371 2.53 0.994297 1.49 0.931888 1.84 0.9671161 2.19 0.985738 2.54 0.994457 1.5 0.933193 1.85 0.9678437 2.2 0.986097 2.55 0.994614 1.51 0.934478 1.86 0.968557 2.21 0.986447 2.56 0.994766 1.52 0.935744 1.87 0.969258 2.22 0.986791 2.57 0.994915 1.53 0.936992 1.88 0.969946 2.23 0.987126 2.58 0.99506 1.54 0.93822 1.89 0.970621 2.24 0.987455 2.59 0.995201 1.55 0.939429 1.9 0.971284 2.25 0.987776 2.6 0.995339 1.56 0.940621 1.91 0.971933 2.26 0.988089 2.61 0.995473 1.57 0.9417928 1.92 0.972571 2.27 0.988396 2.62 0.995603 1.58 0.942947 1.93 0.973197 2.28 0.988696 2.63 0.995731 1.59 0.9440839 1.94 0.97381 2.29 0.988989 2.64 0.995855 1.6 0.9452013 1.95 0.974412 2.3 0.989276 2.65 0.995975 1.61 0.9463014 1.96 0.975002 2.31 0.989556 2.66 0.996093 1.62 0.947384 1.97 0.975581 2.32 0.98983 2.67 0.996207 Continued 230 Table A.1 Continued zF(z) zF(z) ZF(z) zF(z) 2.68 0.996319 3.03 0.998777 3.38 0.999638 3.73 0.999904 2.69 0.996427 3.04 0.998817 3.39 0.99965 3.74 0.999908 2.7 0.996533 3.05 0.998856 3.4 0.999663 3.75 0.999912 2.71 0.996636 3.06 0.998893 3.41 0.999675 3.76 0.999915 2.72 0.996736 3.07 0.99893 3.42 0.999687 3.77 0.999918 2.73 0.996833 3.08 0.998965 3.43 0.999698 3.78 0.999922 2.74 0.996928 3.09 0.998999 3.44 0.999709 3.79 0.999925 2.75 0.99702 3.1 0.999032 3.45 0.99972 3.8 0.999928 2.76 0.99711 3.11 0.999064 3.46 0.99973 3.81 0.99993 2.77 0.997197 3.12 0.999096 3.47 0.99974 3.82 0.999933 2.78 0.997282 3.13 0.999126 3.48 0.999749 3.83 0.999936 2.79 0.997365 3.14 0.999155 3.49 0.999758 3.84 0.999938 2.8 0.997445 3.15 0.999184 3.5 0.999767 3.85 0.999941 2.81 0.997523 3.16 0.999211 3.51 0.999776 3.86 0.999943 2.82 0.997599 3.17 0.999238 3.52 0.999784 3.87 0.999946 2.83 0.997673 3.18 0.999264 3.53 0.999792 3.88 0.999948 2.84 0.997744 3.19 0.999289 3.54 0.9998 3.89 0.99995 2.85 0.997814 3.2 0.999313 3.55 0.999807 3.9 0.999952 2.86 0.997882 3.21 0.999336 3.56 0.999815 3.91 0.999954 2.87 0.997948 3.22 0.999359 3.57 0.999821 3.92 0.999956 2.88 0.998012 3.23 0.999381 3.58 0.999828 3.93 0.999958 2.89 0.998074 3.24 0.999402 3.59 0.999835 3.94 0.999959 2.9 0.998134 3.25 0.999423 3.6 0.999841 3.95 0.999961 2.91 0.998193 3.26 0.999443 3.61 0.999847 3.96 0.999963 2.92 0.99825 3.27 0.999462 3.62 0.999853 3.97 0.999964 2.93 0.998305 3.28 0.999481 3.63 0.999858 3.98 0.999966 2.94 0.998359 3.29 0.999499 3.64 0.999864 3.99 0.999967 2.95 0.998411 3.3 0.999517 3.65 0.999869 4 0.999968 2.96 0.998462 3.31 0.999533 3.66 0.999874 2.97 0.998511 3.32 0.99955 3.67 0.999879 2.98 0.998559 3.33 0.999566 3.68 0.999883 2.99 0.998605 3.34 0.999581 3.69 0.999888 3 0.99865 3.35 0.999596 3.7 0.999892 3.01 0.998694 3.36 0.99961 3.71 0.999896 3.02 0.998736 3.37 0.999624 3.72 0.9999 231 Table A.2 Cut off points for the chi-square distribution Degrees of α freedom 0.005 0.01 0.025 0.05 0.1 1 7.88 6.63 5.02 3.84 2.71 2 10.60 9.21 7.38 5.99 4.61 3 12.84 11.34 9.35 7.81 6.25 4 14.86 13.28 11.14 9.49 7.78 5 16.75 15.09 12.83 11.07 9.24 6 18.55 16.81 14.45 12.59 10.64 7 20.28 18.48 16.01 14.07 12.02 8 21.95 20.09 17.53 15.51 13.36 9 23.59 21.67 19.02 16.92 14.68 10 25.19 23.21 20.48 18.31 15.99 11 26.76 24.73 21.92 19.68 17.28 12 28.30 26.22 23.34 21.03 18.55 13 29.82 27.69 24.74 22.36 19.81 14 31.32 29.14 26.12 23.68 21.06 15 32.80 30.58 27.49 25.00 22.31 16 34.27 32.00 28.85 26.30 23.54 17 35.72 33.41 30.19 27.59 24.77 18 37.16 34.81 31.53 28.87 25.99 19 38.58 36.19 32.85 30.14 27.20 20 40.00 37.57 34.17 31.41 28.41 21 41.40 38.93 35.48 32.67 29.62 22 42.80 40.29 36.78 33.92 30.81 23 44.18 41.64 38.08 35.17 32.01 24 45.56 42.98 39.36 36.42 33.20 25 46.93 44.31 40.65 37.65 34.38 26 48.29 45.64 41.92 38.89 35.56 27 49.65 46.96 43.19 40.11 36.74 28 50.99 48.28 44.46 41.34 37.92 29 52.34 49.59 45.72 42.56 39.09 30 53.67 50.89 46.98 43.77 40.26 31 55.00 52.19 48.23 44.99 41.42 32 56.33 53.49 49.48 46.19 42.58 33 57.65 54.78 50.73 47.40 43.75 34 58.96 56.06 51.97 48.60 44.90 Continued 232 Table A.2 Continued Degrees of α freedom 0.005 0.01 0.025 0.05 0.1 35 60.27 57.34 53.20 49.80 46.06 36 61.58 58.62 54.44 51.00 47.21 37 62.88 59.89 55.67 52.19 48.36 38 64.18 61.16 56.90 53.38 49.51 39 65.48 62.43 58.12 54.57 50.66 40 66.77 63.69 59.34 55.76 51.81 41 68.05 64.95 60.56 56.94 52.95 42 69.34 66.21 61.78 58.12 54.09 43 70.62 67.46 62.99 59.30 55.23 44 71.89 68.71 64.20 60.48 56.37 45 73.17 69.96 65.41 61.66 57.51 46 74.44 71.20 66.62 62.83 58.64 47 75.70 72.44 67.82 64.00 59.77 48 76.97 73.68 69.02 65.17 60.91 49 78.23 74.92 70.22 66.34 62.04 50 79.49 76.15 71.42 67.50 63.17 51 80.75 77.39 72.62 68.67 64.30 52 82.00 78.62 73.81 69.83 65.42 53 83.25 79.84 75.00 70.99 66.55 54 84.50 81.07 76.19 72.15 67.67 55 85.75 82.29 77.38 73.31 68.80 56 86.99 83.51 78.57 74.47 69.92 57 88.24 84.73 79.75 75.62 71.04 58 89.48 85.95 80.94 76.78 72.16 59 90.72 87.17 82.12 77.93 73.28 60 91.95 88.38 83.30 79.08 74.40 61 93.19 89.59 84.48 80.23 75.51 62 94.42 90.80 85.65 81.38 76.63 63 95.65 92.01 86.83 82.53 77.75 64 96.88 93.22 88.00 83.68 78.86 65 98.10 94.42 89.18 84.82 79.97 66 99.33 95.63 90.35 85.96 81.09 67 100.55 96.83 91.52 87.11 82.20 68 101.78 98.03 92.69 88.25 83.31 Continued 233 Table A.2 Continued Degrees of α freedom 0.005 0.01 0.025 0.05 0.1 69 103.00 99.23 93.86 89.39 84.42 70 104.21 100.43 95.02 90.53 85.53 71 105.43 101.62 96.19 91.67 86.64 72 106.65 102.82 97.35 92.81 87.74 73 107.86 104.01 98.52 93.95 88.85 74 109.07 105.20 99.68 95.08 89.96 75 110.29 106.39 100.84 96.22 91.06 76 111.50 107.58 102.00 97.35 92.17 77 112.70 108.77 103.16 98.48 93.27 78 113.91 109.96 104.32 99.62 94.37 79 115.12 111.14 105.47 100.75 95.48 80 116.32 112.33 106.63 101.88 96.58 81 117.52 113.51 107.78 103.01 97.68 82