# University of California Davis Construct a Confidence Interval Statistics Questions

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Sacramento State University ENGR 115 Test No. 3 (show your work) For this exam, no electronic devices other than an FE-approved calculator are allowed unless specifically noted. Except for communicating your answers to the instructor, devices such as laptops, cell phones, iPads, and the like may not be used. You may not use computer software as a substitute calculator. No part of the test may be distributed at any time. Most importantly, no collaboration of any kind is permitted. This includes classmates, students not taking the class, or online answer services. Honor Code Statement: I certify that I have adhered to the statement above, that I have neither given nor received assistance to complete the exam, and that the exam represents my work alone. I understand the consequences of receiving assistance will result in a significant grade reduction to the point of likely failing the exam, notifying the Department Chair, and reporting the incident to the Office of Student Conduct. SIGNATURE: __________________________ NAME (PRINT):________________________ STUDENT ID #:________________________ ENGR 115 2 TEST 3 1- Assume that the mean hourly cost to operate a commercial airplane follows a normal distribution with a mean of \$2100 per hour and a standard deviation of \$250. Determine the lowest 6% of the operating cost. (20 pts) 2- Rainfall duration at a location along the Gulf Coast follows an exponential distribution with a mean value of 2.725 hours. What is the probability that a duration of a particular rainfall event is between 135 to 190 minutes? (20 pts) 3- What is the confidence level for the interval 𝑥𝑥̅ ± 1.47 𝜎𝜎 √𝑛𝑛 ? (10 pts) ENGR 115 3 TEST 3 4- The article “Ultimate Local Capacities of Expansion Anchor Bolts” (J. of Energy Engr., 1993: 139158) gave the following summary data on shear strength (kip) for a sample of 3/8-in anchor bolts: n= 23, mean is 4.50, and a sample standard deviation of 1.5. Calculate a 95% confidence interval for true average shear strength. Interpret your finding. (15 pts) 5-Seventy percent of 50 randomly selected students from college of science and engineering were happy with their college experience. Construct a 92% confidence interval for the true proportion of “happy” students and interpret your finding. (15 pts) 6- A state legislator wishes to survey residents of her district to see what proportion of the registered voters are aware of her position on using state funds to pay for a new Entertainment Center. What sample size is necessary if the 90% CI for population proportion to be within 7% of the true proportion? (10 pts) 7-Assume that helium porosity (in percentage) of coal samples taken from any particular seams is normally distributed with a true standard deviation of 0.75. How large a sample size is necessary if the width of the 95% interval is to be 0.5? (10 pts) Table: Cumulative Binomial probabilities c n  P[X ≤ c ] = ∑   p x (1 − p )n− x x =0  x  p n=1 n=2 n=3 n=4 n=5 n=6 n=7 c 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0 0.950 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.050 1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.903 0.810 0.640 0.490 0.360 0.250 0.160 0.090 0.040 0.010 0.003 1 0.998 0.990 0.960 0.910 0.840 0.750 0.640 0.510 0.360 0.190 0.098 2 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.857 0.729 0.512 0.343 0.216 0.125 0.064 0.027 0.008 0.001 0.000 1 0.993 0.972 0.896 0.784 0.648 0.500 0.352 0.216 0.104 0.028 0.007 2 1.000 0.999 0.992 0.973 0.936 0.875 0.784 0.657 0.488 0.271 0.143 3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.815 0.656 0.410 0.240 0.130 0.063 0.026 0.008 0.002 0.000 0.000 1 0.986 0.948 0.819 0.652 0.475 0.313 0.179 0.084 0.027 0.004 0.000 2 1.000 0.996 0.973 0.916 0.821 0.688 0.525 0.348 0.181 0.052 0.014 3 1.000 1.000 0.998 0.992 0.974 0.938 0.870 0.760 0.590 0.344 0.185 4 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.774 0.590 0.328 0.168 0.078 0.031 0.010 0.002 0.000 0.000 0.000 1 0.977 0.919 0.737 0.528 0.337 0.188 0.087 0.031 0.007 0.000 0.000 2 0.999 0.991 0.942 0.837 0.683 0.500 0.317 0.163 0.058 0.009 0.001 3 1.000 1.000 0.993 0.969 0.913 0.813 0.663 0.472 0.263 0.081 0.023 4 1.000 1.000 1.000 0.998 0.990 0.969 0.922 0.832 0.672 0.410 0.226 5 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.735 0.531 0.262 0.118 0.047 0.016 0.004 0.001 0.000 0.000 0.000 1 0.967 0.886 0.655 0.420 0.233 0.109 0.041 0.011 0.002 0.000 0.000 2 0.998 0.984 0.901 0.744 0.544 0.344 0.179 0.070 0.017 0.001 0.000 3 1.000 0.999 0.983 0.930 0.821 0.656 0.456 0.256 0.099 0.016 0.002 4 1.000 1.000 0.998 0.989 0.959 0.891 0.767 0.580 0.345 0.114 0.033 5 1.000 1.000 1.000 0.999 0.996 0.984 0.953 0.882 0.738 0.469 0.265 6 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.698 0.478 0.210 0.082 0.028 0.008 0.002 0.000 0.000 0.000 0.000 1 0.956 0.850 0.577 0.329 0.159 0.063 0.019 0.004 0.000 0.000 0.000 2 0.996 0.974 0.852 0.647 0.420 0.227 0.096 0.029 0.005 0.000 0.000 3 1.000 0.997 0.967 0.874 0.710 0.500 0.290 0.126 0.033 0.003 0.000 4 1.000 1.000 0.995 0.971 0.904 0.773 0.580 0.353 0.148 0.026 0.004 5 1.000 1.000 1.000 0.996 0.981 0.938 0.841 0.671 0.423 0.150 0.044 6 1.000 1.000 1.000 1.000 0.998 0.992 0.972 0.918 0.790 0.522 0.302 7 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1 Table: Cumulative Binomial probabilities (continued) p n=8 n=9 n = 10 n = 11 c 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0 0.663 0.430 0.168 0.058 0.017 0.004 0.001 0.000 0.000 0.000 0.000 1 0.943 0.813 0.503 0.255 0.106 0.035 0.009 0.001 0.000 0.000 0.000 2 0.994 0.962 0.797 0.552 0.315 0.145 0.050 0.011 0.001 0.000 0.000 3 1.000 0.995 0.944 0.806 0.594 0.363 0.174 0.058 0.010 0.000 0.000 4 1.000 1.000 0.990 0.942 0.826 0.637 0.406 0.194 0.056 0.005 0.000 5 1.000 1.000 0.999 0.989 0.950 0.855 0.685 0.448 0.203 0.038 0.006 6 1.000 1.000 1.000 0.999 0.991 0.965 0.894 0.745 0.497 0.187 0.057 7 1.000 1.000 1.000 1.000 0.999 0.996 0.983 0.942 0.832 0.570 0.337 8 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.630 0.387 0.134 0.040 0.010 0.002 0.000 0.000 0.000 0.000 0.000 1 0.929 0.775 0.436 0.196 0.071 0.020 0.004 0.000 0.000 0.000 0.000 2 0.992 0.947 0.738 0.463 0.232 0.090 0.025 0.004 0.000 0.000 0.000 3 0.999 0.992 0.914 0.730 0.483 0.254 0.099 0.025 0.003 0.000 0.000 4 1.000 0.999 0.980 0.901 0.733 0.500 0.267 0.099 0.020 0.001 0.000 5 1.000 1.000 0.997 0.975 0.901 0.746 0.517 0.270 0.086 0.008 0.001 6 1.000 1.000 1.000 0.996 0.975 0.910 0.768 0.537 0.262 0.053 0.008 7 1.000 1.000 1.000 1.000 0.996 0.980 0.929 0.804 0.564 0.225 0.071 8 1.000 1.000 1.000 1.000 1.000 0.998 0.990 0.960 0.866 0.613 0.370 9 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.599 0.349 0.107 0.028 0.006 0.001 0.000 0.000 0.000 0.000 0.000 1 0.914 0.736 0.376 0.149 0.046 0.011 0.002 0.000 0.000 0.000 0.000 2 0.988 0.930 0.678 0.383 0.167 0.055 0.012 0.002 0.000 0.000 0.000 3 0.999 0.987 0.879 0.650 0.382 0.172 0.055 0.011 0.001 0.000 0.000 4 1.000 0.998 0.967 0.850 0.633 0.377 0.166 0.047 0.006 0.000 0.000 5 1.000 1.000 0.994 0.953 0.834 0.623 0.367 0.150 0.033 0.002 0.000 6 1.000 1.000 0.999 0.989 0.945 0.828 0.618 0.350 0.121 0.013 0.001 7 1.000 1.000 1.000 0.998 0.988 0.945 0.833 0.617 0.322 0.070 0.012 8 1.000 1.000 1.000 1.000 0.998 0.989 0.954 0.851 0.624 0.264 0.086 9 1.000 1.000 1.000 1.000 1.000 0.999 0.994 0.972 0.893 0.651 0.401 10 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.569 0.314 0.086 0.020 0.004 0.000 0.000 0.000 0.000 0.000 0.000 1 0.898 0.697 0.322 0.113 0.030 0.006 0.001 0.000 0.000 0.000 0.000 2 0.985 0.910 0.617 0.313 0.119 0.033 0.006 0.001 0.000 0.000 0.000 3 0.998 0.981 0.839 0.570 0.296 0.113 0.029 0.004 0.000 0.000 0.000 4 1.000 0.997 0.950 0.790 0.533 0.274 0.099 0.022 0.002 0.000 0.000 5 1.000 1.000 0.988 0.922 0.753 0.500 0.247 0.078 0.012 0.000 0.000 6 1.000 1.000 0.998 0.978 0.901 0.726 0.467 0.210 0.050 0.003 0.000 7 1.000 1.000 1.000 0.996 0.971 0.887 0.704 0.430 0.161 0.019 0.002 8 1.000 1.000 1.000 0.999 0.994 0.967 0.881 0.687 0.383 0.090 0.015 9 1.000 1.000 1.000 1.000 0.999 0.994 0.970 0.887 0.678 0.303 0.102 10 1.000 1.000 1.000 1.000 1.000 1.000 0.996 0.980 0.914 0.686 0.431 11 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2 Table: Cumulative Binomial probabilities (continued) p n = 12 n = 13 n = 14 c 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0 0.540 0.282 0.069 0.014 0.002 0.000 0.000 0.000 0.000 0.000 0.000 1 0.882 0.659 0.275 0.085 0.020 0.003 0.000 0.000 0.000 0.000 0.000 2 0.980 0.889 0.558 0.253 0.083 0.019 0.003 0.000 0.000 0.000 0.000 3 0.998 0.974 0.795 0.493 0.225 0.073 0.015 0.002 0.000 0.000 0.000 4 1.000 0.996 0.927 0.724 0.438 0.194 0.057 0.009 0.001 0.000 0.000 5 1.000 0.999 0.981 0.882 0.665 0.387 0.158 0.039 0.004 0.000 0.000 6 1.000 1.000 0.996 0.961 0.842 0.613 0.335 0.118 0.019 0.001 0.000 7 1.000 1.000 0.999 0.991 0.943 0.806 0.562 0.276 0.073 0.004 0.000 8 1.000 1.000 1.000 0.998 0.985 0.927 0.775 0.507 0.205 0.026 0.002 9 1.000 1.000 1.000 1.000 0.997 0.981 0.917 0.747 0.442 0.111 0.020 10 1.000 1.000 1.000 1.000 1.000 0.997 0.980 0.915 0.725 0.341 0.118 11 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.986 0.931 0.718 0.460 12 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.513 0.254 0.055 0.010 0.001 0.000 0.000 0.000 0.000 0.000 0.000 1 0.865 0.621 0.234 0.064 0.013 0.002 0.000 0.000 0.000 0.000 0.000 2 0.975 0.866 0.502 0.202 0.058 0.011 0.001 0.000 0.000 0.000 0.000 3 0.997 0.966 0.747 0.421 0.169 0.046 0.008 0.001 0.000 0.000 0.000 4 1.000 0.994 0.901 0.654 0.353 0.133 0.032 0.004 0.000 0.000 0.000 5 1.000 0.999 0.970 0.835 0.574 0.291 0.098 0.018 0.001 0.000 0.000 6 1.000 1.000 0.993 0.938 0.771 0.500 0.229 0.062 0.007 0.000 0.000 7 1.000 1.000 0.999 0.982 0.902 0.709 0.426 0.165 0.030 0.001 0.000 8 1.000 1.000 1.000 0.996 0.968 0.867 0.647 0.346 0.099 0.006 0.000 9 1.000 1.000 1.000 0.999 0.992 0.954 0.831 0.579 0.253 0.034 0.003 10 1.000 1.000 1.000 1.000 0.999 0.989 0.942 0.798 0.498 0.134 0.025 11 1.000 1.000 1.000 1.000 1.000 0.998 0.987 0.936 0.766 0.379 0.135 12 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.990 0.945 0.746 0.487 13 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.488 0.229 0.044 0.007 0.001 0.000 0.000 0.000 0.000 0.000 0.000 1 0.847 0.585 0.198 0.047 0.008 0.001 0.000 0.000 0.000 0.000 0.000 2 0.970 0.842 0.448 0.161 0.040 0.006 0.001 0.000 0.000 0.000 0.000 3 0.996 0.956 0.698 0.355 0.124 0.029 0.004 0.000 0.000 0.000 0.000 4 1.000 0.991 0.870 0.584 0.279 0.090 0.018 0.002 0.000 0.000 0.000 5 1.000 0.999 0.956 0.781 0.486 0.212 0.058 0.008 0.000 0.000 0.000 6 1.000 1.000 0.988 0.907 0.692 0.395 0.150 0.031 0.002 0.000 0.000 7 1.000 1.000 0.998 0.969 0.850 0.605 0.308 0.093 0.012 0.000 0.000 8 1.000 1.000 1.000 0.992 0.942 0.788 0.514 0.219 0.044 0.001 0.000 9 1.000 1.000 1.000 0.998 0.982 0.910 0.721 0.416 0.130 0.009 0.000 10 1.000 1.000 1.000 1.000 0.996 0.971 0.876 0.645 0.302 0.044 0.004 11 1.000 1.000 1.000 1.000 0.999 0.994 0.960 0.839 0.552 0.158 0.030 12 1.000 1.000 1.000 1.000 1.000 0.999 0.992 0.953 0.802 0.415 0.153 13 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.993 0.956 0.771 0.512 14 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 3 Table: Cumulative Binomial probabilities (continued) p n = 15 n = 16 c 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0 0.463 0.206 0.035 0.005 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.829 0.549 0.167 0.035 0.005 0.000 0.000 0.000 0.000 0.000 0.000 2 0.964 0.816 0.398 0.127 0.027 0.004 0.000 0.000 0.000 0.000 0.000 3 0.995 0.944 0.648 0.297 0.091 0.018 0.002 0.000 0.000 0.000 0.000 4 0.999 0.987 0.836 0.515 0.217 0.059 0.009 0.001 0.000 0.000 0.000 5 1.000 0.998 0.939 0.722 0.403 0.151 0.034 0.004 0.000 0.000 0.000 6 1.000 1.000 0.982 0.869 0.610 0.304 0.095 0.015 0.001 0.000 0.000 7 1.000 1.000 0.996 0.950 0.787 0.500 0.213 0.050 0.004 0.000 0.000 8 1.000 1.000 0.999 0.985 0.905 0.696 0.390 0.131 0.018 0.000 0.000 9 1.000 1.000 1.000 0.996 0.966 0.849 0.597 0.278 0.061 0.002 0.000 10 1.000 1.000 1.000 0.999 0.991 0.941 0.783 0.485 0.164 0.013 0.001 11 1.000 1.000 1.000 1.000 0.998 0.982 0.909 0.703 0.352 0.056 0.005 12 1.000 1.000 1.000 1.000 1.000 0.996 0.973 0.873 0.602 0.184 0.036 13 1.000 1.000 1.000 1.000 1.000 1.000 0.995 0.965 0.833 0.451 0.171 14 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.995 0.965 0.794 0.537 15 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.440 0.185 0.028 0.003 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.811 0.515 0.141 0.026 0.003 0.000 0.000 0.000 0.000 0.000 0.000 2 0.957 0.789 0.352 0.099 0.018 0.002 0.000 0.000 0.000 0.000 0.000 3 0.993 0.932 0.598 0.246 0.065 0.011 0.001 0.000 0.000 0.000 0.000 4 0.999 0.983 0.798 0.450 0.167 0.038 0.005 0.000 0.000 0.000 0.000 5 1.000 0.997 0.918 0.660 0.329 0.105 0.019 0.002 0.000 0.000 0.000 6 1.000 0.999 0.973 0.825 0.527 0.227 0.058 0.007 0.000 0.000 0.000 7 1.000 1.000 0.993 0.926 0.716 0.402 0.142 0.026 0.001 0.000 0.000 8 1.000 1.000 0.999 0.974 0.858 0.598 0.284 0.074 0.007 0.000 0.000 9 1.000 1.000 1.000 0.993 0.942 0.773 0.473 0.175 0.027 0.001 0.000 10 1.000 1.000 1.000 0.998 0.981 0.895 0.671 0.340 0.082 0.003 0.000 11 1.000 1.000 1.000 1.000 0.995 0.962 0.833 0.550 0.202 0.017 0.001 12 1.000 1.000 1.000 1.000 0.999 0.989 0.935 0.754 0.402 0.068 0.007 13 1.000 1.000 1.000 1.000 1.000 0.998 0.982 0.901 0.648 0.211 0.043 14 1.000 1.000 1.000 1.000 1.000 1.000 0.997 0.974 0.859 0.485 0.189 15 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.997 0.972 0.815 0.560 16 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 4 Table: Cumulative Binomial probabilities (continued) p n = 17 n = 18 c 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0 0.418 0.167 0.023 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.792 0.482 0.118 0.019 0.002 0.000 0.000 0.000 0.000 0.000 0.000 2 0.950 0.762 0.310 0.077 0.012 0.001 0.000 0.000 0.000 0.000 0.000 3 0.991 0.917 0.549 0.202 0.046 0.006 0.000 0.000 0.000 0.000 0.000 4 0.999 0.978 0.758 0.389 0.126 0.025 0.003 0.000 0.000 0.000 0.000 5 1.000 0.995 0.894 0.597 0.264 0.072 0.011 0.001 0.000 0.000 0.000 6 1.000 0.999 0.962 0.775 0.448 0.166 0.035 0.003 0.000 0.000 0.000 7 1.000 1.000 0.989 0.895 0.641 0.315 0.092 0.013 0.000 0.000 0.000 8 1.000 1.000 0.997 0.960 0.801 0.500 0.199 0.040 0.003 0.000 0.000 9 1.000 1.000 1.000 0.987 0.908 0.685 0.359 0.105 0.011 0.000 0.000 10 1.000 1.000 1.000 0.997 0.965 0.834 0.552 0.225 0.038 0.001 0.000 11 1.000 1.000 1.000 0.999 0.989 0.928 0.736 0.403 0.106 0.005 0.000 12 1.000 1.000 1.000 1.000 0.997 0.975 0.874 0.611 0.242 0.022 0.001 13 1.000 1.000 1.000 1.000 1.000 0.994 0.954 0.798 0.451 0.083 0.009 14 1.000 1.000 1.000 1.000 1.000 0.999 0.988 0.923 0.690 0.238 0.050 15 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.981 0.882 0.518 0.208 16 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.977 0.833 0.582 17 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.397 0.150 0.018 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.774 0.450 0.099 0.014 0.001 0.000 0.000 0.000 0.000 0.000 0.000 2 0.942 0.734 0.271 0.060 0.008 0.001 0.000 0.000 0.000 0.000 0.000 3 0.989 0.902 0.501 0.165 0.033 0.004 0.000 0.000 0.000 0.000 0.000 4 0.998 0.972 0.716 0.333 0.094 0.015 0.001 0.000 0.000 0.000 0.000 5 1.000 0.994 0.867 0.534 0.209 0.048 0.006 0.000 0.000 0.000 0.000 6 1.000 0.999 0.949 0.722 0.374 0.119 0.020 0.001 0.000 0.000 0.000 7 1.000 1.000 0.984 0.859 0.563 0.240 0.058 0.006 0.000 0.000 0.000 8 1.000 1.000 0.996 0.940 0.737 0.407 0.135 0.021 0.001 0.000 0.000 9 1.000 1.000 0.999 0.979 0.865 0.593 0.263 0.060 0.004 0.000 0.000 10 1.000 1.000 1.000 0.994 0.942 0.760 0.437 0.141 0.016 0.000 0.000 11 1.000 1.000 1.000 0.999 0.980 0.881 0.626 0.278 0.051 0.001 0.000 12 1.000 1.000 1.000 1.000 0.994 0.952 0.791 0.466 0.133 0.006 0.000 13 1.000 1.000 1.000 1.000 0.999 0.985 0.906 0.667 0.284 0.028 0.002 14 1.000 1.000 1.000 1.000 1.000 0.996 0.967 0.835 0.499 0.098 0.011 15 1.000 1.000 1.000 1.000 1.000 0.999 0.992 0.940 0.729 0.266 0.058 16 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.986 0.901 0.550 0.226 17 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.982 0.850 0.603 18 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 5 Table: Cumulative Binomial probabilities (continued) p n = 19 n = 20 c 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0 0.377 0.135 0.014 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.755 0.420 0.083 0.010 0.001 0.000 0.000 0.000 0.000 0.000 0.000 2 0.933 0.705 0.237 0.046 0.005 0.000 0.000 0.000 0.000 0.000 0.000 3 0.987 0.885 0.455 0.133 0.023 0.002 0.000 0.000 0.000 0.000 0.000 4 0.998 0.965 0.673 0.282 0.070 0.010 0.001 0.000 0.000 0.000 0.000 5 1.000 0.991 0.837 0.474 0.163 0.032 0.003 0.000 0.000 0.000 0.000 6 1.000 0.998 0.932 0.666 0.308 0.084 0.012 0.001 0.000 0.000 0.000 7 1.000 1.000 0.977 0.818 0.488 0.180 0.035 0.003 0.000 0.000 0.000 8 1.000 1.000 0.993 0.916 0.667 0.324 0.088 0.011 0.000 0.000 0.000 9 1.000 1.000 0.998 0.967 0.814 0.500 0.186 0.033 0.002 0.000 0.000 10 1.000 1.000 1.000 0.989 0.912 0.676 0.333 0.084 0.007 0.000 0.000 11 1.000 1.000 1.000 0.997 0.965 0.820 0.512 0.182 0.023 0.000 0.000 12 1.000 1.000 1.000 0.999 0.988 0.916 0.692 0.334 0.068 0.002 0.000 13 1.000 1.000 1.000 1.000 0.997 0.968 0.837 0.526 0.163 0.009 0.000 14 1.000 1.000 1.000 1.000 0.999 0.990 0.930 0.718 0.327 0.035 0.002 15 1.000 1.000 1.000 1.000 1.000 0.998 0.977 0.867 0.545 0.115 0.013 16 1.000 1.000 1.000 1.000 1.000 1.000 0.995 0.954 0.763 0.295 0.067 17 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.990 0.917 0.580 0.245 18 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.986 0.865 0.623 19 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0 0.358 0.122 0.012 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.736 0.392 0.069 0.008 0.001 0.000 0.000 0.000 0.000 0.000 0.000 2 0.925 0.677 0.206 0.035 0.004 0.000 0.000 0.000 0.000 0.000 0.000 3 0.984 0.867 0.411 0.107 0.016 0.001 0.000 0.000 0.000 0.000 0.000 4 0.997 0.957 0.630 0.238 0.051 0.006 0.000 0.000 0.000 0.000 0.000 5 1.000 0.989 0.804 0.416 0.126 0.021 0.002 0.000 0.000 0.000 0.000 6 1.000 0.998 0.913 0.608 0.250 0.058 0.006 0.000 0.000 0.000 0.000 7 1.000 1.000 0.968 0.772 0.416 0.132 0.021 0.001 0.000 0.000 0.000 8 1.000 1.000 0.990 0.887 0.596 0.252 0.057 0.005 0.000 0.000 0.000 9 1.000 1.000 0.997 0.952 0.755 0.412 0.128 0.017 0.001 0.000 0.000 10 1.000 1.000 0.999 0.983 0.872 0.588 0.245 0.048 0.003 0.000 0.000 11 1.000 1.000 1.000 0.995 0.943 0.748 0.404 0.113 0.010 0.000 0.000 12 1.000 1.000 1.000 0.999 0.979 0.868 0.584 0.228 0.032 0.000 0.000 13 1.000 1.000 1.000 1.000 0.994 0.942 0.750 0.392 0.087 0.002 0.000 14 1.000 1.000 1.000 1.000 0.998 0.979 0.874 0.584 0.196 0.011 0.000 15 1.000 1.000 1.000 1.000 1.000 0.994 0.949 0.762 0.370 0.043 0.003 16 1.000 1.000 1.000 1.000 1.000 0.999 0.984 0.893 0.589 0.133 0.016 17 1.000 1.000 1.000 1.000 1.000 1.000 0.996 0.965 0.794 0.323 0.075 18 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.992 0.931 0.608 0.264 19 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.988 0.878 0.642 20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 6 Table: Cumulative Binomial probabilities (continued) p n = 25 c 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0 0.277 0.072 0.004 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.642 0.271 0.027 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 0.873 0.537 0.098 0.009 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 0.966 0.764 0.234 0.033 0.002 0.000 0.000 0.000 0.000 0.000 0.000 4 0.993 0.902 0.421 0.090 0.009 0.000 0.000 0.000 0.000 0.000 0.000 5 0.999 0.967 0.617 0.193 0.029 0.002 0.000 0.000 0.000 0.000 0.000 6 1.000 0.991 0.780 0.341 0.074 0.007 0.000 0.000 0.000 0.000 0.000 7 1.000 0.998 0.891 0.512 0.154 0.022 0.001 0.000 0.000 0.000 0.000 8 1.000 1.000 0.953 0.677 0.274 0.054 0.004 0.000 0.000 0.000 0.000 9 1.000 1.000 0.983 0.811 0.425 0.115 0.013 0.000 0.000 0.000 0.000 10 1.000 1.000 0.994 0.902 0.586 0.212 0.034 0.002 0.000 0.000 0.000 11 1.000 1.000 0.998 0.956 0.732 0.345 0.078 0.006 0.000 0.000 0.000 12 1.000 1.000 1.000 0.983 0.846 0.500 0.154 0.017 0.000 0.000 0.000 13 1.000 1.000 1.000 0.994 0.922 0.655 0.268 0.044 0.002 0.000 0.000 14 1.000 1.000 1.000 0.998 0.966 0.788 0.414 0.098 0.006 0.000 0.000 15 1.000 1.000 1.000 1.000 0.987 0.885 0.575 0.189 0.017 0.000 0.000 16 1.000 1.000 1.000 1.000 0.996 0.946 0.726 0.323 0.047 0.000 0.000 17 1.000 1.000 1.000 1.000 0.999 0.978 0.846 0.488 0.109 0.002 0.000 18 1.000 1.000 1.000 1.000 1.000 0.993 0.926 0.659 0.220 0.009 0.000 19 1.000 1.000 1.000 1.000 1.000 0.998 0.971 0.807 0.383 0.033 0.001 20 1.000 1.000 1.000 1.000 1.000 1.000 0.991 0.910 0.579 0.098 0.007 21 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.967 0.766 0.236 0.034 22 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.991 0.902 0.463 0.127 23 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.998 0.973 0.729 0.358 24 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.996 0.928 0.723 25 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 7 Tables of the Poisson Cumulative Distribution The table below gives the probability of that a Poisson random variable X with mean = λ is less than or equal to x. That is, the table gives e −λ P ( X ≤ x) = ∑ λ r! r =0 x r λ= x= λ= x= 0 1 2 3 4 5 6 7 8 9 0.1 0.9048 0.9953 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.2 0.8187 0.9825 0.9989 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3 0.7408 0.9631 0.9964 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.4 0.6703 0.9384 0.9921 0.9992 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 0.5 0.6065 0.9098 0.9856 0.9982 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 0.6 0.5488 0.8781 0.9769 0.9966 0.9996 1.0000 1.0000 1.0000 1.0000 1.0000 0.7 0.4966 0.8442 0.9659 0.9942 0.9992 0.9999 1.0000 1.0000 1.0000 1.0000 0.8 0.4493 0.8088 0.9526 0.9909 0.9986 0.9998 1.0000 1.0000 1.0000 1.0000 0.9 0.4066 0.7725 0.9371 0.9865 0.9977 0.9997 1.0000 1.0000 1.0000 1.0000 1.0 0.3679 0.7358 0.9197 0.9810 0.9963 0.9994 0.9999 1.0000 1.0000 1.0000 1.2 0.3012 0.6626 0.8795 0.9662 0.9923 0.9985 0.9997 1.0000 1.0000 1.0000 1.4 0.2466 0.5918 0.8335 0.9463 0.9857 0.9968 0.9994 0.9999 1.0000 1.0000 1.6 0.2019 0.5249 0.7834 0.9212 0.9763 0.9940 0.9987 0.9997 1.0000 1.0000 1.8 0.1653 0.4628 0.7306 0.8913 0.9636 0.9896 0.9974 0.9994 0.9999 1.0000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 2.0 0.1353 0.4060 0.6767 0.8571 0.9473 0.9834 0.9955 0.9989 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.2 0.1108 0.3546 0.6227 0.8194 0.9275 0.9751 0.9925 0.9980 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.4 0.0907 0.3084 0.5697 0.7787 0.9041 0.9643 0.9884 0.9967 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.6 0.0743 0.2674 0.5184 0.7360 0.8774 0.9510 0.9828 0.9947 0.9985 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2.8 0.0608 0.2311 0.4695 0.6919 0.8477 0.9349 0.9756 0.9919 0.9976 0.9993 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 3.0 0.0498 0.1991 0.4232 0.6472 0.8153 0.9161 0.9665 0.9881 0.9962 0.9989 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 3.2 0.0408 0.1712 0.3799 0.6025 0.7806 0.8946 0.9554 0.9832 0.9943 0.9982 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 3.4 0.0334 0.1468 0.3397 0.5584 0.7442 0.8705 0.9421 0.9769 0.9917 0.9973 0.9992 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 3.6 0.0273 0.1257 0.3027 0.5152 0.7064 0.8441 0.9267 0.9692 0.9883 0.9960 0.9987 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 3.8 0.0224 0.1074 0.2689 0.4735 0.6678 0.8156 0.9091 0.9599 0.9840 0.9942 0.9981 0.9994 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 4.0 0.0183 0.0916 0.2381 0.4335 0.6288 0.7851 0.8893 0.9489 0.9786 0.9919 0.9972 0.9991 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 4.5 0.0111 0.0611 0.1736 0.3423 0.5321 0.7029 0.8311 0.9134 0.9597 0.9829 0.9933 0.9976 0.9992 0.9997 0.9999 1.0000 1.0000 1.0000 5.0 0.0067 0.0404 0.1247 0.2650 0.4405 0.6160 0.7622 0.8666 0.9319 0.9682 0.9863 0.9945 0.9980 0.9993 0.9998 0.9999 1.0000 1.0000 5.5 0.0041 0.0266 0.0884 0.2017 0.3575 0.5289 0.6860 0.8095 0.8944 0.9462 0.9747 0.9890 0.9955 0.9983 0.9994 0.9998 0.9999 1.0000 λ= x= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 11.0 10.0 12.0 14.0 15.0 0.0025 0.0174 0.0620 0.1512 0.2851 0.4457 0.6063 0.7440 0.8472 0.9161 0.9574 0.9799 0.9912 0.9964 0.9986 0.9995 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0015 0.0113 0.0430 0.1118 0.2237 0.3690 0.5265 0.6728 0.7916 0.8774 0.9332 0.9661 0.9840 0.9929 0.9970 0.9988 0.9996 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0009 0.0073 0.0296 0.0818 0.1730 0.3007 0.4497 0.5987 0.7291 0.8305 0.9015 0.9467 0.9730 0.9872 0.9943 0.9976 0.9990 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0006 0.0047 0.0203 0.0591 0.1321 0.2414 0.3782 0.5246 0.6620 0.7764 0.8622 0.9208 0.9573 0.9784 0.9897 0.9954 0.9980 0.9992 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0003 0.0030 0.0138 0.0424 0.0996 0.1912 0.3134 0.4530 0.5925 0.7166 0.8159 0.8881 0.9362 0.9658 0.9827 0.9918 0.9963 0.9984 0.9993 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0002 0.0019 0.0093 0.0301 0.0744 0.1496 0.2562 0.3856 0.5231 0.6530 0.7634 0.8487 0.9091 0.9486 0.9726 0.9862 0.9934 0.9970 0.9987 0.9995 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0001 0.0012 0.0062 0.0212 0.0550 0.1157 0.2068 0.3239 0.4557 0.5874 0.7060 0.8030 0.8758 0.9261 0.9585 0.9780 0.9889 0.9947 0.9976 0.9989 0.9996 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0001 0.0008 0.0042 0.0149 0.0403 0.0885 0.1649 0.2687 0.3918 0.5218 0.6453 0.7520 0.8364 0.8981 0.9400 0.9665 0.9823 0.9911 0.9957 0.9980 0.9991 0.9996 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0005 0.0028 0.0103 0.0293 0.0671 0.1301 0.2202 0.3328 0.4579 0.5830 0.6968 0.7916 0.8645 0.9165 0.9513 0.9730 0.9857 0.9928 0.9965 0.9984 0.9993 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0002 0.0012 0.0049 0.0151 0.0375 0.0786 0.1432 0.2320 0.3405 0.4599 0.5793 0.6887 0.7813 0.8540 0.9074 0.9441 0.9678 0.9823 0.9907 0.9953 0.9977 0.9990 0.9995 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0005 0.0028 0.0103 0.0293 0.0671 0.1301 0.2202 0.3328 0.4579 0.5830 0.6968 0.7916 0.8645 0.9165 0.9513 0.9730 0.9857 0.9928 0.9965 0.9984 0.9993 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0001 0.0005 0.0023 0.0076 0.0203 0.0458 0.0895 0.1550 0.2424 0.3472 0.4616 0.5760 0.6815 0.7720 0.8444 0.8987 0.9370 0.9626 0.9787 0.9884 0.9939 0.9970 0.9985 0.9993 0.9997 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0001 0.0005 0.0018 0.0055 0.0142 0.0316 0.0621 0.1094 0.1757 0.2600 0.3585 0.4644 0.5704 0.6694 0.7559 0.8272 0.8826 0.9235 0.9521 0.9712 0.9833 0.9907 0.9950 0.9974 0.9987 0.9994 0.9997 0.9999 0.9999 1.0000 1.0000 0.0000 0.0000 0.0000 0.0002 0.0009 0.0028 0.0076 0.0180 0.0374 0.0699 0.1185 0.1848 0.2676 0.3632 0.4657 0.5681 0.6641 0.7489 0.8195 0.8752 0.9170 0.9469 0.9673 0.9805 0.9888 0.9938 0.9967 0.9983 0.9991 0.9996 0.9998 0.9999 1.0000 Standard Normal Probabilities Table entry Table entry for z is the area under the standard normal curve to the left of z. z z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 –3.4 –3.3 –3.2 –3.1 –3.0 –2.9 –2.8 –2.7 –2.6 –2.5 –2.4 –2.3 –2.2 –2.1 –2.0 –1.9 –1.8 –1.7 –1.6 –1.5 –1.4 –1.3 –1.2 –1.1 –1.0 –0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1 –0.0 .0003 .0005 .0007 .0010 .0013 .0019 .0026 .0035 .0047 .0062 .0082 .0107 .0139 .0179 .0228 .0287 .0359 .0446 .0548 .0668 .0808 .0968 .1151 .1357 .1587 .1841 .2119 .2420 .2743 .3085 .3446 .3821 .4207 .4602 .5000 .0003 .0005 .0007 .0009 .0013 .0018 .0025 .0034 .0045 .0060 .0080 .0104 .0136 .0174 .0222 .0281 .0351 .0436 .0537 .0655 .0793 .0951 .1131 .1335 .1562 .1814 .2090 .2389 .2709 .3050 .3409 .3783 .4168 .4562 .4960 .0003 .0005 .0006 .0009 .0013 .0018 .0024 .0033 .0044 .0059 .0078 .0102 .0132 .0170 .0217 .0274 .0344 .0427 .0526 .0643 .0778 .0934 .1112 .1314 .1539 .1788 .2061 .2358 .2676 .3015 .3372 .3745 .4129 .4522 .4920 .0003 .0004 .0006 .0009 .0012 .0017 .0023 .0032 .0043 .0057 .0075 .0099 .0129 .0166 .0212 .0268 .0336 .0418 .0516 .0630 .0764 .0918 .1093 .1292 .1515 .1762 .2033 .2327 .2643 .2981 .3336 .3707 .4090 .4483 .4880 .0003 .0004 .0006 .0008 .0012 .0016 .0023 .0031 .0041 .0055 .0073 .0096 .0125 .0162 .0207 .0262 .0329 .0409 .0505 .0618 .0749 .0901 .1075 .1271 .1492 .1736 .2005 .2296 .2611 .2946 .3300 .3669 .4052 .4443 .4840 .0003 .0004 .0006 .0008 .0011 .0016 .0022 .0030 .0040 .0054 .0071 .0094 .0122 .0158 .0202 .0256 .0322 .0401 .0495 .0606 .0735 .0885 .1056 .1251 .1469 .1711 .1977 .2266 .2578 .2912 .3264 .3632 .4013 .4404 .4801 .0003 .0004 .0006 .0008 .0011 .0015 .0021 .0029 .0039 .0052 .0069 .0091 .0119 .0154 .0197 .0250 .0314 .0392 .0485 .0594 .0721 .0869 .1038 .1230 .1446 .1685 .1949 .2236 .2546 .2877 .3228 .3594 .3974 .4364 .4761 .0003 .0004 .0005 .0008 .0011 .0015 .0021 .0028 .0038 .0051 .0068 .0089 .0116 .0150 .0192 .0244 .0307 .0384 .0475 .0582 .0708 .0853 .1020 .1210 .1423 .1660 .1922 .2206 .2514 .2843 .3192 .3557 .3936 .4325 .4721 .0003 .0004 .0005 .0007 .0010 .0014 .0020 .0027 .0037 .0049 .0066 .0087 .0113 .0146 .0188 .0239 .0301 .0375 .0465 .0571 .0694 .0838 .1003 .1190 .1401 .1635 .1894 .2177 .2483 .2810 .3156 .3520 .3897 .4286 .4681 .0002 .0003 .0005 .0007 .0010 .0014 .0019 .0026 .0036 .0048 .0064 .0084 .0110 .0143 .0183 .0233 .0294 .0367 .0455 .0559 .0681 .0823 .0985 .1170 .1379 .1611 .1867 .2148 .2451 .2776 .3121 .3483 .3859 .4247 .4641 Standard Normal Probabilities Table entry Table entry for z is the area under the standard normal curve to the left of z. z z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .5000 .5398 .5793 .6179 .6554 .6915 .7257 .7580 .7881 .8159 .8413 .8643 .8849 .9032 .9192 .9332 .9452 .9554 .9641 .9713 .9772 .9821 .9861 .9893 .9918 .9938 .9953 .9965 .9974 .9981 .9987 .9990 .9993 .9995 .9997 .5040 .5438 .5832 .6217 .6591 .6950 .7291 .7611 .7910 .8186 .8438 .8665 .8869 .9049 .9207 .9345 .9463 .9564 .9649 .9719 .9778 .9826 .9864 .9896 .9920 .9940 .9955 .9966 .9975 .9982 .9987 .9991 .9993 .9995 .9997 .5080 .5478 .5871 .6255 .6628 .6985 .7324 .7642 .7939 .8212 .8461 .8686 .8888 .9066 .9222 .9357 .9474 .9573 .9656 .9726 .9783 .9830 .9868 .9898 .9922 .9941 .9956 .9967 .9976 .9982 .9987 .9991 .9994 .9995 .9997 .5120 .5517 .5910 .6293 .6664 .7019 .7357 .7673 .7967 .8238 .8485 .8708 .8907 .9082 .9236 .9370 .9484 .9582 .9664 .9732 .9788 .9834 .9871 .9901 .9925 .9943 .9957 .9968 .9977 .9983 .9988 .9991 .9994 .9996 .9997 .5160 .5557 .5948 .6331 .6700 .7054 .7389 .7704 .7995 .8264 .8508 .8729 .8925 .9099 .9251 .9382 .9495 .9591 .9671 .9738 .9793 .9838 .9875 .9904 .9927 .9945 .9959 .9969 .9977 .9984 .9988 .9992 .9994 .9996 .9997 .5199 .5596 .5987 .6368 .6736 .7088 .7422 .7734 .8023 .8289 .8531 .8749 .8944 .9115 .9265 .9394 .9505 .9599 .9678 .9744 .9798 .9842 .9878 .9906 .9929 .9946 .9960 .9970 .9978 .9984 .9989 .9992 .9994 .9996 .9997 .5239 .5636 .6026 .6406 .6772 .7123 .7454 .7764 .8051 .8315 .8554 .8770 .8962 .9131 .9279 .9406 .9515 .9608 .9686 .9750 .9803 .9846 .9881 .9909 .9931 .9948 .9961 .9971 .9979 .9985 .9989 .9992 .9994 .9996 .9997 .5279 .5675 .6064 .6443 .6808 .7157 .7486 .7794 .8078 .8340 .8577 .8790 .8980 .9147 .9292 .9418 .9525 .9616 .9693 .9756 .9808 .9850 .9884 .9911 .9932 .9949 .9962 .9972 .9979 .9985 .9989 .9992 .9995 .9996 .9997 .5319 .5714 .6103 .6480 .6844 .7190 .7517 .7823 .8106 .8365 .8599 .8810 .8997 .9162 .9306 .9429 .9535 .9625 .9699 .9761 .9812 .9854 .9887 .9913 .9934 .9951 .9963 .9973 .9980 .9986 .9990 .9993 .9995 .9996 .9997 .5359 .5753 .6141 .6517 .6879 .7224 .7549 .7852 .8133 .8389 .8621 .8830 .9015 .9177 .9319 .9441 .9545 .9633 .9706 .9767 .9817 .9857 .9890 .9916 .9936 .9952 .9964 .9974 .9981 .9986 .9990 .9993 .9995 .9997 .9998 Table of the Student's t-distribution α The table gives the values of tα ; ν where Pr(Tν > t α; ν ) = α , with ν degrees of freedom α tα ; ν 0.1 0.05 0.025 0.01 0.005 0.001 0.0005 1 2 3 4 5 3.078 1.886 1.638 1.533 1.476 6.314 2.920 2.353 2.132 2.015 12.076 4.303 3.182 2.776 2.571 31.821 6.965 4.541 3.747 3.365 63.657 9.925 5.841 4.604 4.032 318.310 22.326 10.213 7.173 5.893 636.620 31.598 12.924 8.610 6.869 6 7 8 9 10 1.440 1.415 1.397 1.383 1.372 1.943 1.895 1.860 1.833 1.812 2.447 2.365 2.306 2.262 2.228 3.143 2.998 2.896 2.821 2.764 3.707 3.499 3.355 3.250 3.169 5.208 4.785 4.501 4.297 4.144 5.959 5.408 5.041 4.781 4.587 11 12 13 14 15 1.363 1.356 1.350 1.345 1.341 1.796 1.782 1.771 1.761 1.753 2.201 2.179 2.160 2.145 2.131 2.718 2.681 2.650 2.624 2.602 3.106 3.055 3.012 2.977 2.947 4.025 3.930 3.852 3.787 3.733 4.437 4.318 4.221 4.140 4.073 16 17 18 19 20 1.337 1.333 1.330 1.328 1.325 1.746 1.740 1.734 1.729 1.725 2.120 2.110 2.101 2.093 2.086 2.583 2.567 2.552 2.539 2.528 2.921 2.898 2.878 2.861 2.845 3.686 3.646 3.610 3.579 3.552 4.015 3.965 3.922 3.883 3.850 21 22 23 24 25 1.323 1.321 1.319 1.318 1.316 1.721 1.717 1.714 1.711 1.708 2.080 2.074 2.069 2.064 2.060 2.518 2.508 2.500 2.492 2.485 2.831 2.819 2.807 2.797 2.787 3.527 3.505 3.485 3.467 3.450 3.819 3.792 3.767 3.745 3.725 26 27 28 29 30 1.315 1.314 1.313 1.311 1.310 1.706 1.703 1.701 1.699 1.697 2.056 2.052 2.048 2.045 2.042 2.479 2.473 2.467 2.462 2.457 2.779 2.771 2.763 2.756 2.750 3.435 3.421 3.408 3.396 3.385 3.707 3.690 3.674 3.659 3.646 40 60 120 ∞ 1.303 1.296 1.289 1.282 1.684 1.671 1.658 1.645 2.021 2.000 1.980 1.960 2.423 2.390 2.358 2.326 2.704 2.660 2.617 2.576 3.307 3.232 3.160 3.090 3.551 3.460 3.373 3.291 ν
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