A APPENDIX BINOMIAL PROBABILITIES Example: P (X 5 3 | n 5 8, π 5 .50) 5 .2188 This table shows P (X 5 x). π n x .01 .02 .05 .10 .15 .20 .30 .40 .50 .60 .70 .80 .85 .90 .95 .98 .99 2 0 1 2 .9801 .0198 .0001 .9604 .0392 .0004 .9025 .0950 .0025 .8100 .1800 .0100 .7225 .2550 .0225 .6400 .3200 .0400 .4900 .4200 .0900 .3600 .4800 .1600 .2500 .5000 .2500 .1600 .4800 .3600 .0900 .4200 .4900 .0400 .3200 .6400 .0225 .2550 .7225 .0100 .1800 .8100 .0025 .0950 .9025 .0004 .0392 .9604 .0001 .0198 .9801 3 0 1 2 3 .9703 .0294 .0003 — .9412 .0576 .0012 — .8574 .1354 .0071 .0001 .7290 .2430 .0270 .0010 .6141 .3251 .0574 .0034 .5120 .3840 .0960 .0080 .3430 .4410 .1890 .0270 .2160 .4320 .2880 .0640 .1250 .3750 .3750 .1250 .0640 .2880 .4320 .2160 .0270 .1890 .4410 .3430 .0080 .0960 .3840 .5120 .0034 .0574 .3251 .6141 .0010 .0270 .2430 .7290 .0001 .0071 .1354 .8574 — .0012 .0576 .9412 — .0003 .0294 .9703 4 0 1 2 3 4 .9606 .0388 .0006 — — .9224 .0753 .0023 — — .8145 .1715 .0135 .0005 — .6561 .2916 .0486 .0036 .0001 .5220 .3685 .0975 .0115 .0005 .4096 .4096 .1536 .0256 .0016 .2401 .4116 .2646 .0756 .0081 .1296 .3456 .3456 .1536 .0256 .0625 .2500 .3750 .2500 .0625 .0256 .1536 .3456 .3456 .1296 .0081 .0756 .2646 .4116 .2401 .0016 .0256 .1536 .4096 .4096 .0005 .0115 .0975 .3685 .5220 .0001 .0036 .0486 .2916 .6561 — .0005 .0135 .1715 .8145 — — .0023 .0753 .9224 — — .0006 .0388 .9606 5 0 1 2 3 4 5 .9510 .0480 .0010 — — — .9039 .0922 .0038 .0001 — — .7738 .2036 .0214 .0011 — — .5905 .3281 .0729 .0081 .0005 — .4437 .3915 .1382 .0244 .0022 .0001 .3277 .4096 .2048 .0512 .0064 .0003 .1681 .3602 .3087 .1323 .0284 .0024 .0778 .2592 .3456 .2304 .0768 .0102 .0313 .1563 .3125 .3125 .1563 .0313 .0102 .0768 .2304 .3456 .2592 .0778 .0024 .0284 .1323 .3087 .3602 .1681 .0003 .0064 .0512 .2048 .4096 .3277 .0001 .0022 .0244 .1382 .3915 .4437 — .0005 .0081 .0729 .3281 .5905 — — .0011 .0214 .2036 .7738 — — .0001 .0038 .0922 .9039 — — — .0010 .0480 .9510 6 0 1 2 3 4 5 6 .9415 .0571 .0014 — — — — .8858 .1085 .0055 .0002 — — — .7351 .2321 .0305 .0021 .0001 — — .5314 .3543 .0984 .0146 .0012 .0001 — .3771 .3993 .1762 .0415 .0055 .0004 — .2621 .3932 .2458 .0819 .0154 .0015 .0001 .1176 .3025 .3241 .1852 .0595 .0102 .0007 .0467 .1866 .3110 .2765 .1382 .0369 .0041 .0156 .0938 .2344 .3125 .2344 .0938 .0156 .0041 .0369 .1382 .2765 .3110 .1866 .0467 .0007 .0102 .0595 .1852 .3241 .3025 .1176 .0001 .0015 .0154 .0819 .2458 .3932 .2621 — .0004 .0055 .0415 .1762 .3993 .3771 — .0001 .0012 .0146 .0984 .3543 .5314 — — .0001 .0021 .0305 .2321 .7351 — — — .0002 .0055 .1085 .8858 — — — — .0014 .0571 .9415 7 0 1 2 3 4 5 6 7 .9321 .0659 .0020 — — — — — .8681 .1240 .0076 .0003 — — — — .6983 .2573 .0406 .0036 .0002 — — — .4783 .3720 .1240 .0230 .0026 .0002 — — .3206 .3960 .2097 .0617 .0109 .0012 .0001 — .2097 .3670 .2753 .1147 .0287 .0043 .0004 — .0824 .2471 .3177 .2269 .0972 .0250 .0036 .0002 .0280 .1306 .2613 .2903 .1935 .0774 .0172 .0016 .0078 .0547 .1641 .2734 .2734 .1641 .0547 .0078 .0016 .0172 .0774 .1935 .2903 .2613 .1306 .0280 .0002 .0036 .0250 .0972 .2269 .3177 .2471 .0824 — .0004 .0043 .0287 .1147 .2753 .3670 .2097 — .0001 .0012 .0109 .0617 .2097 .3960 .3206 — — .0002 .0026 .0230 .1240 .3720 .4783 — — — .0002 .0036 .0406 .2573 .6983 — — — — .0003 .0076 .1240 .8681 — — — — — .0020 .0659 .9321 8 0 1 2 3 4 5 6 7 8 .9227 .0746 .0026 .0001 — — — — — .8508 .1389 .0099 .0004 — — — — — .6634 .2793 .0515 .0054 .0004 — — — — .4305 .3826 .1488 .0331 .0046 .0004 — — — .2725 .3847 .2376 .0839 .0185 .0026 .0002 — — .1678 .3355 .2936 .1468 .0459 .0092 .0011 .0001 — .0576 .1977 .2965 .2541 .1361 .0467 .0100 .0012 .0001 .0168 .0896 .2090 .2787 .2322 .1239 .0413 .0079 .0007 .0039 .0313 .1094 .2188 .2734 .2188 .1094 .0313 .0039 .0007 .0079 .0413 .1239 .2322 .2787 .2090 .0896 .0168 .0001 .0012 .0100 .0467 .1361 .2541 .2965 .1977 .0576 — .0001 .0011 .0092 .0459 .1468 .2936 .3355 .1678 — — .0002 .0026 .0185 .0839 .2376 .3847 .2725 — — — .0004 .0046 .0331 .1488 .3826 .4305 — — — — .0004 .0054 .0515 .2793 .6634 — — — — — .0004 .0099 .1389 .8508 — — — — — .0001 .0026 .0746 .9227 9 0 1 2 3 4 5 6 7 8 9 .9135 .0830 .0034 .0001 — — — — — — .8337 .1531 .0125 .0006 — — — — — — .6302 .2985 .0629 .0077 .0006 — — — — — .3874 .3874 .1722 .0446 .0074 .0008 .0001 — — — .2316 .3679 .2597 .1069 .0283 .0050 .0006 — — — .1342 .3020 .3020 .1762 .0661 .0165 .0028 .0003 — — .0404 .1556 .2668 .2668 .1715 .0735 .0210 .0039 .0004 — .0101 .0605 .1612 .2508 .2508 .1672 .0743 .0212 .0035 .0003 .0020 .0176 .0703 .1641 .2461 .2461 .1641 .0703 .0176 .0020 .0003 .0035 .0212 .0743 .1672 .2508 .2508 .1612 .0605 .0101 — .0004 .0039 .0210 .0735 .1715 .2668 .2668 .1556 .0404 — — .0003 .0028 .0165 .0661 .1762 .3020 .3020 .1342 — — — .0006 .0050 .0283 .1069 .2597 .3679 .2316 — — — .0001 .0008 .0074 .0446 .1722 .3874 .3874 — — — — — .0006 .0077 .0629 .2985 .6302 — — — — — — .0006 .0125 .1531 .8337 — — — — — — .0001 .0034 .0830 .9135 762 Appendix A 763 π n x .01 .02 .05 .10 .15 .20 .30 .40 .50 .60 .70 .80 .85 .90 .95 .98 .99 10 0 1 2 3 4 5 6 7 8 9 10 .9044 .0914 .0042 .0001 — — — — — — — .8171 .1667 .0153 .0008 — — — — — — — .5987 .3151 .0746 .0105 .0010 .0001 — — — — — .3487 .3874 .1937 .0574 .0112 .0015 .0001 — — — — .1969 .3474 .2759 .1298 .0401 .0085 .0012 .0001 — — — .1074 .2684 .3020 .2013 .0881 .0264 .0055 .0008 .0001 — — .0282 .1211 .2335 .2668 .2001 .1029 .0368 .0090 .0014 .0001 — .0060 .0403 .1209 .2150 .2508 .2007 .1115 .0425 .0106 .0016 .0001 .0010 .0098 .0439 .1172 .2051 .2461 .2051 .1172 .0439 .0098 .0010 .0001 .0016 .0106 .0425 .1115 .2007 .2508 .2150 .1209 .0403 .0060 — .0001 .0014 .0090 .0368 .1029 .2001 .2668 .2335 .1211 .0282 — — .0001 .0008 .0055 .0264 .0881 .2013 .3020 .2684 .1074 — — — .0001 .0012 .0085 .0401 .1298 .2759 .3474 .1969 — — — — .0001 .0015 .0112 .0574 .1937 .3874 .3487 — — — — — .0001 .0010 .0105 .0746 .3151 .5987 — — — — — — — .0008 .0153 .1667 .8171 — — — — — — — .0001 .0042 .0914 .9044 12 0 1 2 3 4 5 6 7 8 9 10 11 12 .8864 .1074 .0060 .0002 — — — — — — — — — .7847 .1922 .0216 .0015 .0001 — — — — — — — — .5404 .3413 .0988 .0173 .0021 .0002 — — — — — — — .2824 .3766 .2301 .0852 .0213 .0038 .0005 — — — — — — .1422 .3012 .2924 .1720 .0683 .0193 .0040 .0006 .0001 — — — — .0687 .2062 .2835 .2362 .1329 .0532 .0155 .0033 .0005 .0001 — — — .0138 .0712 .1678 .2397 .2311 .1585 .0792 .0291 .0078 .0015 .0002 — — .0022 .0174 .0639 .1419 .2128 .2270 .1766 .1009 .0420 .0125 .0025 .0003 — .0002 .0029 .0161 .0537 .1208 .1934 .2256 .1934 .1208 .0537 .0161 .0029 .0002 — .0003 .0025 .0125 .0420 .1009 .1766 .2270 .2128 .1419 .0639 .0174 .0022 — — .0002 .0015 .0078 .0291 .0792 .1585 .2311 .2397 .1678 .0712 .0138 — — — .0001 .0005 .0033 .0155 .0532 .1329 .2362 .2835 .2062 .0687 — — — — .0001 .0006 .0040 .0193 .0683 .1720 .2924 .3012 .1422 — — — — — — .0005 .0038 .0213 .0852 .2301 .3766 .2824 — — — — — — — .0002 .0021 .0173 .0988 .3413 .5404 — — — — — — — — .0001 .0015 .0216 .1922 .7847 — — — — — — — — — .0002 .0060 .1074 .8864 14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 .8687 .1229 .0081 .0003 — — — — — — — — — — — .7536 .2153 .0286 .0023 .0001 — — — — — — — — — — .4877 .3593 .1229 .0259 .0037 .0004 — — — — — — — — — .2288 .3559 .2570 .1142 .0349 .0078 .0013 .0002 — — — — — — — .1028 .2539 .2912 .2056 .0998 .0352 .0093 .0019 .0003 — — — — — — .0440 .1539 .2501 .2501 .1720 .0860 .0322 .0092 .0020 .0003 — — — — — .0068 .0407 .1134 .1943 .2290 .1963 .1262 .0618 .0232 .0066 .0014 .0002 — — — .0008 .0073 .0317 .0845 .1549 .2066 .2066 .1574 .0918 .0408 .0136 .0033 .0005 .0001 — .0001 .0009 .0056 .0222 .0611 .1222 .1833 .2095 .1833 .1222 .0611 .0222 .0056 .0009 .0001 — .0001 .0005 .0033 .0136 .0408 .0918 .1574 .2066 .2066 .1549 .0845 .0317 .0073 .0008 — — — .0002 .0014 .0066 .0232 .0618 .1262 .1963 .2290 .1943 .1134 .0407 .0068 — — — — — .0003 .0020 .0092 .0322 .0860 .1720 .2501 .2501 .1539 .0440 — — — — — — .0003 .0019 .0093 .0352 .0998 .2056 .2912 .2539 .1028 — — — — — — — .0002 .0013 .0078 .0349 .1142 .2570 .3559 .2288 — — — — — — — — — .0004 .0037 .0259 .1229 .3593 .4877 — — — — — — — — — — .0001 .0023 .0286 .2153 .7536 — — — — — — — — — — — .0003 .0081 .1229 .8687 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .8515 .1376 .0104 .0005 — — — — — — — — — — — — — .7238 .2363 .0362 .0034 .0002 — — — — — — — — — — — — .4401 .3706 .1463 .0359 .0061 .0008 .0001 — — — — — — — — — — .1853 .3294 .2745 .1423 .0514 .0137 .0028 .0004 .0001 — — — — — — — — .0743 .2097 .2775 .2285 .1311 .0555 .0180 .0045 .0009 .0001 — — — — — — — .0281 .1126 .2111 .2463 .2001 .1201 .0550 .0197 .0055 .0012 .0002 — — — — — — .0033 .0228 .0732 .1465 .2040 .2099 .1649 .1010 .0487 .0185 .0056 .0013 .0002 — — — — .0003 .0030 .0150 .0468 .1014 .1623 .1983 .1889 .1417 .0840 .0392 .0142 .0040 .0008 .0001 — — — .0002 .0018 .0085 .0278 .0667 .1222 .1746 .1964 .1746 .1222 .0667 .0278 .0085 .0018 .0002 — — — .0001 .0008 .0040 .0142 .0392 .0840 .1417 .1889 .1983 .1623 .1014 .0468 .0150 .0030 .0003 — — — — .0002 .0013 .0056 .0185 .0487 .1010 .1649 .2099 .2040 .1465 .0732 .0228 .0033 — — — — — — .0002 .0012 .0055 .0197 .0550 .1201 .2001 .2463 .2111 .1126 .0281 — — — — — — — .0001 .0009 .0045 .0180 .0555 .1311 .2285 .2775 .2097 .0743 — — — — — — — — .0001 .0004 .0028 .0137 .0514 .1423 .2745 .3294 .1853 — — — — — — — — — — .0001 .0008 .0061 .0359 .1463 .3706 .4401 — — — — — — — — — — — — .0002 .0034 .0362 .2363 .7238 — — — — — — — — — — — — — .0005 .0104 .1376 .8515 APPENDIX B POISSON PROBABILITIES Example: P (X 5 3 | λ 5 2.3) 5 .2033 This table shows P (X 5 x). λ x 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 0 1 2 3 4 5 6 7 8 .9048 .0905 .0045 .0002 — — — — — .8187 .1637 .0164 .0011 .0001 — — — — .7408 .2222 .0333 .0033 .0003 — — — — .6703 .2681 .0536 .0072 .0007 .0001 — — — .6065 .3033 .0758 .0126 .0016 .0002 — — — .5488 .3293 .0988 .0198 .0030 .0004 — — — .4966 .3476 .1217 .0284 .0050 .0007 .0001 — — .4493 .3595 .1438 .0383 .0077 .0012 .0002 — — .4066 .3659 .1647 .0494 .0111 .0020 .0003 — — .3679 .3679 .1839 .0613 .0153 .0031 .0005 .0001 — .3329 .3662 .2014 .0738 .0203 .0045 .0008 .0001 — .3012 .3614 .2169 .0867 .0260 .0062 .0012 .0002 — .2725 .3543 .2303 .0998 .0324 .0084 .0018 .0003 .0001 .2466 .3452 .2417 .1128 .0395 .0111 .0026 .0005 .0001 .2231 .3347 .2510 .1255 .0471 .0141 .0035 .0008 .0001 λ x 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 0 1 2 3 4 5 6 7 8 9 10 11 12 .2019 .3230 .2584 .1378 .0551 .0176 .0047 .0011 .0002 — — — — .1827 .3106 .2640 .1496 .0636 .0216 .0061 .0015 .0003 .0001 — — — .1653 .2975 .2678 .1607 .0723 .0260 .0078 .0020 .0005 .0001 — — — .1496 .2842 .2700 .1710 .0812 .0309 .0098 .0027 .0006 .0001 — — — .1353 .2707 .2707 .1804 .0902 .0361 .0120 .0034 .0009 .0002 — — — .1225 .2572 .2700 .1890 .0992 .0417 .0146 .0044 .0011 .0003 .0001 — — .1108 .2438 .2681 .1966 .1082 .0476 .0174 .0055 .0015 .0004 .0001 — — .1003 .2306 .2652 .2033 .1169 .0538 .0206 .0068 .0019 .0005 .0001 — — .0907 .2177 .2613 .2090 .1254 .0602 .0241 .0083 .0025 .0007 .0002 — — .0821 .2052 .2565 .2138 .1336 .0668 .0278 .0099 .0031 .0009 .0002 — — .0743 .1931 .2510 .2176 .1414 .0735 .0319 .0118 .0038 .0011 .0003 .0001 — .0672 .1815 .2450 .2205 .1488 .0804 .0362 .0139 .0047 .0014 .0004 .0001 — .0608 .1703 .2384 .2225 .1557 .0872 .0407 .0163 .0057 .0018 .0005 .0001 — .0550 .1596 .2314 .2237 .1622 .0940 .0455 .0188 .0068 .0022 .0006 .0002 — .0498 .1494 .2240 .2240 .1680 .1008 .0504 .0216 .0081 .0027 .0008 .0002 .0001 λ x 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .0450 .1397 .2165 .2237 .1733 .1075 .0555 .0246 .0095 .0033 .0010 .0003 .0001 — — — .0408 .1304 .2087 .2226 .1781 .1140 .0608 .0278 .0111 .0040 .0013 .0004 .0001 — — — .0369 .1217 .2008 .2209 .1823 .1203 .0662 .0312 .0129 .0047 .0016 .0005 .0001 — — — .0334 .1135 .1929 .2186 .1858 .1264 .0716 .0348 .0148 .0056 .0019 .0006 .0002 — — — .0302 .1057 .1850 .2158 .1888 .1322 .0771 .0385 .0169 .0066 .0023 .0007 .0002 .0001 — — .0273 .0984 .1771 .2125 .1912 .1377 .0826 .0425 .0191 .0076 .0028 .0009 .0003 .0001 — — .0247 .0915 .1692 .2087 .1931 .1429 .0881 .0466 .0215 .0089 .0033 .0011 .0003 .0001 — — .0224 .0850 .1615 .2046 .1944 .1477 .0936 .0508 .0241 .0102 .0039 .0013 .0004 .0001 — — .0202 .0789 .1539 .2001 .1951 .1522 .0989 .0551 .0269 .0116 .0045 .0016 .0005 .0002 — — .0183 .0733 .1465 .1954 .1954 .1563 .1042 .0595 .0298 .0132 .0053 .0019 .0006 .0002 .0001 — .0166 .0679 .1393 .1904 .1951 .1600 .1093 .0640 .0328 .0150 .0061 .0023 .0008 .0002 .0001 — .0150 .0630 .1323 .1852 .1944 .1633 .1143 .0686 .0360 .0168 .0071 .0027 .0009 .0003 .0001 — .0136 .0583 .1254 .1798 .1933 .1662 .1191 .0732 .0393 .0188 .0081 .0032 .0011 .0004 .0001 — .0123 .0540 .1188 .1743 .1917 .1687 .1237 .0778 .0428 .0209 .0092 .0037 .0013 .0005 .0001 — .0111 .0500 .1125 .1687 .1898 .1708 .1281 .0824 .0463 .0232 .0104 .0043 .0016 .0006 .0002 .0001 764 Appendix B 765 λ x 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 .0101 .0462 .1063 .1631 .1875 .1725 .1323 .0869 .0500 .0255 .0118 .0049 .0019 .0007 .0002 .0001 — — .0091 .0427 .1005 .1574 .1849 .1738 .1362 .0914 .0537 .0281 .0132 .0056 .0022 .0008 .0003 .0001 — — .0082 .0395 .0948 .1517 .1820 .1747 .1398 .0959 .0575 .0307 .0147 .0064 .0026 .0009 .0003 .0001 — — .0074 .0365 .0894 .1460 .1789 .1753 .1432 .1002 .0614 .0334 .0164 .0073 .0030 .0011 .0004 .0001 — — .0067 .0337 .0842 .1404 .1755 .1755 .1462 .1044 .0653 .0363 .0181 .0082 .0034 .0013 .0005 .0002 — — .0061 .0311 .0793 .1348 .1719 .1753 .1490 .1086 .0692 .0392 .0200 .0093 .0039 .0015 .0006 .0002 .0001 — .0055 .0287 .0746 .1293 .1681 .1748 .1515 .1125 .0731 .0423 .0220 .0104 .0045 .0018 .0007 .0002 .0001 — .0050 .0265 .0701 .1239 .1641 .1740 .1537 .1163 .0771 .0454 .0241 .0116 .0051 .0021 .0008 .0003 .0001 — .0045 .0244 .0659 .1185 .1600 .1728 .1555 .1200 .0810 .0486 .0262 .0129 .0058 .0024 .0009 .0003 .0001 — .0041 .0225 .0618 .1133 .1558 .1714 .1571 .1234 .0849 .0519 .0285 .0143 .0065 .0028 .0011 .0004 .0001 — .0037 .0207 .0580 .1082 .1515 .1697 .1584 .1267 .0887 .0552 .0309 .0157 .0073 .0032 .0013 .0005 .0002 .0001 .0033 .0191 .0544 .1033 .1472 .1678 .1594 .1298 .0925 .0586 .0334 .0173 .0082 .0036 .0015 .0006 .0002 .0001 .0030 .0176 .0509 .0985 .1428 .1656 .1601 .1326 .0962 .0620 .0359 .0190 .0092 .0041 .0017 .0007 .0002 .0001 .0027 .0162 .0477 .0938 .1383 .1632 .1605 .1353 .0998 .0654 .0386 .0207 .0102 .0046 .0019 .0008 .0003 .0001 .0025 .0149 .0446 .0892 .1339 .1606 .1606 .1377 .1033 .0688 .0413 .0225 .0113 .0052 .0022 .0009 .0003 .0001 λ x 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .0022 .0137 .0417 .0848 .1294 .1579 .1605 .1399 .1066 .0723 .0441 .0244 .0124 .0058 .0025 .0010 .0004 .0001 — — — .0020 .0126 .0390 .0806 .1249 .1549 .1601 .1418 .1099 .0757 .0469 .0265 .0137 .0065 .0029 .0012 .0005 .0002 .0001 — — .0018 .0116 .0364 .0765 .1205 .1519 .1595 .1435 .1130 .0791 .0498 .0285 .0150 .0073 .0033 .0014 .0005 .0002 .0001 — — .0017 .0106 .0340 .0726 .1162 .1487 .1586 .1450 .1160 .0825 .0528 .0307 .0164 .0081 .0037 .0016 .0006 .0002 .0001 — — .0015 .0098 .0318 .0688 .1118 .1454 .1575 .1462 .1188 .0858 .0558 .0330 .0179 .0089 .0041 .0018 .0007 .0003 .0001 — — .0014 .0090 .0296 .0652 .1076 .1420 .1562 .1472 .1215 .0891 .0588 .0353 .0194 .0099 .0046 .0020 .0008 .0003 .0001 — — .0012 .0082 .0276 .0617 .1034 .1385 .1546 .1480 .1240 .0923 .0618 .0377 .0210 .0108 .0052 .0023 .0010 .0004 .0001 .0001 — .0011 .0076 .0258 .0584 .0992 .1349 .1529 .1486 .1263 .0954 .0649 .0401 .0227 .0119 .0058 .0026 .0011 .0004 .0002 .0001 — .0010 .0070 .0240 .0552 .0952 .1314 .1511 .1489 .1284 .0985 .0679 .0426 .0245 .0130 .0064 .0029 .0013 .0005 .0002 .0001 — .0009 .0064 .0223 .0521 .0912 .1277 .1490 .1490 .1304 .1014 .0710 .0452 .0263 .0142 .0071 .0033 .0014 .0006 .0002 .0001 — .0008 .0059 .0208 .0492 .0874 .1241 .1468 .1489 .1321 .1042 .0740 .0478 .0283 .0154 .0078 .0037 .0016 .0007 .0003 .0001 — .0007 .0054 .0194 .0464 .0836 .1204 .1445 .1486 .1337 .1070 .0770 .0504 .0303 .0168 .0086 .0041 .0019 .0008 .0003 .0001 — .0007 .0049 .0180 .0438 .0799 .1167 .1420 .1481 .1351 .1096 .0800 .0531 .0323 .0181 .0095 .0046 .0021 .0009 .0004 .0001 .0001 .0006 .0045 .0167 .0413 .0764 .1130 .1394 .1474 .1363 .1121 .0829 .0558 .0344 .0196 .0104 .0051 .0024 .0010 .0004 .0002 .0001 .0006 .0041 .0156 .0389 .0729 .1094 .1367 .1465 .1373 .1144 .0858 .0585 .0366 .0211 .0113 .0057 .0026 .0012 .0005 .0002 .0001 766 Appendix B (continued ) λ x 8.0 8.5 9.0 9.5 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 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 33 34 35 36 37 38 39 .0003 .0027 .0107 .0286 .0573 .0916 .1221 .1396 .1396 .1241 .0993 .0722 .0481 .0296 .0169 .0090 .0045 .0021 .0009 .0004 .0002 .0001 — — — — — — — — — — — — — — — — — — .0002 .0017 .0074 .0208 .0443 .0752 .1066 .1294 .1375 .1299 .1104 .0853 .0604 .0395 .0240 .0136 .0072 .0036 .0017 .0008 .0003 .0001 .0001 — — — — — — — — — — — — — — — — — .0001 .0011 .0050 .0150 .0337 .0607 .0911 .1171 .1318 .1318 .1186 .0970 .0728 .0504 .0324 .0194 .0109 .0058 .0029 .0014 .0006 .0003 .0001 — — — — — — — — — — — — — — — — — .0001 .0007 .0034 .0107 .0254 .0483 .0764 .1037 .1232 .1300 .1235 .1067 .0844 .0617 .0419 .0265 .0157 .0088 .0046 .0023 .0011 .0005 .0002 .0001 — — — — — — — — — — — — — — — — — .0005 .0023 .0076 .0189 .0378 .0631 .0901 .1126 .1251 .1251 .1137 .0948 .0729 .0521 .0347 .0217 .0128 .0071 .0037 .0019 .0009 .0004 .0002 .0001 — — — — — — — — — — — — — — — — .0002 .0010 .0037 .0102 .0224 .0411 .0646 .0888 .1085 .1194 .1194 .1094 .0926 .0728 .0534 .0367 .0237 .0145 .0084 .0046 .0024 .0012 .0006 .0003 .0001 — — — — — — — — — — — — — — — .0001 .0004 .0018 .0053 .0127 .0255 .0437 .0655 .0874 .1048 .1144 .1144 .1056 .0905 .0724 .0543 .0383 .0255 .0161 .0097 .0055 .0030 .0016 .0008 .0004 .0002 .0001 — — — — — — — — — — — — — — .0002 .0008 .0027 .0070 .0152 .0281 .0457 .0661 .0859 .1015 .1099 .1099 .1021 .0885 .0719 .0550 .0397 .0272 .0177 .0109 .0065 .0037 .0020 .0010 .0005 .0002 .0001 .0001 — — — — — — — — — — — — .0001 .0004 .0013 .0037 .0087 .0174 .0304 .0473 .0663 .0844 .0984 .1060 .1060 .0989 .0866 .0713 .0554 .0409 .0286 .0191 .0121 .0074 .0043 .0024 .0013 .0007 .0003 .0002 .0001 — — — — — — — — — — — — .0002 .0006 .0019 .0048 .0104 .0194 .0324 .0486 .0663 .0829 .0956 .1024 .1024 .0960 .0847 .0706 .0557 .0418 .0299 .0204 .0133 .0083 .0050 .0029 .0016 .0009 .0004 .0002 .0001 .0001 — — — — — — — — — — .0001 .0003 .0010 .0026 .0060 .0120 .0213 .0341 .0496 .0661 .0814 .0930 .0992 .0992 .0934 .0830 .0699 .0559 .0426 .0310 .0216 .0144 .0092 .0057 .0034 .0019 .0011 .0006 .0003 .0001 .0001 — — — — — — — — — — .0001 .0005 .0014 .0034 .0072 .0135 .0230 .0355 .0504 .0658 .0800 .0906 .0963 .0963 .0909 .0814 .0692 .0560 .0433 .0320 .0226 .0154 .0101 .0063 .0038 .0023 .0013 .0007 .0004 .0002 .0001 — — — — — — — — — .0001 .0002 .0007 .0019 .0042 .0083 .0150 .0245 .0368 .0509 .0655 .0786 .0884 .0936 .0936 .0887 .0798 .0684 .0560 .0438 .0328 .0237 .0164 .0109 .0070 .0044 .0026 .0015 .0009 .0005 .0002 .0001 .0001 — — — — — — — — .0001 .0004 .0010 .0024 .0050 .0095 .0164 .0259 .0378 .0514 .0650 .0772 .0863 .0911 .0911 .0866 .0783 .0676 .0559 .0442 .0336 .0246 .0173 .0117 .0077 .0049 .0030 .0018 .0010 .0006 .0003 .0002 .0001 — — — — — — — .0001 .0002 .0005 .0013 .0029 .0058 .0106 .0176 .0271 .0387 .0516 .0646 .0760 .0844 .0888 .0888 .0846 .0769 .0669 .0557 .0446 .0343 .0254 .0181 .0125 .0083 .0054 .0034 .0020 .0012 .0007 .0004 .0002 .0001 .0001 APPENDIX C-1 A STANDARD NORMAL AREAS Example: P (0 # z # 1.96) 5 .4750 This table shows the normal area between 0 and z. 0 z z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 .0000 .0398 .0793 .1179 .1554 .0040 .0438 .0832 .1217 .1591 .0080 .0478 .0871 .1255 .1628 .0120 .0517 .0910 .1293 .1664 .0160 .0557 .0948 .1331 .1700 .0199 .0596 .0987 .1368 .1736 .0239 .0636 .1026 .1406 .1772 .0279 .0675 .1064 .1443 .1808 .0319 .0714 .1103 .1480 .1844 .0359 .0753 .1141 .1517 .1879 0.5 0.6 0.7 0.8 0.9 .1915 .2257 .2580 .2881 .3159 .1950 .2291 .2611 .2910 .3186 .1985 .2324 .2642 .2939 .3212 .2019 .2357 .2673 .2967 .3238 .2054 .2389 .2704 .2995 .3264 .2088 .2422 .2734 .3023 .3289 .2123 .2454 .2764 .3051 .3315 .2157 .2486 .2794 .3078 .3340 .2190 .2517 .2823 .3106 .3365 .2224 .2549 .2852 .3133 .3389 1.0 1.1 1.2 1.3 1.4 .3413 .3643 .3849 .4032 .4192 .3438 .3665 .3869 .4049 .4207 .3461 .3686 .3888 .4066 .4222 .3485 .3708 .3907 .4082 .4236 .3508 .3729 .3925 .4099 .4251 .3531 .3749 .3944 .4115 .4265 .3554 .3770 .3962 .4131 .4279 .3577 .3790 .3980 .4147 .4292 .3599 .3810 .3997 .4162 .4306 .3621 .3830 .4015 .4177 .4319 1.5 1.6 1.7 1.8 1.9 .4332 .4452 .4554 .4641 .4713 .4345 .4463 .4564 .4649 .4719 .4357 .4474 .4573 .4656 .4726 .4370 .4484 .4582 .4664 .4732 .4382 .4495 .4591 .4671 .4738 .4394 .4505 .4599 .4678 .4744 .4406 .4515 .4608 .4686 .4750 .4418 .4525 .4616 .4693 .4756 .4429 .4535 .4625 .4699 .4761 .4441 .4545 .4633 .4706 .4767 2.0 2.1 2.2 2.3 2.4 .4772 .4821 .4861 .4893 .4918 .4778 .4826 .4864 .4896 .4920 .4783 .4830 .4868 .4898 .4922 .4788 .4834 .4871 .4901 .4925 .4793 .4838 .4875 .4904 .4927 .4798 .4842 .4878 .4906 .4929 .4803 .4846 .4881 .4909 .4931 .4808 .4850 .4884 .4911 .4932 .4812 .4854 .4887 .4913 .4934 .4817 .4857 .4890 .4916 .4936 2.5 2.6 2.7 2.8 2.9 .4938 .4953 .4965 .4974 .4981 .4940 .4955 .4966 .4975 .4982 .4941 .4956 .4967 .4976 .4982 .4943 .4957 .4968 .4977 .4983 .4945 .4959 .4969 .4977 .4984 .4946 .4960 .4970 .4978 .4984 .4948 .4961 .4971 .4979 .4985 .4949 .4962 .4972 .4979 .4985 .4951 .4963 .4973 .4980 .4986 .4952 .4964 .4974 .4981 .4986 3.0 3.1 3.2 3.3 3.4 .49865 .49903 .49931 .49952 .49966 .49869 .49906 .49934 .49953 .49968 .49874 .49910 .49936 .49955 .49969 .49878 .49913 .49938 .49957 .49970 .49882 .49916 .49940 .49958 .49971 .49886 .49918 .49942 .49960 .49972 .49889 .49921 .49944 .49961 .49973 .49893 .49924 .49946 .49962 .49974 .49896 .49926 .49948 .49964 .49975 .49900 .49929 .49950 .49965 .49976 3.5 3.6 3.7 .49977 .49984 .49989 .49978 .49985 .49990 .49978 .49985 .49990 .49979 .49986 .49990 .49980 .49986 .49991 .49981 .49987 .49991 .49981 .49987 .49992 .49982 .49988 .49992 .49983 .49988 .49992 .49983 .49989 .49992 767 C-2 APPENDIX CUMULATIVE STANDARD NORMAL DISTRIBUTION Example: P (z # 21.96) 5 .0250 This table shows the normal area less than z. z z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 23.7 23.6 23.5 .00011 .00016 .00023 .00010 .00015 .00022 .00010 .00015 .00022 .00010 .00014 .00021 .00009 .00014 .00020 .00009 .00013 .00019 .00008 .00013 .00019 .00008 .00012 .00018 .00008 .00012 .00017 .00008 .00011 .00017 23.4 23.3 23.2 23.1 23.0 .00034 .00048 .00069 .00097 .00135 .00032 .00047 .00066 .00094 .00131 .00031 .00045 .00064 .00090 .00126 .00030 .00043 .00062 .00087 .00122 .00029 .00042 .00060 .00084 .00118 .00028 .00040 .00058 .00082 .00114 .00027 .00039 .00056 .00079 .00111 .00026 .00038 .00054 .00076 .00107 .00025 .00036 .00052 .00074 .00104 .00024 .00035 .00050 .00071 .00100 22.9 22.8 22.7 22.6 22.5 .0019 .0026 .0035 .0047 .0062 .0018 .0025 .0034 .0045 .0060 .0018 .0024 .0033 .0044 .0059 .0017 .0023 .0032 .0043 .0057 .0016 .0023 .0031 .0041 .0055 .0016 .0022 .0030 .0040 .0054 .0015 .0021 .0029 .0039 .0052 .0015 .0021 .0028 .0038 .0051 .0014 .0020 .0027 .0037 .0049 .0014 .0019 .0026 .0036 .0048 22.4 22.3 22.2 22.1 22.0 .0082 .0107 .0139 .0179 .0228 .0080 .0104 .0136 .0174 .0222 .0078 .0102 .0132 .0170 .0217 .0075 .0099 .0129 .0166 .0212 .0073 .0096 .0125 .0162 .0207 .0071 .0094 .0122 .0158 .0202 .0069 .0091 .0119 .0154 .0197 .0068 .0089 .0116 .0150 .0192 .0066 .0087 .0113 .0146 .0188 .0064 .0084 .0110 .0143 .0183 21.9 21.8 21.7 21.6 21.5 .0287 .0359 .0446 .0548 .0668 .0281 .0351 .0436 .0537 .0655 .0274 .0344 .0427 .0526 .0643 .0268 .0336 .0418 .0516 .0630 .0262 .0329 .0409 .0505 .0618 .0256 .0322 .0401 .0495 .0606 .0250 .0314 .0392 .0485 .0594 .0244 .0307 .0384 .0475 .0582 .0239 .0301 .0375 .0465 .0571 .0233 .0294 .0367 .0455 .0559 21.4 21.3 21.2 21.1 21.0 .0808 .0968 .1151 .1357 .1587 .0793 .0951 .1131 .1335 .1562 .0778 .0934 .1112 .1314 .1539 .0764 .0918 .1093 .1292 .1515 .0749 .0901 .1075 .1271 .1492 .0735 .0885 .1056 .1251 .1469 .0721 .0869 .1038 .1230 .1446 .0708 .0853 .1020 .1210 .1423 .0694 .0838 .1003 .1190 .1401 .0681 .0823 .0985 .1170 .1379 20.9 20.8 20.7 20.6 20.5 .1841 .2119 .2420 .2743 .3085 .1814 .2090 .2389 .2709 .3050 .1788 .2061 .2358 .2676 .3015 .1762 .2033 .2327 .2643 .2981 .1736 .2005 .2296 .2611 .2946 .1711 .1977 .2266 .2578 .2912 .1685 .1949 .2236 .2546 .2877 .1660 .1922 .2206 .2514 .2843 .1635 .1894 .2177 .2483 .2810 .1611 .1867 .2148 .2451 .2776 20.4 20.3 20.2 20.1 20.0 .3446 .3821 .4207 .4602 .5000 .3409 .3783 .4168 .4562 .4960 .3372 .3745 .4129 .4522 .4920 .3336 .3707 .4090 .4483 .4880 .3300 .3669 .4052 .4443 .4841 .3264 .3632 .4013 .4404 .4801 .3228 .3594 .3974 .4364 .4761 .3192 .3557 .3936 .4325 .4721 .3156 .3520 .3897 .4286 .4681 .3121 .3483 .3859 .4247 .4641 768 Appendix C-2 This table shows the normal area less than z. 769 z z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 0.4 .5000 .5398 .5793 .6179 .6554 .5040 .5438 .5832 .6217 .6591 .5080 .5478 .5871 .6255 .6628 .5120 .5517 .5910 .6293 .6664 .5160 .5557 .5948 .6331 .6700 .5199 .5596 .5987 .6368 .6736 .5239 .5636 .6026 .6406 .6772 .5279 .5675 .6064 .6443 .6808 .5319 .5714 .6103 .6480 .6844 .5359 .5753 .6141 .6517 .6879 0.5 0.6 0.7 0.8 0.9 .6915 .7257 .7580 .7881 .8159 .6950 .7291 .7611 .7910 .8186 .6985 .7324 .7642 .7939 .8212 .7019 .7357 .7673 .7967 .8238 .7054 .7389 .7704 .7995 .8264 .7088 .7422 .7734 .8023 .8289 .7123 .7454 .7764 .8051 .8315 .7157 .7486 .7794 .8078 .8340 .7190 .7517 .7823 .8106 .8365 .7224 .7549 .7852 .8133 .8389 1.0 1.1 1.2 1.3 1.4 .8413 .8643 .8849 .9032 .9192 .8438 .8665 .8869 .9049 .9207 .8461 .8686 .8888 .9066 .9222 .8485 .8708 .8907 .9082 .9236 .8508 .8729 .8925 .9099 .9251 .8531 .8749 .8944 .9115 .9265 .8554 .8770 .8962 .9131 .9279 .8577 .8790 .8980 .9147 .9292 .8599 .8810 .8997 .9162 .9306 .8621 .8830 .9015 .9177 .9319 1.5 1.6 1.7 1.8 1.9 .9332 .9452 .9554 .9641 .9713 .9345 .9463 .9564 .9649 .9719 .9357 .9474 .9573 .9656 .9726 .9370 .9484 .9582 .9664 .9732 .9382 .9495 .9591 .9671 .9738 .9394 .9505 .9599 .9678 .9744 .9406 .9515 .9608 .9686 .9750 .9418 .9525 .9616 .9693 .9756 .9429 .9535 .9625 .9699 .9761 .9441 .9545 .9633 .9706 .9767 2.0 2.1 2.2 2.3 2.4 .9772 .9821 .9861 .9893 .9918 .9778 .9826 .9864 .9896 .9920 .9783 .9830 .9868 .9898 .9922 .9788 .9834 .9871 .9901 .9925 .9793 .9838 .9875 .9904 .9927 .9798 .9842 .9878 .9906 .9929 .9803 .9846 .9881 .9909 .9931 .9808 .9850 .9884 .9911 .9932 .9812 .9854 .9887 .9913 .9934 .9817 .9857 .9890 .9916 .9936 2.5 2.6 2.7 2.8 2.9 .9938 .9953 .9965 .9974 .9981 .9940 .9955 .9966 .9975 .9982 .9941 .9956 .9967 .9976 .9982 .9943 .9957 .9968 .9977 .9983 .9945 .9959 .9969 .9977 .9984 .9946 .9960 .9970 .9978 .9984 .9948 .9961 .9971 .9979 .9985 .9949 .9962 .9972 .9979 .9985 .9951 .9963 .9973 .9980 .9986 .9952 .9964 .9974 .9981 .9986 3.0 3.1 3.2 3.3 3.4 .99865 .99903 .99931 .99952 .99966 .99869 .99906 .99934 .99953 .99968 .99874 .99910 .99936 .99955 .99969 .99878 .99913 .99938 .99957 .99970 .99882 .99916 .99940 .99958 .99971 .99886 .99918 .99942 .99960 .99972 .99889 .99921 .99944 .99961 .99973 .99893 .99924 .99946 .99962 .99974 .99896 .99926 .99948 .99964 .99975 .99900 .99929 .99950 .99965 .99976 3.5 3.6 3.7 .99977 .99984 .99989 .99978 .99985 .99990 .99978 .99985 .99990 .99979 .99986 .99990 .99980 .99986 .99991 .99981 .99987 .99991 .99981 .99987 .99992 .99982 .99988 .99992 .99983 .99988 .99992 .99983 .99989 .99992 APPENDIX D STUDENT’S t CRITICAL VALUES This table shows the t-value that defines the area for the stated degrees of freedom (d.f.). Confidence Level .95 .98 Confidence Level .95 .98 t .99 .80 .20 Significance Level for Two-Tailed Test .10 .05 .02 .01 .20 Significance Level for Two-Tailed Test .10 .05 .02 .01 d.f. .10 Significance Level for One-Tailed Test .05 .025 .01 .005 .10 Significance Level for One-Tailed Test .05 .025 .01 .005 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 12.706 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 36 37 38 39 40 1.306 1.305 1.304 1.304 1.303 1.688 1.687 1.686 1.685 1.684 2.028 2.026 2.024 2.023 2.021 2.434 2.431 2.429 2.426 2.423 2.719 2.715 2.712 2.708 2.704 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 41 42 43 44 45 1.303 1.302 1.302 1.301 1.301 1.683 1.682 1.681 1.680 1.679 2.020 2.018 2.017 2.015 2.014 2.421 2.418 2.416 2.414 2.412 2.701 2.698 2.695 2.692 2.690 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 46 47 48 49 50 1.300 1.300 1.299 1.299 1.299 1.679 1.678 1.677 1.677 1.676 2.013 2.012 2.011 2.010 2.009 2.410 2.408 2.407 2.405 2.403 2.687 2.685 2.682 2.680 2.678 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 55 60 65 70 75 1.297 1.296 1.295 1.294 1.293 1.673 1.671 1.669 1.667 1.665 2.004 2.000 1.997 1.994 1.992 2.396 2.390 2.385 2.381 2.377 2.668 2.660 2.654 2.648 2.643 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 80 85 90 95 100 1.292 1.292 1.291 1.291 1.290 1.664 1.663 1.662 1.661 1.660 1.990 1.988 1.987 1.985 1.984 2.374 2.371 2.368 2.366 2.364 2.639 2.635 2.632 2.629 2.626 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 2.750 110 120 130 140 150 1.289 1.289 1.288 1.288 1.287 1.659 1.658 1.657 1.656 1.655 1.982 1.980 1.978 1.977 1.976 2.361 2.358 2.355 2.353 2.351 2.621 2.617 2.614 2.611 2.609 31 32 33 34 35 1.309 1.309 1.308 1.307 1.306 1.696 1.694 1.692 1.691 1.690 2.040 2.037 2.035 2.032 2.030 2.453 2.449 2.445 2.441 2.438 2.744 2.738 2.733 2.728 2.724 ∞ 1.282 1.645 1.960 2.326 2.576 .80 .90 d.f. .90 0 .99 Note:As n increases, critical values of Student’s t approach the z-values in the last line of this table. A common rule of thumb is to use z when n . 30, but that is not conservative. 770 APPENDIX A E Example for d.f. 5 4 CHI-SQUARE CRITICAL VALUES .05 This table shows the critical value of chi-square for each desired right-tail area and degrees of freedom (d.f.) 0 9.488 Area in Upper Tail d.f. .995 .990 .975 .95 .90 .10 .05 .025 .01 .005 1 2 3 4 5 0.000 0.010 0.072 0.207 0.412 0.000 0.020 0.115 0.297 0.554 0.001 0.051 0.216 0.484 0.831 0.004 0.103 0.352 0.711 1.145 0.016 0.211 0.584 1.064 1.610 2.706 4.605 6.251 7.779 9.236 3.841 5.991 7.815 9.488 11.07 5.024 7.378 9.348 11.14 12.83 6.635 9.210 11.34 13.28 15.09 7.879 10.60 12.84 14.86 16.75 6 7 8 9 10 0.676 0.989 1.344 1.735 2.156 0.872 1.239 1.646 2.088 2.558 1.237 1.690 2.180 2.700 3.247 1.635 2.167 2.733 3.325 3.940 2.204 2.833 3.490 4.168 4.865 10.64 12.02 13.36 14.68 15.99 12.59 14.07 15.51 16.92 18.31 14.45 16.01 17.53 19.02 20.48 16.81 18.48 20.09 21.67 23.21 18.55 20.28 21.95 23.59 25.19 11 12 13 14 15 2.603 3.074 3.565 4.075 4.601 3.053 3.571 4.107 4.660 5.229 3.816 4.404 5.009 5.629 6.262 4.575 5.226 5.892 6.571 7.261 5.578 6.304 7.042 7.790 8.547 17.28 18.55 19.81 21.06 22.31 19.68 21.03 22.36 23.68 25.00 21.92 23.34 24.74 26.12 27.49 24.72 26.22 27.69 29.14 30.58 26.76 28.30 29.82 31.32 32.80 16 17 18 19 20 5.142 5.697 6.265 6.844 7.434 5.812 6.408 7.015 7.633 8.260 6.908 7.564 8.231 8.907 9.591 7.962 8.672 9.390 10.12 10.85 9.312 10.09 10.86 11.65 12.44 23.54 24.77 25.99 27.20 28.41 26.30 27.59 28.87 30.14 31.41 28.85 30.19 31.53 32.85 34.17 32.00 33.41 34.81 36.19 37.57 34.27 35.72 37.16 38.58 40.00 21 22 23 24 25 8.034 8.643 9.260 9.886 10.52 8.897 9.542 10.20 10.86 11.52 10.28 10.98 11.69 12.40 13.12 11.59 12.34 13.09 13.85 14.61 13.24 14.04 14.85 15.66 16.47 29.62 30.81 32.01 33.20 34.38 32.67 33.92 35.17 36.42 37.65 35.48 36.78 38.08 39.36 40.65 38.93 40.29 41.64 42.98 44.31 41.40 42.80 44.18 45.56 46.93 26 27 28 29 30 11.16 11.81 12.46 13.12 13.79 12.20 12.88 13.56 14.26 14.95 13.84 14.57 15.31 16.05 16.79 15.38 16.15 16.93 17.71 18.49 17.29 18.11 18.94 19.77 20.60 35.56 36.74 37.92 39.09 40.26 38.89 40.11 41.34 42.56 43.77 41.92 43.19 44.46 45.72 46.98 45.64 46.96 48.28 49.59 50.89 48.29 49.64 50.99 52.34 53.67 31 32 33 34 35 14.46 15.13 15.82 16.50 17.19 15.66 16.36 17.07 17.79 18.51 17.54 18.29 19.05 19.81 20.57 19.28 20.07 20.87 21.66 22.47 21.43 22.27 23.11 23.95 24.80 41.42 42.58 43.75 44.90 46.06 44.99 46.19 47.40 48.60 49.80 48.23 49.48 50.73 51.97 53.20 52.19 53.49 54.78 56.06 57.34 55.00 56.33 57.65 58.96 60.27 36 37 38 39 40 17.89 18.59 19.29 20.00 20.71 19.23 19.96 20.69 21.43 22.16 21.34 22.11 22.88 23.65 24.43 23.27 24.07 24.88 25.70 26.51 25.64 26.49 27.34 28.20 29.05 47.21 48.36 49.51 50.66 51.81 51.00 52.19 53.38 54.57 55.76 54.44 55.67 56.90 58.12 59.34 58.62 59.89 61.16 62.43 63.69 61.58 62.88 64.18 65.48 66.77 50 60 70 80 90 100 27.99 35.53 43.28 51.17 59.20 67.33 29.71 37.48 45.44 53.54 61.75 70.06 32.36 40.48 48.76 57.15 65.65 74.22 34.76 43.19 51.74 60.39 69.13 77.93 37.69 46.46 55.33 64.28 73.29 82.36 63.17 74.40 85.53 96.58 107.6 118.5 67.50 79.08 90.53 101.9 113.1 124.3 71.42 83.30 95.02 106.6 118.1 129.6 76.15 88.38 100.4 112.3 124.1 135.8 79.49 91.95 104.2 116.3 128.3 140.2 Note: For d.f. . 100, use the Excel function =CHISQ.INV.RT(α,degrees of freedom). 771 F APPENDIX CRITICAL VALUES OF F.10 This table shows the 10 percent right-tail critical values of F for the stated degrees of freedom (d.f.). 0 F Denominator Degrees of Freedom (df2 ) 1 2 3 4 5 6 7 8 9 10 12 1 2 3 4 5 39.86 8.53 5.54 4.54 4.06 49.50 9.00 5.46 4.32 3.78 53.59 9.16 5.39 4.19 3.62 55.83 9.24 5.34 4.11 3.52 57.24 9.29 5.31 4.05 3.45 58.20 9.33 5.28 4.01 3.40 58.91 9.35 5.27 3.98 3.37 59.44 9.37 5.25 3.95 3.34 59.86 9.38 5.24 3.94 3.32 60.19 9.39 5.23 3.92 3.30 60.71 9.41 5.22 3.90 3.27 6 7 8 9 10 3.78 3.59 3.46 3.36 3.29 3.46 3.26 3.11 3.01 2.92 3.29 3.07 2.92 2.81 2.73 3.18 2.96 2.81 2.69 2.61 3.11 2.88 2.73 2.61 2.52 3.05 2.83 2.67 2.55 2.46 3.01 2.78 2.62 2.51 2.41 2.98 2.75 2.59 2.47 2.38 2.96 2.72 2.56 2.44 2.35 2.94 2.70 2.54 2.42 2.32 2.90 2.67 2.50 2.38 2.28 11 12 13 14 15 3.23 3.18 3.14 3.10 3.07 2.86 2.81 2.76 2.73 2.70 2.66 2.61 2.56 2.52 2.49 2.54 2.48 2.43 2.39 2.36 2.45 2.39 2.35 2.31 2.27 2.39 2.33 2.28 2.24 2.21 2.34 2.28 2.23 2.19 2.16 2.30 2.24 2.20 2.15 2.12 2.27 2.21 2.16 2.12 2.09 2.25 2.19 2.14 2.10 2.06 2.21 2.15 2.10 2.05 2.02 16 17 18 19 20 3.05 3.03 3.01 2.99 2.97 2.67 2.64 2.62 2.61 2.59 2.46 2.44 2.42 2.40 2.38 2.33 2.31 2.29 2.27 2.25 2.24 2.22 2.20 2.18 2.16 2.18 2.15 2.13 2.11 2.09 2.13 2.10 2.08 2.06 2.04 2.09 2.06 2.04 2.02 2.00 2.06 2.03 2.00 1.98 1.96 2.03 2.00 1.98 1.96 1.94 1.99 1.96 1.93 1.91 1.89 21 22 23 24 25 2.96 2.95 2.94 2.93 2.92 2.57 2.56 2.55 2.54 2.53 2.36 2.35 2.34 2.33 2.32 2.23 2.22 2.21 2.19 2.18 2.14 2.13 2.11 2.10 2.09 2.08 2.06 2.05 2.04 2.02 2.02 2.01 1.99 1.98 1.97 1.98 1.97 1.95 1.94 1.93 1.95 1.93 1.92 1.91 1.89 1.92 1.90 1.89 1.88 1.87 1.87 1.86 1.84 1.83 1.82 26 27 28 29 30 2.91 2.90 2.89 2.89 2.88 2.52 2.51 2.50 2.50 2.49 2.31 2.30 2.29 2.28 2.28 2.17 2.17 2.16 2.15 2.14 2.08 2.07 2.06 2.06 2.05 2.01 2.00 2.00 1.99 1.98 1.96 1.95 1.94 1.93 1.93 1.92 1.91 1.90 1.89 1.88 1.88 1.87 1.87 1.86 1.85 1.86 1.85 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 40 50 60 120 200 2.84 2.81 2.79 2.75 2.73 2.44 2.41 2.39 2.35 2.33 2.23 2.20 2.18 2.13 2.11 2.09 2.06 2.04 1.99 1.97 2.00 1.97 1.95 1.90 1.88 1.93 1.90 1.87 1.82 1.80 1.87 1.84 1.82 1.77 1.75 1.83 1.80 1.77 1.72 1.70 1.79 1.76 1.74 1.68 1.66 1.76 1.73 1.71 1.65 1.63 1.71 1.68 1.66 1.60 1.58 ` 2.71 2.30 2.08 1.94 1.85 1.77 1.72 1.67 1.63 1.60 1.55 772 Numerator Degrees of Freedom (df1 ) Appendix F 773 Denominator Degrees of Freedom (df2 ) 15 20 25 30 35 40 50 60 120 200 ∞ 1 2 3 4 5 61.22 9.42 5.20 3.87 3.24 61.74 9.44 5.18 3.84 3.21 62.05 9.45 5.17 3.83 3.19 62.26 9.46 5.17 3.82 3.17 62.42 9.46 5.16 3.81 3.16 62.53 9.47 5.16 3.80 3.16 62.69 9.47 5.15 3.80 3.15 62.79 9.47 5.15 3.79 3.14 63.06 9.48 5.14 3.78 3.12 63.17 9.49 5.14 3.77 3.12 63.32 9.49 5.13 3.76 3.11 6 7 8 9 10 2.87 2.63 2.46 2.34 2.24 2.84 2.59 2.42 2.30 2.20 2.81 2.57 2.40 2.27 2.17 2.80 2.56 2.38 2.25 2.16 2.79 2.54 2.37 2.24 2.14 2.78 2.54 2.36 2.23 2.13 2.77 2.52 2.35 2.22 2.12 2.76 2.51 2.34 2.21 2.11 2.74 2.49 2.32 2.18 2.08 2.73 2.48 2.31 2.17 2.07 2.72 2.47 2.29 2.16 2.06 11 12 13 14 15 2.17 2.10 2.05 2.01 1.97 2.12 2.06 2.01 1.96 1.92 2.10 2.03 1.98 1.93 1.89 2.08 2.01 1.96 1.91 1.87 2.06 2.00 1.94 1.90 1.86 2.05 1.99 1.93 1.89 1.85 2.04 1.97 1.92 1.87 1.83 2.03 1.96 1.90 1.86 1.82 2.00 1.93 1.88 1.83 1.79 1.99 1.92 1.86 1.82 1.77 1.97 1.90 1.85 1.80 1.76 16 17 18 19 20 1.94 1.91 1.89 1.86 1.84 1.89 1.86 1.84 1.81 1.79 1.86 1.83 1.80 1.78 1.76 1.84 1.81 1.78 1.76 1.74 1.82 1.79 1.77 1.74 1.72 1.81 1.78 1.75 1.73 1.71 1.79 1.76 1.74 1.71 1.69 1.78 1.75 1.72 1.70 1.68 1.75 1.72 1.69 1.67 1.64 1.74 1.71 1.68 1.65 1.63 1.72 1.69 1.66 1.63 1.61 21 22 23 24 25 1.83 1.81 1.80 1.78 1.77 1.78 1.76 1.74 1.73 1.72 1.74 1.73 1.71 1.70 1.68 1.72 1.70 1.69 1.67 1.66 1.70 1.68 1.67 1.65 1.64 1.69 1.67 1.66 1.64 1.63 1.67 1.65 1.64 1.62 1.61 1.66 1.64 1.62 1.61 1.59 1.62 1.60 1.59 1.57 1.56 1.61 1.59 1.57 1.56 1.54 1.59 1.57 1.55 1.53 1.52 26 27 28 29 30 1.76 1.75 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.67 1.66 1.65 1.64 1.63 1.65 1.64 1.63 1.62 1.61 1.63 1.62 1.61 1.60 1.59 1.61 1.60 1.59 1.58 1.57 1.59 1.58 1.57 1.56 1.55 1.58 1.57 1.56 1.55 1.54 1.54 1.53 1.52 1.51 1.50 1.53 1.52 1.50 1.49 1.48 1.50 1.49 1.48 1.47 1.46 40 50 60 120 200 1.66 1.63 1.60 1.55 1.52 1.61 1.57 1.54 1.48 1.46 1.57 1.53 1.50 1.44 1.41 1.54 1.50 1.48 1.41 1.38 1.52 1.48 1.45 1.39 1.36 1.51 1.46 1.44 1.37 1.34 1.48 1.44 1.41 1.34 1.31 1.47 1.42 1.40 1.32 1.29 1.42 1.38 1.35 1.26 1.23 1.41 1.36 1.33 1.24 1.20 1.38 1.33 1.29 1.19 1.15 ` 1.49 1.42 1.38 1.34 1.32 1.30 1.26 1.24 1.17 1.13 1.00 Numerator Degrees of Freedom (df1 ) 774 Appendix F CRITICAL VALUES OF F.05 This table shows the 5 percent right-tail critical values of F for the stated degrees of freedom (d.f.). Denominator Degrees of Freedom (df2 ) 0 F Numerator Degrees of Freedom (df1 ) 1 2 3 4 5 6 7 8 9 10 12 1 2 3 4 5 161.4 18.51 10.13 7.71 6.61 199.5 19.00 9.55 6.94 5.79 215.7 19.16 9.28 6.59 5.41 224.6 19.25 9.12 6.39 5.19 230.2 19.30 9.01 6.26 5.05 234.0 19.33 8.94 6.16 4.95 236.8 19.35 8.89 6.09 4.88 238.9 19.37 8.85 6.04 4.82 240.5 19.38 8.81 6.00 4.77 241.9 19.40 8.79 5.96 4.74 243.9 19.41 8.74 5.91 4.68 6 7 8 9 10 5.99 5.59 5.32 5.12 4.96 5.14 4.74 4.46 4.26 4.10 4.76 4.35 4.07 3.86 3.71 4.53 4.12 3.84 3.63 3.48 4.39 3.97 3.69 3.48 3.33 4.28 3.87 3.58 3.37 3.22 4.21 3.79 3.50 3.29 3.14 4.15 3.73 3.44 3.23 3.07 4.10 3.68 3.39 3.18 3.02 4.06 3.64 3.35 3.14 2.98 4.00 3.57 3.28 3.07 2.91 11 12 13 14 15 4.84 4.75 4.67 4.60 4.54 3.98 3.89 3.81 3.74 3.68 3.59 3.49 3.41 3.34 3.29 3.36 3.26 3.18 3.11 3.06 3.20 3.11 3.03 2.96 2.90 3.09 3.00 2.92 2.85 2.79 3.01 2.91 2.83 2.76 2.71 2.95 2.85 2.77 2.70 2.64 2.90 2.80 2.71 2.65 2.59 2.85 2.75 2.67 2.60 2.54 2.79 2.69 2.60 2.53 2.48 16 17 18 19 20 4.49 4.45 4.41 4.38 4.35 3.63 3.59 3.55 3.52 3.49 3.24 3.20 3.16 3.13 3.10 3.01 2.96 2.93 2.90 2.87 2.85 2.81 2.77 2.74 2.71 2.74 2.70 2.66 2.63 2.60 2.66 2.61 2.58 2.54 2.51 2.59 2.55 2.51 2.48 2.45 2.54 2.49 2.46 2.42 2.39 2.49 2.45 2.41 2.38 2.35 2.42 2.38 2.34 2.31 2.28 21 22 23 24 25 4.32 4.30 4.28 4.26 4.24 3.47 3.44 3.42 3.40 3.39 3.07 3.05 3.03 3.01 2.99 2.84 2.82 2.80 2.78 2.76 2.68 2.66 2.64 2.62 2.60 2.57 2.55 2.53 2.51 2.49 2.49 2.46 2.44 2.42 2.40 2.42 2.40 2.37 2.36 2.34 2.37 2.34 2.32 2.30 2.28 2.32 2.30 2.27 2.25 2.24 2.25 2.23 2.20 2.18 2.16 26 27 28 29 30 4.23 4.21 4.20 4.18 4.17 3.37 3.35 3.34 3.33 3.32 2.98 2.96 2.95 2.93 2.92 2.74 2.73 2.71 2.70 2.69 2.59 2.57 2.56 2.55 2.53 2.47 2.46 2.45 2.43 2.42 2.39 2.37 2.36 2.35 2.33 2.32 2.31 2.29 2.28 2.27 2.27 2.25 2.24 2.22 2.21 2.22 2.20 2.19 2.18 2.16 2.15 2.13 2.12 2.10 2.09 40 50 60 120 200 4.08 4.03 4.00 3.92 3.89 3.23 3.18 3.15 3.07 3.04 2.84 2.79 2.76 2.68 2.65 2.61 2.56 2.53 2.45 2.42 2.45 2.40 2.37 2.29 2.26 2.34 2.29 2.25 2.18 2.14 2.25 2.20 2.17 2.09 2.06 2.18 2.13 2.10 2.02 1.98 2.12 2.07 2.04 1.96 1.93 2.08 2.03 1.99 1.91 1.88 2.00 1.95 1.92 1.83 1.80 ` 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 Appendix F Denominator Degrees of Freedom (df2 ) 775 Numerator Degrees of Freedom (df1 ) 15 20 25 30 35 40 50 60 120 200 ∞ 1 2 3 4 5 245.9 19.43 8.70 5.86 4.62 248.0 19.45 8.66 5.80 4.56 249.3 19.46 8.63 5.77 4.52 250.1 19.46 8.62 5.75 4.50 250.7 19.47 8.60 5.73 4.48 251.1 19.47 8.59 5.72 4.46 251.8 19.48 8.58 5.70 4.44 252.2 19.48 8.57 5.69 4.43 253.3 19.49 8.55 5.66 4.40 253.7 19.49 8.54 5.65 4.39 254.3 19.50 8.53 5.63 4.37 6 7 8 9 10 3.94 3.51 3.22 3.01 2.85 3.87 3.44 3.15 2.94 2.77 3.83 3.40 3.11 2.89 2.73 3.81 3.38 3.08 2.86 2.70 3.79 3.36 3.06 2.84 2.68 3.77 3.34 3.04 2.83 2.66 3.75 3.32 3.02 2.80 2.64 3.74 3.30 3.01 2.79 2.62 3.70 3.27 2.97 2.75 2.58 3.69 3.25 2.95 2.73 2.56 3.67 3.23 2.93 2.71 2.54 11 12 13 14 15 2.72 2.62 2.53 2.46 2.40 2.65 2.54 2.46 2.39 2.33 2.60 2.50 2.41 2.34 2.28 2.57 2.47 2.38 2.31 2.25 2.55 2.44 2.36 2.28 2.22 2.53 2.43 2.34 2.27 2.20 2.51 2.40 2.31 2.24 2.18 2.49 2.38 2.30 2.22 2.16 2.45 2.34 2.25 2.18 2.11 2.43 2.32 2.23 2.16 2.10 2.41 2.30 2.21 2.13 2.07 16 17 18 19 20 2.35 2.31 2.27 2.23 2.20 2.28 2.23 2.19 2.16 2.12 2.23 2.18 2.14 2.11 2.07 2.19 2.15 2.11 2.07 2.04 2.17 2.12 2.08 2.05 2.01 2.15 2.10 2.06 2.03 1.99 2.12 2.08 2.04 2.00 1.97 2.11 2.06 2.02 1.98 1.95 2.06 2.01 1.97 1.93 1.90 2.04 1.99 1.95 1.91 1.88 2.01 1.96 1.92 1.88 1.84 21 22 23 24 25 2.18 2.15 2.13 2.11 2.09 2.10 2.07 2.05 2.03 2.01 2.05 2.02 2.00 1.97 1.96 2.01 1.98 1.96 1.94 1.92 1.98 1.96 1.93 1.91 1.89 1.96 1.94 1.91 1.89 1.87 1.94 1.91 1.88 1.86 1.84 1.92 1.89 1.86 1.84 1.82 1.87 1.84 1.81 1.79 1.77 1.84 1.82 1.79 1.77 1.75 1.81 1.78 1.76 1.73 1.71 26 27 28 29 30 2.07 2.06 2.04 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.94 1.92 1.91 1.89 1.88 1.90 1.88 1.87 1.85 1.84 1.87 1.86 1.84 1.83 1.81 1.85 1.84 1.82 1.81 1.79 1.82 1.81 1.79 1.77 1.76 1.80 1.79 1.77 1.75 1.74 1.75 1.73 1.71 1.70 1.68 1.73 1.71 1.69 1.67 1.66 1.69 1.67 1.66 1.64 1.62 40 50 60 120 200 1.92 1.87 1.84 1.75 1.72 1.84 1.78 1.75 1.66 1.62 1.78 1.73 1.69 1.60 1.56 1.74 1.69 1.65 1.55 1.52 1.72 1.66 1.62 1.52 1.48 1.69 1.63 1.59 1.50 1.46 1.66 1.60 1.56 1.46 1.41 1.64 1.58 1.53 1.43 1.39 1.58 1.51 1.47 1.35 1.30 1.55 1.48 1.44 1.32 1.26 1.51 1.44 1.39 1.26 1.19 ` 1.67 1.57 1.51 1.46 1.42 1.39 1.35 1.32 1.22 1.17 1.00 776 Appendix F CRITICAL VALUES OF F.025 This table shows the 2.5 percent right-tail critical values of F for the stated degrees of freedom (d.f.). Denominator Degrees of Freedom (df2 ) 0 F Numerator Degrees of Freedom (df1 ) 1 2 3 4 5 6 7 8 9 10 12 1 2 3 4 5 647.8 38.51 17.44 12.22 10.01 799.5 39.00 16.04 10.65 8.43 864.2 39.17 15.44 9.98 7.76 899.6 39.25 15.10 9.60 7.39 921.8 39.30 14.88 9.36 7.15 937.1 39.33 14.73 9.20 6.98 948.2 39.36 14.62 9.07 6.85 956.6 39.37 14.54 8.98 6.76 963.3 39.39 14.47 8.90 6.68 968.6 39.40 14.42 8.84 6.62 976.7 39.41 14.34 8.75 6.52 6 7 8 9 10 8.81 8.07 7.57 7.21 6.94 7.26 6.54 6.06 5.71 5.46 6.60 5.89 5.42 5.08 4.83 6.23 5.52 5.05 4.72 4.47 5.99 5.29 4.82 4.48 4.24 5.82 5.12 4.65 4.32 4.07 5.70 4.99 4.53 4.20 3.95 5.60 4.90 4.43 4.10 3.85 5.52 4.82 4.36 4.03 3.78 5.46 4.76 4.30 3.96 3.72 5.37 4.67 4.20 3.87 3.62 11 12 13 14 15 6.72 6.55 6.41 6.30 6.20 5.26 5.10 4.97 4.86 4.77 4.63 4.47 4.35 4.24 4.15 4.28 4.12 4.00 3.89 3.80 4.04 3.89 3.77 3.66 3.58 3.88 3.73 3.60 3.50 3.41 3.76 3.61 3.48 3.38 3.29 3.66 3.51 3.39 3.29 3.20 3.59 3.44 3.31 3.21 3.12 3.53 3.37 3.25 3.15 3.06 3.43 3.28 3.15 3.05 2.96 16 17 18 19 20 6.12 6.04 5.98 5.92 5.87 4.69 4.62 4.56 4.51 4.46 4.08 4.01 3.95 3.90 3.86 3.73 3.66 3.61 3.56 3.51 3.50 3.44 3.38 3.33 3.29 3.34 3.28 3.22 3.17 3.13 3.22 3.16 3.10 3.05 3.01 3.12 3.06 3.01 2.96 2.91 3.05 2.98 2.93 2.88 2.84 2.99 2.92 2.87 2.82 2.77 2.89 2.82 2.77 2.72 2.68 21 22 23 24 25 5.83 5.79 5.75 5.72 5.69 4.42 4.38 4.35 4.32 4.29 3.82 3.78 3.75 3.72 3.69 3.48 3.44 3.41 3.38 3.35 3.25 3.22 3.18 3.15 3.13 3.09 3.05 3.02 2.99 2.97 2.97 2.93 2.90 2.87 2.85 2.87 2.84 2.81 2.78 2.75 2.80 2.76 2.73 2.70 2.68 2.73 2.70 2.67 2.64 2.61 2.64 2.60 2.57 2.54 2.51 26 27 28 29 30 5.66 5.63 5.61 5.59 5.57 4.27 4.24 4.22 4.20 4.18 3.67 3.65 3.63 3.61 3.59 3.33 3.31 3.29 3.27 3.25 3.10 3.08 3.06 3.04 3.03 2.94 2.92 2.90 2.88 2.87 2.82 2.80 2.78 2.76 2.75 2.73 2.71 2.69 2.67 2.65 2.65 2.63 2.61 2.59 2.57 2.59 2.57 2.55 2.53 2.51 2.49 2.47 2.45 2.43 2.41 40 50 60 120 200 5.42 5.34 5.29 5.15 5.10 4.05 3.97 3.93 3.80 3.76 3.46 3.39 3.34 3.23 3.18 3.13 3.05 3.01 2.89 2.85 2.90 2.83 2.79 2.67 2.63 2.74 2.67 2.63 2.52 2.47 2.62 2.55 2.51 2.39 2.35 2.53 2.46 2.41 2.30 2.26 2.45 2.38 2.33 2.22 2.18 2.39 2.32 2.27 2.16 2.11 2.29 2.22 2.17 2.05 2.01 ` 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.94 Appendix F Denominator Degrees of Freedom (df2 ) 777 Numerator Degrees of Freedom (df1 ) 15 20 25 30 35 40 50 60 120 200 ∞ 1 2 3 4 5 984.9 39.43 14.25 8.66 6.43 993.1 39.45 14.17 8.56 6.33 998.1 39.46 14.12 8.50 6.27 1001 39.46 14.08 8.46 6.23 1004 39.47 14.06 8.43 6.20 1006 39.47 14.04 8.41 6.18 1008 39.48 14.01 8.38 6.14 1010 39.48 13.99 8.36 6.12 1014 39.49 13.95 8.31 6.07 1016 39.49 13.93 8.29 6.05 1018 39.50 13.90 8.26 6.02 6 7 8 9 10 5.27 4.57 4.10 3.77 3.52 5.17 4.47 4.00 3.67 3.42 5.11 4.40 3.94 3.60 3.35 5.07 4.36 3.89 3.56 3.31 5.04 4.33 3.86 3.53 3.28 5.01 4.31 3.84 3.51 3.26 4.98 4.28 3.81 3.47 3.22 4.96 4.25 3.78 3.45 3.20 4.90 4.20 3.73 3.39 3.14 4.88 4.18 3.70 3.37 3.12 4.85 4.14 3.67 3.33 3.08 11 12 13 14 15 3.33 3.18 3.05 2.95 2.86 3.23 3.07 2.95 2.84 2.76 3.16 3.01 2.88 2.78 2.69 3.12 2.96 2.84 2.73 2.64 3.09 2.93 2.80 2.70 2.61 3.06 2.91 2.78 2.67 2.59 3.03 2.87 2.74 2.64 2.55 3.00 2.85 2.72 2.61 2.52 2.94 2.79 2.66 2.55 2.46 2.92 2.76 2.63 2.53 2.44 2.88 2.73 2.60 2.49 2.40 16 17 18 19 20 2.79 2.72 2.67 2.62 2.57 2.68 2.62 2.56 2.51 2.46 2.61 2.55 2.49 2.44 2.40 2.57 2.50 2.44 2.39 2.35 2.53 2.47 2.41 2.36 2.31 2.51 2.44 2.38 2.33 2.29 2.47 2.41 2.35 2.30 2.25 2.45 2.38 2.32 2.27 2.22 2.38 2.32 2.26 2.20 2.16 2.36 2.29 2.23 2.18 2.13 2.32 2.25 2.19 2.13 2.09 21 22 23 24 25 2.53 2.50 2.47 2.44 2.41 2.42 2.39 2.36 2.33 2.30 2.36 2.32 2.29 2.26 2.23 2.31 2.27 2.24 2.21 2.18 2.27 2.24 2.20 2.17 2.15 2.25 2.21 2.18 2.15 2.12 2.21 2.17 2.14 2.11 2.08 2.18 2.14 2.11 2.08 2.05 2.11 2.08 2.04 2.01 1.98 2.09 2.05 2.01 1.98 1.95 2.04 2.01 1.97 1.94 1.91 26 27 28 29 30 2.39 2.36 2.34 2.32 2.31 2.28 2.25 2.23 2.21 2.20 2.21 2.18 2.16 2.14 2.12 2.16 2.13 2.11 2.09 2.07 2.12 2.10 2.08 2.06 2.04 2.09 2.07 2.05 2.03 2.01 2.05 2.03 2.01 1.99 1.97 2.03 2.00 1.98 1.96 1.94 1.95 1.93 1.91 1.89 1.87 1.92 1.90 1.88 1.86 1.84 1.88 1.85 1.83 1.81 1.79 40 50 60 120 200 2.18 2.11 2.06 1.94 1.90 2.07 1.99 1.94 1.82 1.78 1.99 1.92 1.87 1.75 1.70 1.94 1.87 1.82 1.69 1.64 1.90 1.83 1.78 1.65 1.60 1.88 1.80 1.74 1.61 1.56 1.83 1.75 1.70 1.56 1.51 1.80 1.72 1.67 1.53 1.47 1.72 1.64 1.58 1.43 1.37 1.69 1.60 1.54 1.39 1.32 1.64 1.55 1.48 1.31 1.23 ` 1.83 1.71 1.63 1.57 1.52 1.48 1.43 1.39 1.27 1.21 1.00 778 Appendix F CRITICAL VALUES OF F.01 This table shows the 1 percent right-tail critical values of F for the stated degrees of freedom (d.f.). Denominator Degrees of Freedom (df2 ) 1 2 3 4 5 0 F Numerator Degrees of Freedom (df1 ) 1 2 3 4 5 6 7 8 9 10 12 4052 4999 5404 5624 5764 5859 5928 5981 6022 6056 6107 98.50 99.00 99.16 99.25 99.30 99.33 99.36 99.38 99.39 99.40 99.42 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.34 27.23 27.05 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 6 7 8 9 10 13.75 12.25 11.26 10.56 10.04 10.92 9.55 8.65 8.02 7.56 9.78 8.45 7.59 6.99 6.55 9.15 7.85 7.01 6.42 5.99 8.75 7.46 6.63 6.06 5.64 8.47 7.19 6.37 5.80 5.39 8.26 6.99 6.18 5.61 5.20 8.10 6.84 6.03 5.47 5.06 7.98 6.72 5.91 5.35 4.94 7.87 6.62 5.81 5.26 4.85 7.72 6.47 5.67 5.11 4.71 11 12 13 14 15 9.65 9.33 9.07 8.86 8.68 7.21 6.93 6.70 6.51 6.36 6.22 5.95 5.74 5.56 5.42 5.67 5.41 5.21 5.04 4.89 5.32 5.06 4.86 4.69 4.56 5.07 4.82 4.62 4.46 4.32 4.89 4.64 4.44 4.28 4.14 4.74 4.50 4.30 4.14 4.00 4.63 4.39 4.19 4.03 3.89 4.54 4.30 4.10 3.94 3.80 4.40 4.16 3.96 3.80 3.67 16 17 18 19 20 8.53 8.40 8.29 8.18 8.10 6.23 6.11 6.01 5.93 5.85 5.29 5.19 5.09 5.01 4.94 4.77 4.67 4.58 4.50 4.43 4.44 4.34 4.25 4.17 4.10 4.20 4.10 4.01 3.94 3.87 4.03 3.93 3.84 3.77 3.70 3.89 3.79 3.71 3.63 3.56 3.78 3.68 3.60 3.52 3.46 3.69 3.59 3.51 3.43 3.37 3.55 3.46 3.37 3.30 3.23 21 22 23 24 25 8.02 7.95 7.88 7.82 7.77 5.78 5.72 5.66 5.61 5.57 4.87 4.82 4.76 4.72 4.68 4.37 4.31 4.26 4.22 4.18 4.04 3.99 3.94 3.90 3.85 3.81 3.76 3.71 3.67 3.63 3.64 3.59 3.54 3.50 3.46 3.51 3.45 3.41 3.36 3.32 3.40 3.35 3.30 3.26 3.22 3.31 3.26 3.21 3.17 3.13 3.17 3.12 3.07 3.03 2.99 26 27 28 29 30 7.72 7.68 7.64 7.60 7.56 5.53 5.49 5.45 5.42 5.39 4.64 4.60 4.57 4.54 4.51 4.14 4.11 4.07 4.04 4.02 3.82 3.78 3.75 3.73 3.70 3.59 3.56 3.53 3.50 3.47 3.42 3.39 3.36 3.33 3.30 3.29 3.26 3.23 3.20 3.17 3.18 3.15 3.12 3.09 3.07 3.09 3.06 3.03 3.00 2.98 2.96 2.93 2.90 2.87 2.84 40 50 60 120 200 7.31 7.17 7.08 6.85 6.76 5.18 5.06 4.98 4.79 4.71 4.31 4.20 4.13 3.95 3.88 3.83 3.72 3.65 3.48 3.41 3.51 3.41 3.34 3.17 3.11 3.29 3.19 3.12 2.96 2.89 3.12 3.02 2.95 2.79 2.73 2.99 2.89 2.82 2.66 2.60 2.89 2.78 2.72 2.56 2.50 2.80 2.70 2.63 2.47 2.41 2.66 2.56 2.50 2.34 2.27 ` 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.18 Appendix F Denominator Degrees of Freedom (df2 ) 1 2 3 4 5 779 Numerator Degrees of Freedom (df1 ) 15 20 25 30 35 40 50 60 120 200 ∞ 6157 6209 6240 6260 6275 6286 6302 6313 6340 6350 6366 99.43 99.45 99.46 99.47 99.47 99.48 99.48 99.48 99.49 99.49 99.50 26.87 26.69 26.58 26.50 26.45 26.41 26.35 26.32 26.22 26.18 26.13 14.20 14.02 13.91 13.84 13.79 13.75 13.69 13.65 13.56 13.52 13.47 9.72 9.55 9.45 9.38 9.33 9.29 9.24 9.20 9.11 9.08 9.02 6 7 8 9 10 7.56 6.31 5.52 4.96 4.56 7.40 6.16 5.36 4.81 4.41 7.30 6.06 5.26 4.71 4.31 7.23 5.99 5.20 4.65 4.25 7.18 5.94 5.15 4.60 4.20 7.14 5.91 5.12 4.57 4.17 7.09 5.86 5.07 4.52 4.12 7.06 5.82 5.03 4.48 4.08 6.97 5.74 4.95 4.40 4.00 6.93 5.70 4.91 4.36 3.96 6.88 5.65 4.86 4.31 3.91 11 12 13 14 15 4.25 4.01 3.82 3.66 3.52 4.10 3.86 3.66 3.51 3.37 4.01 3.76 3.57 3.41 3.28 3.94 3.70 3.51 3.35 3.21 3.89 3.65 3.46 3.30 3.17 3.86 3.62 3.43 3.27 3.13 3.81 3.57 3.38 3.22 3.08 3.78 3.54 3.34 3.18 3.05 3.69 3.45 3.25 3.09 2.96 3.66 3.41 3.22 3.06 2.92 3.60 3.36 3.17 3.01 2.87 16 17 18 19 20 3.41 3.31 3.23 3.15 3.09 3.26 3.16 3.08 3.00 2.94 3.16 3.07 2.98 2.91 2.84 3.10 3.00 2.92 2.84 2.78 3.05 2.96 2.87 2.80 2.73 3.02 2.92 2.84 2.76 2.69 2.97 2.87 2.78 2.71 2.64 2.93 2.83 2.75 2.67 2.61 2.84 2.75 2.66 2.58 2.52 2.81 2.71 2.62 2.55 2.48 2.76 2.66 2.57 2.49 2.42 21 22 23 24 25 3.03 2.98 2.93 2.89 2.85 2.88 2.83 2.78 2.74 2.70 2.79 2.73 2.69 2.64 2.60 2.72 2.67 2.62 2.58 2.54 2.67 2.62 2.57 2.53 2.49 2.64 2.58 2.54 2.49 2.45 2.58 2.53 2.48 2.44 2.40 2.55 2.50 2.45 2.40 2.36 2.46 2.40 2.35 2.31 2.27 2.42 2.36 2.32 2.27 2.23 2.36 2.31 2.26 2.21 2.17 26 27 28 29 30 2.81 2.78 2.75 2.73 2.70 2.66 2.63 2.60 2.57 2.55 2.57 2.54 2.51 2.48 2.45 2.50 2.47 2.44 2.41 2.39 2.45 2.42 2.39 2.36 2.34 2.42 2.38 2.35 2.33 2.30 2.36 2.33 2.30 2.27 2.25 2.33 2.29 2.26 2.23 2.21 2.23 2.20 2.17 2.14 2.11 2.19 2.16 2.13 2.10 2.07 2.13 2.10 2.07 2.04 2.01 40 50 60 120 200 2.52 2.42 2.35 2.19 2.13 2.37 2.27 2.20 2.03 1.97 2.27 2.17 2.10 1.93 1.87 2.20 2.10 2.03 1.86 1.79 2.15 2.05 1.98 1.81 1.74 2.11 2.01 1.94 1.76 1.69 2.06 1.95 1.88 1.70 1.63 2.02 1.91 1.84 1.66 1.58 1.92 1.80 1.73 1.53 1.45 1.87 1.76 1.68 1.48 1.39 1.81 1.69 1.60 1.38 1.28 ` 2.04 1.88 1.77 1.70 1.64 1.59 1.52 1.47 1.32 1.25 1.00 APPENDIX A J Excel Statistical Functions Descriptive Statistics Pre-2010 Excel* 2013 Excel Number of data items Largest data value Smallest data value Mean Median Mode (returns first mode only) Mode (array function for multiple modes; highlight output range and use Ctrl-Shift-Enter) Geometric mean (positive data values only) Quartile k (old Excel method),* e.g., k 5 3 for Q3 Quartile k (mainstream),* e.g., k 5 3 for Q3 Percentile p (old Excel method),* e.g., p 5 .25 for Q1 Percentile p (mainstream),* e.g., p 5 .25 for Q1 Sample standard deviation Sample covariance for (X,Y ) data pairs Population standard deviation Population variance for (X,Y ) data pairs Standardize an X value (use sample mean and standard deviation if μ and σ unknown) Correlation coefficient for (X,Y ) data pairs Average deviation around the mean Slope of simple X-Y regression Intercept of simple X-Y regression R-squared for simple X-Y regression COUNT(Data) MAX(Data) MIN(Data) AVERAGE(Data) MEDIAN(Data) MODE(Data) ---------------- COUNT(Data) MAX(Data) MIN(Data) AVERAGE(Data) MEDIAN(Data) MODE.SNGL(Data) {MODE.MULT(Data)} GEOMEAN(Data) QUARTILE(Data, k) ---------------PERCENTILE(Data, p) ---------------STDEV(Data) ---------------STDEVP(Data) COVAR(XData, YData) STANDARDIZE(Data, μ, σ) GEOMEAN(Data) QUARTILE.INC(Data, k) QUARTILE.EXC(Data, k) PERCENTILE.INC(Data, p) PERCENTILE.EXC(Data, p) STDEV.S(Data) COVARIANCE.S(XData, YData) STDEV.P(Data) COVARIANCE.P(XDdata, YData) STANDARDIZE(Data, μ, σ) CORREL(XData, YData) AVEDEV(Data) SLOPE(XData, YData) INTERCEPT(XData, YData) RSQ(XData, YData) CORREL(XData, YData) AVEDEV(Data) SLOPE(XData, YData) INTERCEPT(XData, YData) RSQ(XData, YData) * In 2010, Excel changed many of its statistical functions. The pre-2010 functions will work in newer versions of Excel, but not vice-versa. For the latest information about Excel statistical functions, see https://support.office.com/ and Search “Excel Functions.” See Chapter 4, Section 4.5 for explanation of interpolation methods for percentiles and quartiles. Excel’s old method was rather unconventional, while its new method agrees with mainstream statistical packages. Discrete Probability Distributions Pre-2010 Excel 2013 Excel Binomial distribution PDF: Returns probability P (X 5 x) CDF: Returns probability P (X # x) Inverse CDF: Returns x for P (X # x) 5 α BINOMDIST(x, n, π, 0) BINOMDIST(x, n, π, 1) CRITBINOM(n, π, α) BINOM.DIST(x, n, π, 0) BINOM.DIST(x, n, π, 1) BINOM.INV(n, π, α) Poisson distribution PDF: Returns probability P (X 5 x) CDF: Returns probability P (X # x) Inverse CDF: Returns x for P (X # x) 5 α POISSON(x, λ, 0) POISSON (x, λ, 1) ---------------- POISSON.DIST(x, λ,0) POISSON.DIST(x, λ,1) ---------------- Hypergeometric distribution PDF: Returns probability P (X 5 x) CDF: Returns probability P (X # x) Inverse CDF: Returns x for P (X # x) 5 α HYPGEOMDIST(x, n, s, N) ------------------------------- HYPGEOM.DIST(x, n, s, N, 0) HYPGEOM.DIST(x, n, s, N, 1) ---------------813 814 Appendix J Continuous Probability Distributions Pre-2010 Excel 2013 Excel Normal distribution PDF: Returns height of f (x) CDF: Returns probability P (X # x) Inverse CDF: Returns x for P (X # x) 5 α NORMDIST(x, μ, σ, 0) NORMDIST(x, μ, σ, 1) NORMINV(α, μ, σ) NORM.DIST(x, μ, σ, 0) NORM.DIST(x, μ, σ, 1) NORM.INV(α, μ, σ) Standard normal distribution PDF: Returns height of f (z) CDF: Returns probability P (Z # z) Inverse CDF: Returns z for P (Z # z) 5 α ---------------NORMSDIST(z) NORMSINV(α) NORM.S.DIST(z, 0) NORM.S.DIST(z, 1) NORM.S.INV(α) Exponential distribution PDF: Returns height of f (x) CDF: Returns probability P (X # x) Inverse CDF: Returns x for P (X # x) 5 α EXPONDIST(x, λ, 0) EXPONDIST(x, λ, 1) ---------------- EXPON.DIST(x, λ, 0) EXPON.DIST(x, λ, 1) ---------------- Student’s t distribution PDF: Returns height of f (t) CDF: Returns probability P (t # t0) Inverse CDF: Returns t0 for P (t # t0) 5 α ---------------1-TDIST(t0, df, 1) only if t0 . 0 5TINV(α, df ) for two-tailed test T.DIST(t, df, 0) T.DIST(t0, df, 1) T.INV(α, df ) F distribution PDF: Returns height of f (x) CDF: Returns probability P (X # x) Inverse CDF: Returns F0 for P (F # F0) 5 α ---------------1-FDIST(x, df1, df2) FINV(1 2 α, df1, df2) F.DIST(x, df1, df2, 0) F.DIST(x, df1, df2, 1) F.INV(α, df1, df2) Common Hypothesis Tests Pre-2010 Excel 2013 Excel Normal distribution* Left-tailed p-value for test statistic zcalc Right-tailed p-value for test statistic zcalc Two-tailed p-value for test statistic zcalc Critical z value for left-tailed test at α Critical z value for right-tailed test at α Critical z values for two-tailed test at α NORMSDIST(zcalc) 1-NORMSDIST(zcalc) 2*(1-NORMSDIST(|zcalc|)) NORMSINV(α) NORMSINV(1 2 α) 6NORMSINV(αy2) NORM.S.DIST(zcalc, 1) 1-NORM.S.DIST(zcalc, 1) 2*(1-NORM.S.DIST(|zcalc|, 1)) NORM.S.INV(α) NORM.S.INV(1 2 α) 6NORM.S.INV(αy2) Student’s t distribution* Left-tailed p-value for test statistic tcalc Right-tailed p-value for test statistic tcalc Two-tailed p-value for test statistic tcalc Critical value of tα for left-tailed test at α Critical value of tα for right-tailed test at α Critical values of tα y2 for two-tailed test at α TDIST(|tcalc|, df, 1) TDIST(tcalc, df, 1) TDIST(|tcalc|, df, 2) 2TINV(2α, df ) TINV(2α, df ) 6TINV(α, df ) T.DIST(tcalc, df, 1) T.DIST.RT(tcalc, df ) T.DIST.2T(|tcalc|, df ) T.INV(α, df ) T.INV(1 2 α, df ) 6T.INV.2T(α, df ) F distribution Left-tailed p-value for test statistic Fcalc , 1 Right-tailed p-value for test statistic Fcalc . 1 Two-tailed p-value for folded Fcalc test Critical value for left-tailed test at α Critical value for right-tailed test at α Critical value for folded F test at α 1-FDIST( Fcalc , df1, df2) FDIST(Fcalc, df1, df2) 2*FDIST(Fcalc, df1, df2) 1yFINV(α, df2, df1) FINV(α, df1, df2) FINV(αy2, df1, df2) F.DIST(Fcalc, df1, df2, 1) F.DIST.RT(Fcalc, df1, df2) 2*F.DIST.RT(Fcalc, df1, df2) F.INV(α, df1, df2) F.INV(1 2 α, df1, df2) F.INV.RT(αy2, df1, df2) Chi-square distribution Left-tailed p-value for test statistic χ2calc Right-tailed p-value for test statistic χ 2calc Two-tailed p-value for test statistic χ 2calc Critical value for left-tailed test at α Critical value for right-tailed test at α Critical value for two-tailed test at α 1-CHIDIST(χ 2calc, df ) CHIDIST(χ 2calc, df ) 2*CHIDIST(χ 2calc, df ) CHIINV(1 2 α, df ) CHIINV(α, df ) CHIINV(αy2, df ) CHISQ.DIST(χ 2calc, df, 1) CHISQ.DIST.RT(χ 2calc, df ) 2*CHISQ.DIST.RT(χ 2calc, df ) CHISQ.INV(α, df ) CHISQ.INV.RT(α, df ) CHISQ.INV(1 2 αy2, df ) *For the normal and Student’s t distributions, the symbols |zcalc| and |tcalc| are used to denote absolute values for functions that require a positive argument. The 6 symbol is used in two-tailed z and t tests to indicate that left- and right-tail critical values are the same except for sign. Appendix J 815 Hypothesis Test Calculations Pre-2010 Excel 2013 Excel t-test for two means: returns a two-tailed p-value for a test of zero difference in two data arrays TTEST(Data1, Data2, Tails, Type) T.TEST(Data1, Data2, Tails, Type) where Tails 5 1 or 2 and Type 1 5 paired (must have n1 5 n2) 2 5 equal variances assumed 3 5 unequal variances assumed where Tails 5 1 or 2 and Type 1 5 paired (must have n1 5 n2) 2 5 equal variances assumed 3 5 unequal variances assumed F-test of two variances: returns a two-tailed p-value for equality of variances in two arrays FTEST(Data1, Data2) F.TEST(Data1, Data2) χ 2 goodness-of-fit test of k frequencies: returns a two-tailed p-value assuming k 2 1 degrees of freedom (assumes no parameters estimated). No warning if array frequencies do not have the same sum (as they should). CHITEST(Data1, Data2) CHISQ.TEST(Data1, Data2) where Data1 is an array of k observed frequencies and Data2 is an array of k expected frequencies where Data1 is an array of k observed frequencies and Data2 is an array of k expected frequencies Other Useful Stats Functions Pre-2010 Excel 2013 Excel Rank (average for ties) ---------------- RANK.AVG(x, Data, k) where x is a cell reference in array Data, k 5 0 (descending), k 5 1 (ascending) Rank (no correction for ties) RANK(x, Data, k) where x is a cell reference in array Data, k 5 0 (descending), k 5 1 (ascending) RANK(x, Data, k) where x is a cell reference in array Data, k 5 0 (descending), k 5 1 (ascending) Random uniform (0 # x , 1) RAND() RAND() Random integers (a # x # b) RANDBETWEEN(a, b) RANDBETWEEN(a, b) Confidence ___ interval half-width 6z σy√(n) (margin of error) using normal distribution with known standard deviation σ and confidence 1 2 α CONFIDENCE(α, σ, n) CONFIDENCE.NORM(α, σ, n) Confidence interval half-width 6t sy√n (margin of error) using Student’s t distribution with unknown standard deviation s and confidence 1 2 α ---------------- CONFIDENCE.T(α, s, n) Sum of squares of an array of data values around their mean DEVSQ(Data) DEVSQ(Data) Frequency of items in a data array using bin upper limits in bin array (highlight output range and use Ctrl-Shift-Enter) {FREQUENCY(Data, Bins)} {FREQUENCY(Data, Bins)} __