Appendix B: Statistical Tables

Table I: Random Number Table
Table II: Area under the standard normal curve to the left of a given [latex]z[/latex] score
Table III: Normal scores (theoretical quantiles) for normal Q-Q plot given a certain sample size [latex]n[/latex]
Table IV: Values of [latex]t_{\alpha}[/latex] of [latex]t[/latex]-distribution for a given degrees of freedom [latex]df[/latex]
Table V: Values of [latex]\chi^2_{\alpha}[/latex] of [latex]\chi^2[/latex]-distribution for a given degrees of freedom [latex]df[/latex]
Table VI: Values of [latex]F_{\alpha}[/latex] of [latex]F[/latex]-distribution for given degrees of freedom [latex]df_n[/latex] and [latex]df_d[/latex]

Acknowledgment

Table II (area under the standard normal), Table IV ([latex]t[/latex]-score table), Table V ([latex]\chi^2[/latex] table), and Table VI (F table) were adopted and modified from the statistical tables created by Haziq Jamil from https://haziqj.github.io/stat-tables.

Table I: Random Number Table
Line Column Number
Number 00-09 10-19 20-29 30-39 40-49
00 1 6 7 0 4 9 9 7 5 7 0 2 0 8 4 1 3 7 0 1 9 1 0 8 5 2 2 0 1 0 8 4 8 7 0 1 8 4 7 7 6 5 6 1 0 8 1 4 5 3
01 8 7 8 5 8 5 0 3 0 8 4 7 3 4 5 9 1 5 2 8 2 8 4 8 3 0 0 4 6 7 7 2 4 1 9 9 8 1 2 5 5 0 4 3 6 4 0 4 5 4
02 3 0 8 6 8 1 1 9 7 3 7 9 3 6 0 0 1 9 7 6 7 8 2 3 9 7 7 8 5 4 7 0 0 5 3 0 2 5 0 3 0 6 7 7 4 7 4 5 8 4
03 3 6 0 7 9 3 4 9 9 7 5 3 9 1 3 1 9 9 6 4 2 2 8 8 9 0 4 3 4 8 1 9 6 4 0 8 4 2 2 3 8 6 6 4 1 8 7 1 8 7
04 3 1 9 7 2 1 8 3 1 3 8 7 7 9 8 6 0 4 5 5 8 2 9 9 4 2 7 9 0 4 5 4 2 0 7 2 4 8 4 3 6 4 6 6 0 7 0 8 2 2
05 8 2 5 2 8 6 7 3 1 4 1 2 7 0 0 9 2 3 2 3 1 9 5 1 3 2 3 9 2 2 4 7 1 3 3 4 5 4 2 0 4 5 6 2 2 6 5 8 2 2
06 6 5 6 2 3 5 8 9 0 8 5 1 8 1 3 0 2 3 8 5 3 2 9 8 9 5 0 5 9 7 9 3 0 0 6 1 8 2 6 2 5 2 9 7 8 5 7 2 4 3
07 5 7 6 7 8 0 8 5 6 9 1 5 1 9 8 8 8 2 1 6 6 6 4 3 8 2 9 0 0 8 8 4 1 6 1 5 0 1 2 0 6 3 1 5 3 7 8 9 8 2
08 6 7 3 5 7 8 7 7 6 3 7 2 5 4 8 6 3 5 7 7 4 7 5 6 2 7 4 4 9 5 0 7 7 5 2 3 0 6 4 8 4 1 0 3 4 1 0 8 2 3
09 6 8 2 2 9 8 4 1 3 4 9 1 0 2 2 3 6 0 7 8 7 3 4 4 1 8 5 3 3 3 9 5 7 2 3 9 0 4 4 5 3 4 3 6 4 8 9 7 4 6
10 1 8 4 2 9 8 8 0 4 1 4 2 3 6 3 8 0 2 9 9 0 5 2 4 1 8 3 5 2 0 4 8 7 8 6 7 7 4 2 8 4 7 5 2 8 1 5 8 1 8
11 9 1 6 7 7 0 4 9 2 0 2 4 2 2 0 7 6 0 2 5 8 8 2 9 6 8 8 2 3 7 9 9 1 4 7 7 1 6 9 1 0 8 4 9 8 9 1 9 9 0
12 9 7 3 2 0 4 4 1 6 4 3 6 0 8 7 0 2 9 7 4 7 8 6 4 6 1 0 8 4 5 6 4 4 5 0 3 9 2 0 5 4 9 4 4 6 8 9 8 3 9
13 7 1 9 5 2 7 1 9 9 0 9 6 9 5 2 8 9 6 4 5 2 1 9 5 3 9 5 6 7 4 0 3 3 0 7 9 4 5 8 0 6 9 3 9 5 5 3 1 6 6
14 7 4 4 3 4 2 3 2 5 1 6 9 3 8 7 1 1 4 3 5 8 5 2 9 7 2 9 3 6 0 9 1 1 6 7 3 1 9 9 9 9 1 9 5 2 9 9 9 7 6
15 7 5 6 2 5 5 4 0 7 2 1 5 5 9 6 1 9 7 6 0 8 2 2 0 5 3 8 6 0 2 8 1 5 7 1 4 2 9 0 5 7 3 0 7 2 2 1 4 9 8
16 7 8 0 6 8 6 1 7 9 9 0 4 1 4 9 6 0 1 8 2 2 9 8 8 6 9 8 3 3 1 1 6 5 2 2 0 2 8 7 7 8 8 4 3 1 8 9 7 8 0
17 1 6 9 0 4 4 5 9 9 8 3 2 4 7 6 6 6 1 9 3 6 1 1 8 8 5 8 1 7 7 1 3 3 7 7 9 3 9 5 4 9 2 1 4 0 9 7 7 1 3
18 0 2 4 2 5 0 0 5 4 8 5 9 4 9 0 2 7 8 6 8 3 6 7 0 0 4 1 3 9 0 5 7 1 5 3 7 7 3 6 1 2 4 6 2 8 5 7 5 3 0
19 7 4 8 1 3 1 9 2 1 5 4 4 6 0 5 2 1 4 6 7 6 2 7 7 6 2 2 8 9 2 0 0 3 9 4 4 2 2 4 9 9 6 6 9 7 0 2 2 4 1
Table II: Area under the standard normal curve for negative [latex]z[/latex]

Second decimal place in [latex]z[/latex]
0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 [latex]z[/latex]
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -3.9
0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.8
0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 -3.7
0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002 0.0002 -3.6
0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 -3.5
0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 -3.4
0.0003 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0005 0.0005 0.0005 -3.3
0.0005 0.0005 0.0005 0.0006 0.0006 0.0006 0.0006 0.0006 0.0007 0.0007 -3.2
0.0007 0.0007 0.0008 0.0008 0.0008 0.0008 0.0009 0.0009 0.0009 0.0010 -3.1
0.0010 0.0010 0.0011 0.0011 0.0011 0.0012 0.0012 0.0013 0.0013 0.0013 -3.0
0.0014 0.0014 0.0015 0.0015 0.0016 0.0016 0.0017 0.0018 0.0018 0.0019 -2.9
0.0019 0.0020 0.0021 0.0021 0.0022 0.0023 0.0023 0.0024 0.0025 0.0026 -2.8
0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 0.0032 0.0033 0.0034 0.0035 -2.7
0.0036 0.0037 0.0038 0.0039 0.0040 0.0041 0.0043 0.0044 0.0045 0.0047 -2.6
0.0048 0.0049 0.0051 0.0052 0.0054 0.0055 0.0057 0.0059 0.0060 0.0062 -2.5
0.0064 0.0066 0.0068 0.0069 0.0071 0.0073 0.0075 0.0078 0.0080 0.0082 -2.4
0.0084 0.0087 0.0089 0.0091 0.0094 0.0096 0.0099 0.0102 0.0104 0.0107 -2.3
0.0110 0.0113 0.0116 0.0119 0.0122 0.0125 0.0129 0.0132 0.0136 0.0139 -2.2
0.0143 0.0146 0.0150 0.0154 0.0158 0.0162 0.0166 0.0170 0.0174 0.0179 -2.1
0.0183 0.0188 0.0192 0.0197 0.0202 0.0207 0.0212 0.0217 0.0222 0.0228 -2.0
0.0233 0.0239 0.0244 0.0250 0.0256 0.0262 0.0268 0.0274 0.0281 0.0287 -1.9
0.0294 0.0301 0.0307 0.0314 0.0322 0.0329 0.0336 0.0344 0.0351 0.0359 -1.8
0.0367 0.0375 0.0384 0.0392 0.0401 0.0409 0.0418 0.0427 0.0436 0.0446 -1.7
0.0455 0.0465 0.0475 0.0485 0.0495 0.0505 0.0516 0.0526 0.0537 0.0548 -1.6
0.0559 0.0571 0.0582 0.0594 0.0606 0.0618 0.0630 0.0643 0.0655 0.0668 -1.5
0.0681 0.0694 0.0708 0.0721 0.0735 0.0749 0.0764 0.0778 0.0793 0.0808 -1.4
0.0823 0.0838 0.0853 0.0869 0.0885 0.0901 0.0918 0.0934 0.0951 0.0968 -1.3
0.0985 0.1003 0.1020 0.1038 0.1056 0.1075 0.1093 0.1112 0.1131 0.1151 -1.2
0.1170 0.1190 0.1210 0.1230 0.1251 0.1271 0.1292 0.1314 0.1335 0.1357 -1.1
0.1379 0.1401 0.1423 0.1446 0.1469 0.1492 0.1515 0.1539 0.1562 0.1587 -1.0
0.1611 0.1635 0.1660 0.1685 0.1711 0.1736 0.1762 0.1788 0.1814 0.1841 -0.9
0.1867 0.1894 0.1922 0.1949 0.1977 0.2005 0.2033 0.2061 0.2090 0.2119 -0.8
0.2148 0.2177 0.2206 0.2236 0.2266 0.2296 0.2327 0.2358 0.2389 0.2420 -0.7
0.2451 0.2483 0.2514 0.2546 0.2578 0.2611 0.2643 0.2676 0.2709 0.2743 -0.6
0.2776 0.2810 0.2843 0.2877 0.2912 0.2946 0.2981 0.3015 0.3050 0.3085 -0.5
0.3121 0.3156 0.3192 0.3228 0.3264 0.3300 0.3336 0.3372 0.3409 0.3446 -0.4
0.3483 0.3520 0.3557 0.3594 0.3632 0.3669 0.3707 0.3745 0.3783 0.3821 -0.3
0.3859 0.3897 0.3936 0.3974 0.4013 0.4052 0.4090 0.4129 0.4168 0.4207 -0.2
0.4247 0.4286 0.4325 0.4364 0.4404 0.4443 0.4483 0.4522 0.4562 0.4602 -0.1
0.4641 0.4681 0.4721 0.4761 0.4801 0.4840 0.4880 0.4920 0.4960 0.5000 -0.0
Table II: Area under the standard normal curve for positive [latex]z[/latex]

Second decimal place in [latex]z[/latex]
[latex]z[/latex] 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Table III: Normal scores for normal Q-Q plot

Sample size [latex]n[/latex]
Rank 5 6 7 8 9 10 11 12 13 14 15 16 17
1 -1.18 -1.28 -1.36 -1.43 -1.49 -1.55 -1.59 -1.64 -1.67 -1.71 -1.74 -1.77 -1.80
2 -0.50 -0.64 -0.76 -0.85 -0.93 -1.00 -1.06 -1.11 -1.16 -1.21 -1.25 -1.28 -1.32
3 0.00 -0.20 -0.35 -0.47 -0.57 -0.66 -0.73 -0.79 -0.85 -0.90 -0.95 -0.99 -1.03
4 0.50 0.20 0.00 -0.15 -0.27 -0.38 -0.46 -0.54 -0.60 -0.66 -0.71 -0.76 -0.81
5 1.18 0.64 0.35 0.15 0.00 -0.12 -0.22 -0.31 -0.39 -0.45 -0.51 -0.57 -0.62
6 1.28 0.76 0.47 0.27 0.12 0.00 -0.10 -0.19 -0.27 -0.33 -0.40 -0.45
7 1.36 0.85 0.57 0.38 0.22 0.10 0.00 -0.09 -0.17 -0.23 -0.29
8 1.43 0.93 0.66 0.46 0.31 0.19 0.09 0.00 -0.08 -0.15
9 1.49 1.00 0.73 0.54 0.39 0.27 0.17 0.08 0.00
10 1.55 1.06 0.79 0.60 0.45 0.33 0.23 0.15
11 1.59 1.11 0.85 0.66 0.51 0.40 0.29
12 1.64 1.16 0.90 0.71 0.57 0.45
13 1.67 1.21 0.95 0.76 0.62
14 1.71 1.25 0.99 0.81
15 1.74 1.28 1.03
16 1.77 1.32
17 1.80
Sample size [latex]n[/latex]
Rank 18 19 20 21 22 23 24 25 26 27 28 29 30
1 -1.82 -1.85 -1.87 -1.89 -1.91 -1.93 -1.95 -1.96 -1.98 -2.00 -2.01 -2.03 -2.04
2 -1.35 -1.38 -1.40 -1.43 -1.45 -1.48 -1.50 -1.52 -1.54 -1.56 -1.58 -1.59 -1.61
3 -1.06 -1.10 -1.13 -1.16 -1.19 -1.21 -1.24 -1.26 -1.28 -1.30 -1.32 -1.34 -1.36
4 -0.85 -0.88 -0.92 -0.95 -0.98 -1.01 -1.04 -1.06 -1.09 -1.11 -1.13 -1.16 -1.18
5 -0.66 -0.71 -0.74 -0.78 -0.81 -0.85 -0.88 -0.90 -0.93 -0.96 -0.98 -1.00 -1.02
6 -0.50 -0.55 -0.59 -0.63 -0.67 -0.70 -0.73 -0.76 -0.79 -0.82 -0.84 -0.87 -0.89
7 -0.35 -0.40 -0.45 -0.49 -0.53 -0.57 -0.60 -0.64 -0.67 -0.70 -0.72 -0.75 -0.78
8 -0.21 -0.26 -0.31 -0.36 -0.41 -0.45 -0.48 -0.52 -0.55 -0.58 -0.61 -0.64 -0.67
9 -0.07 -0.13 -0.19 -0.24 -0.29 -0.33 -0.37 -0.41 -0.44 -0.48 -0.51 -0.54 -0.57
10 0.07 0.00 -0.06 -0.12 -0.17 -0.22 -0.26 -0.30 -0.34 -0.38 -0.41 -0.44 -0.47
11 0.21 0.13 0.06 0.00 -0.06 -0.11 -0.16 -0.20 -0.24 -0.28 -0.32 -0.35 -0.38
12 0.35 0.26 0.19 0.12 0.06 0.00 -0.05 -0.10 -0.14 -0.19 -0.22 -0.26 -0.29
13 0.50 0.40 0.31 0.24 0.17 0.11 0.05 0.00 -0.05 -0.09 -0.13 -0.17 -0.21
14 0.66 0.55 0.45 0.36 0.29 0.22 0.16 0.10 0.05 0.00 -0.04 -0.09 -0.12
15 0.85 0.71 0.59 0.49 0.41 0.33 0.26 0.20 0.14 0.09 0.04 0.00 -0.04
16 1.06 0.88 0.74 0.63 0.53 0.45 0.37 0.30 0.24 0.19 0.13 0.09 0.04
17 1.35 1.10 0.92 0.78 0.67 0.57 0.48 0.41 0.34 0.28 0.22 0.17 0.12
18 1.82 1.38 1.13 0.95 0.81 0.70 0.60 0.52 0.44 0.38 0.32 0.26 0.21
19 1.85 1.40 1.16 0.98 0.85 0.73 0.64 0.55 0.48 0.41 0.35 0.29
20 1.87 1.43 1.19 1.01 0.88 0.76 0.67 0.58 0.51 0.44 0.38
21 1.89 1.45 1.21 1.04 0.90 0.79 0.70 0.61 0.54 0.47
22 1.91 1.48 1.24 1.06 0.93 0.82 0.72 0.64 0.57
23 1.93 1.50 1.26 1.09 0.96 0.84 0.75 0.67
24 1.95 1.52 1.28 1.11 0.98 0.87 0.78
25 1.96 1.54 1.30 1.13 1.00 0.89
26 1.98 1.56 1.32 1.16 1.02
27 2.00 1.58 1.34 1.18
28 2.01 1.59 1.36
29 2.03 1.61
30 2.04
Table IV: Values of [latex]t_{\alpha}[/latex] of [latex]t[/latex]-distribution

[latex]\alpha[/latex]: Area to the Right of [latex]t_{\alpha}[/latex]
[latex]df[/latex] 0.40 0.30 0.20 0.15 0.10 0.05 0.025 0.010 0.0075 0.0050 0.0025 0.0005
1 0.325 0.727 1.376 1.963 3.078 6.314 12.706 31.821 42.433 63.657 127.321 636.619
2 0.289 0.617 1.061 1.386 1.886 2.920 4.303 6.965 8.073 9.925 14.089 31.599
3 0.277 0.584 0.978 1.250 1.638 2.353 3.182 4.541 5.047 5.841 7.453 12.924
4 0.271 0.569 0.941 1.190 1.533 2.132 2.776 3.747 4.088 4.604 5.598 8.610
5 0.267 0.559 0.920 1.156 1.476 2.015 2.571 3.365 3.634 4.032 4.773 6.869
6 0.265 0.553 0.906 1.134 1.440 1.943 2.447 3.143 3.372 3.707 4.317 5.959
7 0.263 0.549 0.896 1.119 1.415 1.895 2.365 2.998 3.203 3.499 4.029 5.408
8 0.262 0.546 0.889 1.108 1.397 1.860 2.306 2.896 3.085 3.355 3.833 5.041
9 0.261 0.543 0.883 1.100 1.383 1.833 2.262 2.821 2.998 3.250 3.690 4.781
10 0.260 0.542 0.879 1.093 1.372 1.812 2.228 2.764 2.932 3.169 3.581 4.587
11 0.260 0.540 0.876 1.088 1.363 1.796 2.201 2.718 2.879 3.106 3.497 4.437
12 0.259 0.539 0.873 1.083 1.356 1.782 2.179 2.681 2.836 3.055 3.428 4.318
13 0.259 0.538 0.870 1.079 1.350 1.771 2.160 2.650 2.801 3.012 3.372 4.221
14 0.258 0.537 0.868 1.076 1.345 1.761 2.145 2.624 2.771 2.977 3.326 4.140
15 0.258 0.536 0.866 1.074 1.341 1.753 2.131 2.602 2.746 2.947 3.286 4.073
16 0.258 0.535 0.865 1.071 1.337 1.746 2.120 2.583 2.724 2.921 3.252 4.015
17 0.257 0.534 0.863 1.069 1.333 1.740 2.110 2.567 2.706 2.898 3.222 3.965
18 0.257 0.534 0.862 1.067 1.330 1.734 2.101 2.552 2.689 2.878 3.197 3.922
19 0.257 0.533 0.861 1.066 1.328 1.729 2.093 2.539 2.674 2.861 3.174 3.883
20 0.257 0.533 0.860 1.064 1.325 1.725 2.086 2.528 2.661 2.845 3.153 3.850
21 0.257 0.532 0.859 1.063 1.323 1.721 2.080 2.518 2.649 2.831 3.135 3.819
22 0.256 0.532 0.858 1.061 1.321 1.717 2.074 2.508 2.639 2.819 3.119 3.792
23 0.256 0.532 0.858 1.060 1.319 1.714 2.069 2.500 2.629 2.807 3.104 3.768
24 0.256 0.531 0.857 1.059 1.318 1.711 2.064 2.492 2.620 2.797 3.091 3.745
25 0.256 0.531 0.856 1.058 1.316 1.708 2.060 2.485 2.612 2.787 3.078 3.725
26 0.256 0.531 0.856 1.058 1.315 1.706 2.056 2.479 2.605 2.779 3.067 3.707
27 0.256 0.531 0.855 1.057 1.314 1.703 2.052 2.473 2.598 2.771 3.057 3.690
28 0.256 0.530 0.855 1.056 1.313 1.701 2.048 2.467 2.592 2.763 3.047 3.674
29 0.256 0.530 0.854 1.055 1.311 1.699 2.045 2.462 2.586 2.756 3.038 3.659
30 0.256 0.530 0.854 1.055 1.310 1.697 2.042 2.457 2.581 2.750 3.030 3.646
31 0.256 0.530 0.853 1.054 1.309 1.696 2.040 2.453 2.576 2.744 3.022 3.633
32 0.255 0.530 0.853 1.054 1.309 1.694 2.037 2.449 2.571 2.738 3.015 3.622
33 0.255 0.530 0.853 1.053 1.308 1.692 2.035 2.445 2.566 2.733 3.008 3.611
34 0.255 0.529 0.852 1.052 1.307 1.691 2.032 2.441 2.562 2.728 3.002 3.601
35 0.255 0.529 0.852 1.052 1.306 1.690 2.030 2.438 2.558 2.724 2.996 3.591
36 0.255 0.529 0.852 1.052 1.306 1.688 2.028 2.434 2.555 2.719 2.990 3.582
37 0.255 0.529 0.851 1.051 1.305 1.687 2.026 2.431 2.551 2.715 2.985 3.574
38 0.255 0.529 0.851 1.051 1.304 1.686 2.024 2.429 2.548 2.712 2.980 3.566
39 0.255 0.529 0.851 1.050 1.304 1.685 2.023 2.426 2.545 2.708 2.976 3.558
40 0.255 0.529 0.851 1.050 1.303 1.684 2.021 2.423 2.542 2.704 2.971 3.551
41 0.255 0.529 0.850 1.050 1.303 1.683 2.020 2.421 2.539 2.701 2.967 3.544
42 0.255 0.528 0.850 1.049 1.302 1.682 2.018 2.418 2.537 2.698 2.963 3.538
43 0.255 0.528 0.850 1.049 1.302 1.681 2.017 2.416 2.534 2.695 2.959 3.532
44 0.255 0.528 0.850 1.049 1.301 1.680 2.015 2.414 2.532 2.692 2.956 3.526
45 0.255 0.528 0.850 1.049 1.301 1.679 2.014 2.412 2.529 2.690 2.952 3.520
46 0.255 0.528 0.850 1.048 1.300 1.679 2.013 2.410 2.527 2.687 2.949 3.515
47 0.255 0.528 0.849 1.048 1.300 1.678 2.012 2.408 2.525 2.685 2.946 3.510
48 0.255 0.528 0.849 1.048 1.299 1.677 2.011 2.407 2.523 2.682 2.943 3.505
49 0.255 0.528 0.849 1.048 1.299 1.677 2.010 2.405 2.521 2.680 2.940 3.500
50 0.255 0.528 0.849 1.047 1.299 1.676 2.009 2.403 2.519 2.678 2.937 3.496
[latex]\alpha[/latex]: Area to the Right of [latex]t_{\alpha}[/latex]
[latex]df[/latex] 0.40 0.30 0.20 0.15 0.10 0.05 0.025 0.010 0.0075 0.0050 0.0025 0.0005
51 0.255 0.528 0.849 1.047 1.298 1.675 2.008 2.402 2.518 2.676 2.934 3.492
52 0.255 0.528 0.849 1.047 1.298 1.675 2.007 2.400 2.516 2.674 2.932 3.488
53 0.255 0.528 0.848 1.047 1.298 1.674 2.006 2.399 2.514 2.672 2.929 3.484
54 0.255 0.528 0.848 1.046 1.297 1.674 2.005 2.397 2.513 2.670 2.927 3.480
55 0.255 0.527 0.848 1.046 1.297 1.673 2.004 2.396 2.511 2.668 2.925 3.476
56 0.255 0.527 0.848 1.046 1.297 1.673 2.003 2.395 2.510 2.667 2.923 3.473
57 0.255 0.527 0.848 1.046 1.297 1.672 2.002 2.394 2.508 2.665 2.920 3.470
58 0.255 0.527 0.848 1.046 1.296 1.672 2.002 2.392 2.507 2.663 2.918 3.466
59 0.254 0.527 0.848 1.046 1.296 1.671 2.001 2.391 2.506 2.662 2.916 3.463
60 0.254 0.527 0.848 1.045 1.296 1.671 2.000 2.390 2.504 2.660 2.915 3.460
61 0.254 0.527 0.848 1.045 1.296 1.670 2.000 2.389 2.503 2.659 2.913 3.457
62 0.254 0.527 0.847 1.045 1.295 1.670 1.999 2.388 2.502 2.657 2.911 3.454
63 0.254 0.527 0.847 1.045 1.295 1.669 1.998 2.387 2.501 2.656 2.909 3.452
64 0.254 0.527 0.847 1.045 1.295 1.669 1.998 2.386 2.500 2.655 2.908 3.449
65 0.254 0.527 0.847 1.045 1.295 1.669 1.997 2.385 2.499 2.654 2.906 3.447
66 0.254 0.527 0.847 1.045 1.295 1.668 1.997 2.384 2.498 2.652 2.904 3.444
67 0.254 0.527 0.847 1.045 1.294 1.668 1.996 2.383 2.497 2.651 2.903 3.442
68 0.254 0.527 0.847 1.044 1.294 1.668 1.995 2.382 2.496 2.650 2.902 3.439
69 0.254 0.527 0.847 1.044 1.294 1.667 1.995 2.382 2.495 2.649 2.900 3.437
70 0.254 0.527 0.847 1.044 1.294 1.667 1.994 2.381 2.494 2.648 2.899 3.435
71 0.254 0.527 0.847 1.044 1.294 1.667 1.994 2.380 2.493 2.647 2.897 3.433
72 0.254 0.527 0.847 1.044 1.293 1.666 1.993 2.379 2.492 2.646 2.896 3.431
73 0.254 0.527 0.847 1.044 1.293 1.666 1.993 2.379 2.491 2.645 2.895 3.429
74 0.254 0.527 0.847 1.044 1.293 1.666 1.993 2.378 2.490 2.644 2.894 3.427
75 0.254 0.527 0.846 1.044 1.293 1.665 1.992 2.377 2.490 2.643 2.892 3.425
76 0.254 0.527 0.846 1.044 1.293 1.665 1.992 2.376 2.489 2.642 2.891 3.423
77 0.254 0.527 0.846 1.043 1.293 1.665 1.991 2.376 2.488 2.641 2.890 3.421
78 0.254 0.527 0.846 1.043 1.292 1.665 1.991 2.375 2.487 2.640 2.889 3.420
79 0.254 0.527 0.846 1.043 1.292 1.664 1.990 2.374 2.487 2.640 2.888 3.418
80 0.254 0.526 0.846 1.043 1.292 1.664 1.990 2.374 2.486 2.639 2.887 3.416
81 0.254 0.526 0.846 1.043 1.292 1.664 1.990 2.373 2.485 2.638 2.886 3.415
82 0.254 0.526 0.846 1.043 1.292 1.664 1.989 2.373 2.485 2.637 2.885 3.413
83 0.254 0.526 0.846 1.043 1.292 1.663 1.989 2.372 2.484 2.636 2.884 3.412
84 0.254 0.526 0.846 1.043 1.292 1.663 1.989 2.372 2.483 2.636 2.883 3.410
85 0.254 0.526 0.846 1.043 1.292 1.663 1.988 2.371 2.483 2.635 2.882 3.409
86 0.254 0.526 0.846 1.043 1.291 1.663 1.988 2.370 2.482 2.634 2.881 3.407
87 0.254 0.526 0.846 1.043 1.291 1.663 1.988 2.370 2.482 2.634 2.880 3.406
88 0.254 0.526 0.846 1.043 1.291 1.662 1.987 2.369 2.481 2.633 2.880 3.405
89 0.254 0.526 0.846 1.043 1.291 1.662 1.987 2.369 2.481 2.632 2.879 3.403
90 0.254 0.526 0.846 1.042 1.291 1.662 1.987 2.368 2.480 2.632 2.878 3.402
91 0.254 0.526 0.846 1.042 1.291 1.662 1.986 2.368 2.479 2.631 2.877 3.401
92 0.254 0.526 0.846 1.042 1.291 1.662 1.986 2.368 2.479 2.630 2.876 3.399
93 0.254 0.526 0.846 1.042 1.291 1.661 1.986 2.367 2.478 2.630 2.876 3.398
94 0.254 0.526 0.845 1.042 1.291 1.661 1.986 2.367 2.478 2.629 2.875 3.397
95 0.254 0.526 0.845 1.042 1.291 1.661 1.985 2.366 2.477 2.629 2.874 3.396
96 0.254 0.526 0.845 1.042 1.290 1.661 1.985 2.366 2.477 2.628 2.873 3.395
97 0.254 0.526 0.845 1.042 1.290 1.661 1.985 2.365 2.476 2.627 2.873 3.394
98 0.254 0.526 0.845 1.042 1.290 1.661 1.984 2.365 2.476 2.627 2.872 3.393
99 0.254 0.526 0.845 1.042 1.290 1.660 1.984 2.365 2.476 2.626 2.871 3.392
100 0.254 0.526 0.845 1.042 1.290 1.660 1.984 2.364 2.475 2.626 2.871 3.390
200 0.254 0.525 0.843 1.039 1.286 1.653 1.972 2.345 2.454 2.601 2.839 3.340
400 0.254 0.525 0.843 1.038 1.284 1.649 1.966 2.336 2.443 2.588 2.823 3.315
1000 0.253 0.525 0.842 1.037 1.282 1.646 1.962 2.330 2.437 2.581 2.813 3.300
[latex]\infty[/latex] 0.253 0.524 0.842 1.036 1.282 1.645 1.960 2.326 2.432 2.576 2.807 3.291
Table V: Values of [latex]\chi^2_{\alpha}[/latex] of [latex]\chi^2[/latex]-distribution

[latex]\alpha[/latex]: Area to the Right of [latex]\chi^2_{\alpha}[/latex]
[latex]df[/latex] 0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005
1 [latex]0.0^4393[/latex] [latex]0.0^3157[/latex] [latex]0.0^3982[/latex] [latex]0.0^2393[/latex] 0.016 2.706 3.841 5.024 6.635 7.879
2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
Table VI: Values of [latex]F_{\alpha}[/latex] of [latex]F[/latex]-distribution

[latex]df_n[/latex]
[latex]\alpha[/latex] 1 2 3 4 5 6 7 8 9 10
[latex]df_d=1[/latex]
0.5 1.000 1.500 1.709 1.823 1.894 1.942 1.977 2.004 2.025 2.042
0.1 39.9 49.5 53.6 55.8 57.2 58.2 58.9 59.4 59.9 60.2
0.05 161 199 216 225 230 234 237 239 241 242
0.025 648 799 864 900 922 937 948 957 963 969
0.01 4052 4999 5403 5625 5764 5859 5928 5981 6022 6056
0.005 16211 19999 21615 22500 23056 23437 23715 23925 24091 24224
0.001 405284 499999 540379 562500 576405 585937 592873 598144 602284 605621
[latex]df_d=2[/latex]
0.5 0.667 1.000 1.135 1.207 1.252 1.282 1.305 1.321 1.334 1.345
0.1 8.526 9.000 9.162 9.243 9.293 9.326 9.349 9.367 9.381 9.392
0.05 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40
0.025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40
0.01 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40
0.005 198.50 199.00 199.17 199.25 199.30 199.33 199.36 199.37 199.39 199.40
0.001 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39 999.40
[latex]df_d=3[/latex]
0.5 0.585 0.881 1.000 1.063 1.102 1.129 1.148 1.163 1.174 1.183
0.1 5.538 5.462 5.391 5.343 5.309 5.285 5.266 5.252 5.240 5.230
0.05 10.128 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786
0.025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42
0.01 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23
0.005 55.6 49.8 47.5 46.2 45.4 44.8 44.4 44.1 43.9 43.7
0.001 167.0 148.5 141.1 137.1 134.6 132.8 131.6 130.6 129.9 129.2
[latex]df_d=4[/latex]
0.5 0.549 0.828 0.941 1.000 1.037 1.062 1.080 1.093 1.104 1.113
0.1 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94 3.92
0.05 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96
0.025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84
0.01 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.7 14.5
0.005 31.3 26.3 24.3 23.2 22.5 22.0 21.6 21.4 21.1 21.0
0.001 74.1 61.2 56.2 53.4 51.7 50.5 49.7 49.0 48.5 48.1
[latex]df_d=5[/latex]
0.5 0.528 0.799 0.907 0.965 1.000 1.024 1.041 1.055 1.065 1.073
0.1 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32 3.30
0.05 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74
0.025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62
0.01 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 10.1
0.005 22.8 18.3 16.5 15.6 14.9 14.5 14.2 14.0 13.8 13.6
0.001 47.2 37.1 33.2 31.1 29.8 28.8 28.2 27.6 27.2 26.9
[latex]df_n[/latex]
[latex]\alpha[/latex] 11 12 13 14 15 20 30 60 120 [latex]\infty[/latex]
[latex]df_d=1[/latex]
0.5 2.06 2.07 2.08 2.09 2.09 2.12 2.15 2.17 2.18 2.20
0.1 60.5 60.7 60.9 61.1 61.2 61.7 62.3 62.8 63.1 63.3
0.05 243 244 245 245 246 248 250 252 253 254
0.025 973 977 980 983 985 993 1001 1010 1014 1018
0.01 6083 6106 6126 6143 6157 6209 6261 6313 6339 6366
0.005 24334 24426 24505 24572 24630 24836 25044 25253 25359 25464
0.001 608368 610668 612622 614303 615764 620908 626099 631337 633972 636619
[latex]df_d=2[/latex]
0.5 1.354 1.361 1.367 1.372 1.377 1.393 1.410 1.426 1.434 1.443
0.1 9.401 9.408 9.415 9.420 9.425 9.441 9.458 9.475 9.483 9.491
0.05 19.40 19.41 19.42 19.42 19.43 19.45 19.46 19.48 19.49 19.50
0.025 39.41 39.41 39.42 39.43 39.43 39.45 39.46 39.48 39.49 39.50
0.01 99.41 99.42 99.42 99.43 99.43 99.45 99.47 99.48 99.49 99.50
0.005 199.41 199.42 199.42 199.43 199.43 199.45 199.47 199.48 199.49 199.50
0.001 999.41 999.42 999.42 999.43 999.43 999.45 999.47 999.48 999.49 999.50
[latex]df_d=3[/latex]
0.5 1.191 1.197 1.203 1.207 1.211 1.225 1.239 1.254 1.261 1.268
0.1 5.222 5.216 5.210 5.205 5.200 5.184 5.168 5.151 5.143 5.134
0.05 8.763 8.745 8.729 8.715 8.703 8.660 8.617 8.572 8.549 8.526
0.025 14.37 14.34 14.30 14.28 14.25 14.17 14.08 13.99 13.95 13.90
0.01 27.13 27.05 26.98 26.92 26.87 26.69 26.50 26.32 26.22 26.13
0.005 43.5 43.4 43.3 43.2 43.1 42.8 42.5 42.1 42.0 41.8
0.001 128.7 128.3 128.0 127.6 127.4 126.4 125.4 124.5 124.0 123.5
[latex]df_d=4[/latex]
0.5 1.120 1.126 1.131 1.135 1.139 1.152 1.165 1.178 1.185 1.192
0.1 3.91 3.90 3.89 3.88 3.87 3.84 3.82 3.79 3.78 3.76
0.05 5.94 5.91 5.89 5.87 5.86 5.80 5.75 5.69 5.66 5.63
0.025 8.79 8.75 8.71 8.68 8.66 8.56 8.46 8.36 8.31 8.26
0.01 14.5 14.4 14.3 14.2 14.2 14.0 13.8 13.7 13.6 13.5
0.005 20.8 20.7 20.6 20.5 20.4 20.2 19.9 19.6 19.5 19.3
0.001 47.7 47.4 47.2 46.9 46.8 46.1 45.4 44.7 44.4 44.1
[latex]df_d=5[/latex]
0.5 1.080 1.085 1.090 1.094 1.098 1.111 1.123 1.136 1.143 1.149
0.1 3.28 3.27 3.26 3.25 3.24 3.21 3.17 3.14 3.12 3.10
0.05 4.70 4.68 4.66 4.64 4.62 4.56 4.50 4.43 4.40 4.36
0.025 6.57 6.52 6.49 6.46 6.43 6.33 6.23 6.12 6.07 6.02
0.01 10.0 9.9 9.8 9.8 9.7 9.6 9.4 9.2 9.1 9.0
0.005 13.5 13.4 13.3 13.2 13.1 12.9 12.7 12.4 12.3 12.1
0.001 26.6 26.4 26.2 26.1 25.9 25.4 24.9 24.3 24.1 23.8
[latex]df_n[/latex]
[latex]\alpha[/latex] 1 2 3 4 5 6 7 8 9 10
[latex]df_d=6[/latex]
0.5 0.515 0.780 0.886 0.942 0.977 1.000 1.017 1.030 1.040 1.048
0.1 3.776 3.463 3.289 3.181 3.108 3.055 3.014 2.983 2.958 2.937
0.05 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060
0.025 8.813 7.260 6.599 6.227 5.988 5.820 5.695 5.600 5.523 5.461
0.01 13.745 10.925 9.780 9.148 8.746 8.466 8.260 8.102 7.976 7.874
0.005 18.635 14.544 12.917 12.028 11.464 11.073 10.786 10.566 10.391 10.250
0.001 35.507 27.000 23.703 21.924 20.803 20.030 19.463 19.030 18.688 18.411
[latex]df_d=7[/latex]
0.5 0.506 0.767 0.871 0.926 0.960 0.983 1.000 1.013 1.022 1.030
0.1 3.589 3.257 3.074 2.961 2.883 2.827 2.785 2.752 2.725 2.703
0.05 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637
0.025 8.073 6.542 5.890 5.523 5.285 5.119 4.995 4.899 4.823 4.761
0.01 12.246 9.547 8.451 7.847 7.460 7.191 6.993 6.840 6.719 6.620
0.005 16.236 12.404 10.882 10.050 9.522 9.155 8.885 8.678 8.514 8.380
0.001 29.245 21.689 18.772 17.198 16.206 15.521 15.019 14.634 14.330 14.083
[latex]df_d=8[/latex]
0.5 0.499 0.757 0.860 0.915 0.948 0.971 0.988 1.000 1.010 1.018
0.1 3.458 3.113 2.924 2.806 2.726 2.668 2.624 2.589 2.561 2.538
0.05 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347
0.025 7.571 6.059 5.416 5.053 4.817 4.652 4.529 4.433 4.357 4.295
0.01 11.259 8.649 7.591 7.006 6.632 6.371 6.178 6.029 5.911 5.814
0.005 14.688 11.042 9.596 8.805 8.302 7.952 7.694 7.496 7.339 7.211
0.001 25.415 18.494 15.829 14.392 13.485 12.858 12.398 12.046 11.767 11.540
[latex]df_d=9[/latex]
0.5 0.494 0.749 0.852 0.906 0.939 0.962 0.978 0.990 1.000 1.008
0.1 3.360 3.006 2.813 2.693 2.611 2.551 2.505 2.469 2.440 2.416
0.05 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137
0.025 7.209 5.715 5.078 4.718 4.484 4.320 4.197 4.102 4.026 3.964
0.01 10.561 8.022 6.992 6.422 6.057 5.802 5.613 5.467 5.351 5.257
0.005 13.614 10.107 8.717 7.956 7.471 7.134 6.885 6.693 6.541 6.417
0.001 22.857 16.387 13.902 12.560 11.714 11.128 10.698 10.368 10.107 9.894
[latex]df_d=10[/latex]
0.5 0.490 0.743 0.845 0.899 0.932 0.954 0.971 0.983 0.992 1.000
0.1 3.285 2.924 2.728 2.605 2.522 2.461 2.414 2.377 2.347 2.323
0.05 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978
0.025 6.937 5.456 4.826 4.468 4.236 4.072 3.950 3.855 3.779 3.717
0.01 10.044 7.559 6.552 5.994 5.636 5.386 5.200 5.057 4.942 4.849
0.005 12.826 9.427 8.081 7.343 6.872 6.545 6.302 6.116 5.968 5.847
0.001 21.040 14.905 12.553 11.283 10.481 9.926 9.517 9.204 8.956 8.754
[latex]df_n[/latex]
[latex]\alpha[/latex] 11 12 13 14 15 20 30 60 120 [latex]\infty[/latex]
[latex]df_d=6[/latex]
0.5 1.054 1.060 1.065 1.069 1.072 1.084 1.097 1.109 1.116 1.122
0.1 2.920 2.905 2.892 2.881 2.871 2.836 2.800 2.762 2.742 2.722
0.05 4.027 4.000 3.976 3.956 3.938 3.874 3.808 3.740 3.705 3.669
0.025 5.410 5.366 5.329 5.297 5.269 5.168 5.065 4.959 4.904 4.849
0.01 7.790 7.718 7.657 7.605 7.559 7.396 7.229 7.057 6.969 6.880
0.005 10.133 10.034 9.950 9.877 9.814 9.589 9.358 9.122 9.001 8.879
0.001 18.182 17.989 17.824 17.682 17.559 17.120 16.672 16.214 15.981 15.745
[latex]df_d=7[/latex]
0.5 1.037 1.042 1.047 1.051 1.054 1.066 1.079 1.091 1.097 1.103
0.1 2.684 2.668 2.654 2.643 2.632 2.595 2.555 2.514 2.493 2.471
0.05 3.603 3.575 3.550 3.529 3.511 3.445 3.376 3.304 3.267 3.230
0.025 4.709 4.666 4.628 4.596 4.568 4.467 4.362 4.254 4.199 4.142
0.01 6.538 6.469 6.410 6.359 6.314 6.155 5.992 5.824 5.737 5.650
0.005 8.270 8.176 8.097 8.028 7.968 7.754 7.534 7.309 7.193 7.076
0.001 13.879 13.707 13.561 13.434 13.324 12.932 12.530 12.119 11.909 11.696
[latex]df_d=8[/latex]
0.5 1.024 1.029 1.034 1.038 1.041 1.053 1.065 1.077 1.083 1.089
0.1 2.519 2.502 2.488 2.475 2.464 2.425 2.383 2.339 2.316 2.293
0.05 3.313 3.284 3.259 3.237 3.218 3.150 3.079 3.005 2.967 2.928
0.025 4.243 4.200 4.162 4.130 4.101 3.999 3.894 3.784 3.728 3.670
0.01 5.734 5.667 5.609 5.559 5.515 5.359 5.198 5.032 4.946 4.859
0.005 7.104 7.015 6.938 6.872 6.814 6.608 6.396 6.177 6.065 5.951
0.001 11.352 11.194 11.060 10.943 10.841 10.480 10.109 9.727 9.532 9.334
[latex]df_d=9[/latex]
0.5 1.014 1.019 1.024 1.028 1.031 1.043 1.055 1.067 1.073 1.079
0.1 2.396 2.379 2.364 2.351 2.340 2.298 2.255 2.208 2.184 2.159
0.05 3.102 3.073 3.048 3.025 3.006 2.936 2.864 2.787 2.748 2.707
0.025 3.912 3.868 3.831 3.798 3.769 3.667 3.560 3.449 3.392 3.333
0.01 5.178 5.111 5.055 5.005 4.962 4.808 4.649 4.483 4.398 4.311
0.005 6.314 6.227 6.153 6.089 6.032 5.832 5.625 5.410 5.300 5.188
0.001 9.718 9.570 9.443 9.334 9.238 8.898 8.548 8.187 8.001 7.813
[latex]df_d=10[/latex]
0.5 1.006 1.012 1.016 1.020 1.023 1.035 1.047 1.059 1.064 1.070
0.1 2.302 2.284 2.269 2.255 2.244 2.201 2.155 2.107 2.082 2.055
0.05 2.943 2.913 2.887 2.865 2.845 2.774 2.700 2.621 2.580 2.538
0.025 3.665 3.621 3.583 3.550 3.522 3.419 3.311 3.198 3.140 3.080
0.01 4.772 4.706 4.650 4.601 4.558 4.405 4.247 4.082 3.996 3.909
0.005 5.746 5.661 5.589 5.526 5.471 5.274 5.071 4.859 4.750 4.639
0.001 8.586 8.445 8.324 8.220 8.129 7.804 7.469 7.122 6.944 6.762
[latex]df_n[/latex]
[latex]\alpha[/latex] 1 2 3 4 5 6 7 8 9 10
[latex]df_d=11[/latex]
0.5 0.486 0.739 0.840 0.893 0.926 0.948 0.964 0.977 0.986 0.994
0.1 3.225 2.860 2.660 2.536 2.451 2.389 2.342 2.304 2.274 2.248
0.05 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854
0.025 6.724 5.256 4.630 4.275 4.044 3.881 3.759 3.664 3.588 3.526
0.01 9.646 7.206 6.217 5.668 5.316 5.069 4.886 4.744 4.632 4.539
0.005 12.226 8.912 7.600 6.881 6.422 6.102 5.865 5.682 5.537 5.418
0.001 19.687 13.812 11.561 10.346 9.578 9.047 8.655 8.355 8.116 7.922
[latex]df_d=12[/latex]
0.5 0.484 0.735 0.835 0.888 0.921 0.943 0.959 0.972 0.981 0.989
0.1 3.177 2.807 2.606 2.480 2.394 2.331 2.283 2.245 2.214 2.188
0.05 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753
0.025 6.554 5.096 4.474 4.121 3.891 3.728 3.607 3.512 3.436 3.374
0.01 9.330 6.927 5.953 5.412 5.064 4.821 4.640 4.499 4.388 4.296
0.005 11.754 8.510 7.226 6.521 6.071 5.757 5.525 5.345 5.202 5.085
0.001 18.643 12.974 10.804 9.633 8.892 8.379 8.001 7.710 7.480 7.292
[latex]df_d=13[/latex]
0.5 0.481 0.731 0.832 0.885 0.917 0.939 0.955 0.967 0.977 0.984
0.1 3.136 2.763 2.560 2.434 2.347 2.283 2.234 2.195 2.164 2.138
0.05 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671
0.025 6.414 4.965 4.347 3.996 3.767 3.604 3.483 3.388 3.312 3.250
0.01 9.074 6.701 5.739 5.205 4.862 4.620 4.441 4.302 4.191 4.100
0.005 11.374 8.186 6.926 6.233 5.791 5.482 5.253 5.076 4.935 4.820
0.001 17.815 12.313 10.209 9.073 8.354 7.856 7.489 7.206 6.982 6.799
[latex]df_d=14[/latex]
0.5 0.479 0.729 0.828 0.881 0.914 0.936 0.952 0.964 0.973 0.981
0.1 3.102 2.726 2.522 2.395 2.307 2.243 2.193 2.154 2.122 2.095
0.05 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602
0.025 6.298 4.857 4.242 3.892 3.663 3.501 3.380 3.285 3.209 3.147
0.01 8.862 6.515 5.564 5.035 4.695 4.456 4.278 4.140 4.030 3.939
0.005 11.060 7.922 6.680 5.998 5.562 5.257 5.031 4.857 4.717 4.603
0.001 17.143 11.779 9.729 8.622 7.922 7.436 7.077 6.802 6.583 6.404
[latex]df_d=15[/latex]
0.5 0.478 0.726 0.826 0.878 0.911 0.933 0.949 0.960 0.970 0.977
0.1 3.073 2.695 2.490 2.361 2.273 2.208 2.158 2.119 2.086 2.059
0.05 4.543 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544
0.025 6.200 4.765 4.153 3.804 3.576 3.415 3.293 3.199 3.123 3.060
0.01 8.683 6.359 5.417 4.893 4.556 4.318 4.142 4.004 3.895 3.805
0.005 10.798 7.701 6.476 5.803 5.372 5.071 4.847 4.674 4.536 4.424
0.001 16.587 11.339 9.335 8.253 7.567 7.092 6.741 6.471 6.256 6.081
[latex]df_n[/latex]
[latex]\alpha[/latex] 11 12 13 14 15 20 30 60 120 [latex]\infty[/latex]
[latex]df_d=11[/latex]
0.5 1.000 1.005 1.010 1.013 1.017 1.028 1.040 1.052 1.058 1.064
0.1 2.227 2.209 2.193 2.179 2.167 2.123 2.076 2.026 2.000 1.972
0.05 2.818 2.788 2.761 2.739 2.719 2.646 2.570 2.490 2.448 2.404
0.025 3.474 3.430 3.392 3.359 3.330 3.226 3.118 3.004 2.944 2.883
0.01 4.462 4.397 4.342 4.293 4.251 4.099 3.941 3.776 3.690 3.602
0.005 5.320 5.236 5.165 5.103 5.049 4.855 4.654 4.445 4.337 4.226
0.001 7.761 7.626 7.509 7.409 7.321 7.008 6.684 6.348 6.175 5.998
[latex]df_d=12[/latex]
0.5 0.995 1.000 1.004 1.008 1.012 1.023 1.035 1.046 1.052 1.058
0.1 2.166 2.147 2.131 2.117 2.105 2.060 2.011 1.960 1.932 1.904
0.05 2.717 2.687 2.660 2.637 2.617 2.544 2.466 2.384 2.341 2.296
0.025 3.321 3.277 3.239 3.206 3.177 3.073 2.963 2.848 2.787 2.725
0.01 4.220 4.155 4.100 4.052 4.010 3.858 3.701 3.535 3.449 3.361
0.005 4.988 4.906 4.836 4.775 4.721 4.530 4.331 4.123 4.015 3.904
0.001 7.136 7.005 6.892 6.794 6.709 6.405 6.090 5.762 5.593 5.420
[latex]df_d=13[/latex]
0.5 0.990 0.996 1.000 1.004 1.007 1.019 1.030 1.042 1.048 1.054
0.1 2.116 2.097 2.080 2.066 2.053 2.007 1.958 1.904 1.876 1.846
0.05 2.635 2.604 2.577 2.554 2.533 2.459 2.380 2.297 2.252 2.206
0.025 3.197 3.153 3.115 3.082 3.053 2.948 2.837 2.720 2.659 2.595
0.01 4.025 3.960 3.905 3.857 3.815 3.665 3.507 3.341 3.255 3.165
0.005 4.724 4.643 4.573 4.513 4.460 4.270 4.073 3.866 3.758 3.647
0.001 6.647 6.519 6.409 6.314 6.231 5.934 5.626 5.305 5.138 4.967
[latex]df_d=14[/latex]
0.5 0.987 0.992 0.996 1.000 1.003 1.015 1.026 1.038 1.044 1.050
0.1 2.073 2.054 2.037 2.022 2.010 1.962 1.912 1.857 1.828 1.797
0.05 2.565 2.534 2.507 2.484 2.463 2.388 2.308 2.223 2.178 2.131
0.025 3.095 3.050 3.012 2.979 2.949 2.844 2.732 2.614 2.552 2.487
0.01 3.864 3.800 3.745 3.698 3.656 3.505 3.348 3.181 3.094 3.004
0.005 4.508 4.428 4.359 4.299 4.247 4.059 3.862 3.655 3.547 3.436
0.001 6.256 6.130 6.023 5.930 5.848 5.557 5.254 4.938 4.773 4.604
[latex]df_d=15[/latex]
0.5 0.983 0.989 0.993 0.997 1.000 1.011 1.023 1.034 1.040 1.046
0.1 2.037 2.017 2.000 1.985 1.972 1.924 1.873 1.817 1.787 1.755
0.05 2.507 2.475 2.448 2.424 2.403 2.328 2.247 2.160 2.114 2.066
0.025 3.008 2.963 2.925 2.891 2.862 2.756 2.644 2.524 2.461 2.395
0.01 3.730 3.666 3.612 3.564 3.522 3.372 3.214 3.047 2.959 2.868
0.005 4.329 4.250 4.181 4.122 4.070 3.883 3.687 3.480 3.372 3.260
0.001 5.935 5.812 5.707 5.615 5.535 5.248 4.950 4.638 4.475 4.307
[latex]df_n[/latex]
[latex]\alpha[/latex] 1 2 3 4 5 6 7 8 9 10
[latex]df_d=20[/latex]
0.5 0.472 0.718 0.816 0.868 0.900 0.922 0.938 0.950 0.959 0.966
0.1 2.975 2.589 2.380 2.249 2.158 2.091 2.040 1.999 1.965 1.937
0.05 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348
0.025 5.871 4.461 3.859 3.515 3.289 3.128 3.007 2.913 2.837 2.774
0.01 8.096 5.849 4.938 4.431 4.103 3.871 3.699 3.564 3.457 3.368
0.005 9.944 6.986 5.818 5.174 4.762 4.472 4.257 4.090 3.956 3.847
0.001 14.819 9.953 8.098 7.096 6.461 6.019 5.692 5.440 5.239 5.075
[latex]df_d=30[/latex]
0.5 0.466 0.709 0.807 0.858 0.890 0.912 0.927 0.939 0.948 0.955
0.1 2.881 2.489 2.276 2.142 2.049 1.980 1.927 1.884 1.849 1.819
0.05 4.171 3.316 2.922 2.690 2.534 2.421 2.334 2.266 2.211 2.165
0.025 5.568 4.182 3.589 3.250 3.026 2.867 2.746 2.651 2.575 2.511
0.01 7.562 5.390 4.510 4.018 3.699 3.473 3.304 3.173 3.067 2.979
0.005 9.180 6.355 5.239 4.623 4.228 3.949 3.742 3.580 3.450 3.344
0.001 13.293 8.773 7.054 6.125 5.534 5.122 4.817 4.581 4.393 4.239
[latex]df_d=60[/latex]
0.5 0.460 0.701 0.798 0.849 0.880 0.901 0.917 0.928 0.937 0.945
0.1 2.791 2.393 2.177 2.041 1.946 1.875 1.819 1.775 1.738 1.707
0.05 4.001 3.150 2.758 2.525 2.368 2.254 2.167 2.097 2.040 1.993
0.025 5.286 3.925 3.343 3.008 2.786 2.627 2.507 2.412 2.334 2.270
0.01 7.077 4.977 4.126 3.649 3.339 3.119 2.953 2.823 2.718 2.632
0.005 8.495 5.795 4.729 4.140 3.760 3.492 3.291 3.134 3.008 2.904
0.001 11.973 7.768 6.171 5.307 4.757 4.372 4.086 3.865 3.687 3.541
[latex]df_d=120[/latex]
0.5 0.458 0.697 0.793 0.844 0.875 0.896 0.912 0.923 0.932 0.939
0.1 2.748 2.347 2.130 1.992 1.896 1.824 1.767 1.722 1.684 1.652
0.05 3.920 3.072 2.680 2.447 2.290 2.175 2.087 2.016 1.959 1.910
0.025 5.152 3.805 3.227 2.894 2.674 2.515 2.395 2.299 2.222 2.157
0.01 6.851 4.787 3.949 3.480 3.174 2.956 2.792 2.663 2.559 2.472
0.005 8.179 5.539 4.497 3.921 3.548 3.285 3.087 2.933 2.808 2.705
0.001 11.380 7.321 5.781 4.947 4.416 4.044 3.767 3.552 3.379 3.237
[latex]df_d=\infty[/latex]
0.5 0.455 0.693 0.789 0.839 0.870 0.891 0.907 0.918 0.927 0.934
0.1 2.706 2.303 2.084 1.945 1.847 1.774 1.717 1.670 1.632 1.599
0.05 3.841 2.996 2.605 2.372 2.214 2.099 2.010 1.938 1.880 1.831
0.025 5.024 3.689 3.116 2.786 2.567 2.408 2.288 2.192 2.114 2.048
0.01 6.635 4.605 3.782 3.319 3.017 2.802 2.639 2.511 2.407 2.321
0.005 7.879 5.298 4.279 3.715 3.350 3.091 2.897 2.744 2.621 2.519
0.001 10.828 6.908 5.422 4.617 4.103 3.743 3.475 3.266 3.097 2.959

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