APPENDIX
Statistical Tables
Table 1
Standard Normal Curve Areas
Table 2
Percentage Points of Student’s t Distribution
Table 3
t Test Type II Error Curves
Table 4
Percentage Points of Sign Test: C␣,n
Table 5
Percentage Points of Wilcoxon Rank Sum Test: TL and TU
Table 6
Percentage Points of Wilcoxon Signed-Rank Test
Table 7
Percentage Points of Chi-Square Distribution: 2␣
Table 8
Percentage Points of F Distribution: F␣
Table 9
Values of 2 Arcsin 兹 ˆ
Table 10 Percentage Points of Studentized Range Distribution: q␣(t, v) Table 11 Percentage Points for Dunnett’s Test: d␣(k, v) Table 12 Percentage Points for Hartley’s Fmax Test: Fmax,␣ Table 13 Random Numbers Table 14 F Test Power Curves for AOV Table 15 Poisson Probabilities: Pr(Y ⴝ y)
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Shaded area = Pr(Z z)
TABLE 1 Standard normal curve areas
0
z
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
−3.4 −3.3 −3.2 −3.1 −3.0
0.0003 0.0005 0.0007 0.0010 0.0013
0.0003 0.0005 0.0007 0.0009 0.0013
0.0003 0.0005 0.0006 0.0009 0.0013
0.0003 0.0004 0.0006 0.0009 0.0012
0.0003 0.0004 0.0006 0.0008 0.0012
0.0003 0.0004 0.0006 0.0008 0.0011
0.0003 0.0004 0.0006 0.0008 0.0011
0.0003 0.0004 0.0005 0.0008 0.0011
0.0003 0.0004 0.0005 0.0007 0.0010
0.0002 0.0003 0.0005 0.0007 0.0010
−2.9 −2.8 −2.7 −2.6 −2.5
0.0019 0.0026 0.0035 0.0047 0.0062
0.0018 0.0025 0.0034 0.0045 0.0060
0.0018 0.0024 0.0033 0.0044 0.0059
0.0017 0.0023 0.0032 0.0043 0.0057
0.0016 0.0023 0.0031 0.0041 0.0055
0.0016 0.0022 0.0030 0.0040 0.0054
0.0015 0.0021 0.0029 0.0039 0.0052
0.0015 0.0021 0.0028 0.0038 0.0051
0.0014 0.0020 0.0027 0.0037 0.0049
0.0014 0.0019 0.0026 0.0036 0.0048
−2.4 −2.3 −2.2 −2.1 −2.0
0.0082 0.0107 0.0139 0.0179 0.0228
0.0080 0.0104 0.0136 0.0174 0.0222
0.0078 0.0102 0.0132 0.0170 0.0217
0.0075 0.0099 0.0129 0.0166 0.0212
0.0073 0.0096 0.0125 0.0162 0.0207
0.0071 0.0094 0.0122 0.0158 0.0202
0.0069 0.0091 0.0119 0.0154 0.0197
0.0068 0.0089 0.0116 0.0150 0.0192
0.0066 0.0087 0.0113 0.0146 0.0188
0.0064 0.0084 0.0110 0.0143 0.0183
−1.9 −1.8 −1.7 −1.6 −1.5
0.0287 0.0359 0.0446 0.0548 0.0668
0.0281 0.0351 0.0436 0.0537 0.0655
0.0274 0.0344 0.0427 0.0526 0.0643
0.0268 0.0336 0.0418 0.0516 0.0630
0.0262 0.0329 0.0409 0.0505 0.0618
0.0256 0.0322 0.0401 0.0495 0.0606
0.0250 0.0314 0.0392 0.0485 0.0594
0.0244 0.0307 0.0384 0.0475 0.0582
0.0239 0.0301 0.0375 0.0465 0.0571
0.0233 0.0294 0.0367 0.0455 0.0559
−1.4 −1.3 −1.2 −1.1 −1.0
0.0808 0.0968 0.1151 0.1357 0.1587
0.0793 0.0951 0.1131 0.1335 0.1562
0.0778 0.0934 0.1112 0.1314 0.1539
0.0764 0.0918 0.1093 0.1292 0.1515
0.0749 0.0901 0.1075 0.1271 0.1492
0.0735 0.0885 0.1056 0.1251 0.1469
0.0721 0.0869 0.1038 0.1230 0.1446
0.0708 0.0853 0.1020 0.1210 0.1423
0.0694 0.0838 0.1003 0.1190 0.1401
0.0681 0.0823 0.0985 0.1170 0.1379
−0.9 −0.8 −0.7 −0.6 −0.5
0.1841 0.2119 0.2420 0.2743 0.3085
0.1814 0.2090 0.2389 0.2709 0.3050
0.1788 0.2061 0.2358 0.2676 0.3015
0.1762 0.2033 0.2327 0.2643 0.2981
0.1736 0.2005 0.2296 0.2611 0.2946
0.1711 0.1977 0.2266 0.2578 0.2912
0.1685 0.1949 0.2236 0.2546 0.2877
0.1660 0.1922 0.2206 0.2514 0.2843
0.1635 0.1894 0.2177 0.2483 0.2810
0.1611 0.1867 0.2148 0.2451 0.2776
−0.4 −0.3 −0.2 −0.1 −0.0
0.3446 0.3821 0.4207 0.4602 0.5000
0.3409 0.3783 0.4168 0.4562 0.4960
0.3372 0.3745 0.4129 0.4522 0.4920
0.3336 0.3707 0.4090 0.4483 0.4880
0.3300 0.3669 0.4052 0.4443 0.4840
0.3264 0.3632 0.4013 0.4404 0.4801
0.3228 0.3594 0.3974 0.4364 0.4761
0.3192 0.3557 0.3936 0.4325 0.4721
0.3156 0.3520 0.3897 0.4286 0.4681
0.3121 0.3483 0.3859 0.4247 0.4641
z
Area
−3.50 −4.00 −4.50 −5.00
0.00023263 0.00003167 0.00000340 0.00000029
Source: Computed by M. Longnecker using Splus.
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TABLE 1 Standard normal curve areas z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0 0.1 0.2 0.3 0.4
0.5000 0.5398 0.5793 0.6179 0.6554
0.5040 0.5438 0.5832 0.6217 0.6591
0.5080 0.5478 0.5871 0.6255 0.6628
0.5120 0.5517 0.5910 0.6293 0.6664
0.5160 0.5557 0.5948 0.6331 0.6700
0.5199 0.5596 0.5987 0.6368 0.6736
0.5239 0.5636 0.6026 0.6406 0.6772
0.5279 0.5675 0.6064 0.6443 0.6808
0.5319 0.5714 0.6103 0.6480 0.6844
0.5359 0.5753 0.6141 0.6517 0.6879
0.5 0.6 0.7 0.8 0.9
0.6915 0.7257 0.7580 0.7881 0.8159
0.6950 0.7291 0.7611 0.7910 0.8186
0.6985 0.7324 0.7642 0.7939 0.8212
0.7019 0.7357 0.7673 0.7967 0.8238
0.7054 0.7389 0.7704 0.7995 0.8264
0.7088 0.7422 0.7734 0.8023 0.8289
0.7123 0.7454 0.7764 0.8051 0.8315
0.7157 0.7486 0.7794 0.8078 0.8340
0.7190 0.7517 0.7823 0.8106 0.8365
0.7224 0.7549 0.7852 0.8133 0.8389
1.0 1.1 1.2 1.3 1.4
0.8413 0.8643 0.8849 0.9032 0.9192
0.8438 0.8665 0.8869 0.9049 0.9207
0.8461 0.8686 0.8888 0.9066 0.9222
0.8485 0.8708 0.8907 0.9082 0.9236
0.8508 0.8729 0.8925 0.9099 0.9251
0.8531 0.8749 0.8944 0.9115 0.9265
0.8554 0.8770 0.8962 0.9131 0.9279
0.8577 0.8790 0.8980 0.9147 0.9292
0.8599 0.8810 0.8997 0.9162 0.9306
0.8621 0.8830 0.9015 0.9177 0.9319
1.5 1.6 1.7 1.8 1.9
0.9332 0.9452 0.9554 0.9641 0.9713
0.9345 0.9463 0.9564 0.9649 0.9719
0.9357 0.9474 0.9573 0.9656 0.9726
0.9370 0.9484 0.9582 0.9664 0.9732
0.9382 0.9495 0.9591 0.9671 0.9738
0.9394 0.9505 0.9599 0.9678 0.9744
0.9406 0.9515 0.9608 0.9686 0.9750
0.9418 0.9525 0.9616 0.9693 0.9756
0.9429 0.9535 0.9625 0.9699 0.9761
0.9441 0.9545 0.9633 0.9706 0.9767
2.0 2.1 2.2 2.3 2.4
0.9772 0.9821 0.9861 0.9893 0.9918
0.9778 0.9826 0.9864 0.9896 0.9920
0.9783 0.9830 0.9868 0.9898 0.9922
0.9788 0.9834 0.9871 0.9901 0.9925
0.9793 0.9838 0.9875 0.9904 0.9927
0.9798 0.9842 0.9878 0.9906 0.9929
0.9803 0.9846 0.9881 0.9909 0.9931
0.9808 0.9850 0.9884 0.9911 0.9932
0.9812 0.9854 0.9887 0.9913 0.9934
0.9817 0.9857 0.9890 0.9916 0.9936
2.5 2.6 2.7 2.8 2.9
0.9938 0.9953 0.9965 0.9974 0.9981
0.9940 0.9955 0.9966 0.9975 0.9982
0.9941 0.9956 0.9967 0.9976 0.9982
0.9943 0.9957 0.9968 0.9977 0.9983
0.9945 0.9959 0.9969 0.9977 0.9984
0.9946 0.9960 0.9970 0.9978 0.9984
0.9948 0.9961 0.9971 0.9979 0.9985
0.9949 0.9962 0.9972 0.9979 0.9985
0.9951 0.9963 0.9973 0.9980 0.9986
0.9952 0.9964 0.9974 0.9981 0.9986
3.0 3.1 3.2 3.3 3.4
0.9987 0.9990 0.9993 0.9995 0.9997
0.9987 0.9991 0.9993 0.9995 0.9997
0.9987 0.9991 0.9994 0.9995 0.9997
0.9988 0.9991 0.9994 0.9996 0.9997
0.9988 0.9992 0.9994 0.9996 0.9997
0.9989 0.9992 0.9994 0.9996 0.9997
0.9989 0.9992 0.9994 0.9996 0.9997
0.9989 0.9992 0.9995 0.9996 0.9997
0.9990 0.9993 0.9995 0.9996 0.9997
0.9990 0.9993 0.9995 0.9997 0.9998
z
Area
3.50 4.00 4.50 5.00
0.99976737 0.99996833 0.99999660 0.99999971
Appendix
1093
Shaded area = ␣
TABLE 2
0
Percentage points of Student’s t distribution df/ ␣ ⴝ
t␣,
.40
.25
.10
.05
.025
.01
.005
.001
.0005
1 2 3 4 5
0.325 0.289 0.277 0.271 0.267
1.000 0.816 0.765 0.741 0.727
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
318.309 22.327 10.215 7.173 5.893
636.619 31.599 12.924 8.610 6.869
6 7 8 9 10
0.265 0.263 0.262 0.261 0.260
0.718 0.711 0.706 0.703 0.700
1.440 1.415 1.397 1.383 1.372
1.943 1.895 1.860 1.833 1.812
2.447 2.365 2.306 2.262 2.228
3.143 2.998 2.896 2.821 2.764
3.707 3.499 3.355 3.250 3.169
5.208 4.785 4.501 4.297 4.144
5.959 5.408 5.041 4.781 4.587
11 12 13 14 15
0.260 0.259 0.259 0.258 0.258
0.697 0.695 0.694 0.692 0.691
1.363 1.356 1.350 1.345 1.341
1.796 1.782 1.771 1.761 1.753
2.201 2.179 2.160 2.145 2.131
2.718 2.681 2.650 2.624 2.602
3.106 3.055 3.012 2.977 2.947
4.025 3.930 3.852 3.787 3.733
4.437 4.318 4.221 4.140 4.073
16 17 18 19 20
0.258 0.257 0.257 0.257 0.257
0.690 0.689 0.688 0.688 0.687
1.337 1.333 1.330 1.328 1.325
1.746 1.740 1.734 1.729 1.725
2.120 2.110 2.101 2.093 2.086
2.583 2.567 2.552 2.539 2.528
2.921 2.898 2.878 2.861 2.845
3.686 3.646 3.610 3.579 3.552
4.015 3.965 3.922 3.883 3.850
21 22 23 24 25
0.257 0.256 0.256 0.256 0.256
0.686 0.686 0.685 0.685 0.684
1.323 1.321 1.319 1.318 1.316
1.721 1.717 1.714 1.711 1.708
2.080 2.074 2.069 2.064 2.060
2.518 2.508 2.500 2.492 2.485
2.831 2.819 2.807 2.797 2.787
3.527 3.505 3.485 3.467 3.450
3.819 3.792 3.768 3.745 3.725
26 27 28 29 30
0.256 0.256 0.256 0.256 0.256
0.684 0.684 0.683 0.683 0.683
1.315 1.314 1.313 1.311 1.310
1.706 1.703 1.701 1.699 1.697
2.056 2.052 2.048 2.045 2.042
2.479 2.473 2.467 2.462 2.457
2.779 2.771 2.763 2.756 2.750
3.435 3.421 3.408 3.396 3.385
3.707 3.690 3.674 3.659 3.646
35 40 50 60 120 inf.
0.255 0.255 0.255 0.254 0.254 0.253
0.682 0.681 0.679 0.679 0.677 0.674
1.306 1.303 1.299 1.296 1.289 1.282
1.690 1.684 1.676 1.671 1.658 1.645
2.030 2.021 2.009 2.000 1.980 1.960
2.438 2.423 2.403 2.390 2.358 2.326
2.724 2.704 2.678 2.660 2.617 2.576
3.340 3.307 3.261 3.232 3.160 3.090
3.591 3.551 3.496 3.460 3.373 3.291
Source: Computed by M. Longnecker using Splus.
1094
Appendix TABLE 3
Probability of Type II error curves for 움 ⫽ .01 (one-sided)
1.0
Probability of Type II error
0.9 0.8 0.7
2
0.6 0.5 3
0.4 0.3
4
0.2 0.1
49 39 29
99
19 14
8
74
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 Difference (d) Source: Computed by M. Longnecker using SAS.
TABLE 3 Probability of Type II error curves for 움 ⫽ .05 (one-sided)
1.0
Probability of Type II error
0.9 0.8 0.7 0.6 0.5 0.4 2
0.3 3
0.2 0.1
39 19 14 74 99 49 29
4 8
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 Difference (d) Source: Computed by M. Longnecker using SAS.
Appendix TABLE 3 Probability of Type II error curves for 움 ⫽ .01 (two-sided)
1.0
Probability of Type II error
0.9 2
0.8 0.7 3
0.6 0.5
4
0.4 0.3 0.2 0.1
8 99
49 74 39
29
19
14
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 Difference (d) Source: Computed by M. Longnecker using SAS.
TABLE 3 Probability of Type II error curves for 움 ⫽ .05 (two-sided)
1.0
Probability of Type II error
0.9 0.8 0.7 0.6 0.5 2
0.4 0.3
3 4
0.2
39
0.1
49 29 99 74
14
8
19
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 Difference (d) Source: Computed by M. Longnecker using SAS.
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Appendix
TABLE 4 Percentage points for confidence intervals on the median and the sign test: C움,n
␣(2) ␣(1)
.20 .10
.10 .05
.05 .025
.02 .01
.01 .005
.005 .0025
.002 .001
n 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
␣(2) ␣(1)
.20 .10
.10 .05
.05 .025
.02 .01
.01 .005
.005 .0025
.002 .001
9 9 10 10 10 11 11 12 12 13 13 14 14 15 15 15 16 16 17 17 18 18 19 19 19
8 8 9 9 10 10 10 11 11 12 12 13 13 13 14 14 15 15 16 16 16 17 17 18 18
7 7 8 8 9 9 9 10 10 11 11 12 12 12 13 13 14 14 15 15 15 16 16 17 17
6 7 7 7 8 8 8 9 9 10 10 10 11 11 12 12 13 13 13 14 14 15 15 15 16
6 6 6 7 7 7 8 8 9 9 9 10 10 11 11 11 12 12 13 13 13 14 14 15 15
5 5 6 6 6 7 7 8 8 8 9 9 9 10 10 11 11 11 12 12 13 13 13 14 14
4 5 5 5 6 6 6 7 7 8 8 8 9 9 9 10 10 11 11 11 12 12 12 13 13
n * * * 0 0 0 1 1 2 2 2 3 3 4 4 4 5 5 6 6 7 7 7 8 8
* * * * 0 0 0 1 1 1 2 2 3 3 3 4 4 5 5 5 6 6 7 7 7
* * * * * 0 0 0 1 1 1 2 2 2 3 3 4 4 4 5 5 5 6 6 7
* * * * * * 0 0 0 0 1 1 1 2 2 2 3 3 4 4 4 5 5 5 6
* * * * * * * 0 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5
* * * * * * * * 0 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5
* * * * * * * * * 0 0 0 0 1 1 1 1 2 2 2 3 3 3 4 4
26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Note: An * means that no test or confidence interval of this level exists. Source: Computed by M. Longnecker using Splus.
Appendix
1097
TABLE 5 Critical values of TL and TU for the Wilcoxon rank sum test: independent samples. Test statistic is rank sum associated with smaller sample (if equal sample sizes, either rank sum can be used). a. 움 ⫽ .025 one-tailed; 움 ⫽ .05 two-tailed n1 n2
3
3 4 5 6 7 8 9 10
4
5
6
7
8
9
10
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
5 6 6 7 7 8 8 9
16 18 21 23 26 28 31 33
6 11 12 12 13 14 15 16
18 25 28 32 35 38 41 44
6 12 18 19 20 21 22 24
21 28 37 41 45 49 53 56
7 12 19 26 28 29 31 32
23 32 41 52 56 61 65 70
7 13 20 28 37 39 41 43
26 35 45 56 68 73 78 83
8 14 21 29 39 49 51 54
28 38 49 61 73 87 93 98
8 15 22 31 41 51 63 66
31 41 53 65 78 93 108 114
9 16 24 32 43 54 66 79
33 44 56 70 83 98 114 131
b. 움 ⫽ .05 one-tailed; 움 ⫽ .10 two-tailed n1 n2
3
3 4 5 6 7 8 9 10
4
5
6
7
8
9
10
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
TL
TU
6 7 7 8 9 9 10 11
15 17 20 22 24 27 29 31
7 12 13 14 15 16 17 18
17 24 27 30 33 36 39 42
7 13 19 20 22 24 25 26
20 27 36 40 43 46 50 54
8 14 20 28 30 32 33 35
22 30 40 50 54 58 63 67
9 15 22 30 39 41 43 46
24 33 43 54 66 71 76 80
9 16 24 32 41 52 54 57
27 36 46 58 71 84 90 95
10 17 25 33 43 54 66 69
29 39 50 63 76 90 105 111
11 18 26 35 46 57 69 83
31 42 54 67 80 95 111 127
Source: From F. Wilcoxon and R. A. Wilcox, Some Rapid Approximate Statistical Procedures (Pearl River, N.Y. Lederle Laboratories, 1964), pp. 20–23. Reproduced with the permission of American Cyanamid Company.
1098
Appendix TABLE 6
Critical values for the Wilcoxon signed-rank test [n ⫽ 5(1)54]
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
nⴝ5 2 0
nⴝ6 3 2 0
nⴝ7 5 3 2 0
nⴝ8 8 5 3 1 0
nⴝ9 10 8 5 3 1 0
n ⴝ 15
n ⴝ 16
n ⴝ 17
n ⴝ 18
n ⴝ 19
36 30 25 19 15 12 8
42 35 29 23 19 15 11
48 41 34 27 23 19 14
55 47 40 32 27 23 18
62 53 46 37 32 27 21
n ⴝ 25
n ⴝ 26
n ⴝ 27
n ⴝ 28
n ⴝ 29
113 100 89 76 68 60 51
124 110 98 84 75 67 58
134 119 107 92 83 74 64
145 130 116 101 91 82 71
157 140 126 110 100 90 79
n ⴝ 35
n ⴝ 36
n ⴝ 37
n ⴝ 38
n ⴝ 39
235 213 195 173 159 146 131
250 227 208 185 171 157 141
265 241 221 198 182 168 151
281 256 235 211 194 180 162
297 271 249 224 207 192 173
n ⴝ 45
n ⴝ 46
n ⴝ 47
n ⴝ 48
n ⴝ 49
402 371 343 312 291 272 249
422 389 361 328 307 287 263
441 407 378 345 322 302 277
462 426 396 362 339 318 292
482 446 415 379 355 334 307
Source: Computed by P. J. Hildebrand.
TABLE 6 (continued)
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
One-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.1 .05 .025 .01 .005 .0025 .001
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Two-Sided p p p p p p p
⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽
.2 .1 .05 .02 .01 .005 .002
Appendix
1099
n ⴝ 10
n ⴝ 11
n ⴝ 12
n ⴝ 13
n ⴝ 14
14 10 8 5 3 1 0
17 13 10 7 5 3 1
21 17 13 9 7 5 2
26 21 17 12 9 7 4
31 25 21 15 12 9 6
n ⴝ 20
n ⴝ 21
n ⴝ 22
n ⴝ 23
n ⴝ 24
69 60 52 43 37 32 26
77 67 58 49 42 37 30
86 75 65 55 48 42 35
94 83 73 62 54 48 40
104 91 81 69 61 54 45
n ⴝ 30
n ⴝ 31
n ⴝ 32
n ⴝ 33
n ⴝ 34
169 151 137 120 109 98 86
181 163 147 130 118 107 94
194 175 159 140 128 116 103
207 187 170 151 138 126 112
221 200 182 162 148 136 121
n ⴝ 40
n ⴝ 41
n ⴝ 42
n ⴝ 43
n ⴝ 44
313 286 264 238 220 204 185
330 302 279 252 233 217 197
348 319 294 266 247 230 209
365 336 310 281 261 244 222
384 353 327 296 276 258 235
n ⴝ 50
n ⴝ 51
n ⴝ 52
n ⴝ 53
n ⴝ 54
503 466 434 397 373 350 323
525 486 453 416 390 367 339
547 507 473 434 408 384 355
569 529 494 454 427 402 372
592 550 514 473 445 420 389
1100
Appendix ␣
TABLE 7
2 ␣
Percentage points of the chi-square distribution df ␣ ⴝ 1 2 3 4 5
.999
.995
.99
.975
.000002 .002001 .02430 .09080 .2102
.000039 .01003 .07172 .2070 .4117
.000157 .02010 .1148 .2971 .5543
.000982 .05064 .2158 .4844 .8312
.95
.90
.003932 .1026 .3518 .7107 1.145
.01579 .2107 .5844 1.064 1.610
6 7 8 9 10
.3811 .5985 .8571 1.152 1.479
.6757 .9893 1.344 1.735 2.156
.8721 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
11 12 13 14 15
1.834 2.214 2.617 3.041 3.483
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
16 17 18 19 20
3.942 4.416 4.905 5.407 5.921
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
21 22 23 24 25
6.447 6.983 7.529 8.085 8.649
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
26 27 28 29 30
9.222 9.803 10.39 10.99 11.59
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 16.06 16.79
15.38 16.15 16.93 17.71 18.49
17.29 18.11 18.94 19.77 20.60
40 50 60 70 80
17.92 24.67 31.74 39.04 46.52
20.71 27.99 35.53 43.28 51.17
22.16 29.71 37.48 45.44 53.54
24.43 32.36 40.48 48.76 57.15
26.51 34.76 43.19 51.74 60.39
29.05 37.69 46.46 55.33 64.28
90 100 120 240
54.16 61.92 77.76 177.95
59.20 67.33 83.85 187.32
61.75 70.06 86.92 191.99
65.65 74.22 91.57 198.98
69.13 77.93 95.70 205.14
73.29 82.36 100.62 212.39
Appendix TABLE 7 (continued )
␣ ⴝ .10
.05
.025
.01
.005
.001
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
10.83 13.82 16.27 18.47 20.52
1 2 3 4 5
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
22.46 24.32 26.12 27.88 29.59
6 7 8 9 10
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
31.27 32.91 34.53 36.12 37.70
11 12 13 14 15
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
39.25 40.79 42.31 43.82 45.31
16 17 18 19 20
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
46.80 48.27 49.73 51.18 52.62
21 22 23 24 25
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 49.65 50.99 52.34 53.67
54.05 55.48 56.89 58.30 59.70
26 27 28 29 30
51.81 63.17 74.40 85.53 96.58
55.76 67.50 79.08 90.53 101.88
59.34 71.42 83.30 95.02 106.63
63.69 76.15 88.38 100.43 112.33
66.77 79.49 91.95 104.21 116.32
73.40 86.66 99.61 112.32 124.84
40 50 60 70 80
107.57 118.50 140.23 268.47
113.15 124.34 146.57 277.14
118.14 129.56 152.21 284.80
124.12 135.81 158.95 293.89
128.30 140.17 163.65 300.18
137.21 149.45 173.62 313.44
90 100 120 240
2.706 4.605 6.251 7.779 9.236
Source: Computed by P. J. Hildebrand.
df
1101
1102
Appendix
␣
TABLE 8
F␣
Percentage points of the F distribution (df2 between 1 and 6) df1 df2
␣
1
2
3
4
5
6
7
8
9
10
1
.25 .10 .05 .025 .01
5.83 39.86 161.4 647.8 4052
7.50 49.50 199.5 799.5 5000
8.20 53.59 215.7 864.2 5403
8.58 55.83 224.6 899.6 5625
8.82 57.24 230.2 921.8 5764
8.98 58.20 234.0 937.1 5859
9.10 58.91 236.8 948.2 5928
9.19 59.44 238.9 956.7 5981
9.26 59.86 240.5 963.3 6022
9.32 60.19 241.9 968.6 6056
2
.25 .10 .05 .025 .01 .005 .001
2.57 8.53 18.51 38.51 98.50 198.5 998.5
3.00 9.00 19.00 39.00 99.00 199.0 999.0
3.15 9.16 19.16 39.17 99.17 199.2 999.2
3.23 9.24 19.25 39.25 99.25 199.2 999.2
3.28 9.29 19.30 39.30 99.30 199.3 999.3
3.31 9.33 19.33 39.33 99.33 199.3 999.3
3.34 9.35 19.35 39.36 99.36 199.4 999.4
3.35 9.37 19.37 39.37 99.37 199.4 999.4
3.37 9.38 19.38 39.39 99.39 199.4 999.4
3.38 9.39 19.40 39.40 99.40 199.4 999.4
3
.25 .10 .05 .025 .01 .005 .001
2.02 5.54 10.13 17.44 34.12 55.55 167.0
2.28 5.46 9.55 16.04 30.82 49.80 148.5
2.36 5.39 9.28 15.44 29.46 47.47 141.1
2.39 5.34 9.12 15.10 28.71 46.19 137.1
2.41 5.31 9.01 14.88 28.24 45.39 134.6
2.42 5.28 8.94 14.73 27.91 44.84 132.8
2.43 5.27 8.89 14.62 27.67 44.43 131.6
2.44 5.25 8.85 14.54 27.49 44.13 130.6
2.44 5.24 8.81 14.47 27.35 43.88 129.9
2.44 5.23 8.79 14.42 27.23 43.69 129.2
4
.25 .10 .05 .025 .01 .005 .001
1.81 4.54 7.71 12.22 21.20 31.33 74.14
2.00 4.32 6.94 10.65 18.00 26.28 61.25
2.05 4.19 6.59 9.98 16.69 24.26 56.18
2.06 4.11 6.39 9.60 15.98 23.15 53.44
2.07 4.05 6.26 9.36 15.52 22.46 51.71
2.08 4.01 6.16 9.20 15.21 21.97 50.53
2.08 3.98 6.09 9.07 14.98 21.62 49.66
2.08 3.95 6.04 8.98 14.80 21.35 49.00
2.08 3.94 6.00 8.90 14.66 21.14 48.47
2.08 3.92 5.96 8.84 14.55 20.97 48.05
5
.25 .10 .05 .025 .01 .005 .001
1.69 4.06 6.61 10.01 16.26 22.78 47.18
1.85 3.78 5.79 8.43 13.27 18.31 37.12
1.88 3.62 5.41 7.76 12.06 16.53 33.20
1.89 3.52 5.19 7.39 11.39 15.56 31.09
1.89 3.45 5.05 7.15 10.97 14.94 29.75
1.89 3.40 4.95 6.98 10.67 14.51 28.83
1.89 3.37 4.88 6.85 10.46 14.20 28.16
1.89 3.34 4.82 6.76 10.29 13.96 27.65
1.89 3.32 4.77 6.68 10.16 13.77 27.24
1.89 3.30 4.74 6.62 10.05 13.62 26.92
6
.25 .10 .05 .025 .01 .005 .001
1.62 3.78 5.99 8.81 13.75 18.63 35.51
1.76 3.46 5.14 7.26 10.92 14.54 27.00
1.78 3.29 4.76 6.60 9.78 12.92 23.70
1.79 3.18 4.53 6.23 9.15 12.03 21.92
1.79 3.11 4.39 5.99 8.75 11.46 20.80
1.78 3.05 4.28 5.82 8.47 11.07 20.03
1.78 3.01 4.21 5.70 8.26 10.79 19.46
1.78 2.98 4.15 5.60 8.10 10.57 19.03
1.77 2.96 4.10 5.52 7.98 10.39 18.69
1.77 2.94 4.06 5.46 7.87 10.25 18.41
Appendix
1103
TABLE 8 (continued ) df1 12
15
20
24
30
40
60
120
240
inf.
␣
df2
9.41 60.71 243.9 976.7 6106
9.49 61.22 245.9 984.9 6157
9.58 61.74 248.0 993.1 6209
9.63 62.00 249.1 997.2 6235
9.67 62.26 250.1 1001 6261
9.71 62.53 251.1 1006 6287
9.76 62.79 252.2 1010 6313
9.80 63.06 253.3 1014 6339
9.83 63.19 253.8 1016 6353
9.85 63.33 254.3 1018 6366
.25 .10 .05 .025 .01
1
3.39 9.41 19.41 39.41 99.42 199.4 999.4
3.41 9.42 19.43 39.43 99.43 199.4 999.4
3.43 9.44 19.45 39.45 99.45 199.4 999.4
3.43 9.45 19.45 39.46 99.46 199.5 999.5
3.44 9.46 19.46 39.46 99.47 199.5 999.5
3.45 9.47 19.47 39.47 99.47 199.5 999.5
3.46 9.47 19.48 39.48 99.48 199.5 999.5
3.47 9.48 19.49 39.49 99.49 199.5 999.5
3.47 9.49 19.49 39.49 99.50 199.5 999.5
3.48 9.49 19.50 39.50 99.50 199.5 999.5
.25 .10 .05 .025 .01 .005 .001
2
2.45 5.22 8.74 14.34 27.05 43.39 128.3
2.46 5.20 8.70 14.25 26.87 43.08 127.4
2.46 5.18 8.66 14.17 26.69 42.78 126.4
2.46 5.18 8.64 14.12 26.60 42.62 125.9
2.47 5.17 8.62 14.08 26.50 42.47 125.4
2.47 5.16 8.59 14.04 26.41 42.31 125.0
2.47 5.15 8.57 13.99 26.32 42.15 124.5
2.47 5.14 8.55 13.95 26.22 41.99 124.0
2.47 5.14 8.54 13.92 26.17 41.91 123.7
2.47 5.13 8.53 13.90 26.13 41.83 123.5
.25 .10 .05 .025 .01 .005 .001
3
2.08 3.90 5.91 8.75 14.37 20.70 47.41
2.08 3.87 5.86 8.66 14.20 20.44 46.76
2.08 3.84 5.80 8.56 14.02 20.17 46.10
2.08 3.83 5.77 8.51 13.93 20.03 45.77
2.08 3.82 5.75 8.46 13.84 19.89 45.43
2.08 3.80 5.72 8.41 13.75 19.75 45.09
2.08 3.79 5.69 8.36 13.65 19.61 44.75
2.08 3.78 5.66 8.31 13.56 19.47 44.40
2.08 3.77 5.64 8.28 13.51 19.40 44.23
2.08 3.76 5.63 8.26 13.46 19.32 44.05
.25 .10 .05 .025 .01 .005 .001
4
1.89 3.27 4.68 6.52 9.89 13.38 26.42
1.89 3.24 4.62 6.43 9.72 13.15 25.91
1.88 3.21 4.56 6.33 9.55 12.90 25.39
1.88 3.19 4.53 6.28 9.47 12.78 25.13
1.88 3.17 4.50 6.23 9.38 12.66 24.87
1.88 3.16 4.46 6.18 9.29 12.53 24.60
1.87 3.14 4.43 6.12 9.20 12.40 24.33
1.87 3.12 4.40 6.07 9.11 12.27 24.06
1.87 3.11 4.38 6.04 9.07 12.21 23.92
1.87 3.10 4.36 6.02 9.02 12.14 23.79
.25 .10 .05 .025 .01 .005 .001
5
1.77 2.90 4.00 5.37 7.72 10.03 17.99
1.76 2.87 3.94 5.27 7.56 9.81 17.56
1.76 2.84 3.87 5.17 7.40 9.59 17.12
1.75 2.82 3.84 5.12 7.31 9.47 16.90
1.75 2.80 3.81 5.07 7.23 9.36 16.67
1.75 2.78 3.77 5.01 7.14 9.24 16.44
1.74 2.76 3.74 4.96 7.06 9.12 16.21
1.74 2.74 3.70 4.90 6.97 9.00 15.98
1.74 2.73 3.69 4.88 6.92 8.94 15.86
1.74 2.72 3.67 4.85 6.88 8.88 15.75
.25 .10 .05 .025 .01 .005 .001
6
1104
Appendix
␣
TABLE 8
F␣
Percentage points of the F distribution (df2 between 7 and 12) df1 df2
␣
1
2
3
4
5
6
7
8
9
10
7
.25 .10 .05 .025 .01 .005 .001
1.57 3.59 5.59 8.07 12.25 16.24 29.25
1.70 3.26 4.74 6.54 9.55 12.40 21.69
1.72 3.07 4.35 5.89 8.45 10.88 18.77
1.72 2.96 4.12 5.52 7.85 10.05 17.20
1.71 2.88 3.97 5.29 7.46 9.52 16.21
1.71 2.83 3.87 5.12 7.19 9.16 15.52
1.70 2.78 3.79 4.99 6.99 8.89 15.02
1.70 2.75 3.73 4.90 6.84 8.68 14.63
1.69 2.72 3.68 4.82 6.72 8.51 14.33
1.69 2.70 3.64 4.76 6.62 8.38 14.08
8
.25 .10 .05 .025 .01 .005 .001
1.54 3.46 5.32 7.57 11.26 14.69 25.41
1.66 3.11 4.46 6.06 8.65 11.04 18.49
1.67 2.92 4.07 5.42 7.59 9.60 15.83
1.66 2.81 3.84 5.05 7.01 8.81 14.39
1.66 2.73 3.69 4.82 6.63 8.30 13.48
1.65 2.67 3.58 4.65 6.37 7.95 12.86
1.64 2.62 3.50 4.53 6.18 7.69 12.40
1.64 2.59 3.44 4.43 6.03 7.50 12.05
1.63 2.56 3.39 4.36 5.91 7.34 11.77
1.63 2.54 3.35 4.30 5.81 7.21 11.54
9
.25 .10 .05 .025 .01 .005 .001
1.51 3.36 5.12 7.21 10.56 13.61 22.86
1.62 3.01 4.26 5.71 8.02 10.11 16.39
1.63 2.81 3.86 5.08 6.99 8.72 13.90
1.63 2.69 3.63 4.72 6.42 7.96 12.56
1.62 2.61 3.48 4.48 6.06 7.47 11.71
1.61 2.55 3.37 4.32 5.80 7.13 11.13
1.60 2.51 3.29 4.20 5.61 6.88 10.70
1.60 2.47 3.23 4.10 5.47 6.69 10.37
1.59 2.44 3.18 4.03 5.35 6.54 10.11
1.59 2.42 3.14 3.96 5.26 6.42 9.89
10
.25 .10 .05 .025 .01 .005 .001
1.49 3.29 4.96 6.94 10.04 12.83 21.04
1.60 2.92 4.10 5.46 7.56 9.43 14.91
1.60 2.73 3.71 4.83 6.55 8.08 12.55
1.59 2.61 3.48 4.47 5.99 7.34 11.28
1.59 2.52 3.33 4.24 5.64 6.87 10.48
1.58 2.46 3.22 4.07 5.39 6.54 9.93
1.57 2.41 3.14 3.95 5.20 6.30 9.52
1.56 2.38 3.07 3.85 5.06 6.12 9.20
1.56 2.35 3.02 3.78 4.94 5.97 8.96
1.55 2.32 2.98 3.72 4.85 5.85 8.75
11
.25 .10 .05 .025 .01 .005 .001
1.47 3.23 4.84 6.72 9.65 12.23 19.69
1.58 2.86 3.98 5.26 7.21 8.91 13.81
1.58 2.66 3.59 4.63 6.22 7.60 11.56
1.57 2.54 3.36 4.28 5.67 6.88 10.35
1.56 2.45 3.20 4.04 5.32 6.42 9.58
1.55 2.39 3.09 3.88 5.07 6.10 9.05
1.54 2.34 3.01 3.76 4.89 5.86 8.66
1.53 2.30 2.95 3.66 4.74 5.68 8.35
1.53 2.27 2.90 3.59 4.63 5.54 8.12
1.52 2.25 2.85 3.53 4.54 5.42 7.92
12
.25 .10 .05 .025 .01 .005 .001
1.46 3.18 4.75 6.55 9.33 11.75 18.64
1.56 2.81 3.89 5.10 6.93 8.51 12.97
1.56 2.61 3.49 4.47 5.95 7.23 10.80
1.55 2.48 3.26 4.12 5.41 6.52 9.63
1.54 2.39 3.11 3.89 5.06 6.07 8.89
1.53 2.33 3.00 3.73 4.82 5.76 8.38
1.52 2.28 2.91 3.61 4.64 5.52 8.00
1.51 2.24 2.85 3.51 4.50 5.35 7.71
1.51 2.21 2.80 3.44 4.39 5.20 7.48
1.50 2.19 2.75 3.37 4.30 5.09 7.29
Appendix
1105
TABLE 8 (continued ) df1 12
15
20
24
30
40
60
120
240
inf.
␣
df2
1.68 2.67 3.57 4.67 6.47 8.18 13.71
1.68 2.63 3.51 4.57 6.31 7.97 13.32
1.67 2.59 3.44 4.47 6.16 7.75 12.93
1.67 2.58 3.41 4.41 6.07 7.64 12.73
1.66 2.56 3.38 4.36 5.99 7.53 12.53
1.66 2.54 3.34 4.31 5.91 7.42 12.33
1.65 2.51 3.30 4.25 5.82 7.31 12.12
1.65 2.49 3.27 4.20 5.74 7.19 11.91
1.65 2.48 3.25 4.17 5.69 7.13 11.80
1.65 2.47 3.23 4.14 5.65 7.08 11.70
.25 .10 .05 .025 .01 .005 .001
7
1.62 2.50 3.28 4.20 5.67 7.01 11.19
1.62 2.46 3.22 4.10 5.52 6.81 10.84
1.61 2.42 3.15 4.00 5.36 6.61 10.48
1.60 2.40 3.12 3.95 5.28 6.50 10.30
1.60 2.38 3.08 3.89 5.20 6.40 10.11
1.59 2.36 3.04 3.84 5.12 6.29 9.92
1.59 2.34 3.01 3.78 5.03 6.18 9.73
1.58 2.32 2.97 3.73 4.95 6.06 9.53
1.58 2.30 2.95 3.70 4.90 6.01 9.43
1.58 2.29 2.93 3.67 4.86 5.95 9.33
.25 .10 .05 .025 .01 .005 .001
8
1.58 2.38 3.07 3.87 5.11 6.23 9.57
1.57 2.34 3.01 3.77 4.96 6.03 9.24
1.56 2.30 2.94 3.67 4.81 5.83 8.90
1.56 2.28 2.90 3.61 4.73 5.73 8.72
1.55 2.25 2.86 3.56 4.65 5.62 8.55
1.54 2.23 2.83 3.51 4.57 5.52 8.37
1.64 2.21 2.79 3.45 4.48 5.41 8.19
1.53 2.18 2.75 3.39 4.40 5.30 8.00
1.53 2.17 2.73 3.36 4.35 5.24 7.91
1.53 2.16 2.71 3.33 4.31 5.19 7.81
.25 .10 .05 .025 .01 .005 .001
9
1.54 2.28 2.91 3.62 4.71 5.66 8.45
1.53 2.24 2.85 3.52 4.56 5.47 8.13
1.52 2.20 2.77 3.42 4.41 5.27 7.80
1.52 2.18 2.74 3.37 4.33 5.17 7.64
1.51 2.16 2.70 3.31 4.25 5.07 7.47
1.51 2.13 2.66 3.26 4.17 4.97 7.30
1.50 2.11 2.62 3.20 4.08 4.86 7.12
1.49 2.08 2.58 3.14 4.00 4.75 6.94
1.49 2.07 2.56 3.11 3.95 4.69 6.85
1.48 2.06 2.54 3.08 3.91 4.64 6.76
.25 .10 .05 .025 .01 .005 .001
10
1.51 2.21 2.79 3.43 4.40 5.24 7.63
1.50 2.17 2.72 3.33 4.25 5.05 7.32
1.49 2.12 2.65 3.23 4.10 4.86 7.01
1.49 2.10 2.61 3.17 4.02 4.76 6.85
1.48 2.08 2.57 3.12 3.94 4.65 6.68
1.47 2.05 2.53 3.06 3.86 4.55 6.52
1.47 2.03 2.49 3.00 3.78 4.45 6.35
1.46 2.00 2.45 2.94 3.69 4.34 6.18
1.45 1.99 2.43 2.91 3.65 4.28 6.09
1.45 1.97 2.40 2.88 3.60 4.23 6.00
.25 .10 .05 .025 .01 .005 .001
11
1.49 2.15 2.69 3.28 4.16 4.91 7.00
1.48 2.10 2.62 3.18 4.01 4.72 6.71
1.47 2.06 2.54 3.07 3.86 4.53 6.40
1.46 2.04 2.51 3.02 3.78 4.43 6.25
1.45 2.01 2.47 2.96 3.70 4.33 6.09
1.45 1.99 2.43 2.91 3.62 4.23 5.93
1.44 1.96 2.38 2.85 3.54 4.12 5.76
1.43 1.93 2.34 2.79 3.45 4.01 5.59
1.43 1.92 2.32 2.76 3.41 3.96 5.51
1.42 1.90 2.30 2.72 3.36 3.90 5.42
.25 .10 .05 .025 .01 .005 .001
12
1106
Appendix
␣
TABLE 8
F␣
Percentage points of the F distribution (df2 between 13 and 18) df1 df2
␣
1
2
3
4
5
6
7
8
9
10
13
.25 .10 .05 .025 .01 .005 .001
1.45 3.14 4.67 6.41 9.07 11.37 17.82
1.55 2.76 3.81 4.97 6.70 8.19 12.31
1.55 2.56 3.41 4.35 5.74 6.93 10.21
1.53 2.43 3.18 4.00 5.21 6.23 9.07
1.52 2.35 3.03 3.77 4.86 5.79 8.35
1.51 2.28 2.92 3.60 4.62 5.48 7.86
1.50 2.23 2.83 3.48 4.44 5.25 7.49
1.49 2.20 2.77 3.39 4.30 5.08 7.21
1.49 2.16 2.71 3.31 4.19 4.94 6.98
1.48 2.14 2.67 3.25 4.10 4.82 6.80
14
.25 .10 .05 .025 .01 .005 .001
1.44 3.10 4.60 6.30 8.86 11.06 17.14
1.53 2.73 3.74 4.86 6.51 7.92 11.78
1.53 2.52 3.34 4.24 5.56 6.68 9.73
1.52 2.39 3.11 3.89 5.04 6.00 8.62
1.51 2.31 2.96 3.66 4.69 5.56 7.92
1.50 2.24 2.85 3.50 4.46 5.26 7.44
1.49 2.19 2.76 3.38 4.28 5.03 7.08
1.48 2.15 2.70 3.29 4.14 4.86 6.80
1.47 2.12 2.65 3.21 4.03 4.72 6.58
1.46 2.10 2.60 3.15 3.94 4.60 6.40
15
.25 .10 .05 .025 .01 .005 .001
1.43 3.07 4.54 6.20 8.68 10.80 16.59
1.52 2.70 3.68 4.77 6.36 7.70 11.34
1.52 2.49 3.29 4.15 5.42 6.48 9.34
1.51 2.36 3.06 3.80 4.89 5.80 8.25
1.49 2.27 2.90 3.58 4.56 5.37 7.57
1.48 2.21 2.79 3.41 4.32 5.07 7.09
1.47 2.16 2.71 3.29 4.14 4.85 6.74
1.46 2.12 2.64 3.20 4.00 4.67 6.47
1.46 2.09 2.59 3.12 3.89 4.54 6.26
1.45 2.06 2.54 3.06 3.80 4.42 6.08
16
.25 .10 .05 .025 .01 .005 .001
1.42 3.05 4.49 6.12 8.53 10.58 16.12
1.51 2.67 3.63 4.69 6.23 7.51 10.97
1.51 2.46 3.24 4.08 5.29 6.30 9.01
1.50 2.33 3.01 3.73 4.77 5.64 7.94
1.48 2.24 2.85 3.50 4.44 5.21 7.27
1.47 2.18 2.74 3.34 4.20 4.91 6.80
1.46 2.13 2.66 3.22 4.03 4.69 6.46
1.45 2.09 2.59 3.12 3.89 4.52 6.19
1.44 2.06 2.54 3.05 3.78 4.38 5.98
1.44 2.03 2.49 2.99 3.69 4.27 5.81
17
.25 .10 .05 .025 .01 .005 .001
1.42 3.03 4.45 6.04 8.40 10.38 15.72
1.51 2.64 3.59 4.62 6.11 7.35 10.66
1.50 2.44 3.20 4.01 5.18 6.16 8.73
1.49 2.31 2.96 3.66 4.67 5.50 7.68
1.47 2.22 2.81 3.44 4.34 5.07 7.02
1.46 2.15 2.70 3.28 4.10 4.78 6.56
1.45 2.10 2.61 3.16 3.93 4.56 6.22
1.44 2.06 2.55 3.06 3.79 4.39 5.96
1.43 2.03 2.49 2.98 3.68 4.25 5.75
1.43 2.00 2.45 2.92 3.59 4.14 5.58
18
.25 .10 .05 .025 .01 .005 .001
1.41 3.01 4.41 5.98 8.29 10.22 15.38
1.50 2.62 3.55 4.56 6.01 7.21 10.39
1.49 2.42 3.16 3.95 5.09 6.03 8.49
1.48 2.29 2.93 3.61 4.58 5.37 7.46
1.46 2.20 2.77 3.38 4.25 4.96 6.81
1.45 2.13 2.66 3.22 4.01 4.66 6.35
1.44 2.08 2.58 3.10 3.84 4.44 6.02
1.43 2.04 2.51 3.01 3.71 4.28 5.76
1.42 2.00 2.46 2.93 3.60 4.14 5.56
1.42 1.98 2.41 2.87 3.51 4.03 5.39
Appendix
1107
TABLE 8 (continued ) df1 12
15
20
24
30
40
60
120
240
inf.
␣
df2
1.47 2.10 2.60 3.15 3.96 4.64 6.52
1.46 2.05 2.53 3.05 3.82 4.46 6.23
1.45 2.01 2.46 2.95 3.66 4.27 5.93
1.44 1.98 2.42 2.89 3.59 4.17 5.78
1.43 1.96 2.38 2.84 3.51 4.07 5.63
1.42 1.93 2.34 2.78 3.43 3.97 5.47
1.42 1.90 2.30 2.72 3.34 3.87 5.30
1.41 1.88 2.25 2.66 3.25 3.76 5.14
1.40 1.86 2.23 2.63 3.21 3.70 5.05
1.40 1.85 2.21 2.60 3.17 3.65 4.97
.25 .10 .05 .025 .01 .005 .001
13
1.45 2.05 2.53 3.05 3.80 4.43 6.13
1.44 2.01 2.46 2.95 3.66 4.25 5.85
1.43 1.96 2.39 2.84 3.51 4.06 5.56
1.42 1.94 2.35 2.79 3.43 3.96 5.41
1.41 1.91 2.31 2.73 3.35 3.86 5.25
1.41 1.89 2.27 2.67 3.27 3.76 5.10
1.40 1.86 2.22 2.61 3.18 3.66 4.94
1.39 1.83 2.18 2.55 3.09 3.55 4.77
1.38 1.81 2.15 2.52 3.05 3.49 4.69
1.38 1.80 2.13 2.49 3.00 3.44 4.60
.25 .10 .05 .025 .01 .005 .001
14
1.44 2.02 2.48 2.96 3.67 4.25 5.81
1.43 1.97 2.40 2.86 3.52 4.07 5.54
1.41 1.92 2.33 2.76 3.37 3.88 5.25
1.41 1.90 2.29 2.70 3.29 3.79 5.10
1.40 1.87 2.25 2.64 3.21 3.69 4.95
1.39 1.85 2.20 2.59 3.13 3.58 4.80
1.38 1.82 2.16 2.52 3.05 3.48 4.64
1.37 1.79 2.11 2.46 2.96 3.37 4.47
1.36 1.77 2.09 2.43 2.91 3.32 4.39
1.36 1.76 2.07 2.40 2.87 3.26 4.31
.25 .10 .05 .025 .01 .005 .001
15
1.43 1.99 2.42 2.89 3.55 4.10 5.55
1.41 1.94 2.35 2.79 3.41 3.92 5.27
1.40 1.89 2.28 2.68 3.26 3.73 4.99
1.39 1.87 2.24 2.63 3.18 3.64 4.85
1.38 1.84 2.19 2.57 3.10 3.54 4.70
1.37 1.81 2.15 2.51 3.02 3.44 4.54
1.36 1.78 2.11 2.45 2.93 3.33 4.39
1.35 1.75 2.06 2.38 2.84 3.22 4.23
1.35 1.73 2.03 2.35 2.80 3.17 4.14
1.34 1.72 2.01 2.32 2.75 3.11 4.06
.25 .10 .05 .025 .01 .005 .001
16
1.41 1.96 2.38 2.82 3.46 3.97 5.32
1.40 1.91 2.31 2.72 3.31 3.79 5.05
1.39 1.86 2.23 2.62 3.16 3.61 4.78
1.38 1.84 2.19 2.56 3.08 3.51 4.63
1.37 1.81 2.15 2.50 3.00 3.41 4.48
1.36 1.78 2.10 2.44 2.92 3.31 4.33
1.35 1.75 2.06 2.38 2.83 3.21 4.18
1.34 1.72 2.01 2.32 2.75 3.10 4.02
1.33 1.70 1.99 2.28 2.70 3.04 3.93
1.33 1.69 1.96 2.25 2.65 2.98 3.85
.25 .10 .05 .025 .01 .005 .001
17
1.40 1.93 2.34 2.77 3.37 3.86 5.13
1.39 1.89 2.27 2.67 3.23 3.68 4.87
1.38 1.84 2.19 2.56 3.08 3.50 4.59
1.37 1.81 2.15 2.50 3.00 3.40 4.45
1.36 1.78 2.11 2.44 2.92 3.30 4.30
1.35 1.75 2.06 2.38 2.84 3.20 4.15
1.34 1.72 2.02 2.32 2.75 3.10 4.00
1.33 1.69 1.97 2.26 2.66 2.99 3.84
1.32 1.67 1.94 2.22 2.61 2.93 3.75
1.32 1.66 1.92 2.19 2.57 2.87 3.67
.25 .10 .05 .025 .01 .005 .001
18
1108
Appendix
␣
TABLE 8
F␣
Percentage points of the F distribution (df2 between 19 and 24) df1 df2
␣
1
2
3
4
5
6
7
8
9
10
19
.25 .10 .05 .025 .01 .005 .001
1.41 2.99 4.38 5.92 8.18 10.07 15.08
1.49 2.61 3.52 4.51 5.93 7.09 10.16
1.49 2.40 3.13 3.90 5.01 5.92 8.28
1.47 2.27 2.90 3.56 4.50 5.27 7.27
1.46 2.18 2.74 3.33 4.17 4.85 6.62
1.44 2.11 2.63 3.17 3.94 4.56 6.18
1.43 2.06 2.54 3.05 3.77 4.34 5.85
1.42 2.02 2.48 2.96 3.63 4.18 5.59
1.41 1.98 2.42 2.88 3.52 4.04 5.39
1.41 1.96 2.38 2.82 3.43 3.93 5.22
20
.25 .10 .05 .025 .01 .005 .001
1.40 2.97 4.35 5.87 8.10 9.94 14.82
1.49 2.59 3.49 4.46 5.85 6.99 9.95
1.48 2.38 3.10 3.86 4.94 5.82 8.10
1.47 2.25 2.87 3.51 4.43 5.17 7.10
1.45 2.16 2.71 3.29 4.10 4.76 6.46
1.44 2.09 2.60 3.13 3.87 4.47 6.02
1.43 2.04 2.51 3.01 3.70 4.26 5.69
1.42 2.00 2.45 2.91 3.56 4.09 5.44
1.41 1.96 2.39 2.84 3.46 3.96 5.24
1.40 1.94 2.35 2.77 3.37 3.85 5.08
21
.25 .10 .05 .025 .01 .005 .001
1.40 2.96 4.32 5.83 8.02 9.83 14.59
1.48 2.57 3.47 4.42 5.78 6.89 9.77
1.48 2.36 3.07 3.82 4.87 5.73 7.94
1.46 2.23 2.84 3.48 4.37 5.09 6.95
1.44 2.14 2.68 3.25 4.04 4.68 6.32
1.43 2.08 2.57 3.09 3.81 4.39 5.88
1.42 2.02 2.49 2.97 3.64 4.18 5.56
1.41 1.98 2.42 2.87 3.51 4.01 5.31
1.40 1.95 2.37 2.80 3.40 3.88 5.11
1.39 1.92 2.32 2.73 3.31 3.77 4.95
22
.25 .10 .05 .025 .01 .005 .001
1.40 2.95 4.30 5.79 7.95 9.73 14.38
1.48 2.56 3.44 4.38 5.72 6.81 9.61
1.47 2.35 3.05 3.78 4.82 5.65 7.80
1.45 2.22 2.82 3.44 4.31 5.02 6.81
1.44 2.13 2.66 3.22 3.99 4.61 6.19
1.42 2.06 2.55 3.05 3.76 4.32 5.76
1.41 2.01 2.46 2.93 3.59 4.11 5.44
1.40 1.97 2.40 2.84 3.45 3.94 5.19
1.39 1.93 2.34 2.76 3.35 3.81 4.99
1.39 1.90 2.30 2.70 3.26 3.70 4.83
23
.25 .10 .05 .025 .01 .005 .001
1.39 2.94 4.28 5.75 7.88 9.63 14.20
1.47 2.55 3.42 4.35 5.66 6.73 9.47
1.47 2.34 3.03 3.75 4.76 5.58 7.67
1.45 2.21 2.80 3.41 4.26 4.95 6.70
1.43 2.11 2.64 3.18 3.94 4.54 6.08
1.42 2.05 2.53 3.02 3.71 4.26 5.65
1.41 1.99 2.44 2.90 3.54 4.05 5.33
1.40 1.95 2.37 2.81 3.41 3.88 5.09
1.39 1.92 2.32 2.73 3.30 3.75 4.89
1.38 1.89 2.27 2.67 3.21 3.64 4.73
24
.25 .10 .05 .025 .01 .005 .001
1.39 2.93 4.26 5.72 7.82 9.55 14.03
1.47 2.54 3.40 4.32 5.61 6.66 9.34
1.46 2.33 3.01 3.72 4.72 5.52 7.55
1.44 2.19 2.78 3.38 4.22 4.89 6.59
1.43 2.10 2.62 3.15 3.90 4.49 5.98
1.41 2.04 2.51 2.99 3.67 4.20 5.55
1.40 1.98 2.42 2.87 3.50 3.99 5.23
1.39 1.94 2.36 2.78 3.36 3.83 4.99
1.38 1.91 2.30 2.70 3.26 3.69 4.80
1.38 1.88 2.25 2.64 3.17 3.59 4.64
Appendix
1109
TABLE 8 (continued ) df1 12
15
20
24
30
40
60
120
240
inf.
␣
df2
1.40 1.91 2.31 2.72 3.30 3.76 4.97
1.38 1.86 2.23 2.62 3.15 3.59 4.70
1.37 1.81 2.16 2.51 3.00 3.40 4.43
1.36 1.79 2.11 2.45 2.92 3.31 4.29
1.35 1.76 2.07 2.39 2.84 3.21 4.14
1.34 1.73 2.03 2.33 2.76 3.11 3.99
1.33 1.70 1.98 2.27 2.67 3.00 3.84
1.32 1.67 1.93 2.20 2.58 2.89 3.68
1.31 1.65 1.90 2.17 2.54 2.83 3.60
1.30 1.63 1.88 2.13 2.49 2.78 3.51
.25 .10 .05 .025 .01 .005 .001
19
1.39 1.89 2.28 2.68 3.23 3.68 4.82
1.37 1.84 2.20 2.57 3.09 3.50 4.56
1.36 1.79 2.12 2.46 2.94 3.32 4.29
1.35 1.77 2.08 2.41 2.86 3.22 4.15
1.34 1.74 2.04 2.35 2.78 3.12 4.00
1.33 1.71 1.99 2.29 2.69 3.02 3.86
1.32 1.68 1.95 2.22 2.61 2.92 3.70
1.31 1.64 1.90 2.16 2.52 2.81 3.54
1.30 1.63 1.87 2.12 2.47 2.75 3.46
1.29 1.61 1.84 2.09 2.42 2.69 3.38
.25 .10 .05 .025 .01 .005 .001
20
1.38 1.87 2.25 2.64 3.17 3.60 4.70
1.37 1.83 2.18 2.53 3.03 3.43 4.44
1.35 1.78 2.10 2.42 2.88 3.24 4.17
1.34 1.75 2.05 2.37 2.80 3.15 4.03
1.33 1.72 2.01 2.31 2.72 3.05 3.88
1.32 1.69 1.96 2.25 2.64 2.95 3.74
1.31 1.66 1.92 2.18 2.55 2.84 3.58
1.30 1.62 1.87 2.11 2.46 2.73 3.42
1.29 1.60 1.84 2.08 2.41 2.67 3.34
1.28 1.59 1.81 2.04 2.36 2.61 3.26
.25 .10 .05 .025 .01 .005 .001
21
1.37 1.86 2.23 2.60 3.12 3.54 4.58
1.36 1.81 2.15 2.50 2.98 3.36 4.33
1.34 1.76 2.07 2.39 2.83 3.18 4.06
1.33 1.73 2.03 2.33 2.75 3.08 3.92
1.32 1.70 1.98 2.27 2.67 2.98 3.78
1.31 1.67 1.94 2.21 2.58 2.88 3.63
1.30 1.64 1.89 2.14 2.50 2.77 3.48
1.29 1.60 1.84 2.08 2.40 2.66 3.32
1.28 1.59 1.81 2.04 2.35 2.60 3.23
1.28 1.57 1.78 2.00 2.31 2.55 3.15
.25 .10 .05 .025 .01 .005 .001
22
1.37 1.84 2.20 2.57 3.07 3.47 4.48
1.35 1.80 2.13 2.47 2.93 3.30 4.23
1.34 1.74 2.05 2.36 2.78 3.12 3.96
1.33 1.72 2.01 2.30 2.70 3.02 3.82
1.32 1.69 1.96 2.24 2.62 2.92 3.68
1.31 1.66 1.91 2.18 2.54 2.82 3.53
1.30 1.62 1.86 2.11 2.45 2.71 3.38
1.28 1.59 1.81 2.04 2.35 2.60 3.22
1.28 1.57 1.79 2.01 2.31 2.54 3.14
1.27 1.55 1.76 1.97 2.26 2.48 3.05
.25 .10 .05 .025 .01 .005 .001
23
1.36 1.83 2.18 2.54 3.03 3.42 4.39
1.35 1.78 2.11 2.44 2.89 3.25 4.14
1.33 1.73 2.03 2.33 2.74 3.06 3.87
1.32 1.70 1.98 2.27 2.66 2.97 3.74
1.31 1.67 1.94 2.21 2.58 2.87 3.59
1.30 1.64 1.89 2.15 2.49 2.77 3.45
1.29 1.61 1.84 2.08 2.40 2.66 3.29
1.28 1.57 1.79 2.01 2.31 2.55 3.14
1.27 1.55 1.76 1.97 2.26 2.49 3.05
1.26 1.53 1.73 1.94 2.21 2.43 2.97
.25 .10 .05 .025 .01 .005 .001
24
1110
Appendix
␣
TABLE 8
F␣
Percentage points of the F distribution (df2 between 25 and 30) df1 df2
␣
1
2
3
4
5
6
7
8
9
10
25
.25 .10 .05 .025 .01 .005 .001
1.39 2.92 4.24 5.69 7.77 9.48 13.88
1.47 2.53 3.39 4.29 5.57 6.60 9.22
1.46 2.32 2.99 3.69 4.68 5.46 7.45
1.44 2.18 2.76 3.35 4.18 4.84 6.49
1.42 2.09 2.60 3.13 3.85 4.43 5.89
1.41 2.02 2.49 2.97 3.63 4.15 5.46
1.40 1.97 2.40 2.85 3.46 3.94 5.15
1.39 1.93 2.34 2.75 3.32 3.78 4.91
1.38 1.89 2.28 2.68 3.22 3.64 4.71
1.37 1.87 2.24 2.61 3.13 3.54 4.56
26
.25 .10 .05 .025 .01 .005 .001
1.38 2.91 4.23 5.66 7.72 9.41 13.74
1.46 2.52 3.37 4.27 5.53 6.54 9.12
1.45 2.31 2.98 3.67 4.64 5.41 7.36
1.44 2.17 2.74 3.33 4.14 4.79 6.41
1.42 2.08 2.59 3.10 3.82 4.38 5.80
1.41 2.01 2.47 2.94 3.59 4.10 5.38
1.39 1.96 2.39 2.82 3.42 3.89 5.07
1.38 1.92 2.32 2.73 3.29 3.73 4.83
1.37 1.88 2.27 2.65 3.18 3.60 4.64
1.37 1.86 2.22 2.59 3.09 3.49 4.48
27
.25 .10 .05 .025 .01 .005 .001
1.38 2.90 4.21 5.63 7.68 9.34 13.61
1.46 2.51 3.35 4.24 5.49 6.49 9.02
1.45 2.30 2.96 3.65 4.60 5.36 7.27
1.43 2.17 2.73 3.31 4.11 4.74 6.33
1.42 2.07 2.57 3.08 3.78 4.34 5.73
1.40 2.00 2.46 2.92 3.56 4.06 5.31
1.39 1.95 2.37 2.80 3.39 3.85 5.00
1.38 1.91 2.31 2.71 3.26 3.69 4.76
1.37 1.87 2.25 2.63 3.15 3.56 4.57
1.36 1.85 2.20 2.57 3.06 3.45 4.41
28
.25 .10 .05 .025 .01 .005 .001
1.38 2.89 4.20 5.61 7.64 9.28 13.50
1.46 2.50 3.34 4.22 5.45 6.44 8.93
1.45 2.29 2.95 3.63 4.57 5.32 7.19
1.43 2.16 2.71 3.29 4.07 4.70 6.25
1.41 2.06 2.56 3.06 3.75 4.30 5.66
1.40 2.00 2.45 2.90 3.53 4.02 5.24
1.39 1.94 2.36 2.78 3.36 3.81 4.93
1.38 1.90 2.29 2.69 3.23 3.65 4.69
1.37 1.87 2.24 2.61 3.12 3.52 4.50
1.36 1.84 2.19 2.55 3.03 3.41 4.35
29
.25 .10 .05 .025 .01 .005 .001
1.38 2.89 4.18 5.59 7.60 9.23 13.39
1.45 2.50 3.33 4.20 5.42 6.40 8.85
1.45 2.28 2.93 3.61 4.54 5.28 7.12
1.43 2.15 2.70 3.27 4.04 4.66 6.19
1.41 2.06 2.55 3.04 3.73 4.26 5.59
1.40 1.99 2.43 2.88 3.50 3.98 5.18
1.38 1.93 2.35 2.76 3.33 3.77 4.87
1.37 1.89 2.28 2.67 3.20 3.61 4.64
1.36 1.86 2.22 2.59 3.09 3.48 4.45
1.35 1.83 2.18 2.53 3.00 3.38 4.29
30
.25 .10 .05 .025 .01 .005 .001
1.38 2.88 4.17 5.57 7.56 9.18 13.29
1.45 2.49 3.32 4.18 5.39 6.35 8.77
1.44 2.28 2.92 3.59 4.51 5.24 7.05
1.42 2.14 2.69 3.25 4.02 4.62 6.12
1.41 2.05 2.53 3.03 3.70 4.23 5.53
1.39 1.98 2.42 2.87 3.47 3.95 5.12
1.38 1.93 2.33 2.75 3.30 3.74 4.82
1.37 1.88 2.27 2.65 3.17 3.58 4.58
1.36 1.85 2.21 2.57 3.07 3.45 4.39
1.35 1.82 2.16 2.51 2.98 3.34 4.24
Appendix
1111
TABLE 8 (continued ) df1 12
15
20
24
30
40
60
120
240
inf.
␣
df2
1.36 1.82 2.16 2.51 2.99 3.37 4.31
1.34 1.77 2.09 2.41 2.85 3.20 4.06
1.33 1.72 2.01 2.30 2.70 3.01 3.79
1.32 1.69 1.96 2.24 2.62 2.92 3.66
1.31 1.66 1.92 2.18 2.54 2.82 3.52
1.29 1.63 1.87 2.12 2.45 2.72 3.37
1.28 1.59 1.82 2.05 2.36 2.61 3.22
1.27 1.56 1.77 1.98 2.27 2.50 3.06
1.26 1.54 1.74 1.94 2.22 2.44 2.98
1.25 1.52 1.71 1.91 2.17 2.38 2.89
.25 .10 .05 .025 .01 .005 .001
25
1.35 1.81 2.15 2.49 2.96 3.33 4.24
1.34 1.76 2.07 2.39 2.81 3.15 3.99
1.32 1.71 1.99 2.28 2.66 2.97 3.72
1.31 1.68 1.95 2.22 2.58 2.87 3.59
1.30 1.65 1.90 2.16 2.50 2.77 3.44
1.29 1.61 1.85 2.09 2.42 2.67 3.30
1.28 1.58 1.80 2.03 2.33 2.56 3.15
1.26 1.54 1.75 1.95 2.23 2.45 2.99
1.26 1.52 1.72 1.92 2.18 2.39 2.90
1.25 1.50 1.69 1.88 2.13 2.33 2.82
.25 .10 .05 .025 .01 .005 .001
26
1.35 1.80 2.13 2.47 2.93 3.28 4.17
1.33 1.75 2.06 2.36 2.78 3.11 3.92
1.32 1.70 1.97 2.25 2.63 2.93 3.66
1.31 1.67 1.93 2.19 2.55 2.83 3.52
1.30 1.64 1.88 2.13 2.47 2.73 3.38
1.28 1.60 1.84 2.07 2.38 2.63 3.23
1.27 1.57 1.79 2.00 2.29 2.52 3.08
1.26 1.53 1.73 1.93 2.20 2.41 2.92
1.25 1.51 1.70 1.89 2.15 2.35 2.84
1.24 1.49 1.67 1.85 2.10 2.29 2.75
.25 .10 .05 .025 .01 .005 .001
27
1.34 1.79 2.12 2.45 2.90 3.25 4.11
1.33 1.74 2.04 2.34 2.75 3.07 3.86
1.31 1.69 1.96 2.23 2.60 2.89 3.60
1.30 1.66 1.91 2.17 2.52 2.79 3.46
1.29 1.63 1.87 2.11 2.44 2.69 3.32
1.28 1.59 1.82 2.05 2.35 2.59 3.18
1.27 1.56 1.77 1.98 2.26 2.48 3.02
1.25 1.52 1.71 1.91 2.17 2.37 2.86
1.24 1.50 1.68 1.87 2.12 2.31 2.78
1.24 1.48 1.65 1.83 2.06 2.25 2.69
.25 .10 .05 .025 .01 .005 .001
28
1.34 1.78 2.10 2.43 2.87 3.21 4.05
1.32 1.73 2.03 2.32 2.73 3.04 3.80
1.31 1.68 1.94 2.21 2.57 2.86 3.54
1.30 1.65 1.90 2.15 2.49 2.76 3.41
1.29 1.62 1.85 2.09 2.41 2.66 3.27
1.27 1.58 1.81 2.03 2.33 2.56 3.12
1.26 1.55 1.75 1.96 2.23 2.45 2.97
1.25 1.51 1.70 1.89 2.14 2.33 2.81
1.24 1.49 1.67 1.85 2.09 2.27 2.73
1.23 1.47 1.64 1.81 2.03 2.21 2.64
.25 .10 .05 .025 .01 .005 .001
29
1.34 1.77 2.09 2.41 2.84 3.18 4.00
1.32 1.72 2.01 2.31 2.70 3.01 3.75
1.30 1.67 1.93 2.20 2.55 2.82 3.49
1.29 1.64 1.89 2.14 2.47 2.73 3.36
1.28 1.61 1.84 2.07 2.39 2.63 3.22
1.27 1.57 1.79 2.01 2.30 2.52 3.07
1.26 1.54 1.74 1.94 2.21 2.42 2.92
1.24 1.50 1.68 1.87 2.11 2.30 2.76
1.23 1.48 1.65 1.83 2.06 2.24 2.68
1.23 1.46 1.62 1.79 2.01 2.18 2.59
.25 .10 .05 .025 .01 .005 .001
30
1112
Appendix
␣
TABLE 8
F␣
Percentage points of the F distribution (df2 at least 40) df1 df2
␣
1
2
3
4
5
6
7
8
9
10
40
.25 .10 .05 .025 .01 .005 .001
1.36 2.84 4.08 5.42 7.31 8.83 12.61
1.44 2.44 3.23 4.05 5.18 6.07 8.25
1.42 2.23 2.84 3.46 4.31 4.98 6.59
1.40 2.09 2.61 3.13 3.83 4.37 5.70
1.39 2.00 2.45 2.90 3.51 3.99 5.13
1.37 1.93 2.34 2.74 3.29 3.71 4.73
1.36 1.87 2.25 2.62 3.12 3.51 4.44
1.35 1.83 2.18 2.53 2.99 3.35 4.21
1.34 1.79 2.12 2.45 2.89 3.22 4.02
1.33 1.76 2.08 2.39 2.80 3.12 3.87
60
.25 .10 .05 .025 .01 .005 .001
1.35 2.79 4.00 5.29 7.08 8.49 11.97
1.42 2.39 3.15 3.93 4.98 5.79 7.77
1.41 2.18 2.76 3.34 4.13 4.73 6.17
1.38 2.04 2.53 3.01 3.65 4.14 5.31
1.37 1.95 2.37 2.79 3.34 3.76 4.76
1.35 1.87 2.25 2.63 3.12 3.49 4.37
1.33 1.82 2.17 2.51 2.95 3.29 4.09
1.32 1.77 2.10 2.41 2.82 3.13 3.86
1.31 1.74 2.04 2.33 2.72 3.01 3.69
1.30 1.71 1.99 2.27 2.63 2.90 3.54
90
.25 .10 .05 .025 .01 .005 .001
1.34 2.76 3.95 5.20 6.93 8.28 11.57
1.41 2.36 3.10 3.84 4.85 5.62 7.47
1.39 2.15 2.71 3.26 4.01 4.57 5.91
1.37 2.01 2.47 2.93 3.53 3.99 5.06
1.35 1.91 2.32 2.71 3.23 3.62 4.53
1.33 1.84 2.20 2.55 3.01 3.35 4.15
1.32 1.78 2.11 2.43 2.84 3.15 3.87
1.31 1.74 2.04 2.34 2.72 3.00 3.65
1.30 1.70 1.99 2.26 2.61 2.87 3.48
1.29 1.67 1.94 2.19 2.52 2.77 3.34
120
.25 .10 .05 .025 .01 .005 .001
1.34 2.75 3.92 5.15 6.85 8.18 11.38
1.40 2.35 3.07 3.80 4.79 5.54 7.32
1.39 2.13 2.68 3.23 3.95 4.50 5.78
1.37 1.99 2.45 2.89 3.48 3.92 4.95
1.35 1.90 2.29 2.67 3.17 3.55 4.42
1.33 1.82 2.18 2.52 2.96 3.28 4.04
1.31 1.77 2.09 2.39 2.79 3.09 3.77
1.30 1.72 2.02 2.30 2.66 2.93 3.55
1.29 1.68 1.96 2.22 2.56 2.81 3.38
1.28 1.65 1.91 2.16 2.47 2.71 3.24
240
.25 .10 .05 .025 .01 .005 .001
1.33 2.73 3.88 5.09 6.74 8.03 11.10
1.39 2.32 3.03 3.75 4.69 5.42 7.11
1.38 2.10 2.64 3.17 3.86 4.38 5.60
1.36 1.97 2.41 2.84 3.40 3.82 4.78
1.34 1.87 2.25 2.62 3.09 3.45 4.25
1.32 1.80 2.14 2.46 2.88 3.19 3.89
1.30 1.74 2.04 2.34 2.71 2.99 3.62
1.29 1.70 1.98 2.25 2.59 2.84 3.41
1.27 1.65 1.92 2.17 2.48 2.71 3.24
1.27 1.63 1.87 2.10 2.40 2.61 3.09
inf.
.25 .10 .05 .025 .01 .005 .001
1.32 2.71 3.84 5.02 6.63 7.88 10.83
1.39 2.30 3.00 3.69 4.61 5.30 6.91
1.37 2.08 2.60 3.12 3.78 4.28 5.42
1.35 1.94 2.37 2.79 3.32 3.72 4.62
1.33 1.85 2.21 2.57 3.02 3.35 4.10
1.31 1.77 2.10 2.41 2.80 3.09 3.74
1.29 1.72 2.01 2.29 2.64 2.90 3.47
1.28 1.67 1.94 2.19 2.51 2.74 3.27
1.27 1.63 1.88 2.11 2.41 2.62 3.10
1.25 1.60 1.83 2.05 2.32 2.52 2.96
Appendix
1113
TABLE 8 (continued ) df1 12
15
20
24
30
40
60
120
240
inf.
␣
df2
1.31 1.71 2.00 2.29 2.66 2.95 3.64
1.30 1.66 1.92 2.18 2.52 2.78 3.40
1.28 1.61 1.84 2.07 2.37 2.60 3.14
1.26 1.57 1.79 2.01 2.29 2.50 3.01
1.25 1.54 1.74 1.94 2.20 2.40 2.87
1.24 1.51 1.69 1.88 2.11 2.30 2.73
1.22 1.47 1.64 1.80 2.02 2.18 2.57
1.21 1.42 1.58 1.72 1.92 2.06 2.41
1.20 1.40 1.54 1.68 1.86 2.00 2.32
1.19 1.38 1.51 1.64 1.80 1.93 2.23
.25 .10 .05 .025 .01 .005 .001
40
1.29 1.66 1.92 2.17 2.50 2.74 3.32
1.27 1.60 1.84 2.06 2.35 2.57 3.08
1.25 1.54 1.75 1.94 2.20 2.39 2.83
1.24 1.51 1.70 1.88 2.12 2.29 2.69
1.22 1.48 1.65 1.82 2.03 2.19 2.55
1.21 1.44 1.59 1.74 1.94 2.08 2.41
1.19 1.40 1.53 1.67 1.84 1.96 2.25
1.17 1.35 1.47 1.58 1.73 1.83 2.08
1.16 1.32 1.43 1.53 1.67 1.76 1.99
1.15 1.29 1.39 1.48 1.60 1.69 1.89
.25 .10 .05 .025 .01 .005 .001
60
1.27 1.62 1.86 2.09 2.39 2.61 3.11
1.25 1.56 1.78 1.98 2.24 2.44 2.88
1.23 1.50 1.69 1.86 2.09 2.25 2.63
1.22 1.47 1.64 1.80 2.00 2.15 2.50
1.20 1.43 1.59 1.73 1.92 2.05 2.36
1.19 1.39 1.53 1.66 1.82 1.94 2.21
1.17 1.35 1.46 1.58 1.72 1.82 2.05
1.15 1.29 1.39 1.48 1.60 1.68 1.87
1.13 1.26 1.35 1.43 1.53 1.61 1.77
1.12 1.23 1.30 1.37 1.46 1.52 1.66
.25 .10 .05 .025 .01 .005 .001
90
1.26 1.60 1.83 2.05 2.34 2.54 3.02
1.24 1.55 1.75 1.94 2.19 2.37 2.78
1.22 1.48 1.66 1.82 2.03 2.19 2.53
1.21 1.45 1.61 1.76 1.95 2.09 2.40
1.19 1.41 1.55 1.69 1.86 1.98 2.26
1.18 1.37 1.50 1.61 1.76 1.87 2.11
1.16 1.32 1.43 1.53 1.66 1.75 1.95
1.13 1.26 1.35 1.43 1.53 1.61 1.77
1.12 1.23 1.31 1.38 1.46 1.52 1.66
1.10 1.19 1.25 1.31 1.38 1.43 1.54
.25 10 .05 .025 .01 .005 .001
120
1.25 1.57 1.79 2.00 2.26 2.45 2.88
1.23 1.52 1.71 1.89 2.11 2.28 2.65
1.21 1.45 1.61 1.77 1.96 2.09 2.40
1.19 1.42 1.56 1.70 1.87 1.99 2.26
1.18 1.38 1.51 1.63 1.78 1.89 2.12
1.16 1.33 1.44 1.55 1.68 1.77 1.97
1.14 1.28 1.37 1.46 1.57 1.64 1.80
1.11 1.22 1.29 1.35 1.43 1.49 1.61
1.09 1.18 1.24 1.29 1.35 1.40 1.49
1.07 1.13 1.17 1.21 1.25 1.28 1.35
.25 10 .05 .025 .01 .005 .001
240
1.24 1.55 1.75 1.94 2.18 2.36 2.74
1.22 1.49 1.67 1.83 2.04 2.19 2.51
1.19 1.42 1.57 1.71 1.88 2.00 2.27
1.18 1.38 1.52 1.64 1.79 1.90 2.13
1.16 1.34 1.46 1.57 1.70 1.79 1.99
1.14 1.30 1.39 1.48 1.59 1.67 1.84
1.12 1.24 1.32 1.39 1.47 1.53 1.66
1.08 1.17 1.22 1.27 1.32 1.36 1.45
1.06 1.12 1.15 1.19 1.22 1.25 1.31
1.00 1.00 1.00 1.00 1.00 1.00 1.00
.25 10 .05 .025 .01 .005 .001
inf.
Source: Computed by P. J. Hildebrand.
1114
Appendix TABLE 9 Values of 2 arcsin 兹앟ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
.001 .002 .003 .004 .005
.0633 .0895 .1096 .1266 .1415
.041 .042 .043 .044 .045
.4078 .4128 .4178 .4227 .4275
.36 .37 .38 .39 .40
1.2870 1.3078 1.3284 1.3490 1.3694
.76 .77 .78 .79 .80
2.1177 2.1412 2.1652 2.1895 2.2143
.971 .972 .973 .974 .975
2.7993 2.8053 2.8115 2.8177 2.8240
.006 .007 .008 .009 .010
.1551 .1675 .1791 .1900 .2003
.046 .047 .048 .049 .050
.4323 .4371 .4418 .4464 .4510
.41 .42 .43 .44 .45
1.3898 1.4101 1.4303 1.4505 1.4706
.81 .82 .83 .84 .85
2.2395 2.2653 2.2916 2.3186 2.3462
.976 .977 .978 .979 .980
2.8305 2.8371 2.8438 2.8507 2.8578
.011 .012 .013 .014 .015
.2101 .2195 .2285 .2372 .2456
.06 .07 .08 .09 .10
.4949 .5355 .5735 .6094 .6435
.46 .47 .48 .49 .50
1.4907 1.5108 1.5308 1.5508 1.5708
.86 .87 .88 .89 .90
2.3746 2.4039 2.4341 2.4655 2.4981
.981 .982 .983 .984 .985
2.8650 2.8725 2.8801 2.8879 2.8960
.016 .017 .018 .019 .020
.2537 .2615 .2691 .2766 .2838
.11 .12 .13 .14 .15
.6761 .7075 .7377 .7670 .7954
.51 .52 .53 .54 .55
1.5908 1.6108 1.6308 1.6509 1.6710
.91 .92 .93 .94 .95
2.5322 2.5681 2.6062 2.6467 2.6906
.986 .987 .988 .989 .990
2.9044 2.9131 2.9221 2.9315 2.9413
.021 .022 .023 .024 .025
.2909 .2978 .3045 .3111 .3176
.16 .17 .18 .19 .20
.8230 .8500 .8763 .9021 .9273
.56 .57 .58 .59 .60
1.6911 1.7113 1.7315 1.7518 1.7722
.951 .952 .953 .954 .955
2.6952 2.6998 2.7045 2.7093 2.7141
.991 .992 .993 .994 .995
2.9516 2.9625 2.9741 2.9865 3.0001
.026 .027 .028 .029 .030
.3239 .3301 .3363 .3423 .3482
.21 .22 .23 .24 .25
.9521 .9764 1.0004 1.0239 1.0472
.61 .62 .63 .64 .65
1.7926 1.8132 1.8338 1.8338 1.8546
.956 .957 .958 .959 .960
2.7189 2.7238 2.7288 2.7338 2.7389
.996 .997 .998 .999
3.0150 3.0320 3.0521 3.0783
.031 .032 .033 .034 .035
.3540 .3597 .3654 .3709 .3764
.26 .27 .28 .29 .30
1.0701 1.0928 1.1152 1.1374 1.1593
.66 .67 .68 .69 .70
1.8965 1.9177 1.9391 1.9606 1.9823
.961 .962 .963 .964 .965
2.7440 2.7492 2.7545 2.7598 2.7652
.036 .037 .038 .039 .040
.3818 .3871 .3924 .3976 .4027
.31 .32 .33 .34 .35
1.1810 1.2025 1.2239 1.2451 1.2661
.71 .72 .73 .74 .75
2.0042 2.0264 2.0488 2.0715 2.0944
.966 .967 .968 .969 .970
2.7707 2.7762 2.7819 2.7876 2.7934
From Experimental Design: Procedures for the Behavioral Sciences, by Roger E. Kirk. Copyright 1968 by Wadsworth Publishing Company, Inc. Reprinted by permission of the publisher, Brooks/Cole, Pacific Grove, Calif.
Appendix
1115
TABLE 10 Percentage points of the Studentized range Error df 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 앝
t ⴝ Number of Treatment Means
␣
2
3
4
5
6
7
8
9
10
11
.05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01
3.64 5.70 3.46 5.24 3.34 4.95 3.26 4.75 3.20 4.60 3.15 4.48 3.11 4.39 3.08 4.32 3.06 4.26 3.03 4.21 3.01 4.17 3.00 4.13 2.98 4.10 2.97 4.07 2.96 4.05 2.95 4.02 2.92 3.96 2.89 3.89 2.86 3.82 2.83 3.76 2.80 3.70 2.77 3.64
4.60 6.98 4.34 6.33 4.16 5.92 4.04 5.64 3.95 5.43 3.88 5.27 3.82 5.15 3.77 5.05 3.73 4.96 3.70 4.89 3.67 4.84 3.65 4.79 3.63 4.74 3.61 4.70 3.59 4.67 3.58 4.64 3.53 4.55 3.49 4.45 3.44 4.37 3.40 4.28 3.36 4.20 3.31 4.12
5.22 7.80 4.90 7.03 4.68 6.54 4.53 6.20 4.41 5.96 4.33 5.77 4.26 5.62 4.20 5.50 4.15 5.40 4.11 5.32 4.08 5.25 4.05 5.19 4.02 5.14 4.00 5.09 3.98 5.05 3.96 5.02 3.90 4.91 3.85 4.80 3.79 4.70 3.74 4.59 3.68 4.50 3.63 4.40
5.67 8.42 5.30 7.56 5.06 7.01 4.89 6.62 4.76 6.35 4.65 6.14 4.57 5.97 4.52 5.84 4.45 5.73 4.41 5.63 4.37 5.56 4.33 5.49 4.30 5.43 4.28 5.38 4.25 5.33 4.23 5.29 4.17 5.17 4.10 5.05 4.04 4.93 3.98 4.82 3.92 4.71 3.86 4.60
6.03 8.91 5.63 7.97 5.36 7.37 5.17 6.96 5.02 6.66 4.91 6.43 4.82 6.25 4.75 6.10 4.69 5.98 4.64 5.88 4.59 5.80 4.56 5.72 4.52 5.66 4.49 5.60 4.47 5.55 4.45 5.51 4.37 5.37 4.30 5.24 4.23 5.11 4.16 4.99 4.10 4.87 4.03 4.76
6.33 9.32 5.90 8.32 5.61 7.68 5.40 7.24 5.24 6.91 5.12 6.67 5.03 6.48 4.95 6.32 4.88 6.19 4.83 6.08 4.78 5.99 4.74 5.92 4.70 5.85 4.67 5.79 4.65 5.73 4.62 5.69 4.54 5.54 4.46 5.40 4.39 5.26 4.31 5.13 4.24 5.01 4.17 4.88
6.58 9.67 6.12 8.61 5.82 7.94 5.60 7.47 5.43 7.13 5.30 6.87 5.30 6.67 5.12 6.51 5.05 6.37 4.99 6.26 4.94 6.16 4.90 6.08 4.86 6.01 4.82 5.94 4.79 5.89 4.77 5.84 4.68 5.69 4.60 5.54 4.52 5.39 4.44 5.25 4.36 5.12 4.29 4.99
6.80 9.97 6.32 8.87 6.00 8.17 5.77 7.68 5.59 7.33 5.46 7.05 5.35 6.84 5.27 6.67 5.19 6.53 5.13 6.41 5.08 6.31 5.03 6.22 4.99 6.15 4.96 6.08 4.92 6.02 4.90 5.97 4.81 5.81 4.72 5.65 4.63 5.50 4.55 5.36 4.47 5.21 4.39 5.08
6.99 10.24 6.49 9.10 6.16 8.37 5.92 7.86 5.74 7.49 5.60 7.21 5.49 6.99 5.39 6.81 5.32 6.67 5.25 6.54 5.20 6.44 5.15 6.35 5.11 6.27 5.07 6.20 5.04 6.14 5.01 6.09 3.92 5.92 4.82 5.76 4.73 5.60 4.65 5.45 4.56 5.30 4.47 5.16
7.17 10.48 6.65 9.30 6.30 8.55 6.05 8.03 5.87 7.65 5.72 7.36 5.61 7.13 5.51 6.94 5.43 6.79 5.36 6.66 5.31 6.55 5.26 6.46 5.21 6.38 5.17 6.31 5.14 6.25 5.11 6.19 5.01 6.02 4.92 5.85 4.82 5.69 4.73 5.53 4.64 5.37 4.55 5.23
This table is abridged from E. S. Pearson and H. O. Hartley, eds., Biometrika Tables for Statisticians, 2d ed., Vol 1 (New York: Cambridge University Press, 1958), Table 29. Reproduced with the permission of the editors and the trustees of Biometrika.
1116
Appendix
TABLE 10 (continued ) t ⴝ Number of Treatment Means Error df 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24 30 40 60 120 앝
12
13
14
15
16
17
18
19
20
␣
7.32 10.70 6.79 9.48 6.43 8.71 6.18 8.18 5.98 7.78 5.83 7.49 5.71 7.25 5.61 7.06 5.53 6.90 5.46 6.77 5.40 6.66 5.35 6.56 5.31 6.48 5.27 6.41 5.23 6.34 5.20 6.28 5.10 6.11 5.00 5.93 4.90 5.76 4.81 5.60 4.71 5.44 4.62 5.29
7.47 10.89 6.92 9.65 6.55 8.86 6.29 8.31 6.09 7.91 5.93 7.60 5.81 7.36 5.71 7.17 5.63 7.01 5.55 6.87 5.49 6.76 5.44 6.66 5.39 6.57 5.35 6.50 5.31 6.43 5.28 6.37 5.18 6.19 5.08 6.01 4.98 5.83 4.88 5.67 4.78 5.50 4.68 5.35
7.60 11.08 7.03 9.81 6.66 9.00 6.39 8.44 6.19 8.03 6.03 7.71 5.90 7.46 5.80 7.26 5.71 7.10 5.64 6.96 5.57 6.84 5.52 6.74 5.47 6.66 5.43 6.58 5.39 6.51 5.36 6.45 5.25 6.26 5.15 6.08 5.04 5.90 4.94 5.73 4.84 5.56 4.74 5.40
7.72 11.24 7.14 9.95 6.76 9.12 6.48 8.55 6.28 8.13 6.11 7.81 5.98 7.56 5.86 7.36 5.79 7.19 5.71 7.05 5.65 6.93 5.59 6.82 5.54 6.73 5.50 6.65 5.46 6.58 5.43 6.52 5.32 6.33 5.21 6.14 5.11 5.96 5.00 5.78 4.90 5.61 4.80 5.45
7.83 11.40 7.24 10.08 6.85 9.24 6.57 8.66 6.36 8.23 6.19 7.91 6.06 7.65 5.95 7.44 5.86 7.27 5.79 7.13 5.72 7.00 5.66 6.90 5.61 6.81 5.57 6.73 5.53 6.65 5.49 6.59 5.38 6.39 5.27 6.20 5.16 6.02 5.06 5.84 4.95 5.66 4.85 5.49
7.93 11.55 7.34 10.21 6.94 9.35 6.65 8.76 6.44 8.33 6.27 7.99 6.13 7.73 6.02 7.52 5.93 7.35 5.85 7.20 5.78 7.07 5.73 6.97 5.67 6.87 5.63 6.79 5.59 6.72 5.55 6.65 5.44 6.45 5.33 6.26 5.22 6.07 5.11 5.89 5.00 5.71 4.89 5.54
8.03 11.68 7.43 10.32 7.02 9.46 6.73 8.85 6.51 8.41 6.34 8.08 6.20 7.81 6.09 7.59 5.99 7.42 5.91 7.27 5.85 7.14 5.79 7.03 5.73 6.94 5.69 6.85 5.65 6.78 5.61 6.71 5.49 6.51 5.38 6.31 5.27 6.12 5.15 5.93 5.04 5.75 4.93 5.57
8.12 11.81 7.51 10.43 7.10 9.55 6.80 8.94 6.58 8.49 6.40 8.15 6.27 7.88 6.15 7.66 6.05 7.48 5.97 7.33 5.90 7.20 5.84 7.09 5.79 7.00 5.74 6.91 5.70 6.84 5.66 6.77 5.55 6.56 5.43 6.36 5.31 6.16 5.20 5.97 5.09 5.79 4.97 5.61
8.21 11.93 7.59 10.54 7.17 9.65 6.87 9.03 6.64 8.57 6.47 8.23 6.33 7.95 6.21 7.73 6.11 7.55 6.03 7.39 5.96 7.26 5.90 7.15 5.84 7.05 5.79 6.97 5.75 6.89 5.71 6.82 5.59 6.61 5.47 6.41 5.36 6.21 5.24 6.01 5.13 5.83 5.01 5.65
.05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01 .05 .01
Appendix
1117
TABLE 11 Percentage points for Dunnett’s test: d움 (k, )
␣ ⴝ 0.05 (one-sided)
kⴝ2
3
4
5
6
7
8
9
10
11
12
15
20
5 6 7 8 9
2.44 2.34 2.27 2.22 2.18
2.68 2.56 2.48 2.42 2.37
2.85 2.71 2.62 2.55 2.50
2.98 2.83 2.73 2.66 2.60
3.08 2.92 2.82 2.74 2.68
3.16 3.00 2.89 2.81 2.75
3.24 3.07 2.95 2.87 2.81
3.30 3.12 3.01 2.92 2.86
3.36 3.17 3.05 2.96 2.90
3.41 3.22 3.10 3.01 2.94
3.45 3.26 3.13 3.04 2.97
3.57 3.37 3.23 3.14 3.06
3.72 3.50 3.36 3.25 3.18
10 11 12 13 14
2.15 2.13 2.11 2.09 2.08
2.34 2.31 2.29 2.27 2.25
2.47 2.44 2.41 2.39 2.37
2.56 2.53 2.50 2.48 2.46
2.64 2.60 2.58 2.55 2.53
2.70 2.67 2.64 2.61 2.59
2.76 2.72 2.69 2.66 2.64
2.81 2.77 2.74 2.71 2.69
2.85 2.81 2.78 2.75 2.72
2.89 2.85 2.81 2.78 2.76
2.92 2.88 2.84 2.82 2.79
3.01 2.96 2.93 2.90 2.87
3.12 3.07 3.03 3.00 2.97
15 16 17 18 19
2.07 2.06 2.05 2.04 2.03
2.24 2.23 2.22 2.21 2.20
2.36 2.34 2.33 2.32 2.31
2.44 2.43 2.42 2.41 2.40
2.51 2.50 2.49 2.48 2.47
2.57 2.56 2.54 2.53 2.52
2.62 2.61 2.59 2.58 2.57
2.67 2.65 2.64 2.62 2.61
2.70 2.69 2.67 2.66 2.65
2.74 2.72 2.71 2.69 2.68
2.77 2.75 2.74 2.72 2.71
2.85 2.83 2.81 2.80 2.79
2.95 2.93 2.91 2.89 2.88
20 24 30 40 60
2.03 2.01 1.99 1.97 1.95
2.19 2.17 2.15 2.13 2.10
2.30 2.28 2.25 2.23 2.21
2.39 2.36 2.33 2.31 2.28
2.46 2.43 2.40 2.37 2.35
2.51 2.48 2.45 2.42 2.39
2.56 2.53 2.50 2.47 2.44
2.60 2.57 2.54 2.51 2.48
2.64 2.60 2.57 2.54 2.51
2.67 2.64 2.60 2.57 2.54
2.70 2.66 2.63 2.60 2.56
2.77 2.74 2.70 2.67 2.63
2.87 2.83 2.79 2.75 2.72
120 앝
1.93 1.92
2.08 2.06
2.18 2.16
2.26 2.23
2.32 2.29
2.37 2.34
2.41 2.38
2.45 2.42
2.48 2.45
2.51 2.48
2.53 2.50
2.60 2.56
2.68 2.64
From C. W. Dunnett (1955), ‘‘A Multiple Comparison Procedure for Comparing Several Treatments with a Control,’’ Journal of the American Statistical Association 50, 1112–1118. Reprinted with permission from Journal of the American Statistical Association. Copyright 1955 by the American Statistical Association. All rights reserved. C. W. Dunnett (1964), ‘‘New Tables for Multiple Comparisons with a Control,’’ Biometrics 20, 482–491. Also additional tables produced by C. W. Dunnett in 1980.
1118
Appendix
TABLE 11 Percentage points for Dunnett’s test: d움 (k, )
␣ ⴝ 0.01 (one-sided)
kⴝ2
3
4
5
6
7
8
9
10
11
12
15
20
5 6 7 8 9
3.90 3.61 3.42 3.29 3.19
4.21 3.88 3.66 3.51 3.40
4.43 4.07 3.83 3.67 3.55
4.60 4.21 3.96 3.79 3.66
4.73 4.33 4.07 3.88 3.75
4.85 4.43 4.15 3.96 3.82
4.94 4.51 4.23 4.03 3.89
5.03 4.59 4.30 4.09 3.94
5.11 4.64 4.35 4.14 3.99
5.17 4.70 4.40 4.19 4.04
5.24 4.76 4.45 4.23 4.08
5.39 4.89 4.57 4.34 4.18
5.59 5.06 4.72 4.48 4.31
10 11 12 13 14
3.11 3.06 3.01 2.97 2.94
3.31 3.25 3.19 3.15 3.11
3.45 3.38 3.32 3.27 3.23
3.56 3.48 3.42 3.37 3.32
3.64 3.56 3.50 3.44 3.40
3.71 3.63 3.56 3.51 3.46
3.78 3.69 3.62 3.56 3.51
3.83 3.74 3.67 3.61 3.56
3.88 3.79 3.71 3.65 3.60
3.92 3.83 3.75 3.69 3.64
3.96 3.86 3.79 3.73 3.67
4.06 3.96 3.88 3.81 3.76
4.18 4.08 3.99 3.92 3.87
15 16 17 18 19
2.91 2.88 2.86 2.84 2.83
3.08 3.05 3.03 3.01 2.99
3.20 3.17 3.14 3.12 3.10
3.29 3.26 3.23 3.21 3.18
3.36 3.33 3.30 3.27 3.25
3.42 3.39 3.36 3.33 3.31
3.47 3.44 3.41 3.38 3.36
3.52 3.48 3.45 3.42 3.40
3.56 3.52 3.49 3.46 3.44
3.60 3.56 3.53 3.50 3.47
3.63 3.59 3.56 3.53 3.50
3.71 3.67 3.64 3.61 3.58
3.82 3.78 3.74 3.71 3.68
20 24 30 40 60
2.81 2.77 2.72 2.68 2.64
2.97 2.92 2.87 2.82 2.78
3.08 3.03 2.97 2.92 2.87
3.17 3.11 3.05 2.99 2.94
3.23 3.17 3.11 3.05 3.00
3.29 3.22 3.16 3.10 3.04
3.34 3.27 3.21 3.14 3.08
3.38 3.31 3.24 3.18 3.12
3.42 3.35 3.28 3.21 3.15
3.45 3.38 3.31 3.24 3.18
3.48 3.41 3.34 3.27 3.20
3.56 3.48 3.41 3.34 3.27
3.65 3.57 3.50 3.42 3.35
120 앝
2.60 2.56
2.73 2.68
2.82 2.77
2.89 2.84
2.94 2.89
2.99 2.93
3.03 2.97
3.06 3.00
3.09 3.03
3.12 3.06
3.14 3.08
3.20 3.14
3.28 3.21
Appendix
1119
TABLE 11 Percentage points for Dunnett’s test: d움 (k, ) (continued )
␣ ⴝ 0.05 (two-sided)
kⴝ2
3
4
5
6
7
8
9
10
11
12
15
20
5 6 7 8 9
3.03 2.86 2.75 2.67 2.61
3.29 3.10 2.97 2.88 2.81
3.48 3.26 3.12 3.02 2.95
3.62 3.39 3.24 3.13 3.05
3.73 3.49 3.33 3.22 3.14
3.82 3.57 3.41 3.29 3.20
3.90 3.64 3.47 3.35 3.26
3.97 3.71 3.53 3.41 3.32
4.03 3.76 3.58 3.46 3.36
4.09 3.81 3.63 3.50 3.40
4.14 3.86 3.67 3.54 3.44
4.26 3.97 3.78 3.64 3.53
4.42 4.11 3.91 3.76 3.65
10 11 12 13 14
2.57 2.53 2.50 2.48 2.46
2.76 2.72 2.68 2.65 2.63
2.89 2.84 2.81 2.78 2.75
2.99 2.94 2.90 2.87 2.84
3.07 3.02 2.98 2.94 2.91
3.14 3.08 3.04 3.00 2.97
3.19 3.14 3.09 3.06 3.02
3.24 3.19 3.14 3.10 3.07
3.29 3.23 3.18 3.14 3.11
3.33 3.27 3.22 3.18 3.14
3.36 3.30 3.25 3.21 3.18
3.45 3.39 3.34 3.29 3.26
3.57 3.50 3.45 3.40 3.36
15 16 17 18 19
2.44 2.42 2.41 2.40 2.39
2.61 2.59 2.58 2.56 2.55
2.73 2.71 2.69 2.68 2.66
2.82 2.80 2.78 2.76 2.75
2.89 2.87 2.85 2.83 2.81
2.95 2.92 2.90 2.89 2.87
3.00 2.97 2.95 2.94 2.92
3.04 3.02 3.00 2.98 2.96
3.08 3.06 3.03 3.01 3.00
3.12 3.09 3.07 3.05 3.03
3.15 3.12 3.10 3.08 3.06
3.23 3.20 3.18 3.16 3.14
3.33 3.30 3.27 3.25 3.23
20 24 30 40 60
2.38 2.35 2.32 2.29 2.27
2.54 2.51 2.47 2.44 2.41
2.65 2.61 2.58 2.54 2.51
2.73 2.70 2.66 2.62 2.58
2.80 2.76 2.72 2.68 2.64
2.86 2.81 2.77 2.73 2.69
2.90 2.86 2.82 2.77 2.73
2.95 2.90 2.86 2.81 2.77
2.98 2.94 2.89 2.85 2.80
3.02 2.97 2.92 2.87 2.83
3.05 3.00 2.95 2.90 2.86
3.12 3.07 3.02 2.97 2.92
3.22 3.16 3.11 3.06 3.00
120 앝
2.24 2.21
2.38 2.35
2.47 2.44
2.55 2.51
2.60 2.57
2.65 2.61
2.69 2.65
2.73 2.69
2.76 2.72
2.79 2.74
2.81 2.77
2.87 2.83
2.95 2.91
1120
Appendix
TABLE 11 Percentage points for Dunnett’s test: d움 (k, ) (continued )
␣ ⴝ 0.01 (two-sided)
kⴝ2
3
4
5
6
7
8
9
10
11
12
15
20
5 6 7 8 9
4.63 4.21 3.95 3.77 3.63
4.98 4.51 4.21 4.00 3.85
5.22 4.71 4.39 4.17 4.01
5.41 4.87 4.53 4.29 4.12
5.56 5.00 4.64 4.40 4.22
5.69 5.10 4.74 4.48 4.30
5.80 5.20 4.82 4.56 4.37
5.89 5.28 4.89 4.62 4.43
5.98 5.35 4.95 4.68 4.48
6.05 5.41 5.01 4.73 4.53
6.12 5.47 5.06 4.78 4.57
6.30 5.62 5.19 4.90 4.68
6.52 5.81 5.36 5.05 4.82
10 11 12 13 14
3.53 3.45 3.39 3.33 3.29
3.74 3.65 3.58 3.52 3.47
3.88 3.79 3.71 3.65 3.59
3.99 3.89 3.81 3.74 3.69
4.08 3.98 3.89 3.82 3.76
4.16 4.05 3.96 3.89 3.83
4.22 4.11 4.02 3.94 3.88
4.28 4.16 4.07 3.99 3.93
4.33 4.21 4.12 4.04 3.97
4.37 4.25 4.16 4.08 4.01
4.42 4.29 4.19 4.11 4.05
4.52 4.39 4.29 4.20 4.13
4.65 4.52 4.41 4.32 4.24
15 16 17 18 19
3.25 3.22 3.19 3.17 3.15
3.43 3.39 3.36 3.33 3.31
3.55 3.51 3.47 3.44 3.42
3.64 3.60 3.56 3.53 3.50
3.71 3.67 3.63 3.60 3.57
3.78 3.73 3.69 3.66 3.63
3.83 3.78 3.74 3.71 3.68
3.88 3.83 3.79 3.75 3.72
3.92 3.87 3.83 3.79 3.76
3.95 3.91 3.86 3.83 3.79
3.99 3.94 3.90 3.86 3.83
4.07 4.02 3.98 3.94 3.90
4.18 4.13 4.08 4.04 4.00
20 24 30 40 60
3.13 3.07 3.01 2.95 2.90
3.29 3.22 3.15 3.09 3.03
3.40 3.32 3.25 3.19 3.12
3.48 3.40 3.33 3.26 3.19
3.55 3.47 3.39 3.32 3.25
3.60 3.52 3.44 3.37 3.29
3.65 3.57 3.49 3.41 3.33
3.69 3.61 3.52 3.44 3.37
3.73 3.64 3.56 3.48 3.40
3.77 3.68 3.59 3.51 3.42
3.80 3.70 3.62 3.53 3.45
3.87 3.78 3.69 3.60 3.51
3.97 3.87 3.78 3.68 3.59
120 앝
2.85 2.79
2.97 2.92
3.06 3.00
3.12 3.06
3.18 3.11
3.22 3.15
3.26 3.19
3.29 3.22
3.32 3.25
3.35 3.27
3.37 3.29
3.43 3.35
3.51 3.42
Appendix
1121
TABLE 12 Percentage points of Fmax ⫽ s 2max /s 2min Upper 5% Points t df2
2
3
4
5
6
7
8
9
10
11
12
2 3 4 5
39.0 15.4 9.60 7.15
87.5 27.8 15.5 10.8
142 39.2 20.6 13.7
202 50.7 25.2 16.3
266 62.0 29.5 18.7
333 72.9 33.6 20.8
403 83.5 37.5 22.9
475 93.9 41.1 24.7
550 104 44.6 26.5
626 114 48.0 28.2
704 124 51.4 29.9
6 7 8 9 10
5.82 4.99 4.43 4.03 3.72
8.38 6.94 6.00 5.34 4.85
10.4 8.44 7.18 6.31 5.67
12.1 9.70 8.12 7.11 6.34
13.7 10.8 9.03 7.80 6.92
15.0 11.8 9.78 8.41 7.42
16.3 12.7 10.5 8.95 7.87
17.5 13.5 11.1 9.45 8.28
18.6 14.3 11.7 9.91 8.66
19.7 15.1 12.2 10.3 9.01
20.7 15.8 12.7 10.7 9.34
12 15 20 30 60 앝
3.28 2.86 2.46 2.07 1.67 1.00
4.16 3.54 2.95 2.40 1.85 1.00
4.79 4.01 3.29 2.61 1.96 1.00
5.30 4.37 3.54 2.78 2.04 1.00
5.72 4.68 3.76 2.91 2.11 1.00
6.09 4.95 3.94 3.02 2.17 1.00
6.42 5.19 4.10 3.12 2.22 1.00
6.72 5.40 4.24 3.21 2.26 1.00
7.00 5.59 4.37 3.29 2.30 1.00
7.25 5.77 4.49 3.36 2.33 1.00
7.48 5.93 4.59 3.39 2.36 1.00
7
8
9
10
11
12
1705 21(6) 79 42
2063 24(9) 89 46
2432 28(1) 97 50
2813 31(0) 106 54
3204 33(7) 113 57
3605 36(1) 120 60
Upper 1% Points t df2
2 2 3 4 5
199 47.5 23.2 14.9
3
4
448 85 37 22
729 120 49 28
5 1036 151 59 33
6 1362 184 69 38
6 7 8 9 10
11.1 8.89 7.50 6.54 5.85
15.5 12.1 9.9 8.5 7.4
19.1 14.5 11.7 9.9 8.6
22 16.5 13.2 11.1 9.6
25 18.4 14.5 12.1 10.4
27 20 15.8 13.1 11.1
30 22 16.6 13.9 11.8
32 23 17.9 14.7 12.4
34 24 18.9 15.3 12.9
36 26 19.8 16.0 13.4
37 27 21 16.6 13.9
12 15 20 30 60 앝
4.91 4.07 3.32 2.63 1.96 1.00
6.1 4.9 3.8 3.0 2.2 1.0
6.9 5.5 4.3 3.3 2.3 1.0
7.6 6.0 4.6 3.4 2.4 1.0
8.2 6.4 4.9 3.6 2.4 1.0
8.7 6.7 5.1 3.7 2.5 1.0
9.1 7.1 5.3 3.8 2.5 1.0
9.5 7.3 5.5 3.9 2.6 1.0
9.9 7.5 5.6 4.0 2.6 1.0
10.2 7.8 5.8 4.1 2.7 1.0
10.6 8.0 5.9 4.2 2.7 1.0
s 2max is the largest and s 2min the smallest in a set of t independent mean squares, each based on df2 ⫽ n ⫺ 1 degrees of freedom. Values in the column t ⫽ 2 and in the rows df2 ⫽ 2 and 앝 are exact. Elsewhere, the third digit may be in error by a few units for the 5% points and several units for the 1% points. The third-digit figures in parentheses for df2 ⫽ 3 are the most uncertain. From Biometrika Tables for Statisticians, 3rd ed., Vol. 1, edited by E. S. Pearson and H. O. Hartley (New York: Cambridge University Press, 1966), Table, p. 202. Reproduced by permission of the Biometrika Trustees.
1122
Appendix
TABLE 13 Random numbers Line/ Col.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
1 2 3 4 5
10480 22368 24130 42167 37570
15011 46573 48360 93093 39975
01536 25595 22527 06243 81837
02011 85393 97265 61680 16656
81647 30995 76393 07856 06121
91646 89198 64809 16376 91782
69179 27982 15179 39440 60468
14194 53402 24830 53537 81305
62590 93965 49340 71341 49684
36207 34095 32081 57004 60672
20969 52666 30680 00849 14110
99570 19174 19655 74917 06927
91291 39615 63348 97758 01263
90700 99505 58629 16379 54613
6 7 8 9 10
77921 99562 96301 89579 85475
06907 72905 91977 14342 36857
11008 56420 05463 63661 53342
42751 69994 07972 10281 53988
27756 98872 18876 17453 53060
53498 31016 20922 18103 59533
18602 71194 94595 57740 38867
70659 18738 56869 84378 62300
90655 44013 69014 25331 08158
15053 48840 60045 12566 17983
21916 63213 18425 58678 16439
81825 21069 84903 44947 11458
44394 10634 42508 05585 18593
42880 12952 32307 56941 64952
11 12 13 14 15
28918 63553 09429 10365 07119
69578 40961 93969 61129 97336
88231 48235 52636 87529 71048
33276 03427 92737 85689 08178
70997 49626 88974 48237 77233
79936 69445 33488 52267 13916
56865 18663 36320 67689 47564
05859 72695 17617 93394 81056
90106 52180 30015 01511 97735
31595 20847 08272 26358 85977
01547 12234 84115 85104 29372
85590 90511 27156 20285 74461
91610 33703 30613 29975 28551
78188 90322 74952 89868 90707
16 17 18 19 20
51085 02368 01011 52162 07056
12765 21382 54092 53916 97628
51821 52404 33362 46369 33787
51259 60268 94904 58586 09998
77452 89368 31273 23216 42698
16308 19885 04146 14513 06691
60756 55322 18594 83149 76988
92144 44819 29852 98736 13602
49442 01188 71585 23495 51851
53900 65255 85030 64350 46104
70960 64835 51132 94738 88916
63990 44919 01915 17752 19509
75601 05944 92747 35156 25625
40719 55157 64951 35749 58104
21 22 23 24 25
48663 54164 32639 29334 02488
91245 58492 32363 27001 33062
85828 22421 05597 87637 28834
14346 74103 24200 87308 07351
09172 47070 13363 58731 19731
30168 25306 38005 00256 92420
90229 76468 94342 45834 60952
04734 26384 28728 15398 61280
59193 58151 35806 46557 50001
22178 06646 06912 41135 67658
30421 21524 17012 10367 32586
61666 15227 64161 07684 86679
99904 96909 18296 36188 50720
32812 44592 22851 18510 94953
Abridged from William H. Beyer, ed., Handbook of Tables for Probability and Statistics, 2nd ed. The Chemical Rubber Co., 1968. Used by permission of CRC Press, Inc.
Appendix TABLE 14 Power of the analysis of variance test (움 ⫽ .05, t ⫽ 3)
60 20 12 v2 = ⬁ 30 15 10 9 8 7 6
0.99 v1 = 2 0.98
Power = 1—
0.97 0.96 0.95 0.94 0.92 0.90
⬁ 30 15 10 8 7 6
0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 1
2
4
3
Power of the analysis of variance test (움 ⫽ .05, t ⫽ 4)
v2 = ⬁ 60 30 20 15 12 10 9
0.99
8
7
6
v1 = 3 0.98
Power = 1—
0.97 0.96 0.95 0.94 0.92 0.90
⬁ 30 15 10 8 7 6
0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 1
2
3
4
1123
Appendix TABLE 14 Power of the analysis of variance test (움 ⫽ .05, t ⫽ 5)
v2 = ⬁ 60 30 20 15 12 10 9
0.99
8
6
7
v1 = 4 0.98
Power = 1—
0.97 0.96 0.95 0.94 0.92 0.90
⬁ 30 15 10 9 8 7 6
0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 1
2
3
Power of the analysis of variance test (움 ⫽ .05, t ⫽ 6)
v2 = ⬁ 60 30 20 15
0.99
12 10 9 8
v1 = 5 0.98 0.97 Power = 1—
1124
0.96 0.95 0.94 0.92 0.90
⬁ 30 15 10 9 8 7 6
0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 1
2
3
7
6
Appendix TABLE 14 Power of the analysis of variance test (움 ⫽ .05, t ⫽ 7)
v2 = ⬁ 60 30 20 15 12 10 9 8
0.99
7
6
v2 = ⬁ 60 30 20 15 12 10 9 8 7
6
v1 = 6 0.98
Power = 1—
0.97 0.96 0.95 0.94 0.92 0.90
⬁ 30 15 10 9 8 7 6
0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 1
2
3
Power of the analysis of variance test (움 ⫽ .05, t ⫽ 8)
0.99 v1 = 7 0.98
Power = 1—
0.97 0.96 0.95 0.94 0.92 0.90
⬁ 60 30 20 15 12 10 9
8 7
6
0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 1
2
3
1125
Appendix TABLE 14 Power of the analysis of variance test (움 ⫽ .05, t ⫽ 9)
v2 = ⬁ 60 30 20 15 12 10 9 8 7
0.99 v1 = 8 0.98 0.97
Power = 1—
1126
0.96 0.95 0.94 0.92 0.90
⬁ 60 30 20 15 12 10 9 8
7
6
0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 1
2
3
6
Appendix
1127
TABLE 15 Poisson probabilities (애 between .1 and 4.0) y
.1
.2
.3
.4
0 1 2 3 4 5 6
.9048 .0905 .0045 .0002 .0000 .0000 .0000
.8187 .1637 .0164 .0011 .0001 .0000 .0000
.7408 .2222 .0333 .0033 .0003 .0000 .0000
.6703 .2681 .0536 .0072 .0007 .0001 .0000
y
1.1
1.2
1.3
0 1 2 3 4 5 6 7 8
.3329 .3662 .2014 .0738 .0203 .0045 .0008 .0001 .0000
.3012 .3614 .2169 .0867 .0260 .0062 .0012 .0002 .0000
y
2.1
0 1 2 3 4 5 6 7 8 9 10 11
.5
.6
.7
.8
.9
1.0
.6065 .3033 .0758 .0126 .0016 .0002 .0000
.5488 .3293 .0988 .0198 .0030 .0004 .0000
.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
1.4
1.5
1.6
1.7
1.8
1.9
2.0
.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
.2019 .3230 .2584 .1378 .0551 .0176 .0047 .0011 .0002
.1827 .3106 .2640 .1496 .0636 .0216 .0061 .0015 .0003
.1653 .2975 .2678 .1607 .0723 .0260 .0078 .0020 .0005
.1496 .2842 .2700 .1710 .0812 .0309 .0098 .0027 .0006
.1353 .2707 .2707 .1804 .0902 .0361 .0120 .0034 .0009
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
.1225 .2572 .2700 .1890 .0992 .0417 .0146 .0044 .0011 .0003 .0001 .0000
.1108 .2438 .2681 .1966 .1082 .0476 .0174 .0055 .0015 .0004 .0001 .0000
.1003 .2306 .2652 .2033 .1169 .0538 .0206 .0068 .0019 .0005 .0001 .0000
.0907 .2177 .2613 .2090 .1254 .0602 .0241 .0083 .0025 .0007 .0002 .0000
.0821 .2052 .2565 .2138 .1336 .0668 .0278 .0099 .0031 .0009 .0002 .0000
.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
y
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
0 1 2 3 4 5 6
.0450 .1397 .2165 .2237 .1733 .1075 .0555
.0408 .1304 .2087 .2226 .1781 .1140 .0608
.0369 .1217 .2008 .2209 .1823 .1203 .0662
.0334 .1135 .1929 .2186 .1858 .1264 .0716
.0302 .1057 .1850 .2158 .1888 .1322 .0771
.0273 .0984 .1771 .2125 .1912 .1377 .0826
.0247 .0915 .1692 .2087 .1931 .1429 .0881
.0224 .0850 .1615 .2046 .1944 .1477 .0936
.0202 .0789 .1539 .2001 .1951 .1522 .0989
.0183 .0733 .1465 .1954 .1954 .1563 .1042
1128
Appendix
TABLE 15 Poisson probabilities (애 between 3.1 and 10.0) y
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
7 8 9 10 11 12 13
.0246 .0095 .0033 .0010 .0003 .0001 .0000
.0278 .0111 .0040 .0013 .0004 .0001 .0000
.0312 .0129 .0047 .0016 .0005 .0001 .0000
.0348 .0148 .0056 .0019 .0006 .0002 .0000
.0385 .0169 .0066 .0023 .0007 .0002 .0001
.0425 .0191 .0076 .0028 .0009 .0003 .0001
.0466 .0215 .0089 .0033 .0011 .0003 .0001
.0508 .0241 .0102 .0039 .0013 .0004 .0001
.0551 .0269 .0116 .0045 .0016 .0005 .0002
.0595 .0298 .0132 .0053 .0019 .0006 .0002
y
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
.0166 .0679 .1393 .1904 .1951 .1600 .1093 .0640 .0328 .0150 .0061 .0023 .0008 .0002 .0001 .0000
.0150 .0630 .1323 .1852 .1944 .1633 .1143 .0686 .0360 .0168 .0071 .0027 .0009 .0003 .0001 .0000
.0136 .0583 .1254 .1798 .1933 .1662 .1191 .0732 .0393 .0188 .0081 .0032 .0011 .0004 .0001 .0000
.0123 .0540 .1188 .1743 .1917 .1687 .1237 .0778 .0428 .0209 .0092 .0037 .0013 .0005 .0001 .0000
.0111 .0500 .1125 .1687 .1898 .1708 .1281 .0824 .0463 .0232 .0104 .0043 .0016 .0006 .0002 .0001
.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
y
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
.0041 .0225 .0618 .1133 .1558 .1714 .1571 .1234 .0849 .0519 .0285 .0143 .0065 .0028 .0011 .0004
.0025 .0149 .0446 .0892 .1339 .1606 .1606 .1377 .1033 .0688 .0413 .0225 .0113 .0052 .0022 .0009
.0015 .0098 .0318 .0688 .1118 .1454 .1575 .1462 .1188 .0858 .0558 .0330 .0179 .0089 .0041 .0018
.0009 .0064 .0223 .0521 .0912 .1277 .1490 .1490 .1304 .1014 .0710 .0452 .0263 .0142 .0071 .0033
.0006 .0041 .0156 .0389 .0729 .1094 .1367 .1465 .1373 .1144 .0858 .0585 .0366 .0211 .0113 .0057
.0003 .0027 .0107 .0286 .0573 .0916 .1221 .1396 .1396 .1241 .0993 .0722 .0481 .0296 .0169 .0090
.0002 .0017 .0074 .0208 .0443 .0752 .1066 .1294 .1375 .1299 .1104 .0853 .0604 .0395 .0240 .0136
.0001 .0011 .0050 .0150 .0337 .0607 .0911 .1171 .1318 .1318 .1186 .0970 .0728 .0504 .0324 .0194
.0001 .0007 .0034 .0107 .0254 .0483 .0764 .1037 .1232 .1300 .1235 .1067 .0844 .0617 .0419 .0265
.0000 .0005 .0023 .0076 .0189 .0378 .0631 .0901 .1126 .1251 .1251 .1137 .0948 .0729 .0521 .0347
Appendix
1129
TABLE 15 Poisson probabilities (애 between 5.5 and 20.0) y
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
16 17 18 19 20 21 22 23
.0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
.0003 .0001 .0000 .0000 .0000 .0000 .0000 .0000
.0007 .0003 .0001 .0000 .0000 .0000 .0000 .0000
.0014 .0006 .0002 .0001 .0000 .0000 .0000 .0000
.0026 .0012 .0005 .0002 .0001 .0000 .0000 .0000
.0045 .0021 .0009 .0004 .0002 .0001 .0000 .0000
.0072 .0036 .0017 .0008 .0003 .0001 .0001 .0000
.0109 .0058 .0029 .0014 .0006 .0003 .0001 .0000
.0157 .0088 .0046 .0023 .0011 .0005 .0002 .0001
.0217 .0128 .0071 .0037 .0019 .0009 .0004 .0002
y
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
.0000 .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 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
.0000 .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 .0000 .0000 .0000 .0000 .0000 .0000
.0000 .0000 .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 .0000 .0000 .0000 .0000
.0000 .0000 .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 .0000 .0000 .0000
.0000 .0000 .0000 .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 0000
.0000 .0000 .0000 .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
.0000 .0000 .0000 .0000 .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
.0000 .0000 .0000 .0000 .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
.0000 .0000 .0000 .0000 .0000 .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
.0000 .0000 .0000 .0000 .0000 .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
Source: Computed by D. K. Hildebrand.
