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)
1090
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PB164-OTT
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February 28, 2002
14:21
Appendix
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.
1091
P1: FNT
PB164-OTT
6346F-OTT
1092
February 28, 2002
14:21
Appendix
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.
1095
1096
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앟̂
␲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.