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T The standard normal distribution Table of the cumulative distribution function P(Z z). The top line of the table refers to the second decimal. Example: P(Z 0) 0. z 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 T Critical values of the distribution Table of the right critical values v 1 2 3 2 0,v 2 4 6 2 0, v 2 0, v 2 0, v 2 ,v 3 5 7 5 7 9 6 9 11 v 29 30 32 . We have P( 2 v 2 ,v ) 2 0, v . 2 0, v 2 0, v 2 0, v 39 40 42 42 43 46 45 46 49 49 50 53 4 7 9 11 13 34 44 48 51 56 5 9 11 12 15 36 47 51 54 58 6 10 12 14 16 38 49 53 56 61 7 12 14 16 18 40 51 55 59 63 8 13 15 17 20 45 57 61 65 69 9 14 16 19 21 50 63 67 71 76 10 15 18 20 23 55 68 73 77 82 11 17 19 21 24 60 74 79 83 88 12 18 21 23 26 65 79 84 89 94 13 14 19 21 22 23 24 26 27 29 70 75 85 91 90 96 95 100 100 106 15 22 25 27 30 80 96 101 106 112 16 23 26 28 32 85 102 107 112 118 17 24 27 30 33 90 107 113 118 124 18 25 28 31 34 95 113 118 12 3 6 129 19 27 30 32 36 100 118 124 129 135 20 28 31 34 37 110 129 135 140 147 21 29 32 35 38 120 140 146 152 158 22 30 33 36 40 130 151 157 163 170 23 32 35 38 41 140 161 168 174 181 24 25 33 34 36 37 39 40 42 160 44 180 183 204 190 212 196 204 219 227 26 35 38 41 45 200 226 233 241 249 27 36 40 43 46 225 252 260 268 277 28 37 41 44 48 250 279 287 295 304 2 T Critical values of the t distribution Table of the right critical values v 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 t0, v t0, v 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t0, v t ,v . We have t0,v v 12 31 29 4 6 30 3 4 32 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 34 36 38 40 45 50 55 60 65 70 75 80 85 90 95 100 110 120 130 140 160 180 200 250 t0, v 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 P(tv t ,v ) . t0, v t0, v t0,v 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 T Critical values of the F distribution, Table of the right critical values 0 . F0,v1 ,v2 . We have P( Fv1 ,v2 F0,v1 ,v2 ) 0 . v1 v2 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 40 50 60 70 80 100 1 2 3 4 5 6 7 8 161 199 215 224 230 233 236 238 18 19 19 19 19 19 19 19 10 9 9 9 9 8 8 8 7 6 6 6 6 6 6 6 6 5 5 5 5 4 4 4 5 5 4 4 4 4 4 4 5 4 4 4 3 3 3 3 5 4 4 3 3 3 3 3 5 4 3 3 3 3 3 3 4 4 3 3 3 3 3 3 4 3 3 3 3 3 3 2 4 3 3 3 3 3 2 2 4 3 3 3 3 2 2 2 4 3 3 3 2 2 2 2 4 3 3 3 2 2 2 2 4 3 3 3 2 2 2 2 4 3 3 2 2 2 2 2 4 3 3 2 2 2 2 2 4 3 3 2 2 2 2 2 4 3 3 2 2 2 2 2 4 3 3 2 2 2 2 2 4 3 3 2 2 2 2 2 4 3 3 2 2 2 2 2 4 3 3 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 4 3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 3 3 2 2 2 2 2 2 3 3 2 2 2 2 2 1 4 0 . T Critical values of the F distribution, Table of the right critical values P( Fv1 ,v2 F0,v1 ,v2 ) F0,v1 ,v2 . We have 0 . v1 v2 1 2 1 2 3 6 7 8 4 5 647 799 864 899 921 937 948 956 38 39 39 39 39 39 39 39 3 17 16 15 15 14 14 14 14 4 12 10 9 9 9 9 9 8 5 10 8 7 7 7 6 6 6 6 8 7 6 6 5 5 5 5 7 8 6 5 5 5 5 4 4 8 7 6 5 5 4 4 4 4 9 7 5 5 4 4 4 4 4 10 11 6 6 5 5 4 4 4 4 4 4 4 3 3 3 3 3 12 13 6 6 5 4 4 4 4 4 3 3 3 3 3 3 3 3 14 6 4 4 3 3 3 3 3 15 6 4 4 3 3 3 3 3 16 6 4 4 3 3 3 3 3 17 6 4 4 3 3 3 3 3 18 5 4 3 3 3 3 3 3 19 5 4 3 3 3 3 3 2 20 5 4 3 3 3 3 3 2 21 5 4 3 3 3 3 2 2 22 5 4 3 3 3 3 2 2 23 5 4 3 3 3 3 2 2 24 25 5 5 4 4 3 3 3 3 3 3 2 2 2 2 2 2 26 5 4 3 3 3 2 2 2 27 5 4 3 3 3 2 2 2 28 5 4 3 3 3 2 2 2 29 5 4 3 3 3 2 2 2 30 5 4 3 3 3 2 2 2 40 5 4 3 3 2 2 2 2 50 5 3 3 3 2 2 2 2 60 5 3 3 3 2 2 2 2 70 80 5 5 3 3 3 3 2 2 2 2 2 2 2 2 2 2 100 5 5 3 3 3 3 2 2 2 2 2 2 2 2 2 2 6 v1 v2 1 2 3 10 12 15 30 40 100 20 24 968 976 984 993 997 1001 1005 1013 39 39 39 39 39 39 39 39 14 14 14 14 14 14 14 13 4 8 8 8 8 8 8 8 8 5 6 6 6 6 6 6 6 6 6 7 5 4 5 4 5 4 5 4 5 4 5 4 5 4 4 4 8 4 4 4 4 3 3 3 3 9 3 3 3 3 3 3 3 3 10 11 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 12 3 3 3 3 3 2 2 2 13 14 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 15 3 2 2 2 2 2 2 2 16 2 2 2 2 2 2 2 2 17 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 19 2 2 2 2 2 2 2 2 20 2 2 2 2 2 2 2 2 21 2 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 23 2 2 2 2 2 2 2 2 24 25 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 26 2 2 2 2 2 2 2 1 27 28 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 29 2 2 2 2 2 2 2 1 30 2 2 2 2 2 2 2 1 40 2 2 2 2 2 1 1 1 50 2 2 2 1 1 1 1 1 60 2 2 2 1 1 1 1 1 70 80 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 100 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 7 Critical Values of the U Testing) n2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .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 n1 3 4 5 0 0 6 1 0 3 1 5 2 7 1 0 5 1 6 3 8 4 8 2 1 6 2 8 4 10 6 13 7 9 2 0 4 1 7 3 10 5 12 7 15 9 17 11 10 3 0 5 2 8 4 11 6 14 9 17 11 20 13 23 16 11 3 0 6 2 9 5 13 7 16 10 19 13 23 16 26 18 30 21 12 4 1 7 3 11 6 14 9 18 12 22 15 26 18 29 21 33 24 37 27 13 4 1 8 3 12 7 16 10 20 13 24 17 28 20 33 24 37 27 41 31 45 34 14 5 1 9 4 13 7 17 11 22 15 26 18 31 22 36 26 40 30 45 34 50 38 55 42 15 5 2 10 5 14 8 19 12 24 16 29 20 34 24 39 29 44 33 49 37 54 42 59 46 64 51 16 6 2 11 5 15 9 21 13 26 18 31 22 37 27 42 31 47 36 53 41 59 45 64 50 70 55 75 60 17 6 2 11 6 17 10 22 15 28 19 34 24 39 29 45 34 51 39 57 44 63 49 69 54 75 60 81 65 87 70 18 7 2 12 6 18 11 24 16 30 21 36 26 42 31 48 37 55 42 61 47 67 53 74 58 80 64 86 70 93 75 99 81 19 20 7 8 3 3 13 14 7 8 19 20 12 13 25 27 17 18 32 34 22 24 38 41 28 30 45 48 33 36 52 55 39 42 58 62 45 48 65 69 51 54 72 76 57 60 78 83 63 67 85 90 69 73 92 98 74 79 99 105 81 86 106 112 87 92 113 119 93 99 127 105 Critical Values of the U Testing) n2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .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 n1 3 4 0 5 1 0 4 1 6 2 1 5 2 7 3 7 2 0 4 1 6 3 8 4 11 6 8 3 0 5 2 8 4 10 6 13 7 15 9 9 3 1 6 3 9 5 12 7 15 9 18 11 21 14 10 4 1 7 3 11 6 14 8 17 11 20 13 24 16 27 19 11 5 1 8 4 12 7 16 9 19 12 23 15 27 18 31 22 34 25 12 5 2 9 5 13 8 17 11 21 14 26 17 30 21 34 24 38 28 42 31 13 6 2 10 5 15 9 19 12 24 16 28 20 33 23 37 27 42 31 47 35 51 39 14 7 2 11 6 16 10 21 13 26 17 31 22 36 26 41 30 46 34 51 38 56 43 61 47 15 7 3 12 7 18 11 23 15 28 19 33 24 39 28 44 33 50 37 55 42 61 47 66 51 72 56 16 8 3 14 7 19 12 25 16 30 21 36 26 42 31 48 36 54 41 60 46 65 51 71 56 77 61 83 66 17 9 4 15 8 20 13 26 18 33 23 39 28 45 33 51 38 57 44 64 49 70 55 77 60 83 66 89 71 96 77 18 19 20 9 10 11 4 4 5 16 17 18 9 9 10 22 23 25 14 15 16 28 30 32 19 20 22 35 37 39 24 26 28 41 44 47 30 32 34 48 51 54 36 38 40 55 58 62 41 44 47 61 65 69 47 50 53 68 72 77 53 56 60 75 80 84 59 63 67 82 87 92 65 69 73 88 94 100 70 75 80 95 101 107 76 82 87 102 109 115 82 88 93 109 116 123 88 94 100 123 130 101 107 138 114

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1
T.1 The standard normal distribution
Table of the cumulative distribution function P(Z < z). The top line of the table refers to the
second decimal. Example: P(Z < 0.36) = 0.6406.
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.0
0.5000
0.5040
0.5080
0.5120
0.5160
0.5199
0.5239
0.5279
0.5319
0.1
0.5398
0.5438
0.5478
0.5517
0.5557
0.5596
0.5636
0.5675
0.5714
0.2
0.5793
0.5832
0.5871
0.5910
0.5948
0.5987
0.6026
0.6064
0.6103
0.3
0.6179
0.6217
0.6255
0.6293
0.6331
0.6368
0.6406
0.6443
0.6480
0.4
0.6554
0.6591
0.6628
0.6664
0.6700
0.6736
0.6772
0.6808
0.6844
0.5
0.6915
0.6950
0.6985
0.7019
0.7054
0.7088
0.7123
0.7157
0.7190
0.6
0.7257
0.7291
0.7324
0.7357
0.7389
0.7422
0.7454
0.7486
0.7517
0.7
0.7580
0.7611
0.7642
0.7673
0.7704
0.7734
0.7764
0.7794
0.7823
0.8
0.7881
0.7910
0.7939
0.7967
0.7995
0.8023
0.8051
0.8078
0.8106
0.9
0.8159
0.8186
0.8212
0.8238
0.8264
0.8289
0.8315
0.8340
0.8365
1.0
0.8413
0.8438
0.8461
0.8485
0.8508
0.8531
0.8554
0.8577
0.8599
1.1
0.8643
0.8665
0.8686
0.8708
0.8729
0.8749
0.8770
0.8790
0.8810
1.2
0.8849
0.8869
0.8888
0.8907
0.8925
0.8944
0.8962
0.8980
0.8997
1.3
0.9032
0.9049
0.9066
0.9082
0.9099
0.9115
0.9131
0.9147
0.9162
1.4
0.9192
0.9207
0.9222
0.9236
0.9251
0.9265
0.9279
0.9292
0.9306
1.5
0.9332
0.9345
0.9357
0.9370
0.9382
0.9394
0.9406
0.9418
0.9429
1.6
0.9452
0.9463
0.9474
0.9484
0.9495
0.9505
0.9515
0.9525
0.9535
1.7
0.9554
0.9564
0.9573
0.9582
0.9591
0.9599
0.9608
0.9616
0.9625
1.8
0.9641
0.9649
0.9656
0.9664
0.9671
0.9678
0.9686
0.9693
0.9699
1.9
0.9713
0.9719
0.9726
0.9732
0.9738
0.9744
0.9750
0.9756
0.9761
2.0
0.9772
0.9778
0.9783
0.9788
0.9793
0.9798
0.9803
0.9808
0.9812
2.1
0.9821
0.9826
0.9830
0.9834
0.9838
0.9842
0.9846
0.9850
0.9854
2.2
0.9861
0.9864
0.9868
0.9871
0.9875
0.9878
0.9881
0.9884
0.9887
2.3
0.9893
0.9896
0.9898
0.9901
0.9904
0.9906
0.9909
0.9911
0.9913
2.4
0.9918
0.9920
0.9922
0.9925
0.9927
0.9929
0.9931
0.9932
0.9934
2.5
0.9938
0.9940
0.9941
0.9943
0.9945
0.9946
0.9948
0.9949
0.9951
2.6
0.9953
0.9955
0.9956
0.9957
0.9959
0.9960
0.9961
0.9962
0.9963
2.7
0.9965
0.9966
0.9967
0.9968
0.9969
0.9970
0.9971
0.9972
0.9973
2.8
0.9974
0.9975
0.9976
0.9977
0.9977
0.9978
0.9979
0.9979
0.9980
2.9
0.9981
0.9982
0.9982
0.9983
0.9984
0.9984
0.9985
0.9985
0.9986
3.0
0.9987
0.9987
0.9987
0.9988
0.9988
0.9989
0.9989
0.9989
0.9990
3.1
0.9990
0.9991
0.9991
0.9991
0.9992
0.9992
0.9992
0.9992
0.9993
3.2
0.9993
0.9993
0.9994
0.9994
0.9994
0.9994
0.9994
0.9995
0.9995
3.3
0.9995
0.9995
0.9995
0.9996
0.9996
0.9996
0.9996
0.9996
0.9996
3.4
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
0.9997
3.5
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
3.6
0.9998
0.9998
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
3.7
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
3.8
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
3.9
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
4.0
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000

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