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Applied Statistics in Business and Economics 5th editon-trang-787-804,838-840

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A
APPENDIX
BINOMIAL PROBABILITIES
Example: P (X 5 3 | n 5 8, π 5 .50) 5 .2188
This table shows P (X 5 x).
π
n
x
.01
.02
.05
.10
.15
.20
.30
.40
.50
.60
.70
.80
.85
.90
.95
.98
.99
2
0
1
2
.9801
.0198
.0001
.9604
.0392
.0004
.9025
.0950
.0025
.8100
.1800
.0100
.7225
.2550
.0225
.6400
.3200
.0400
.4900
.4200
.0900
.3600
.4800
.1600
.2500
.5000
.2500
.1600
.4800
.3600
.0900
.4200
.4900
.0400
.3200
.6400
.0225
.2550
.7225
.0100
.1800
.8100
.0025
.0950
.9025
.0004
.0392
.9604
.0001
.0198
.9801
3
0
1
2
3
.9703
.0294
.0003
—
.9412
.0576
.0012
—
.8574
.1354
.0071
.0001
.7290
.2430
.0270
.0010
.6141
.3251
.0574
.0034
.5120
.3840
.0960
.0080
.3430
.4410
.1890
.0270
.2160
.4320
.2880
.0640
.1250
.3750
.3750
.1250
.0640
.2880
.4320
.2160
.0270
.1890
.4410
.3430
.0080
.0960
.3840
.5120
.0034
.0574
.3251
.6141
.0010
.0270
.2430
.7290
.0001
.0071
.1354
.8574
—
.0012
.0576
.9412
—
.0003
.0294
.9703
4
0
1
2
3
4
.9606
.0388
.0006
—
—
.9224
.0753
.0023
—
—
.8145
.1715
.0135
.0005
—
.6561
.2916
.0486
.0036
.0001
.5220
.3685
.0975
.0115
.0005
.4096
.4096
.1536
.0256
.0016
.2401
.4116
.2646
.0756
.0081
.1296
.3456
.3456
.1536
.0256
.0625
.2500
.3750
.2500
.0625
.0256
.1536
.3456
.3456
.1296
.0081
.0756
.2646
.4116
.2401
.0016
.0256
.1536
.4096
.4096
.0005
.0115
.0975
.3685
.5220
.0001
.0036
.0486
.2916
.6561
—
.0005
.0135
.1715
.8145
—
—
.0023
.0753
.9224
—
—
.0006
.0388
.9606
5
0
1
2
3
4
5
.9510
.0480
.0010
—
—
—
.9039
.0922
.0038
.0001
—
—
.7738
.2036
.0214
.0011
—
—
.5905
.3281
.0729
.0081
.0005
—
.4437
.3915
.1382
.0244
.0022
.0001
.3277
.4096
.2048
.0512
.0064
.0003
.1681
.3602
.3087
.1323
.0284
.0024
.0778
.2592
.3456
.2304
.0768
.0102
.0313
.1563
.3125
.3125
.1563
.0313
.0102
.0768
.2304
.3456
.2592
.0778
.0024
.0284
.1323
.3087
.3602
.1681
.0003
.0064
.0512
.2048
.4096
.3277
.0001
.0022
.0244
.1382
.3915
.4437
—
.0005
.0081
.0729
.3281
.5905
—
—
.0011
.0214
.2036
.7738
—
—
.0001
.0038
.0922
.9039
—
—
—
.0010
.0480
.9510
6
0
1
2
3
4
5
6
.9415
.0571
.0014
—
—
—
—
.8858
.1085
.0055
.0002
—
—
—
.7351
.2321
.0305
.0021
.0001
—
—
.5314
.3543
.0984
.0146
.0012
.0001
—
.3771
.3993
.1762
.0415
.0055
.0004
—
.2621
.3932
.2458
.0819
.0154
.0015
.0001
.1176
.3025
.3241
.1852
.0595
.0102
.0007
.0467
.1866
.3110
.2765
.1382
.0369
.0041
.0156
.0938
.2344
.3125
.2344
.0938
.0156
.0041
.0369
.1382
.2765
.3110
.1866
.0467
.0007
.0102
.0595
.1852
.3241
.3025
.1176
.0001
.0015
.0154
.0819
.2458
.3932
.2621
—
.0004
.0055
.0415
.1762
.3993
.3771
—
.0001
.0012
.0146
.0984
.3543
.5314
—
—
.0001
.0021
.0305
.2321
.7351
—
—
—
.0002
.0055
.1085
.8858
—
—
—
—
.0014
.0571
.9415
7
0
1
2
3
4
5
6
7
.9321
.0659
.0020
—
—
—
—
—
.8681
.1240
.0076
.0003
—
—
—
—
.6983
.2573
.0406
.0036
.0002
—
—
—
.4783
.3720
.1240
.0230
.0026
.0002
—
—
.3206
.3960
.2097
.0617
.0109
.0012
.0001
—
.2097
.3670
.2753
.1147
.0287
.0043
.0004
—
.0824
.2471
.3177
.2269
.0972
.0250
.0036
.0002
.0280
.1306
.2613
.2903
.1935
.0774
.0172
.0016
.0078
.0547
.1641
.2734
.2734
.1641
.0547
.0078
.0016
.0172
.0774
.1935
.2903
.2613
.1306
.0280
.0002
.0036
.0250
.0972
.2269
.3177
.2471
.0824
—
.0004
.0043
.0287
.1147
.2753
.3670
.2097
—
.0001
.0012
.0109
.0617
.2097
.3960
.3206
—
—
.0002
.0026
.0230
.1240
.3720
.4783
—
—
—
.0002
.0036
.0406
.2573
.6983
—
—
—
—
.0003
.0076
.1240
.8681
—
—
—
—
—
.0020
.0659
.9321
8
0
1
2
3
4
5
6
7
8
.9227
.0746
.0026
.0001
—
—
—
—
—
.8508
.1389
.0099
.0004
—
—
—
—
—
.6634
.2793
.0515
.0054
.0004
—
—
—
—
.4305
.3826
.1488
.0331
.0046
.0004
—
—
—
.2725
.3847
.2376
.0839
.0185
.0026
.0002
—
—
.1678
.3355
.2936
.1468
.0459
.0092
.0011
.0001
—
.0576
.1977
.2965
.2541
.1361
.0467
.0100
.0012
.0001
.0168
.0896
.2090
.2787
.2322
.1239
.0413
.0079
.0007
.0039
.0313
.1094
.2188
.2734
.2188
.1094
.0313
.0039
.0007
.0079
.0413
.1239
.2322
.2787
.2090
.0896
.0168
.0001
.0012
.0100
.0467
.1361
.2541
.2965
.1977
.0576
—
.0001
.0011
.0092
.0459
.1468
.2936
.3355
.1678
—
—
.0002
.0026
.0185
.0839
.2376
.3847
.2725
—
—
—
.0004
.0046
.0331
.1488
.3826
.4305
—
—
—
—
.0004
.0054
.0515
.2793
.6634
—
—
—
—
—
.0004
.0099
.1389
.8508
—
—
—
—
—
.0001
.0026
.0746
.9227
9
0
1
2
3
4
5
6
7
8
9
.9135
.0830
.0034
.0001
—
—
—
—
—
—
.8337
.1531
.0125
.0006
—
—
—
—
—
—
.6302
.2985
.0629
.0077
.0006
—
—
—
—
—
.3874
.3874
.1722
.0446
.0074
.0008
.0001
—
—
—
.2316
.3679
.2597
.1069
.0283
.0050
.0006
—
—
—
.1342
.3020
.3020
.1762
.0661
.0165
.0028
.0003
—
—
.0404
.1556
.2668
.2668
.1715
.0735
.0210
.0039
.0004
—
.0101
.0605
.1612
.2508
.2508
.1672
.0743
.0212
.0035
.0003
.0020
.0176
.0703
.1641
.2461
.2461
.1641
.0703
.0176
.0020
.0003
.0035
.0212
.0743
.1672
.2508
.2508
.1612
.0605
.0101
—
.0004
.0039
.0210
.0735
.1715
.2668
.2668
.1556
.0404
—
—
.0003
.0028
.0165
.0661
.1762
.3020
.3020
.1342
—
—
—
.0006
.0050
.0283
.1069
.2597
.3679
.2316
—
—
—
.0001
.0008
.0074
.0446
.1722
.3874
.3874
—
—
—
—
—
.0006
.0077
.0629
.2985
.6302
—
—
—
—
—
—
.0006
.0125
.1531
.8337
—
—
—
—
—
—
.0001
.0034
.0830
.9135
762
Appendix A
763
π
n
x
.01
.02
.05
.10
.15
.20
.30
.40
.50
.60
.70
.80
.85
.90
.95
.98
.99
10
0
1
2
3
4
5
6
7
8
9
10
.9044
.0914
.0042
.0001
—
—
—
—
—
—
—
.8171
.1667
.0153
.0008
—
—
—
—
—
—
—
.5987
.3151
.0746
.0105
.0010
.0001
—
—
—
—
—
.3487
.3874
.1937
.0574
.0112
.0015
.0001
—
—
—
—
.1969
.3474
.2759
.1298
.0401
.0085
.0012
.0001
—
—
—
.1074
.2684
.3020
.2013
.0881
.0264
.0055
.0008
.0001
—
—
.0282
.1211
.2335
.2668
.2001
.1029
.0368
.0090
.0014
.0001
—
.0060
.0403
.1209
.2150
.2508
.2007
.1115
.0425
.0106
.0016
.0001
.0010
.0098
.0439
.1172
.2051
.2461
.2051
.1172
.0439
.0098
.0010
.0001
.0016
.0106
.0425
.1115
.2007
.2508
.2150
.1209
.0403
.0060
—
.0001
.0014
.0090
.0368
.1029
.2001
.2668
.2335
.1211
.0282
—
—
.0001
.0008
.0055
.0264
.0881
.2013
.3020
.2684
.1074
—
—
—
.0001
.0012
.0085
.0401
.1298
.2759
.3474
.1969
—
—
—
—
.0001
.0015
.0112
.0574
.1937
.3874
.3487
—
—
—
—
—
.0001
.0010
.0105
.0746
.3151
.5987
—
—
—
—
—
—
—
.0008
.0153
.1667
.8171
—
—
—
—
—
—
—
.0001
.0042
.0914
.9044
12
0
1
2
3
4
5
6
7
8
9
10
11
12
.8864
.1074
.0060
.0002
—
—
—
—
—
—
—
—
—
.7847
.1922
.0216
.0015
.0001
—
—
—
—
—
—
—
—
.5404
.3413
.0988
.0173
.0021
.0002
—
—
—
—
—
—
—
.2824
.3766
.2301
.0852
.0213
.0038
.0005
—
—
—
—
—
—
.1422
.3012
.2924
.1720
.0683
.0193
.0040
.0006
.0001
—
—
—
—
.0687
.2062
.2835
.2362
.1329
.0532
.0155
.0033
.0005
.0001
—
—
—
.0138
.0712
.1678
.2397
.2311
.1585
.0792
.0291
.0078
.0015
.0002
—
—
.0022
.0174
.0639
.1419
.2128
.2270
.1766
.1009
.0420
.0125
.0025
.0003
—
.0002
.0029
.0161
.0537
.1208
.1934
.2256
.1934
.1208
.0537
.0161
.0029
.0002
—
.0003
.0025
.0125
.0420
.1009
.1766
.2270
.2128
.1419
.0639
.0174
.0022
—
—
.0002
.0015
.0078
.0291
.0792
.1585
.2311
.2397
.1678
.0712
.0138
—
—
—
.0001
.0005
.0033
.0155
.0532
.1329
.2362
.2835
.2062
.0687
—
—
—
—
.0001
.0006
.0040
.0193
.0683
.1720
.2924
.3012
.1422
—
—
—
—
—
—
.0005
.0038
.0213
.0852
.2301
.3766
.2824
—
—
—
—
—
—
—
.0002
.0021
.0173
.0988
.3413
.5404
—
—
—
—
—
—
—
—
.0001
.0015
.0216
.1922
.7847
—
—
—
—
—
—
—
—
—
.0002
.0060
.1074
.8864
14
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
.8687
.1229
.0081
.0003
—
—
—
—
—
—
—
—
—
—
—
.7536
.2153
.0286
.0023
.0001
—
—
—
—
—
—
—
—
—
—
.4877
.3593
.1229
.0259
.0037
.0004
—
—
—
—
—
—
—
—
—
.2288
.3559
.2570
.1142
.0349
.0078
.0013
.0002
—
—
—
—
—
—
—
.1028
.2539
.2912
.2056
.0998
.0352
.0093
.0019
.0003
—
—
—
—
—
—
.0440
.1539
.2501
.2501
.1720
.0860
.0322
.0092
.0020
.0003
—
—
—
—
—
.0068
.0407
.1134
.1943
.2290
.1963
.1262
.0618
.0232
.0066
.0014
.0002
—
—
—
.0008
.0073
.0317
.0845
.1549
.2066
.2066
.1574
.0918
.0408
.0136
.0033
.0005
.0001
—
.0001
.0009
.0056
.0222
.0611
.1222
.1833
.2095
.1833
.1222
.0611
.0222
.0056
.0009
.0001
—
.0001
.0005
.0033
.0136
.0408
.0918
.1574
.2066
.2066
.1549
.0845
.0317
.0073
.0008
—
—
—
.0002
.0014
.0066
.0232
.0618
.1262
.1963
.2290
.1943
.1134
.0407
.0068
—
—
—
—
—
.0003
.0020
.0092
.0322
.0860
.1720
.2501
.2501
.1539
.0440
—
—
—
—
—
—
.0003
.0019
.0093
.0352
.0998
.2056
.2912
.2539
.1028
—
—
—
—
—
—
—
.0002
.0013
.0078
.0349
.1142
.2570
.3559
.2288
—
—
—
—
—
—
—
—
—
.0004
.0037
.0259
.1229
.3593
.4877
—
—
—
—
—
—
—
—
—
—
.0001
.0023
.0286
.2153
.7536
—
—
—
—
—
—
—
—
—
—
—
.0003
.0081
.1229
.8687
16
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
.8515
.1376
.0104
.0005
—
—
—
—
—
—
—
—
—
—
—
—
—
.7238
.2363
.0362
.0034
.0002
—
—
—
—
—
—
—
—
—
—
—
—
.4401
.3706
.1463
.0359
.0061
.0008
.0001
—
—
—
—
—
—
—
—
—
—
.1853
.3294
.2745
.1423
.0514
.0137
.0028
.0004
.0001
—
—
—
—
—
—
—
—
.0743
.2097
.2775
.2285
.1311
.0555
.0180
.0045
.0009
.0001
—
—
—
—
—
—
—
.0281
.1126
.2111
.2463
.2001
.1201
.0550
.0197
.0055
.0012
.0002
—
—
—
—
—
—
.0033
.0228
.0732
.1465
.2040
.2099
.1649
.1010
.0487
.0185
.0056
.0013
.0002
—
—
—
—
.0003
.0030
.0150
.0468
.1014
.1623
.1983
.1889
.1417
.0840
.0392
.0142
.0040
.0008
.0001
—
—
—
.0002
.0018
.0085
.0278
.0667
.1222
.1746
.1964
.1746
.1222
.0667
.0278
.0085
.0018
.0002
—
—
—
.0001
.0008
.0040
.0142
.0392
.0840
.1417
.1889
.1983
.1623
.1014
.0468
.0150
.0030
.0003
—
—
—
—
.0002
.0013
.0056
.0185
.0487
.1010
.1649
.2099
.2040
.1465
.0732
.0228
.0033
—
—
—
—
—
—
.0002
.0012
.0055
.0197
.0550
.1201
.2001
.2463
.2111
.1126
.0281
—
—
—
—
—
—
—
.0001
.0009
.0045
.0180
.0555
.1311
.2285
.2775
.2097
.0743
—
—
—
—
—
—
—
—
.0001
.0004
.0028
.0137
.0514
.1423
.2745
.3294
.1853
—
—
—
—
—
—
—
—
—
—
.0001
.0008
.0061
.0359
.1463
.3706
.4401
—
—
—
—
—
—
—
—
—
—
—
—
.0002
.0034
.0362
.2363
.7238
—
—
—
—
—
—
—
—
—
—
—
—
—
.0005
.0104
.1376
.8515
APPENDIX
B
POISSON PROBABILITIES
Example: P (X 5 3 | λ 5 2.3) 5 .2033
This table shows P (X 5 x).
λ
x
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0
1
2
3
4
5
6
7
8
.9048
.0905
.0045
.0002
—
—
—
—
—
.8187
.1637
.0164
.0011
.0001
—
—
—
—
.7408
.2222
.0333
.0033
.0003
—
—
—
—
.6703
.2681
.0536
.0072
.0007
.0001
—
—
—
.6065
.3033
.0758
.0126
.0016
.0002
—
—
—
.5488
.3293
.0988
.0198
.0030
.0004
—
—
—
.4966
.3476
.1217
.0284
.0050
.0007
.0001
—
—
.4493
.3595
.1438
.0383
.0077
.0012
.0002
—
—
.4066
.3659
.1647
.0494
.0111
.0020
.0003
—
—
.3679
.3679
.1839
.0613
.0153
.0031
.0005
.0001
—
.3329
.3662
.2014
.0738
.0203
.0045
.0008
.0001
—
.3012
.3614
.2169
.0867
.0260
.0062
.0012
.0002
—
.2725
.3543
.2303
.0998
.0324
.0084
.0018
.0003
.0001
.2466
.3452
.2417
.1128
.0395
.0111
.0026
.0005
.0001
.2231
.3347
.2510
.1255
.0471
.0141
.0035
.0008
.0001
λ
x
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
0
1
2
3
4
5
6
7
8
9
10
11
12
.2019
.3230
.2584
.1378
.0551
.0176
.0047
.0011
.0002
—
—
—
—
.1827
.3106
.2640
.1496
.0636
.0216
.0061
.0015
.0003
.0001
—
—
—
.1653
.2975
.2678
.1607
.0723
.0260
.0078
.0020
.0005
.0001
—
—
—
.1496
.2842
.2700
.1710
.0812
.0309
.0098
.0027
.0006
.0001
—
—
—
.1353
.2707
.2707
.1804
.0902
.0361
.0120
.0034
.0009
.0002
—
—
—
.1225
.2572
.2700
.1890
.0992
.0417
.0146
.0044
.0011
.0003
.0001
—
—
.1108
.2438
.2681
.1966
.1082
.0476
.0174
.0055
.0015
.0004
.0001
—
—
.1003
.2306
.2652
.2033
.1169
.0538
.0206
.0068
.0019
.0005
.0001
—
—
.0907
.2177
.2613
.2090
.1254
.0602
.0241
.0083
.0025
.0007
.0002
—
—
.0821
.2052
.2565
.2138
.1336
.0668
.0278
.0099
.0031
.0009
.0002
—
—
.0743
.1931
.2510
.2176
.1414
.0735
.0319
.0118
.0038
.0011
.0003
.0001
—
.0672
.1815
.2450
.2205
.1488
.0804
.0362
.0139
.0047
.0014
.0004
.0001
—
.0608
.1703
.2384
.2225
.1557
.0872
.0407
.0163
.0057
.0018
.0005
.0001
—
.0550
.1596
.2314
.2237
.1622
.0940
.0455
.0188
.0068
.0022
.0006
.0002
—
.0498
.1494
.2240
.2240
.1680
.1008
.0504
.0216
.0081
.0027
.0008
.0002
.0001
λ
x
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
.0450
.1397
.2165
.2237
.1733
.1075
.0555
.0246
.0095
.0033
.0010
.0003
.0001
—
—
—
.0408
.1304
.2087
.2226
.1781
.1140
.0608
.0278
.0111
.0040
.0013
.0004
.0001
—
—
—
.0369
.1217
.2008
.2209
.1823
.1203
.0662
.0312
.0129
.0047
.0016
.0005
.0001
—
—
—
.0334
.1135
.1929
.2186
.1858
.1264
.0716
.0348
.0148
.0056
.0019
.0006
.0002
—
—
—
.0302
.1057
.1850
.2158
.1888
.1322
.0771
.0385
.0169
.0066
.0023
.0007
.0002
.0001
—
—
.0273
.0984
.1771
.2125
.1912
.1377
.0826
.0425
.0191
.0076
.0028
.0009
.0003
.0001
—
—
.0247
.0915
.1692
.2087
.1931
.1429
.0881
.0466
.0215
.0089
.0033
.0011
.0003
.0001
—
—
.0224
.0850
.1615
.2046
.1944
.1477
.0936
.0508
.0241
.0102
.0039
.0013
.0004
.0001
—
—
.0202
.0789
.1539
.2001
.1951
.1522
.0989
.0551
.0269
.0116
.0045
.0016
.0005
.0002
—
—
.0183
.0733
.1465
.1954
.1954
.1563
.1042
.0595
.0298
.0132
.0053
.0019
.0006
.0002
.0001
—
.0166
.0679
.1393
.1904
.1951
.1600
.1093
.0640
.0328
.0150
.0061
.0023
.0008
.0002
.0001
—
.0150
.0630
.1323
.1852
.1944
.1633
.1143
.0686
.0360
.0168
.0071
.0027
.0009
.0003
.0001
—
.0136
.0583
.1254
.1798
.1933
.1662
.1191
.0732
.0393
.0188
.0081
.0032
.0011
.0004
.0001
—
.0123
.0540
.1188
.1743
.1917
.1687
.1237
.0778
.0428
.0209
.0092
.0037
.0013
.0005
.0001
—
.0111
.0500
.1125
.1687
.1898
.1708
.1281
.0824
.0463
.0232
.0104
.0043
.0016
.0006
.0002
.0001
764
Appendix B
765
λ
x
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
.0101
.0462
.1063
.1631
.1875
.1725
.1323
.0869
.0500
.0255
.0118
.0049
.0019
.0007
.0002
.0001
—
—
.0091
.0427
.1005
.1574
.1849
.1738
.1362
.0914
.0537
.0281
.0132
.0056
.0022
.0008
.0003
.0001
—
—
.0082
.0395
.0948
.1517
.1820
.1747
.1398
.0959
.0575
.0307
.0147
.0064
.0026
.0009
.0003
.0001
—
—
.0074
.0365
.0894
.1460
.1789
.1753
.1432
.1002
.0614
.0334
.0164
.0073
.0030
.0011
.0004
.0001
—
—
.0067
.0337
.0842
.1404
.1755
.1755
.1462
.1044
.0653
.0363
.0181
.0082
.0034
.0013
.0005
.0002
—
—
.0061
.0311
.0793
.1348
.1719
.1753
.1490
.1086
.0692
.0392
.0200
.0093
.0039
.0015
.0006
.0002
.0001
—
.0055
.0287
.0746
.1293
.1681
.1748
.1515
.1125
.0731
.0423
.0220
.0104
.0045
.0018
.0007
.0002
.0001
—
.0050
.0265
.0701
.1239
.1641
.1740
.1537
.1163
.0771
.0454
.0241
.0116
.0051
.0021
.0008
.0003
.0001
—
.0045
.0244
.0659
.1185
.1600
.1728
.1555
.1200
.0810
.0486
.0262
.0129
.0058
.0024
.0009
.0003
.0001
—
.0041
.0225
.0618
.1133
.1558
.1714
.1571
.1234
.0849
.0519
.0285
.0143
.0065
.0028
.0011
.0004
.0001
—
.0037
.0207
.0580
.1082
.1515
.1697
.1584
.1267
.0887
.0552
.0309
.0157
.0073
.0032
.0013
.0005
.0002
.0001
.0033
.0191
.0544
.1033
.1472
.1678
.1594
.1298
.0925
.0586
.0334
.0173
.0082
.0036
.0015
.0006
.0002
.0001
.0030
.0176
.0509
.0985
.1428
.1656
.1601
.1326
.0962
.0620
.0359
.0190
.0092
.0041
.0017
.0007
.0002
.0001
.0027
.0162
.0477
.0938
.1383
.1632
.1605
.1353
.0998
.0654
.0386
.0207
.0102
.0046
.0019
.0008
.0003
.0001
.0025
.0149
.0446
.0892
.1339
.1606
.1606
.1377
.1033
.0688
.0413
.0225
.0113
.0052
.0022
.0009
.0003
.0001
λ
x
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
.0022
.0137
.0417
.0848
.1294
.1579
.1605
.1399
.1066
.0723
.0441
.0244
.0124
.0058
.0025
.0010
.0004
.0001
—
—
—
.0020
.0126
.0390
.0806
.1249
.1549
.1601
.1418
.1099
.0757
.0469
.0265
.0137
.0065
.0029
.0012
.0005
.0002
.0001
—
—
.0018
.0116
.0364
.0765
.1205
.1519
.1595
.1435
.1130
.0791
.0498
.0285
.0150
.0073
.0033
.0014
.0005
.0002
.0001
—
—
.0017
.0106
.0340
.0726
.1162
.1487
.1586
.1450
.1160
.0825
.0528
.0307
.0164
.0081
.0037
.0016
.0006
.0002
.0001
—
—
.0015
.0098
.0318
.0688
.1118
.1454
.1575
.1462
.1188
.0858
.0558
.0330
.0179
.0089
.0041
.0018
.0007
.0003
.0001
—
—
.0014
.0090
.0296
.0652
.1076
.1420
.1562
.1472
.1215
.0891
.0588
.0353
.0194
.0099
.0046
.0020
.0008
.0003
.0001
—
—
.0012
.0082
.0276
.0617
.1034
.1385
.1546
.1480
.1240
.0923
.0618
.0377
.0210
.0108
.0052
.0023
.0010
.0004
.0001
.0001
—
.0011
.0076
.0258
.0584
.0992
.1349
.1529
.1486
.1263
.0954
.0649
.0401
.0227
.0119
.0058
.0026
.0011
.0004
.0002
.0001
—
.0010
.0070
.0240
.0552
.0952
.1314
.1511
.1489
.1284
.0985
.0679
.0426
.0245
.0130
.0064
.0029
.0013
.0005
.0002
.0001
—
.0009
.0064
.0223
.0521
.0912
.1277
.1490
.1490
.1304
.1014
.0710
.0452
.0263
.0142
.0071
.0033
.0014
.0006
.0002
.0001
—
.0008
.0059
.0208
.0492
.0874
.1241
.1468
.1489
.1321
.1042
.0740
.0478
.0283
.0154
.0078
.0037
.0016
.0007
.0003
.0001
—
.0007
.0054
.0194
.0464
.0836
.1204
.1445
.1486
.1337
.1070
.0770
.0504
.0303
.0168
.0086
.0041
.0019
.0008
.0003
.0001
—
.0007
.0049
.0180
.0438
.0799
.1167
.1420
.1481
.1351
.1096
.0800
.0531
.0323
.0181
.0095
.0046
.0021
.0009
.0004
.0001
.0001
.0006
.0045
.0167
.0413
.0764
.1130
.1394
.1474
.1363
.1121
.0829
.0558
.0344
.0196
.0104
.0051
.0024
.0010
.0004
.0002
.0001
.0006
.0041
.0156
.0389
.0729
.1094
.1367
.1465
.1373
.1144
.0858
.0585
.0366
.0211
.0113
.0057
.0026
.0012
.0005
.0002
.0001
766
Appendix B
(continued )
λ
x
8.0
8.5
9.0
9.5
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
.0003
.0027
.0107
.0286
.0573
.0916
.1221
.1396
.1396
.1241
.0993
.0722
.0481
.0296
.0169
.0090
.0045
.0021
.0009
.0004
.0002
.0001
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.0002
.0017
.0074
.0208
.0443
.0752
.1066
.1294
.1375
.1299
.1104
.0853
.0604
.0395
.0240
.0136
.0072
.0036
.0017
.0008
.0003
.0001
.0001
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.0001
.0011
.0050
.0150
.0337
.0607
.0911
.1171
.1318
.1318
.1186
.0970
.0728
.0504
.0324
.0194
.0109
.0058
.0029
.0014
.0006
.0003
.0001
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.0001
.0007
.0034
.0107
.0254
.0483
.0764
.1037
.1232
.1300
.1235
.1067
.0844
.0617
.0419
.0265
.0157
.0088
.0046
.0023
.0011
.0005
.0002
.0001
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.0005
.0023
.0076
.0189
.0378
.0631
.0901
.1126
.1251
.1251
.1137
.0948
.0729
.0521
.0347
.0217
.0128
.0071
.0037
.0019
.0009
.0004
.0002
.0001
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.0002
.0010
.0037
.0102
.0224
.0411
.0646
.0888
.1085
.1194
.1194
.1094
.0926
.0728
.0534
.0367
.0237
.0145
.0084
.0046
.0024
.0012
.0006
.0003
.0001
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.0001
.0004
.0018
.0053
.0127
.0255
.0437
.0655
.0874
.1048
.1144
.1144
.1056
.0905
.0724
.0543
.0383
.0255
.0161
.0097
.0055
.0030
.0016
.0008
.0004
.0002
.0001
—
—
—
—
—
—
—
—
—
—
—
—
—
—
.0002
.0008
.0027
.0070
.0152
.0281
.0457
.0661
.0859
.1015
.1099
.1099
.1021
.0885
.0719
.0550
.0397
.0272
.0177
.0109
.0065
.0037
.0020
.0010
.0005
.0002
.0001
.0001
—
—
—
—
—
—
—
—
—
—
—
—
.0001
.0004
.0013
.0037
.0087
.0174
.0304
.0473
.0663
.0844
.0984
.1060
.1060
.0989
.0866
.0713
.0554
.0409
.0286
.0191
.0121
.0074
.0043
.0024
.0013
.0007
.0003
.0002
.0001
—
—
—
—
—
—
—
—
—
—
—
—
.0002
.0006
.0019
.0048
.0104
.0194
.0324
.0486
.0663
.0829
.0956
.1024
.1024
.0960
.0847
.0706
.0557
.0418
.0299
.0204
.0133
.0083
.0050
.0029
.0016
.0009
.0004
.0002
.0001
.0001
—
—
—
—
—
—
—
—
—
—
.0001
.0003
.0010
.0026
.0060
.0120
.0213
.0341
.0496
.0661
.0814
.0930
.0992
.0992
.0934
.0830
.0699
.0559
.0426
.0310
.0216
.0144
.0092
.0057
.0034
.0019
.0011
.0006
.0003
.0001
.0001
—
—
—
—
—
—
—
—
—
—
.0001
.0005
.0014
.0034
.0072
.0135
.0230
.0355
.0504
.0658
.0800
.0906
.0963
.0963
.0909
.0814
.0692
.0560
.0433
.0320
.0226
.0154
.0101
.0063
.0038
.0023
.0013
.0007
.0004
.0002
.0001
—
—
—
—
—
—
—
—
—
.0001
.0002
.0007
.0019
.0042
.0083
.0150
.0245
.0368
.0509
.0655
.0786
.0884
.0936
.0936
.0887
.0798
.0684
.0560
.0438
.0328
.0237
.0164
.0109
.0070
.0044
.0026
.0015
.0009
.0005
.0002
.0001
.0001
—
—
—
—
—
—
—
—
.0001
.0004
.0010
.0024
.0050
.0095
.0164
.0259
.0378
.0514
.0650
.0772
.0863
.0911
.0911
.0866
.0783
.0676
.0559
.0442
.0336
.0246
.0173
.0117
.0077
.0049
.0030
.0018
.0010
.0006
.0003
.0002
.0001
—
—
—
—
—
—
—
.0001
.0002
.0005
.0013
.0029
.0058
.0106
.0176
.0271
.0387
.0516
.0646
.0760
.0844
.0888
.0888
.0846
.0769
.0669
.0557
.0446
.0343
.0254
.0181
.0125
.0083
.0054
.0034
.0020
.0012
.0007
.0004
.0002
.0001
.0001
APPENDIX
C-1
A
STANDARD NORMAL AREAS
Example: P (0 # z # 1.96) 5 .4750
This table shows the normal area between 0 and z.
0
z
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
.0000
.0398
.0793
.1179
.1554
.0040
.0438
.0832
.1217
.1591
.0080
.0478
.0871
.1255
.1628
.0120
.0517
.0910
.1293
.1664
.0160
.0557
.0948
.1331
.1700
.0199
.0596
.0987
.1368
.1736
.0239
.0636
.1026
.1406
.1772
.0279
.0675
.1064
.1443
.1808
.0319
.0714
.1103
.1480
.1844
.0359
.0753
.1141
.1517
.1879
0.5
0.6
0.7
0.8
0.9
.1915
.2257
.2580
.2881
.3159
.1950
.2291
.2611
.2910
.3186
.1985
.2324
.2642
.2939
.3212
.2019
.2357
.2673
.2967
.3238
.2054
.2389
.2704
.2995
.3264
.2088
.2422
.2734
.3023
.3289
.2123
.2454
.2764
.3051
.3315
.2157
.2486
.2794
.3078
.3340
.2190
.2517
.2823
.3106
.3365
.2224
.2549
.2852
.3133
.3389
1.0
1.1
1.2
1.3
1.4
.3413
.3643
.3849
.4032
.4192
.3438
.3665
.3869
.4049
.4207
.3461
.3686
.3888
.4066
.4222
.3485
.3708
.3907
.4082
.4236
.3508
.3729
.3925
.4099
.4251
.3531
.3749
.3944
.4115
.4265
.3554
.3770
.3962
.4131
.4279
.3577
.3790
.3980
.4147
.4292
.3599
.3810
.3997
.4162
.4306
.3621
.3830
.4015
.4177
.4319
1.5
1.6
1.7
1.8
1.9
.4332
.4452
.4554
.4641
.4713
.4345
.4463
.4564
.4649
.4719
.4357
.4474
.4573
.4656
.4726
.4370
.4484
.4582
.4664
.4732
.4382
.4495
.4591
.4671
.4738
.4394
.4505
.4599
.4678
.4744
.4406
.4515
.4608
.4686
.4750
.4418
.4525
.4616
.4693
.4756
.4429
.4535
.4625
.4699
.4761
.4441
.4545
.4633
.4706
.4767
2.0
2.1
2.2
2.3
2.4
.4772
.4821
.4861
.4893
.4918
.4778
.4826
.4864
.4896
.4920
.4783
.4830
.4868
.4898
.4922
.4788
.4834
.4871
.4901
.4925
.4793
.4838
.4875
.4904
.4927
.4798
.4842
.4878
.4906
.4929
.4803
.4846
.4881
.4909
.4931
.4808
.4850
.4884
.4911
.4932
.4812
.4854
.4887
.4913
.4934
.4817
.4857
.4890
.4916
.4936
2.5
2.6
2.7
2.8
2.9
.4938
.4953
.4965
.4974
.4981
.4940
.4955
.4966
.4975
.4982
.4941
.4956
.4967
.4976
.4982
.4943
.4957
.4968
.4977
.4983
.4945
.4959
.4969
.4977
.4984
.4946
.4960
.4970
.4978
.4984
.4948
.4961
.4971
.4979
.4985
.4949
.4962
.4972
.4979
.4985
.4951
.4963
.4973
.4980
.4986
.4952
.4964
.4974
.4981
.4986
3.0
3.1
3.2
3.3
3.4
.49865
.49903
.49931
.49952
.49966
.49869
.49906
.49934
.49953
.49968
.49874
.49910
.49936
.49955
.49969
.49878
.49913
.49938
.49957
.49970
.49882
.49916
.49940
.49958
.49971
.49886
.49918
.49942
.49960
.49972
.49889
.49921
.49944
.49961
.49973
.49893
.49924
.49946
.49962
.49974
.49896
.49926
.49948
.49964
.49975
.49900
.49929
.49950
.49965
.49976
3.5
3.6
3.7
.49977
.49984
.49989
.49978
.49985
.49990
.49978
.49985
.49990
.49979
.49986
.49990
.49980
.49986
.49991
.49981
.49987
.49991
.49981
.49987
.49992
.49982
.49988
.49992
.49983
.49988
.49992
.49983
.49989
.49992
767
C-2
APPENDIX
CUMULATIVE STANDARD NORMAL DISTRIBUTION
Example: P (z # 21.96) 5 .0250
This table shows the normal area less than z.
z
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
23.7
23.6
23.5
.00011
.00016
.00023
.00010
.00015
.00022
.00010
.00015
.00022
.00010
.00014
.00021
.00009
.00014
.00020
.00009
.00013
.00019
.00008
.00013
.00019
.00008
.00012
.00018
.00008
.00012
.00017
.00008
.00011
.00017
23.4
23.3
23.2
23.1
23.0
.00034
.00048
.00069
.00097
.00135
.00032
.00047
.00066
.00094
.00131
.00031
.00045
.00064
.00090
.00126
.00030
.00043
.00062
.00087
.00122
.00029
.00042
.00060
.00084
.00118
.00028
.00040
.00058
.00082
.00114
.00027
.00039
.00056
.00079
.00111
.00026
.00038
.00054
.00076
.00107
.00025
.00036
.00052
.00074
.00104
.00024
.00035
.00050
.00071
.00100
22.9
22.8
22.7
22.6
22.5
.0019
.0026
.0035
.0047
.0062
.0018
.0025
.0034
.0045
.0060
.0018
.0024
.0033
.0044
.0059
.0017
.0023
.0032
.0043
.0057
.0016
.0023
.0031
.0041
.0055
.0016
.0022
.0030
.0040
.0054
.0015
.0021
.0029
.0039
.0052
.0015
.0021
.0028
.0038
.0051
.0014
.0020
.0027
.0037
.0049
.0014
.0019
.0026
.0036
.0048
22.4
22.3
22.2
22.1
22.0
.0082
.0107
.0139
.0179
.0228
.0080
.0104
.0136
.0174
.0222
.0078
.0102
.0132
.0170
.0217
.0075
.0099
.0129
.0166
.0212
.0073
.0096
.0125
.0162
.0207
.0071
.0094
.0122
.0158
.0202
.0069
.0091
.0119
.0154
.0197
.0068
.0089
.0116
.0150
.0192
.0066
.0087
.0113
.0146
.0188
.0064
.0084
.0110
.0143
.0183
21.9
21.8
21.7
21.6
21.5
.0287
.0359
.0446
.0548
.0668
.0281
.0351
.0436
.0537
.0655
.0274
.0344
.0427
.0526
.0643
.0268
.0336
.0418
.0516
.0630
.0262
.0329
.0409
.0505
.0618
.0256
.0322
.0401
.0495
.0606
.0250
.0314
.0392
.0485
.0594
.0244
.0307
.0384
.0475
.0582
.0239
.0301
.0375
.0465
.0571
.0233
.0294
.0367
.0455
.0559
21.4
21.3
21.2
21.1
21.0
.0808
.0968
.1151
.1357
.1587
.0793
.0951
.1131
.1335
.1562
.0778
.0934
.1112
.1314
.1539
.0764
.0918
.1093
.1292
.1515
.0749
.0901
.1075
.1271
.1492
.0735
.0885
.1056
.1251
.1469
.0721
.0869
.1038
.1230
.1446
.0708
.0853
.1020
.1210
.1423
.0694
.0838
.1003
.1190
.1401
.0681
.0823
.0985
.1170
.1379
20.9
20.8
20.7
20.6
20.5
.1841
.2119
.2420
.2743
.3085
.1814
.2090
.2389
.2709
.3050
.1788
.2061
.2358
.2676
.3015
.1762
.2033
.2327
.2643
.2981
.1736
.2005
.2296
.2611
.2946
.1711
.1977
.2266
.2578
.2912
.1685
.1949
.2236
.2546
.2877
.1660
.1922
.2206
.2514
.2843
.1635
.1894
.2177
.2483
.2810
.1611
.1867
.2148
.2451
.2776
20.4
20.3
20.2
20.1
20.0
.3446
.3821
.4207
.4602
.5000
.3409
.3783
.4168
.4562
.4960
.3372
.3745
.4129
.4522
.4920
.3336
.3707
.4090
.4483
.4880
.3300
.3669
.4052
.4443
.4841
.3264
.3632
.4013
.4404
.4801
.3228
.3594
.3974
.4364
.4761
.3192
.3557
.3936
.4325
.4721
.3156
.3520
.3897
.4286
.4681
.3121
.3483
.3859
.4247
.4641
768
Appendix C-2
This table shows the normal area less than z.
769
z
z
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
.5000
.5398
.5793
.6179
.6554
.5040
.5438
.5832
.6217
.6591
.5080
.5478
.5871
.6255
.6628
.5120
.5517
.5910
.6293
.6664
.5160
.5557
.5948
.6331
.6700
.5199
.5596
.5987
.6368
.6736
.5239
.5636
.6026
.6406
.6772
.5279
.5675
.6064
.6443
.6808
.5319
.5714
.6103
.6480
.6844
.5359
.5753
.6141
.6517
.6879
0.5
0.6
0.7
0.8
0.9
.6915
.7257
.7580
.7881
.8159
.6950
.7291
.7611
.7910
.8186
.6985
.7324
.7642
.7939
.8212
.7019
.7357
.7673
.7967
.8238
.7054
.7389
.7704
.7995
.8264
.7088
.7422
.7734
.8023
.8289
.7123
.7454
.7764
.8051
.8315
.7157
.7486
.7794
.8078
.8340
.7190
.7517
.7823
.8106
.8365
.7224
.7549
.7852
.8133
.8389
1.0
1.1
1.2
1.3
1.4
.8413
.8643
.8849
.9032
.9192
.8438
.8665
.8869
.9049
.9207
.8461
.8686
.8888
.9066
.9222
.8485
.8708
.8907
.9082
.9236
.8508
.8729
.8925
.9099
.9251
.8531
.8749
.8944
.9115
.9265
.8554
.8770
.8962
.9131
.9279
.8577
.8790
.8980
.9147
.9292
.8599
.8810
.8997
.9162
.9306
.8621
.8830
.9015
.9177
.9319
1.5
1.6
1.7
1.8
1.9
.9332
.9452
.9554
.9641
.9713
.9345
.9463
.9564
.9649
.9719
.9357
.9474
.9573
.9656
.9726
.9370
.9484
.9582
.9664
.9732
.9382
.9495
.9591
.9671
.9738
.9394
.9505
.9599
.9678
.9744
.9406
.9515
.9608
.9686
.9750
.9418
.9525
.9616
.9693
.9756
.9429
.9535
.9625
.9699
.9761
.9441
.9545
.9633
.9706
.9767
2.0
2.1
2.2
2.3
2.4
.9772
.9821
.9861
.9893
.9918
.9778
.9826
.9864
.9896
.9920
.9783
.9830
.9868
.9898
.9922
.9788
.9834
.9871
.9901
.9925
.9793
.9838
.9875
.9904
.9927
.9798
.9842
.9878
.9906
.9929
.9803
.9846
.9881
.9909
.9931
.9808
.9850
.9884
.9911
.9932
.9812
.9854
.9887
.9913
.9934
.9817
.9857
.9890
.9916
.9936
2.5
2.6
2.7
2.8
2.9
.9938
.9953
.9965
.9974
.9981
.9940
.9955
.9966
.9975
.9982
.9941
.9956
.9967
.9976
.9982
.9943
.9957
.9968
.9977
.9983
.9945
.9959
.9969
.9977
.9984
.9946
.9960
.9970
.9978
.9984
.9948
.9961
.9971
.9979
.9985
.9949
.9962
.9972
.9979
.9985
.9951
.9963
.9973
.9980
.9986
.9952
.9964
.9974
.9981
.9986
3.0
3.1
3.2
3.3
3.4
.99865
.99903
.99931
.99952
.99966
.99869
.99906
.99934
.99953
.99968
.99874
.99910
.99936
.99955
.99969
.99878
.99913
.99938
.99957
.99970
.99882
.99916
.99940
.99958
.99971
.99886
.99918
.99942
.99960
.99972
.99889
.99921
.99944
.99961
.99973
.99893
.99924
.99946
.99962
.99974
.99896
.99926
.99948
.99964
.99975
.99900
.99929
.99950
.99965
.99976
3.5
3.6
3.7
.99977
.99984
.99989
.99978
.99985
.99990
.99978
.99985
.99990
.99979
.99986
.99990
.99980
.99986
.99991
.99981
.99987
.99991
.99981
.99987
.99992
.99982
.99988
.99992
.99983
.99988
.99992
.99983
.99989
.99992
APPENDIX
D
STUDENT’S t CRITICAL VALUES
This table shows the t-value that defines the area for the stated degrees of freedom (d.f.).
Confidence Level
.95
.98
Confidence Level
.95
.98
t
.99
.80
.20
Significance Level for Two-Tailed Test
.10
.05
.02
.01
.20
Significance Level for Two-Tailed Test
.10
.05
.02
.01
d.f.
.10
Significance Level for One-Tailed Test
.05
.025
.01
.005
.10
Significance Level for One-Tailed Test
.05
.025
.01
.005
1
2
3
4
5
3.078
1.886
1.638
1.533
1.476
6.314
2.920
2.353
2.132
2.015
12.706
4.303
3.182
2.776
2.571
31.821
6.965
4.541
3.747
3.365
63.657
9.925
5.841
4.604
4.032
36
37
38
39
40
1.306
1.305
1.304
1.304
1.303
1.688
1.687
1.686
1.685
1.684
2.028
2.026
2.024
2.023
2.021
2.434
2.431
2.429
2.426
2.423
2.719
2.715
2.712
2.708
2.704
6
7
8
9
10
1.440
1.415
1.397
1.383
1.372
1.943
1.895
1.860
1.833
1.812
2.447
2.365
2.306
2.262
2.228
3.143
2.998
2.896
2.821
2.764
3.707
3.499
3.355
3.250
3.169
41
42
43
44
45
1.303
1.302
1.302
1.301
1.301
1.683
1.682
1.681
1.680
1.679
2.020
2.018
2.017
2.015
2.014
2.421
2.418
2.416
2.414
2.412
2.701
2.698
2.695
2.692
2.690
11
12
13
14
15
1.363
1.356
1.350
1.345
1.341
1.796
1.782
1.771
1.761
1.753
2.201
2.179
2.160
2.145
2.131
2.718
2.681
2.650
2.624
2.602
3.106
3.055
3.012
2.977
2.947
46
47
48
49
50
1.300
1.300
1.299
1.299
1.299
1.679
1.678
1.677
1.677
1.676
2.013
2.012
2.011
2.010
2.009
2.410
2.408
2.407
2.405
2.403
2.687
2.685
2.682
2.680
2.678
16
17
18
19
20
1.337
1.333
1.330
1.328
1.325
1.746
1.740
1.734
1.729
1.725
2.120
2.110
2.101
2.093
2.086
2.583
2.567
2.552
2.539
2.528
2.921
2.898
2.878
2.861
2.845
55
60
65
70
75
1.297
1.296
1.295
1.294
1.293
1.673
1.671
1.669
1.667
1.665
2.004
2.000
1.997
1.994
1.992
2.396
2.390
2.385
2.381
2.377
2.668
2.660
2.654
2.648
2.643
21
22
23
24
25
1.323
1.321
1.319
1.318
1.316
1.721
1.717
1.714
1.711
1.708
2.080
2.074
2.069
2.064
2.060
2.518
2.508
2.500
2.492
2.485
2.831
2.819
2.807
2.797
2.787
80
85
90
95
100
1.292
1.292
1.291
1.291
1.290
1.664
1.663
1.662
1.661
1.660
1.990
1.988
1.987
1.985
1.984
2.374
2.371
2.368
2.366
2.364
2.639
2.635
2.632
2.629
2.626
26
27
28
29
30
1.315
1.314
1.313
1.311
1.310
1.706
1.703
1.701
1.699
1.697
2.056
2.052
2.048
2.045
2.042
2.479
2.473
2.467
2.462
2.457
2.779
2.771
2.763
2.756
2.750
110
120
130
140
150
1.289
1.289
1.288
1.288
1.287
1.659
1.658
1.657
1.656
1.655
1.982
1.980
1.978
1.977
1.976
2.361
2.358
2.355
2.353
2.351
2.621
2.617
2.614
2.611
2.609
31
32
33
34
35
1.309
1.309
1.308
1.307
1.306
1.696
1.694
1.692
1.691
1.690
2.040
2.037
2.035
2.032
2.030
2.453
2.449
2.445
2.441
2.438
2.744
2.738
2.733
2.728
2.724
∞
1.282
1.645
1.960
2.326
2.576
.80
.90
d.f.
.90
0
.99
Note:As n increases, critical values of Student’s t approach the z-values in the last line of this table. A common rule of thumb is to use z when n . 30, but that is not conservative.
770
APPENDIX
A
E
Example for d.f. 5 4
CHI-SQUARE CRITICAL VALUES
.05
This table shows the critical value of chi-square for each desired right-tail area and degrees
of freedom (d.f.)
0
9.488
Area in Upper Tail
d.f.
.995
.990
.975
.95
.90
.10
.05
.025
.01
.005
1
2
3
4
5
0.000
0.010
0.072
0.207
0.412
0.000
0.020
0.115
0.297
0.554
0.001
0.051
0.216
0.484
0.831
0.004
0.103
0.352
0.711
1.145
0.016
0.211
0.584
1.064
1.610
2.706
4.605
6.251
7.779
9.236
3.841
5.991
7.815
9.488
11.07
5.024
7.378
9.348
11.14
12.83
6.635
9.210
11.34
13.28
15.09
7.879
10.60
12.84
14.86
16.75
6
7
8
9
10
0.676
0.989
1.344
1.735
2.156
0.872
1.239
1.646
2.088
2.558
1.237
1.690
2.180
2.700
3.247
1.635
2.167
2.733
3.325
3.940
2.204
2.833
3.490
4.168
4.865
10.64
12.02
13.36
14.68
15.99
12.59
14.07
15.51
16.92
18.31
14.45
16.01
17.53
19.02
20.48
16.81
18.48
20.09
21.67
23.21
18.55
20.28
21.95
23.59
25.19
11
12
13
14
15
2.603
3.074
3.565
4.075
4.601
3.053
3.571
4.107
4.660
5.229
3.816
4.404
5.009
5.629
6.262
4.575
5.226
5.892
6.571
7.261
5.578
6.304
7.042
7.790
8.547
17.28
18.55
19.81
21.06
22.31
19.68
21.03
22.36
23.68
25.00
21.92
23.34
24.74
26.12
27.49
24.72
26.22
27.69
29.14
30.58
26.76
28.30
29.82
31.32
32.80
16
17
18
19
20
5.142
5.697
6.265
6.844
7.434
5.812
6.408
7.015
7.633
8.260
6.908
7.564
8.231
8.907
9.591
7.962
8.672
9.390
10.12
10.85
9.312
10.09
10.86
11.65
12.44
23.54
24.77
25.99
27.20
28.41
26.30
27.59
28.87
30.14
31.41
28.85
30.19
31.53
32.85
34.17
32.00
33.41
34.81
36.19
37.57
34.27
35.72
37.16
38.58
40.00
21
22
23
24
25
8.034
8.643
9.260
9.886
10.52
8.897
9.542
10.20
10.86
11.52
10.28
10.98
11.69
12.40
13.12
11.59
12.34
13.09
13.85
14.61
13.24
14.04
14.85
15.66
16.47
29.62
30.81
32.01
33.20
34.38
32.67
33.92
35.17
36.42
37.65
35.48
36.78
38.08
39.36
40.65
38.93
40.29
41.64
42.98
44.31
41.40
42.80
44.18
45.56
46.93
26
27
28
29
30
11.16
11.81
12.46
13.12
13.79
12.20
12.88
13.56
14.26
14.95
13.84
14.57
15.31
16.05
16.79
15.38
16.15
16.93
17.71
18.49
17.29
18.11
18.94
19.77
20.60
35.56
36.74
37.92
39.09
40.26
38.89
40.11
41.34
42.56
43.77
41.92
43.19
44.46
45.72
46.98
45.64
46.96
48.28
49.59
50.89
48.29
49.64
50.99
52.34
53.67
31
32
33
34
35
14.46
15.13
15.82
16.50
17.19
15.66
16.36
17.07
17.79
18.51
17.54
18.29
19.05
19.81
20.57
19.28
20.07
20.87
21.66
22.47
21.43
22.27
23.11
23.95
24.80
41.42
42.58
43.75
44.90
46.06
44.99
46.19
47.40
48.60
49.80
48.23
49.48
50.73
51.97
53.20
52.19
53.49
54.78
56.06
57.34
55.00
56.33
57.65
58.96
60.27
36
37
38
39
40
17.89
18.59
19.29
20.00
20.71
19.23
19.96
20.69
21.43
22.16
21.34
22.11
22.88
23.65
24.43
23.27
24.07
24.88
25.70
26.51
25.64
26.49
27.34
28.20
29.05
47.21
48.36
49.51
50.66
51.81
51.00
52.19
53.38
54.57
55.76
54.44
55.67
56.90
58.12
59.34
58.62
59.89
61.16
62.43
63.69
61.58
62.88
64.18
65.48
66.77
50
60
70
80
90
100
27.99
35.53
43.28
51.17
59.20
67.33
29.71
37.48
45.44
53.54
61.75
70.06
32.36
40.48
48.76
57.15
65.65
74.22
34.76
43.19
51.74
60.39
69.13
77.93
37.69
46.46
55.33
64.28
73.29
82.36
63.17
74.40
85.53
96.58
107.6
118.5
67.50
79.08
90.53
101.9
113.1
124.3
71.42
83.30
95.02
106.6
118.1
129.6
76.15
88.38
100.4
112.3
124.1
135.8
79.49
91.95
104.2
116.3
128.3
140.2
Note: For d.f. . 100, use the Excel function =CHISQ.INV.RT(α,degrees of freedom).
771
F
APPENDIX
CRITICAL VALUES OF F.10
This table shows the 10 percent right-tail critical values of F for the stated degrees of freedom (d.f.).
0
F
Denominator
Degrees of
Freedom
(df2 )
1
2
3
4
5
6
7
8
9
10
12
1
2
3
4
5
39.86
8.53
5.54
4.54
4.06
49.50
9.00
5.46
4.32
3.78
53.59
9.16
5.39
4.19
3.62
55.83
9.24
5.34
4.11
3.52
57.24
9.29
5.31
4.05
3.45
58.20
9.33
5.28
4.01
3.40
58.91
9.35
5.27
3.98
3.37
59.44
9.37
5.25
3.95
3.34
59.86
9.38
5.24
3.94
3.32
60.19
9.39
5.23
3.92
3.30
60.71
9.41
5.22
3.90
3.27
6
7
8
9
10
3.78
3.59
3.46
3.36
3.29
3.46
3.26
3.11
3.01
2.92
3.29
3.07
2.92
2.81
2.73
3.18
2.96
2.81
2.69
2.61
3.11
2.88
2.73
2.61
2.52
3.05
2.83
2.67
2.55
2.46
3.01
2.78
2.62
2.51
2.41
2.98
2.75
2.59
2.47
2.38
2.96
2.72
2.56
2.44
2.35
2.94
2.70
2.54
2.42
2.32
2.90
2.67
2.50
2.38
2.28
11
12
13
14
15
3.23
3.18
3.14
3.10
3.07
2.86
2.81
2.76
2.73
2.70
2.66
2.61
2.56
2.52
2.49
2.54
2.48
2.43
2.39
2.36
2.45
2.39
2.35
2.31
2.27
2.39
2.33
2.28
2.24
2.21
2.34
2.28
2.23
2.19
2.16
2.30
2.24
2.20
2.15
2.12
2.27
2.21
2.16
2.12
2.09
2.25
2.19
2.14
2.10
2.06
2.21
2.15
2.10
2.05
2.02
16
17
18
19
20
3.05
3.03
3.01
2.99
2.97
2.67
2.64
2.62
2.61
2.59
2.46
2.44
2.42
2.40
2.38
2.33
2.31
2.29
2.27
2.25
2.24
2.22
2.20
2.18
2.16
2.18
2.15
2.13
2.11
2.09
2.13
2.10
2.08
2.06
2.04
2.09
2.06
2.04
2.02
2.00
2.06
2.03
2.00
1.98
1.96
2.03
2.00
1.98
1.96
1.94
1.99
1.96
1.93
1.91
1.89
21
22
23
24
25
2.96
2.95
2.94
2.93
2.92
2.57
2.56
2.55
2.54
2.53
2.36
2.35
2.34
2.33
2.32
2.23
2.22
2.21
2.19
2.18
2.14
2.13
2.11
2.10
2.09
2.08
2.06
2.05
2.04
2.02
2.02
2.01
1.99
1.98
1.97
1.98
1.97
1.95
1.94
1.93
1.95
1.93
1.92
1.91
1.89
1.92
1.90
1.89
1.88
1.87
1.87
1.86
1.84
1.83
1.82
26
27
28
29
30
2.91
2.90
2.89
2.89
2.88
2.52
2.51
2.50
2.50
2.49
2.31
2.30
2.29
2.28
2.28
2.17
2.17
2.16
2.15
2.14
2.08
2.07
2.06
2.06
2.05
2.01
2.00
2.00
1.99
1.98
1.96
1.95
1.94
1.93
1.93
1.92
1.91
1.90
1.89
1.88
1.88
1.87
1.87
1.86
1.85
1.86
1.85
1.84
1.83
1.82
1.81
1.80
1.79
1.78
1.77
40
50
60
120
200
2.84
2.81
2.79
2.75
2.73
2.44
2.41
2.39
2.35
2.33
2.23
2.20
2.18
2.13
2.11
2.09
2.06
2.04
1.99
1.97
2.00
1.97
1.95
1.90
1.88
1.93
1.90
1.87
1.82
1.80
1.87
1.84
1.82
1.77
1.75
1.83
1.80
1.77
1.72
1.70
1.79
1.76
1.74
1.68
1.66
1.76
1.73
1.71
1.65
1.63
1.71
1.68
1.66
1.60
1.58
`
2.71
2.30
2.08
1.94
1.85
1.77
1.72
1.67
1.63
1.60
1.55
772
Numerator Degrees of Freedom (df1 )
Appendix F
773
Denominator
Degrees of
Freedom
(df2 )
15
20
25
30
35
40
50
60
120
200
∞
1
2
3
4
5
61.22
9.42
5.20
3.87
3.24
61.74
9.44
5.18
3.84
3.21
62.05
9.45
5.17
3.83
3.19
62.26
9.46
5.17
3.82
3.17
62.42
9.46
5.16
3.81
3.16
62.53
9.47
5.16
3.80
3.16
62.69
9.47
5.15
3.80
3.15
62.79
9.47
5.15
3.79
3.14
63.06
9.48
5.14
3.78
3.12
63.17
9.49
5.14
3.77
3.12
63.32
9.49
5.13
3.76
3.11
6
7
8
9
10
2.87
2.63
2.46
2.34
2.24
2.84
2.59
2.42
2.30
2.20
2.81
2.57
2.40
2.27
2.17
2.80
2.56
2.38
2.25
2.16
2.79
2.54
2.37
2.24
2.14
2.78
2.54
2.36
2.23
2.13
2.77
2.52
2.35
2.22
2.12
2.76
2.51
2.34
2.21
2.11
2.74
2.49
2.32
2.18
2.08
2.73
2.48
2.31
2.17
2.07
2.72
2.47
2.29
2.16
2.06
11
12
13
14
15
2.17
2.10
2.05
2.01
1.97
2.12
2.06
2.01
1.96
1.92
2.10
2.03
1.98
1.93
1.89
2.08
2.01
1.96
1.91
1.87
2.06
2.00
1.94
1.90
1.86
2.05
1.99
1.93
1.89
1.85
2.04
1.97
1.92
1.87
1.83
2.03
1.96
1.90
1.86
1.82
2.00
1.93
1.88
1.83
1.79
1.99
1.92
1.86
1.82
1.77
1.97
1.90
1.85
1.80
1.76
16
17
18
19
20
1.94
1.91
1.89
1.86
1.84
1.89
1.86
1.84
1.81
1.79
1.86
1.83
1.80
1.78
1.76
1.84
1.81
1.78
1.76
1.74
1.82
1.79
1.77
1.74
1.72
1.81
1.78
1.75
1.73
1.71
1.79
1.76
1.74
1.71
1.69
1.78
1.75
1.72
1.70
1.68
1.75
1.72
1.69
1.67
1.64
1.74
1.71
1.68
1.65
1.63
1.72
1.69
1.66
1.63
1.61
21
22
23
24
25
1.83
1.81
1.80
1.78
1.77
1.78
1.76
1.74
1.73
1.72
1.74
1.73
1.71
1.70
1.68
1.72
1.70
1.69
1.67
1.66
1.70
1.68
1.67
1.65
1.64
1.69
1.67
1.66
1.64
1.63
1.67
1.65
1.64
1.62
1.61
1.66
1.64
1.62
1.61
1.59
1.62
1.60
1.59
1.57
1.56
1.61
1.59
1.57
1.56
1.54
1.59
1.57
1.55
1.53
1.52
26
27
28
29
30
1.76
1.75
1.74
1.73
1.72
1.71
1.70
1.69
1.68
1.67
1.67
1.66
1.65
1.64
1.63
1.65
1.64
1.63
1.62
1.61
1.63
1.62
1.61
1.60
1.59
1.61
1.60
1.59
1.58
1.57
1.59
1.58
1.57
1.56
1.55
1.58
1.57
1.56
1.55
1.54
1.54
1.53
1.52
1.51
1.50
1.53
1.52
1.50
1.49
1.48
1.50
1.49
1.48
1.47
1.46
40
50
60
120
200
1.66
1.63
1.60
1.55
1.52
1.61
1.57
1.54
1.48
1.46
1.57
1.53
1.50
1.44
1.41
1.54
1.50
1.48
1.41
1.38
1.52
1.48
1.45
1.39
1.36
1.51
1.46
1.44
1.37
1.34
1.48
1.44
1.41
1.34
1.31
1.47
1.42
1.40
1.32
1.29
1.42
1.38
1.35
1.26
1.23
1.41
1.36
1.33
1.24
1.20
1.38
1.33
1.29
1.19
1.15
`
1.49
1.42
1.38
1.34
1.32
1.30
1.26
1.24
1.17
1.13
1.00
Numerator Degrees of Freedom (df1 )
774
Appendix F
CRITICAL VALUES OF F.05
This table shows the 5 percent right-tail critical values of F for the stated degrees of freedom (d.f.).
Denominator
Degrees of
Freedom
(df2 )
0
F
Numerator Degrees of Freedom (df1 )
1
2
3
4
5
6
7
8
9
10
12
1
2
3
4
5
161.4
18.51
10.13
7.71
6.61
199.5
19.00
9.55
6.94
5.79
215.7
19.16
9.28
6.59
5.41
224.6
19.25
9.12
6.39
5.19
230.2
19.30
9.01
6.26
5.05
234.0
19.33
8.94
6.16
4.95
236.8
19.35
8.89
6.09
4.88
238.9
19.37
8.85
6.04
4.82
240.5
19.38
8.81
6.00
4.77
241.9
19.40
8.79
5.96
4.74
243.9
19.41
8.74
5.91
4.68
6
7
8
9
10
5.99
5.59
5.32
5.12
4.96
5.14
4.74
4.46
4.26
4.10
4.76
4.35
4.07
3.86
3.71
4.53
4.12
3.84
3.63
3.48
4.39
3.97
3.69
3.48
3.33
4.28
3.87
3.58
3.37
3.22
4.21
3.79
3.50
3.29
3.14
4.15
3.73
3.44
3.23
3.07
4.10
3.68
3.39
3.18
3.02
4.06
3.64
3.35
3.14
2.98
4.00
3.57
3.28
3.07
2.91
11
12
13
14
15
4.84
4.75
4.67
4.60
4.54
3.98
3.89
3.81
3.74
3.68
3.59
3.49
3.41
3.34
3.29
3.36
3.26
3.18
3.11
3.06
3.20
3.11
3.03
2.96
2.90
3.09
3.00
2.92
2.85
2.79
3.01
2.91
2.83
2.76
2.71
2.95
2.85
2.77
2.70
2.64
2.90
2.80
2.71
2.65
2.59
2.85
2.75
2.67
2.60
2.54
2.79
2.69
2.60
2.53
2.48
16
17
18
19
20
4.49
4.45
4.41
4.38
4.35
3.63
3.59
3.55
3.52
3.49
3.24
3.20
3.16
3.13
3.10
3.01
2.96
2.93
2.90
2.87
2.85
2.81
2.77
2.74
2.71
2.74
2.70
2.66
2.63
2.60
2.66
2.61
2.58
2.54
2.51
2.59
2.55
2.51
2.48
2.45
2.54
2.49
2.46
2.42
2.39
2.49
2.45
2.41
2.38
2.35
2.42
2.38
2.34
2.31
2.28
21
22
23
24
25
4.32
4.30
4.28
4.26
4.24
3.47
3.44
3.42
3.40
3.39
3.07
3.05
3.03
3.01
2.99
2.84
2.82
2.80
2.78
2.76
2.68
2.66
2.64
2.62
2.60
2.57
2.55
2.53
2.51
2.49
2.49
2.46
2.44
2.42
2.40
2.42
2.40
2.37
2.36
2.34
2.37
2.34
2.32
2.30
2.28
2.32
2.30
2.27
2.25
2.24
2.25
2.23
2.20
2.18
2.16
26
27
28
29
30
4.23
4.21
4.20
4.18
4.17
3.37
3.35
3.34
3.33
3.32
2.98
2.96
2.95
2.93
2.92
2.74
2.73
2.71
2.70
2.69
2.59
2.57
2.56
2.55
2.53
2.47
2.46
2.45
2.43
2.42
2.39
2.37
2.36
2.35
2.33
2.32
2.31
2.29
2.28
2.27
2.27
2.25
2.24
2.22
2.21
2.22
2.20
2.19
2.18
2.16
2.15
2.13
2.12
2.10
2.09
40
50
60
120
200
4.08
4.03
4.00
3.92
3.89
3.23
3.18
3.15
3.07
3.04
2.84
2.79
2.76
2.68
2.65
2.61
2.56
2.53
2.45
2.42
2.45
2.40
2.37
2.29
2.26
2.34
2.29
2.25
2.18
2.14
2.25
2.20
2.17
2.09
2.06
2.18
2.13
2.10
2.02
1.98
2.12
2.07
2.04
1.96
1.93
2.08
2.03
1.99
1.91
1.88
2.00
1.95
1.92
1.83
1.80
`
3.84
3.00
2.60
2.37
2.21
2.10
2.01
1.94
1.88
1.83
1.75
Appendix F
Denominator
Degrees of
Freedom
(df2 )
775
Numerator Degrees of Freedom (df1 )
15
20
25
30
35
40
50
60
120
200
∞
1
2
3
4
5
245.9
19.43
8.70
5.86
4.62
248.0
19.45
8.66
5.80
4.56
249.3
19.46
8.63
5.77
4.52
250.1
19.46
8.62
5.75
4.50
250.7
19.47
8.60
5.73
4.48
251.1
19.47
8.59
5.72
4.46
251.8
19.48
8.58
5.70
4.44
252.2
19.48
8.57
5.69
4.43
253.3
19.49
8.55
5.66
4.40
253.7
19.49
8.54
5.65
4.39
254.3
19.50
8.53
5.63
4.37
6
7
8
9
10
3.94
3.51
3.22
3.01
2.85
3.87
3.44
3.15
2.94
2.77
3.83
3.40
3.11
2.89
2.73
3.81
3.38
3.08
2.86
2.70
3.79
3.36
3.06
2.84
2.68
3.77
3.34
3.04
2.83
2.66
3.75
3.32
3.02
2.80
2.64
3.74
3.30
3.01
2.79
2.62
3.70
3.27
2.97
2.75
2.58
3.69
3.25
2.95
2.73
2.56
3.67
3.23
2.93
2.71
2.54
11
12
13
14
15
2.72
2.62
2.53
2.46
2.40
2.65
2.54
2.46
2.39
2.33
2.60
2.50
2.41
2.34
2.28
2.57
2.47
2.38
2.31
2.25
2.55
2.44
2.36
2.28
2.22
2.53
2.43
2.34
2.27
2.20
2.51
2.40
2.31
2.24
2.18
2.49
2.38
2.30
2.22
2.16
2.45
2.34
2.25
2.18
2.11
2.43
2.32
2.23
2.16
2.10
2.41
2.30
2.21
2.13
2.07
16
17
18
19
20
2.35
2.31
2.27
2.23
2.20
2.28
2.23
2.19
2.16
2.12
2.23
2.18
2.14
2.11
2.07
2.19
2.15
2.11
2.07
2.04
2.17
2.12
2.08
2.05
2.01
2.15
2.10
2.06
2.03
1.99
2.12
2.08
2.04
2.00
1.97
2.11
2.06
2.02
1.98
1.95
2.06
2.01
1.97
1.93
1.90
2.04
1.99
1.95
1.91
1.88
2.01
1.96
1.92
1.88
1.84
21
22
23
24
25
2.18
2.15
2.13
2.11
2.09
2.10
2.07
2.05
2.03
2.01
2.05
2.02
2.00
1.97
1.96
2.01
1.98
1.96
1.94
1.92
1.98
1.96
1.93
1.91
1.89
1.96
1.94
1.91
1.89
1.87
1.94
1.91
1.88
1.86
1.84
1.92
1.89
1.86
1.84
1.82
1.87
1.84
1.81
1.79
1.77
1.84
1.82
1.79
1.77
1.75
1.81
1.78
1.76
1.73
1.71
26
27
28
29
30
2.07
2.06
2.04
2.03
2.01
1.99
1.97
1.96
1.94
1.93
1.94
1.92
1.91
1.89
1.88
1.90
1.88
1.87
1.85
1.84
1.87
1.86
1.84
1.83
1.81
1.85
1.84
1.82
1.81
1.79
1.82
1.81
1.79
1.77
1.76
1.80
1.79
1.77
1.75
1.74
1.75
1.73
1.71
1.70
1.68
1.73
1.71
1.69
1.67
1.66
1.69
1.67
1.66
1.64
1.62
40
50
60
120
200
1.92
1.87
1.84
1.75
1.72
1.84
1.78
1.75
1.66
1.62
1.78
1.73
1.69
1.60
1.56
1.74
1.69
1.65
1.55
1.52
1.72
1.66
1.62
1.52
1.48
1.69
1.63
1.59
1.50
1.46
1.66
1.60
1.56
1.46
1.41
1.64
1.58
1.53
1.43
1.39
1.58
1.51
1.47
1.35
1.30
1.55
1.48
1.44
1.32
1.26
1.51
1.44
1.39
1.26
1.19
`
1.67
1.57
1.51
1.46
1.42
1.39
1.35
1.32
1.22
1.17
1.00
776
Appendix F
CRITICAL VALUES OF F.025
This table shows the 2.5 percent right-tail critical values of F for the stated degrees of freedom (d.f.).
Denominator
Degrees of
Freedom
(df2 )
0
F
Numerator Degrees of Freedom (df1 )
1
2
3
4
5
6
7
8
9
10
12
1
2
3
4
5
647.8
38.51
17.44
12.22
10.01
799.5
39.00
16.04
10.65
8.43
864.2
39.17
15.44
9.98
7.76
899.6
39.25
15.10
9.60
7.39
921.8
39.30
14.88
9.36
7.15
937.1
39.33
14.73
9.20
6.98
948.2
39.36
14.62
9.07
6.85
956.6
39.37
14.54
8.98
6.76
963.3
39.39
14.47
8.90
6.68
968.6
39.40
14.42
8.84
6.62
976.7
39.41
14.34
8.75
6.52
6
7
8
9
10
8.81
8.07
7.57
7.21
6.94
7.26
6.54
6.06
5.71
5.46
6.60
5.89
5.42
5.08
4.83
6.23
5.52
5.05
4.72
4.47
5.99
5.29
4.82
4.48
4.24
5.82
5.12
4.65
4.32
4.07
5.70
4.99
4.53
4.20
3.95
5.60
4.90
4.43
4.10
3.85
5.52
4.82
4.36
4.03
3.78
5.46
4.76
4.30
3.96
3.72
5.37
4.67
4.20
3.87
3.62
11
12
13
14
15
6.72
6.55
6.41
6.30
6.20
5.26
5.10
4.97
4.86
4.77
4.63
4.47
4.35
4.24
4.15
4.28
4.12
4.00
3.89
3.80
4.04
3.89
3.77
3.66
3.58
3.88
3.73
3.60
3.50
3.41
3.76
3.61
3.48
3.38
3.29
3.66
3.51
3.39
3.29
3.20
3.59
3.44
3.31
3.21
3.12
3.53
3.37
3.25
3.15
3.06
3.43
3.28
3.15
3.05
2.96
16
17
18
19
20
6.12
6.04
5.98
5.92
5.87
4.69
4.62
4.56
4.51
4.46
4.08
4.01
3.95
3.90
3.86
3.73
3.66
3.61
3.56
3.51
3.50
3.44
3.38
3.33
3.29
3.34
3.28
3.22
3.17
3.13
3.22
3.16
3.10
3.05
3.01
3.12
3.06
3.01
2.96
2.91
3.05
2.98
2.93
2.88
2.84
2.99
2.92
2.87
2.82
2.77
2.89
2.82
2.77
2.72
2.68
21
22
23
24
25
5.83
5.79
5.75
5.72
5.69
4.42
4.38
4.35
4.32
4.29
3.82
3.78
3.75
3.72
3.69
3.48
3.44
3.41
3.38
3.35
3.25
3.22
3.18
3.15
3.13
3.09
3.05
3.02
2.99
2.97
2.97
2.93
2.90
2.87
2.85
2.87
2.84
2.81
2.78
2.75
2.80
2.76
2.73
2.70
2.68
2.73
2.70
2.67
2.64
2.61
2.64
2.60
2.57
2.54
2.51
26
27
28
29
30
5.66
5.63
5.61
5.59
5.57
4.27
4.24
4.22
4.20
4.18
3.67
3.65
3.63
3.61
3.59
3.33
3.31
3.29
3.27
3.25
3.10
3.08
3.06
3.04
3.03
2.94
2.92
2.90
2.88
2.87
2.82
2.80
2.78
2.76
2.75
2.73
2.71
2.69
2.67
2.65
2.65
2.63
2.61
2.59
2.57
2.59
2.57
2.55
2.53
2.51
2.49
2.47
2.45
2.43
2.41
40
50
60
120
200
5.42
5.34
5.29
5.15
5.10
4.05
3.97
3.93
3.80
3.76
3.46
3.39
3.34
3.23
3.18
3.13
3.05
3.01
2.89
2.85
2.90
2.83
2.79
2.67
2.63
2.74
2.67
2.63
2.52
2.47
2.62
2.55
2.51
2.39
2.35
2.53
2.46
2.41
2.30
2.26
2.45
2.38
2.33
2.22
2.18
2.39
2.32
2.27
2.16
2.11
2.29
2.22
2.17
2.05
2.01
`
5.02
3.69
3.12
2.79
2.57
2.41
2.29
2.19
2.11
2.05
1.94
Appendix F
Denominator
Degrees of
Freedom
(df2 )
777
Numerator Degrees of Freedom (df1 )
15
20
25
30
35
40
50
60
120
200
∞
1
2
3
4
5
984.9
39.43
14.25
8.66
6.43
993.1
39.45
14.17
8.56
6.33
998.1
39.46
14.12
8.50
6.27
1001
39.46
14.08
8.46
6.23
1004
39.47
14.06
8.43
6.20
1006
39.47
14.04
8.41
6.18
1008
39.48
14.01
8.38
6.14
1010
39.48
13.99
8.36
6.12
1014
39.49
13.95
8.31
6.07
1016
39.49
13.93
8.29
6.05
1018
39.50
13.90
8.26
6.02
6
7
8
9
10
5.27
4.57
4.10
3.77
3.52
5.17
4.47
4.00
3.67
3.42
5.11
4.40
3.94
3.60
3.35
5.07
4.36
3.89
3.56
3.31
5.04
4.33
3.86
3.53
3.28
5.01
4.31
3.84
3.51
3.26
4.98
4.28
3.81
3.47
3.22
4.96
4.25
3.78
3.45
3.20
4.90
4.20
3.73
3.39
3.14
4.88
4.18
3.70
3.37
3.12
4.85
4.14
3.67
3.33
3.08
11
12
13
14
15
3.33
3.18
3.05
2.95
2.86
3.23
3.07
2.95
2.84
2.76
3.16
3.01
2.88
2.78
2.69
3.12
2.96
2.84
2.73
2.64
3.09
2.93
2.80
2.70
2.61
3.06
2.91
2.78
2.67
2.59
3.03
2.87
2.74
2.64
2.55
3.00
2.85
2.72
2.61
2.52
2.94
2.79
2.66
2.55
2.46
2.92
2.76
2.63
2.53
2.44
2.88
2.73
2.60
2.49
2.40
16
17
18
19
20
2.79
2.72
2.67
2.62
2.57
2.68
2.62
2.56
2.51
2.46
2.61
2.55
2.49
2.44
2.40
2.57
2.50
2.44
2.39
2.35
2.53
2.47
2.41
2.36
2.31
2.51
2.44
2.38
2.33
2.29
2.47
2.41
2.35
2.30
2.25
2.45
2.38
2.32
2.27
2.22
2.38
2.32
2.26
2.20
2.16
2.36
2.29
2.23
2.18
2.13
2.32
2.25
2.19
2.13
2.09
21
22
23
24
25
2.53
2.50
2.47
2.44
2.41
2.42
2.39
2.36
2.33
2.30
2.36
2.32
2.29
2.26
2.23
2.31
2.27
2.24
2.21
2.18
2.27
2.24
2.20
2.17
2.15
2.25
2.21
2.18
2.15
2.12
2.21
2.17
2.14
2.11
2.08
2.18
2.14
2.11
2.08
2.05
2.11
2.08
2.04
2.01
1.98
2.09
2.05
2.01
1.98
1.95
2.04
2.01
1.97
1.94
1.91
26
27
28
29
30
2.39
2.36
2.34
2.32
2.31
2.28
2.25
2.23
2.21
2.20
2.21
2.18
2.16
2.14
2.12
2.16
2.13
2.11
2.09
2.07
2.12
2.10
2.08
2.06
2.04
2.09
2.07
2.05
2.03
2.01
2.05
2.03
2.01
1.99
1.97
2.03
2.00
1.98
1.96
1.94
1.95
1.93
1.91
1.89
1.87
1.92
1.90
1.88
1.86
1.84
1.88
1.85
1.83
1.81
1.79
40
50
60
120
200
2.18
2.11
2.06
1.94
1.90
2.07
1.99
1.94
1.82
1.78
1.99
1.92
1.87
1.75
1.70
1.94
1.87
1.82
1.69
1.64
1.90
1.83
1.78
1.65
1.60
1.88
1.80
1.74
1.61
1.56
1.83
1.75
1.70
1.56
1.51
1.80
1.72
1.67
1.53
1.47
1.72
1.64
1.58
1.43
1.37
1.69
1.60
1.54
1.39
1.32
1.64
1.55
1.48
1.31
1.23
`
1.83
1.71
1.63
1.57
1.52
1.48
1.43
1.39
1.27
1.21
1.00
778
Appendix F
CRITICAL VALUES OF F.01
This table shows the 1 percent right-tail critical values of F for the stated degrees of freedom (d.f.).
Denominator
Degrees of
Freedom
(df2 )
1
2
3
4
5
0
F
Numerator Degrees of Freedom (df1 )
1
2
3
4
5
6
7
8
9
10
12
4052
4999
5404
5624
5764
5859
5928
5981
6022
6056
6107
98.50
99.00
99.16
99.25
99.30
99.33
99.36
99.38
99.39
99.40
99.42
34.12
30.82
29.46
28.71
28.24
27.91
27.67
27.49
27.34
27.23
27.05
21.20
18.00
16.69
15.98
15.52
15.21
14.98
14.80
14.66
14.55
14.37
16.26
13.27
12.06
11.39
10.97
10.67
10.46
10.29
10.16
10.05
9.89
6
7
8
9
10
13.75
12.25
11.26
10.56
10.04
10.92
9.55
8.65
8.02
7.56
9.78
8.45
7.59
6.99
6.55
9.15
7.85
7.01
6.42
5.99
8.75
7.46
6.63
6.06
5.64
8.47
7.19
6.37
5.80
5.39
8.26
6.99
6.18
5.61
5.20
8.10
6.84
6.03
5.47
5.06
7.98
6.72
5.91
5.35
4.94
7.87
6.62
5.81
5.26
4.85
7.72
6.47
5.67
5.11
4.71
11
12
13
14
15
9.65
9.33
9.07
8.86
8.68
7.21
6.93
6.70
6.51
6.36
6.22
5.95
5.74
5.56
5.42
5.67
5.41
5.21
5.04
4.89
5.32
5.06
4.86
4.69
4.56
5.07
4.82
4.62
4.46
4.32
4.89
4.64
4.44
4.28
4.14
4.74
4.50
4.30
4.14
4.00
4.63
4.39
4.19
4.03
3.89
4.54
4.30
4.10
3.94
3.80
4.40
4.16
3.96
3.80
3.67
16
17
18
19
20
8.53
8.40
8.29
8.18
8.10
6.23
6.11
6.01
5.93
5.85
5.29
5.19
5.09
5.01
4.94
4.77
4.67
4.58
4.50
4.43
4.44
4.34
4.25
4.17
4.10
4.20
4.10
4.01
3.94
3.87
4.03
3.93
3.84
3.77
3.70
3.89
3.79
3.71
3.63
3.56
3.78
3.68
3.60
3.52
3.46
3.69
3.59
3.51
3.43
3.37
3.55
3.46
3.37
3.30
3.23
21
22
23
24
25
8.02
7.95
7.88
7.82
7.77
5.78
5.72
5.66
5.61
5.57
4.87
4.82
4.76
4.72
4.68
4.37
4.31
4.26
4.22
4.18
4.04
3.99
3.94
3.90
3.85
3.81
3.76
3.71
3.67
3.63
3.64
3.59
3.54
3.50
3.46
3.51
3.45
3.41
3.36
3.32
3.40
3.35
3.30
3.26
3.22
3.31
3.26
3.21
3.17
3.13
3.17
3.12
3.07
3.03
2.99
26
27
28
29
30
7.72
7.68
7.64
7.60
7.56
5.53
5.49
5.45
5.42
5.39
4.64
4.60
4.57
4.54
4.51
4.14
4.11
4.07
4.04
4.02
3.82
3.78
3.75
3.73
3.70
3.59
3.56
3.53
3.50
3.47
3.42
3.39
3.36
3.33
3.30
3.29
3.26
3.23
3.20
3.17
3.18
3.15
3.12
3.09
3.07
3.09
3.06
3.03
3.00
2.98
2.96
2.93
2.90
2.87
2.84
40
50
60
120
200
7.31
7.17
7.08
6.85
6.76
5.18
5.06
4.98
4.79
4.71
4.31
4.20
4.13
3.95
3.88
3.83
3.72
3.65
3.48
3.41
3.51
3.41
3.34
3.17
3.11
3.29
3.19
3.12
2.96
2.89
3.12
3.02
2.95
2.79
2.73
2.99
2.89
2.82
2.66
2.60
2.89
2.78
2.72
2.56
2.50
2.80
2.70
2.63
2.47
2.41
2.66
2.56
2.50
2.34
2.27
`
6.63
4.61
3.78
3.32
3.02
2.80
2.64
2.51
2.41
2.32
2.18
Appendix F
Denominator
Degrees of
Freedom
(df2 )
1
2
3
4
5
779
Numerator Degrees of Freedom (df1 )
15
20
25
30
35
40
50
60
120
200
∞
6157
6209
6240
6260
6275
6286
6302
6313
6340
6350
6366
99.43
99.45
99.46
99.47
99.47
99.48
99.48
99.48
99.49
99.49
99.50
26.87
26.69
26.58
26.50
26.45
26.41
26.35
26.32
26.22
26.18
26.13
14.20
14.02
13.91
13.84
13.79
13.75
13.69
13.65
13.56
13.52
13.47
9.72
9.55
9.45
9.38
9.33
9.29
9.24
9.20
9.11
9.08
9.02
6
7
8
9
10
7.56
6.31
5.52
4.96
4.56
7.40
6.16
5.36
4.81
4.41
7.30
6.06
5.26
4.71
4.31
7.23
5.99
5.20
4.65
4.25
7.18
5.94
5.15
4.60
4.20
7.14
5.91
5.12
4.57
4.17
7.09
5.86
5.07
4.52
4.12
7.06
5.82
5.03
4.48
4.08
6.97
5.74
4.95
4.40
4.00
6.93
5.70
4.91
4.36
3.96
6.88
5.65
4.86
4.31
3.91
11
12
13
14
15
4.25
4.01
3.82
3.66
3.52
4.10
3.86
3.66
3.51
3.37
4.01
3.76
3.57
3.41
3.28
3.94
3.70
3.51
3.35
3.21
3.89
3.65
3.46
3.30
3.17
3.86
3.62
3.43
3.27
3.13
3.81
3.57
3.38
3.22
3.08
3.78
3.54
3.34
3.18
3.05
3.69
3.45
3.25
3.09
2.96
3.66
3.41
3.22
3.06
2.92
3.60
3.36
3.17
3.01
2.87
16
17
18
19
20
3.41
3.31
3.23
3.15
3.09
3.26
3.16
3.08
3.00
2.94
3.16
3.07
2.98
2.91
2.84
3.10
3.00
2.92
2.84
2.78
3.05
2.96
2.87
2.80
2.73
3.02
2.92
2.84
2.76
2.69
2.97
2.87
2.78
2.71
2.64
2.93
2.83
2.75
2.67
2.61
2.84
2.75
2.66
2.58
2.52
2.81
2.71
2.62
2.55
2.48
2.76
2.66
2.57
2.49
2.42
21
22
23
24
25
3.03
2.98
2.93
2.89
2.85
2.88
2.83
2.78
2.74
2.70
2.79
2.73
2.69
2.64
2.60
2.72
2.67
2.62
2.58
2.54
2.67
2.62
2.57
2.53
2.49
2.64
2.58
2.54
2.49
2.45
2.58
2.53
2.48
2.44
2.40
2.55
2.50
2.45
2.40
2.36
2.46
2.40
2.35
2.31
2.27
2.42
2.36
2.32
2.27
2.23
2.36
2.31
2.26
2.21
2.17
26
27
28
29
30
2.81
2.78
2.75
2.73
2.70
2.66
2.63
2.60
2.57
2.55
2.57
2.54
2.51
2.48
2.45
2.50
2.47
2.44
2.41
2.39
2.45
2.42
2.39
2.36
2.34
2.42
2.38
2.35
2.33
2.30
2.36
2.33
2.30
2.27
2.25
2.33
2.29
2.26
2.23
2.21
2.23
2.20
2.17
2.14
2.11
2.19
2.16
2.13
2.10
2.07
2.13
2.10
2.07
2.04
2.01
40
50
60
120
200
2.52
2.42
2.35
2.19
2.13
2.37
2.27
2.20
2.03
1.97
2.27
2.17
2.10
1.93
1.87
2.20
2.10
2.03
1.86
1.79
2.15
2.05
1.98
1.81
1.74
2.11
2.01
1.94
1.76
1.69
2.06
1.95
1.88
1.70
1.63
2.02
1.91
1.84
1.66
1.58
1.92
1.80
1.73
1.53
1.45
1.87
1.76
1.68
1.48
1.39
1.81
1.69
1.60
1.38
1.28
`
2.04
1.88
1.77
1.70
1.64
1.59
1.52
1.47
1.32
1.25
1.00
APPENDIX
A
J
Excel Statistical
Functions
Descriptive Statistics
Pre-2010 Excel*
2013 Excel
Number of data items
Largest data value
Smallest data value
Mean
Median
Mode (returns first mode only)
Mode (array function for multiple modes;
highlight output range and use Ctrl-Shift-Enter)
Geometric mean (positive data values only)
Quartile k (old Excel method),* e.g., k 5 3 for Q3
Quartile k (mainstream),* e.g., k 5 3 for Q3
Percentile p (old Excel method),* e.g., p 5 .25 for Q1
Percentile p (mainstream),* e.g., p 5 .25 for Q1
Sample standard deviation
Sample covariance for (X,Y ) data pairs
Population standard deviation
Population variance for (X,Y ) data pairs
Standardize an X value (use sample mean and
standard deviation if μ and σ unknown)
Correlation coefficient for (X,Y ) data pairs
Average deviation around the mean
Slope of simple X-Y regression
Intercept of simple X-Y regression
R-squared for simple X-Y regression
COUNT(Data)
MAX(Data)
MIN(Data)
AVERAGE(Data)
MEDIAN(Data)
MODE(Data)
----------------
COUNT(Data)
MAX(Data)
MIN(Data)
AVERAGE(Data)
MEDIAN(Data)
MODE.SNGL(Data)
{MODE.MULT(Data)}
GEOMEAN(Data)
QUARTILE(Data, k)
---------------PERCENTILE(Data, p)
---------------STDEV(Data)
---------------STDEVP(Data)
COVAR(XData, YData)
STANDARDIZE(Data, μ, σ)
GEOMEAN(Data)
QUARTILE.INC(Data, k)
QUARTILE.EXC(Data, k)
PERCENTILE.INC(Data, p)
PERCENTILE.EXC(Data, p)
STDEV.S(Data)
COVARIANCE.S(XData, YData)
STDEV.P(Data)
COVARIANCE.P(XDdata, YData)
STANDARDIZE(Data, μ, σ)
CORREL(XData, YData)
AVEDEV(Data)
SLOPE(XData, YData)
INTERCEPT(XData, YData)
RSQ(XData, YData)
CORREL(XData, YData)
AVEDEV(Data)
SLOPE(XData, YData)
INTERCEPT(XData, YData)
RSQ(XData, YData)
* In 2010, Excel changed many of its statistical functions. The pre-2010 functions will work in newer versions of Excel, but not vice-versa. For the latest information about Excel statistical
functions, see https://support.office.com/ and Search “Excel Functions.” See Chapter 4, Section 4.5 for explanation of interpolation methods for percentiles and quartiles. Excel’s old method
was rather unconventional, while its new method agrees with mainstream statistical packages.
Discrete Probability Distributions
Pre-2010 Excel
2013 Excel
Binomial distribution
PDF: Returns probability P (X 5 x)
CDF: Returns probability P (X # x)
Inverse CDF: Returns x for P (X # x) 5 α
BINOMDIST(x, n, π, 0)
BINOMDIST(x, n, π, 1)
CRITBINOM(n, π, α)
BINOM.DIST(x, n, π, 0)
BINOM.DIST(x, n, π, 1)
BINOM.INV(n, π, α)
Poisson distribution
PDF: Returns probability P (X 5 x)
CDF: Returns probability P (X # x)
Inverse CDF: Returns x for P (X # x) 5 α
POISSON(x, λ, 0)
POISSON (x, λ, 1)
----------------
POISSON.DIST(x, λ,0)
POISSON.DIST(x, λ,1)
----------------
Hypergeometric distribution
PDF: Returns probability P (X 5 x)
CDF: Returns probability P (X # x)
Inverse CDF: Returns x for P (X # x) 5 α
HYPGEOMDIST(x, n, s, N)
-------------------------------
HYPGEOM.DIST(x, n, s, N, 0)
HYPGEOM.DIST(x, n, s, N, 1)
---------------813
814
Appendix J
Continuous Probability Distributions
Pre-2010 Excel
2013 Excel
Normal distribution
PDF: Returns height of f (x)
CDF: Returns probability P (X # x)
Inverse CDF: Returns x for P (X # x) 5 α
NORMDIST(x, μ, σ, 0)
NORMDIST(x, μ, σ, 1)
NORMINV(α, μ, σ)
NORM.DIST(x, μ, σ, 0)
NORM.DIST(x, μ, σ, 1)
NORM.INV(α, μ, σ)
Standard normal distribution
PDF: Returns height of f (z)
CDF: Returns probability P (Z # z)
Inverse CDF: Returns z for P (Z # z) 5 α
---------------NORMSDIST(z)
NORMSINV(α)
NORM.S.DIST(z, 0)
NORM.S.DIST(z, 1)
NORM.S.INV(α)
Exponential distribution
PDF: Returns height of f (x)
CDF: Returns probability P (X # x)
Inverse CDF: Returns x for P (X # x) 5 α
EXPONDIST(x, λ, 0)
EXPONDIST(x, λ, 1)
----------------
EXPON.DIST(x, λ, 0)
EXPON.DIST(x, λ, 1)
----------------
Student’s t distribution
PDF: Returns height of f (t)
CDF: Returns probability P (t # t0)
Inverse CDF: Returns t0 for P (t # t0) 5 α
---------------1-TDIST(t0, df, 1) only if t0 . 0
5TINV(α, df ) for two-tailed test
T.DIST(t, df, 0)
T.DIST(t0, df, 1)
T.INV(α, df )
F distribution
PDF: Returns height of f (x)
CDF: Returns probability P (X # x)
Inverse CDF: Returns F0 for P (F # F0) 5 α
---------------1-FDIST(x, df1, df2)
FINV(1 2 α, df1, df2)
F.DIST(x, df1, df2, 0)
F.DIST(x, df1, df2, 1)
F.INV(α, df1, df2)
Common Hypothesis Tests
Pre-2010 Excel
2013 Excel
Normal distribution*
Left-tailed p-value for test statistic zcalc
Right-tailed p-value for test statistic zcalc
Two-tailed p-value for test statistic zcalc
Critical z value for left-tailed test at α
Critical z value for right-tailed test at α
Critical z values for two-tailed test at α
NORMSDIST(zcalc)
1-NORMSDIST(zcalc)
2*(1-NORMSDIST(|zcalc|))
NORMSINV(α)
NORMSINV(1 2 α)
6NORMSINV(αy2)
NORM.S.DIST(zcalc, 1)
1-NORM.S.DIST(zcalc, 1)
2*(1-NORM.S.DIST(|zcalc|, 1))
NORM.S.INV(α)
NORM.S.INV(1 2 α)
6NORM.S.INV(αy2)
Student’s t distribution*
Left-tailed p-value for test statistic tcalc
Right-tailed p-value for test statistic tcalc
Two-tailed p-value for test statistic tcalc
Critical value of tα for left-tailed test at α
Critical value of tα for right-tailed test at α
Critical values of tα y2 for two-tailed test at α
TDIST(|tcalc|, df, 1)
TDIST(tcalc, df, 1)
TDIST(|tcalc|, df, 2)
2TINV(2α, df )
TINV(2α, df )
6TINV(α, df )
T.DIST(tcalc, df, 1)
T.DIST.RT(tcalc, df )
T.DIST.2T(|tcalc|, df )
T.INV(α, df )
T.INV(1 2 α, df )
6T.INV.2T(α, df )
F distribution
Left-tailed p-value for test statistic Fcalc , 1
Right-tailed p-value for test statistic Fcalc . 1
Two-tailed p-value for folded Fcalc test
Critical value for left-tailed test at α
Critical value for right-tailed test at α
Critical value for folded F test at α
1-FDIST( Fcalc , df1, df2)
FDIST(Fcalc, df1, df2)
2*FDIST(Fcalc, df1, df2)
1yFINV(α, df2, df1)
FINV(α, df1, df2)
FINV(αy2, df1, df2)
F.DIST(Fcalc, df1, df2, 1)
F.DIST.RT(Fcalc, df1, df2)
2*F.DIST.RT(Fcalc, df1, df2)
F.INV(α, df1, df2)
F.INV(1 2 α, df1, df2)
F.INV.RT(αy2, df1, df2)
Chi-square distribution
Left-tailed p-value for test statistic χ2calc
Right-tailed p-value for test statistic χ 2calc
Two-tailed p-value for test statistic χ 2calc
Critical value for left-tailed test at α
Critical value for right-tailed test at α
Critical value for two-tailed test at α
1-CHIDIST(χ 2calc, df )
CHIDIST(χ 2calc, df )
2*CHIDIST(χ 2calc, df )
CHIINV(1 2 α, df )
CHIINV(α, df )
CHIINV(αy2, df )
CHISQ.DIST(χ 2calc, df, 1)
CHISQ.DIST.RT(χ 2calc, df )
2*CHISQ.DIST.RT(χ 2calc, df )
CHISQ.INV(α, df )
CHISQ.INV.RT(α, df )
CHISQ.INV(1 2 αy2, df )
*For the normal and Student’s t distributions, the symbols |zcalc| and |tcalc| are used to denote absolute values for functions that require a positive argument. The 6 symbol is used in two-tailed
z and t tests to indicate that left- and right-tail critical values are the same except for sign.
Appendix J
815
Hypothesis Test Calculations
Pre-2010 Excel
2013 Excel
t-test for two means: returns a
two-tailed p-value for a test of zero
difference in two data arrays
TTEST(Data1, Data2, Tails, Type)
T.TEST(Data1, Data2, Tails, Type)
where Tails 5 1 or 2 and Type
1 5 paired (must have n1 5 n2)
2 5 equal variances assumed
3 5 unequal variances assumed
where Tails 5 1 or 2 and Type
1 5 paired (must have n1 5 n2)
2 5 equal variances assumed
3 5 unequal variances assumed
F-test of two variances: returns a
two-tailed p-value for equality of
variances in two arrays
FTEST(Data1, Data2)
F.TEST(Data1, Data2)
χ 2 goodness-of-fit test of k frequencies:
returns a two-tailed p-value assuming
k 2 1 degrees of freedom (assumes no
parameters estimated). No warning if
array frequencies do not have the same
sum (as they should).
CHITEST(Data1, Data2)
CHISQ.TEST(Data1, Data2)
where Data1 is an array of
k observed frequencies and
Data2 is an array of k expected
frequencies
where Data1 is an array of
k observed frequencies and
Data2 is an array of k expected
frequencies
Other Useful Stats Functions
Pre-2010 Excel
2013 Excel
Rank (average for ties)
----------------
RANK.AVG(x, Data, k) where x is a
cell reference in array Data, k 5 0
(descending), k 5 1 (ascending)
Rank (no correction for ties)
RANK(x, Data, k) where x is a cell
reference in array Data, k 5 0
(descending), k 5 1 (ascending)
RANK(x, Data, k) where x is a cell
reference in array Data, k 5 0
(descending), k 5 1 (ascending)
Random uniform (0 # x , 1)
RAND()
RAND()
Random integers (a # x # b)
RANDBETWEEN(a, b)
RANDBETWEEN(a, b)
Confidence
___ interval half-width
6z σy√(n) (margin of error) using normal
distribution with known standard deviation σ
and confidence 1 2 α
CONFIDENCE(α, σ, n)
CONFIDENCE.NORM(α, σ, n)
Confidence interval half-width 6t sy√n
(margin of error) using Student’s t distribution
with unknown standard deviation s and
confidence 1 2 α
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CONFIDENCE.T(α, s, n)
Sum of squares of an array of data values
around their mean
DEVSQ(Data)
DEVSQ(Data)
Frequency of items in a data array using bin
upper limits in bin array (highlight output
range and use Ctrl-Shift-Enter)
{FREQUENCY(Data, Bins)}
{FREQUENCY(Data, Bins)}
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