Free Essay

Sir Nicolas

In:

Submitted By nicolashabib
Words 12879
Pages 52
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A P P E N D I X B : TA B L E S

B.1 Binomial Probability Distribution

n51
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1

0.950
0.050

0.900
0.100

0.800
0.200

0.700
0.300

0.600
0.400

0.500
0.500

0.400
0.600

0.300
0.700

0.200
0.800

0.100
0.900

0.050
0.950

n52
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2

0.903
0.095
0.003

0.810
0.180
0.010

0.640
0.320
0.040

0.490
0.420
0.090

0.360
0.480
0.160

0.250
0.500
0.250

0.160
0.480
0.360

0.090
0.420
0.490

0.040
0.320
0.640

0.010
0.180
0.810

0.003
0.095
0.903

n53
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3

0.857
0.135
0.007
0.000

0.729
0.243
0.027
0.001

0.512
0.384
0.096
0.008

0.343
0.441
0.189
0.027

0.216
0.432
0.288
0.064

0.125
0.375
0.375
0.125

0.064
0.288
0.432
0.216

0.027
0.189
0.441
0.343

0.008
0.096
0.384
0.512

0.001
0.027
0.243
0.729

0.000
0.007
0.135
0.857

n54
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.815
0.171
0.014
0.000
0.000

0.656
0.292
0.049
0.004
0.000

0.410
0.410
0.154
0.026
0.002

0.240
0.412
0.265
0.076
0.008

0.130
0.346
0.346
0.154
0.026

0.063
0.250
0.375
0.250
0.063

0.026
0.154
0.346
0.346
0.130

0.008
0.076
0.265
0.412
0.240

0.002
0.026
0.154
0.410
0.410

0.000
0.004
0.049
0.292
0.656

0.000
0.000
0.014
0.171
0.815

n55
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4
5

0.774
0.204
0.021
0.001
0.000
0.000

0.590
0.328
0.073
0.008
0.000
0.000

0.328
0.410
0.205
0.051
0.006
0.000

0.168
0.360
0.309
0.132
0.028
0.002

0.078
0.259
0.346
0.230
0.077
0.010

0.031
0.156
0.313
0.313
0.156
0.031

0.010
0.077
0.230
0.346
0.259
0.078

0.002
0.028
0.132
0.309
0.360
0.168

0.000
0.006
0.051
0.205
0.410
0.328

0.000
0.000
0.008
0.073
0.328
0.590

0.000
0.000
0.001
0.021
0.204
0.774
(continued )

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B.1 Binomial Probability Distribution (continued)

n56
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.735
0.232
0.031
0.002
0.000

0.531
0.354
0.098
0.015
0.001

0.262
0.393
0.246
0.082
0.015

0.118
0.303
0.324
0.185
0.060

0.047
0.187
0.311
0.276
0.138

0.016
0.094
0.234
0.313
0.234

0.004
0.037
0.138
0.276
0.311

0.001
0.010
0.060
0.185
0.324

0.000
0.002
0.015
0.082
0.246

0.000
0.000
0.001
0.015
0.098

0.000
0.000
0.000
0.002
0.031

5
6

0.000
0.000

0.000
0.000

0.002
0.000

0.010
0.001

0.037
0.004

0.094
0.016

0.187
0.047

0.303
0.118

0.393
0.262

0.354
0.531

0.232
0.735

n57
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.698
0.257
0.041
0.004
0.000

0.478
0.372
0.124
0.023
0.003

0.210
0.367
0.275
0.115
0.029

0.082
0.247
0.318
0.227
0.097

0.028
0.131
0.261
0.290
0.194

0.008
0.055
0.164
0.273
0.273

0.002
0.017
0.077
0.194
0.290

0.000
0.004
0.025
0.097
0.227

0.000
0.000
0.004
0.029
0.115

0.000
0.000
0.000
0.003
0.023

0.000
0.000
0.000
0.000
0.004

5
6
7

0.000
0.000
0.000

0.000
0.000
0.000

0.004
0.000
0.000

0.025
0.004
0.000

0.077
0.017
0.002

0.164
0.055
0.008

0.261
0.131
0.028

0.318
0.247
0.082

0.275
0.367
0.210

0.124
0.372
0.478

0.041
0.257
0.698

n58
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.663
0.279
0.051
0.005
0.000

0.430
0.383
0.149
0.033
0.005

0.168
0.336
0.294
0.147
0.046

0.058
0.198
0.296
0.254
0.136

0.017
0.090
0.209
0.279
0.232

0.004
0.031
0.109
0.219
0.273

0.001
0.008
0.041
0.124
0.232

0.000
0.001
0.010
0.047
0.136

0.000
0.000
0.001
0.009
0.046

0.000
0.000
0.000
0.000
0.005

0.000
0.000
0.000
0.000
0.000

5
6
7
8

0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000

0.009
0.001
0.000
0.000

0.047
0.010
0.001
0.000

0.124
0.041
0.008
0.001

0.219
0.109
0.031
0.004

0.279
0.209
0.090
0.017

0.254
0.296
0.198
0.058

0.147
0.294
0.336
0.168

0.033
0.149
0.383
0.430

0.005
0.051
0.279
0.663
(continued )

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B.1 Binomial Probability Distribution (continued)

n59
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.630
0.299
0.063
0.008
0.001

0.387
0.387
0.172
0.045
0.007

0.134
0.302
0.302
0.176
0.066

0.040
0.156
0.267
0.267
0.172

0.010
0.060
0.161
0.251
0.251

0.002
0.018
0.070
0.164
0.246

0.000
0.004
0.021
0.074
0.167

0.000
0.000
0.004
0.021
0.074

0.000
0.000
0.000
0.003
0.017

0.000
0.000
0.000
0.000
0.001

0.000
0.000
0.000
0.000
0.000

5
6
7
8
9

0.000
0.000
0.000
0.000
0.000

0.001
0.000
0.000
0.000
0.000

0.017
0.003
0.000
0.000
0.000

0.074
0.021
0.004
0.000
0.000

0.167
0.074
0.021
0.004
0.000

0.246
0.164
0.070
0.018
0.002

0.251
0.251
0.161
0.060
0.010

0.172
0.267
0.267
0.156
0.040

0.066
0.176
0.302
0.302
0.134

0.007
0.045
0.172
0.387
0.387

0.001
0.008
0.063
0.299
0.630

n 5 10
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.599
0.315
0.075
0.010
0.001

0.349
0.387
0.194
0.057
0.011

0.107
0.268
0.302
0.201
0.088

0.028
0.121
0.233
0.267
0.200

0.006
0.040
0.121
0.215
0.251

0.001
0.010
0.044
0.117
0.205

0.000
0.002
0.011
0.042
0.111

0.000
0.000
0.001
0.009
0.037

0.000
0.000
0.000
0.001
0.006

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

5
6
7
8
9

0.000
0.000
0.000
0.000
0.000

0.001
0.000
0.000
0.000
0.000

0.026
0.006
0.001
0.000
0.000

0.103
0.037
0.009
0.001
0.000

0.201
0.111
0.042
0.011
0.002

0.246
0.205
0.117
0.044
0.010

0.201
0.251
0.215
0.121
0.040

0.103
0.200
0.267
0.233
0.121

0.026
0.088
0.201
0.302
0.268

0.001
0.011
0.057
0.194
0.387

0.000
0.001
0.010
0.075
0.315

10

0.000

0.000

0.000

0.000

0.000

0.001

0.006

0.028

0.107

0.349

0.599

n 5 11
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.569
0.329
0.087
0.014
0.001

0.314
0.384
0.213
0.071
0.016

0.086
0.236
0.295
0.221
0.111

0.020
0.093
0.200
0.257
0.220

0.004
0.027
0.089
0.177
0.236

0.000
0.005
0.027
0.081
0.161

0.000
0.001
0.005
0.023
0.070

0.000
0.000
0.001
0.004
0.017

0.000
0.000
0.000
0.000
0.002

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

5
6
7
8
9

0.000
0.000
0.000
0.000
0.000

0.002
0.000
0.000
0.000
0.000

0.039
0.010
0.002
0.000
0.000

0.132
0.057
0.017
0.004
0.001

0.221
0.147
0.070
0.023
0.005

0.226
0.226
0.161
0.081
0.027

0.147
0.221
0.236
0.177
0.089

0.057
0.132
0.220
0.257
0.200

0.010
0.039
0.111
0.221
0.295

0.000
0.002
0.016
0.071
0.213

0.000
0.000
0.001
0.014
0.087

10
11

0.000
0.000

0.000
0.000

0.000
0.000

0.000
0.000

0.001
0.000

0.005
0.000

0.027
0.004

0.093
0.020

0.236
0.086

0.384
0.314

0.329
0.569
(continued )

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B.1 Binomial Probability Distribution (continued)

n 5 12
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.540
0.341
0.099
0.017
0.002

0.282
0.377
0.230
0.085
0.021

0.069
0.206
0.283
0.236
0.133

0.014
0.071
0.168
0.240
0.231

0.002
0.017
0.064
0.142
0.213

0.000
0.003
0.016
0.054
0.121

0.000
0.000
0.002
0.012
0.042

0.000
0.000
0.000
0.001
0.008

0.000
0.000
0.000
0.000
0.001

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

5
6
7
8
9

0.000
0.000
0.000
0.000
0.000

0.004
0.000
0.000
0.000
0.000

0.053
0.016
0.003
0.001
0.000

0.158
0.079
0.029
0.008
0.001

0.227
0.177
0.101
0.042
0.012

0.193
0.226
0.193
0.121
0.054

0.101
0.177
0.227
0.213
0.142

0.029
0.079
0.158
0.231
0.240

0.003
0.016
0.053
0.133
0.236

0.000
0.000
0.004
0.021
0.085

0.000
0.000
0.000
0.002
0.017

10
11
12

0.000
0.000
0.000

0.000
0.000
0.000

0.000
0.000
0.000

0.000
0.000
0.000

0.002
0.000
0.000

0.016
0.003
0.000

0.064
0.017
0.002

0.168
0.071
0.014

0.283
0.206
0.069

0.230
0.377
0.282

0.099
0.341
0.540

n 5 13
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.513
0.351
0.111
0.021
0.003

0.254
0.367
0.245
0.100
0.028

0.055
0.179
0.268
0.246
0.154

0.010
0.054
0.139
0.218
0.234

0.001
0.011
0.045
0.111
0.184

0.000
0.002
0.010
0.035
0.087

0.000
0.000
0.001
0.006
0.024

0.000
0.000
0.000
0.001
0.003

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

5
6
7
8
9

0.000
0.000
0.000
0.000
0.000

0.006
0.001
0.000
0.000
0.000

0.069
0.023
0.006
0.001
0.000

0.180
0.103
0.044
0.014
0.003

0.221
0.197
0.131
0.066
0.024

0.157
0.209
0.209
0.157
0.087

0.066
0.131
0.197
0.221
0.184

0.014
0.044
0.103
0.180
0.234

0.001
0.006
0.023
0.069
0.154

0.000
0.000
0.001
0.006
0.028

0.000
0.000
0.000
0.000
0.003

10
11
12
13

0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000

0.001
0.000
0.000
0.000

0.006
0.001
0.000
0.000

0.035
0.010
0.002
0.000

0.111
0.045
0.011
0.001

0.218
0.139
0.054
0.010

0.246
0.268
0.179
0.055

0.100
0.245
0.367
0.254

0.021
0.111
0.351
0.513
(continued )

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B.1 Binomial Probability Distribution (concluded)

n 5 14
Probability
x

0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0
1
2
3
4

0.488
0.359
0.123
0.026
0.004

0.229
0.356
0.257
0.114
0.035

0.044
0.154
0.250
0.250
0.172

0.007
0.041
0.113
0.194
0.229

0.001
0.007
0.032
0.085
0.155

0.000
0.001
0.006
0.022
0.061

0.000
0.000
0.001
0.003
0.014

0.000
0.000
0.000
0.000
0.001

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

5
6
7
8
9

0.000
0.000
0.000
0.000
0.000

0.008
0.001
0.000
0.000
0.000

0.086
0.032
0.009
0.002
0.000

0.196
0.126
0.062
0.023
0.007

0.207
0.207
0.157
0.092
0.041

0.122
0.183
0.209
0.183
0.122

0.041
0.092
0.157
0.207
0.207

0.007
0.023
0.062
0.126
0.196

0.000
0.002
0.009
0.032
0.086

0.000
0.000
0.000
0.001
0.008

0.000
0.000
0.000
0.000
0.000

10
11
12
13
14

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

0.001
0.000
0.000
0.000
0.000

0.014
0.003
0.001
0.000
0.000

0.061
0.022
0.006
0.001
0.000

0.155
0.085
0.032
0.007
0.001

0.229
0.194
0.113
0.041
0.007

0.172
0.250
0.250
0.154
0.044

0.035
0.114
0.257
0.356
0.229

0.004
0.026
0.123
0.359
0.488

n 5 15
Probability
x

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0.463
0.366
0.135
0.031
0.005

0.206
0.343
0.267
0.129
0.043

0.035
0.132
0.231
0.250
0.188

0.005
0.031
0.092
0.170
0.219

0.000
0.005
0.022
0.063
0.127

0.000
0.000
0.003
0.014
0.042

0.000
0.000
0.000
0.002
0.007

0.000
0.000
0.000
0.000
0.001

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

5
6
7
8
9

0.001
0.000
0.000
0.000
0.000

0.010
0.002
0.000
0.000
0.000

0.103
0.043
0.014
0.003
0.001

0.206
0.147
0.081
0.035
0.012

0.186
0.207
0.177
0.118
0.061

0.092
0.153
0.196
0.196
0.153

0.024
0.061
0.118
0.177
0.207

0.003
0.012
0.035
0.081
0.147

0.000
0.001
0.003
0.014
0.043

0.000
0.000
0.000
0.000
0.002

0.000
0.000
0.000
0.000
0.000

10
11
12
13
14

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

0.000
0.000
0.000
0.000
0.000

0.003
0.001
0.000
0.000
0.000

0.024
0.007
0.002
0.000
0.000

0.092
0.042
0.014
0.003
0.000

0.186
0.127
0.063
0.022
0.005

0.206
0.219
0.170
0.092
0.031

0.103
0.188
0.250
0.231
0.132

0.010
0.043
0.129
0.267
0.343

0.001
0.005
0.031
0.135
0.366

15

730

0.05

0
1
2
3
4

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.005

0.035

0.206

0.463

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B.2 Poisson Distribution

m x 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0
1
2
3
4

0.9048
0.0905
0.0045
0.0002
0.0000

0.8187
0.1637
0.0164
0.0011
0.0001

0.7408
0.2222
0.0333
0.0033
0.0003

0.6703
0.2681
0.0536
0.0072
0.0007

0.6065
0.3033
0.0758
0.0126
0.0016

0.5488
0.3293
0.0988
0.0198
0.0030

0.4966
0.3476
0.1217
0.0284
0.0050

0.4493
0.3595
0.1438
0.0383
0.0077

0.4066
0.3659
0.1647
0.0494
0.0111

5
6
7

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0001
0.0000
0.0000

0.0002
0.0000
0.0000

0.0004
0.0000
0.0000

0.0007
0.0001
0.0000

0.0012
0.0002
0.0000

0.0020
0.0003
0.0000

x

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

0
1
2
3
4

0.3679
0.3679
0.1839
0.0613
0.0153

0.1353
0.2707
0.2707
0.1804
0.0902

0.0498
0.1494
0.2240
0.2240
0.1680

0.0183
0.0733
0.1465
0.1954
0.1954

0.0067
0.0337
0.0842
0.1404
0.1755

0.0025
0.0149
0.0446
0.0892
0.1339

0.0009
0.0064
0.0223
0.0521
0.0912

0.0003
0.0027
0.0107
0.0286
0.0573

0.0001
0.0011
0.0050
0.0150
0.0337

5
6
7
8
9

0.0031
0.0005
0.0001
0.0000
0.0000

0.0361
0.0120
0.0034
0.0009
0.0002

0.1008
0.0504
0.0216
0.0081
0.0027

0.1563
0.1042
0.0595
0.0298
0.0132

0.1755
0.1462
0.1044
0.0653
0.0363

0.1606
0.1606
0.1377
0.1033
0.0688

0.1277
0.1490
0.1490
0.1304
0.1014

0.0916
0.1221
0.1396
0.1396
0.1241

0.0607
0.0911
0.1171
0.1318
0.1318

10
11
12
13
14

0.0000
0.0000
0.0000
0.0000
0.0000

0.0000
0.0000
0.0000
0.0000
0.0000

0.0008
0.0002
0.0001
0.0000
0.0000

0.0053
0.0019
0.0006
0.0002
0.0001

0.0181
0.0082
0.0034
0.0013
0.0005

0.0413
0.0225
0.0113
0.0052
0.0022

0.0710
0.0452
0.0263
0.0142
0.0071

0.0993
0.0722
0.0481
0.0296
0.0169

0.1186
0.0970
0.0728
0.0504
0.0324

15
16
17
18
19

0.0000
0.0000
0.0000
0.0000
0.0000

0.0000
0.0000
0.0000
0.0000
0.0000

0.0000
0.0000
0.0000
0.0000
0.0000

0.0000
0.0000
0.0000
0.0000
0.0000

0.0002
0.0000
0.0000
0.0000
0.0000

0.0009
0.0003
0.0001
0.0000
0.0000

0.0033
0.0014
0.0006
0.0002
0.0001

0.0090
0.0045
0.0021
0.0009
0.0004

0.0194
0.0109
0.0058
0.0029
0.0014

20
21
22

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0000
0.0000
0.0000

0.0002
0.0001
0.0000

0.0006
0.0003
0.0001

m

731

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B.3 Areas under the Normal Curve

Example:
If z = 1.96, then
P(0 to z) = 0.4750.

0.4750

z

0

1.96

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

0.0000
0.0398
0.0793
0.1179
0.1554

0.0040
0.0438
0.0832
0.1217
0.1591

0.0080
0.0478
0.0871
0.1255
0.1628

0.0120
0.0517
0.0910
0.1293
0.1664

0.0160
0.0557
0.0948
0.1331
0.1700

0.0199
0.0596
0.0987
0.1368
0.1736

0.0239
0.0636
0.1026
0.1406
0.1772

0.0279
0.0675
0.1064
0.1443
0.1808

0.0319
0.0714
0.1103
0.1480
0.1844

0.0359
0.0753
0.1141
0.1517
0.1879

0.5
0.6
0.7
0.8
0.9

0.1915
0.2257
0.2580
0.2881
0.3159

0.1950
0.2291
0.2611
0.2910
0.3186

0.1985
0.2324
0.2642
0.2939
0.3212

0.2019
0.2357
0.2673
0.2967
0.3238

0.2054
0.2389
0.2704
0.2995
0.3264

0.2088
0.2422
0.2734
0.3023
0.3289

0.2123
0.2454
0.2764
0.3051
0.3315

0.2157
0.2486
0.2794
0.3078
0.3340

0.2190
0.2517
0.2823
0.3106
0.3365

0.2224
0.2549
0.2852
0.3133
0.3389

1.0
1.1
1.2
1.3
1.4

0.3413
0.3643
0.3849
0.4032
0.4192

0.3438
0.3665
0.3869
0.4049
0.4207

0.3461
0.3686
0.3888
0.4066
0.4222

0.3485
0.3708
0.3907
0.4082
0.4236

0.3508
0.3729
0.3925
0.4099
0.4251

0.3531
0.3749
0.3944
0.4115
0.4265

0.3554
0.3770
0.3962
0.4131
0.4279

0.3577
0.3790
0.3980
0.4147
0.4292

0.3599
0.3810
0.3997
0.4162
0.4306

0.3621
0.3830
0.4015
0.4177
0.4319

1.5
1.6
1.7
1.8
1.9

0.4332
0.4452
0.4554
0.4641
0.4713

0.4345
0.4463
0.4564
0.4649
0.4719

0.4357
0.4474
0.4573
0.4656
0.4726

0.4370
0.4484
0.4582
0.4664
0.4732

0.4382
0.4495
0.4591
0.4671
0.4738

0.4394
0.4505
0.4599
0.4678
0.4744

0.4406
0.4515
0.4608
0.4686
0.4750

0.4418
0.4525
0.4616
0.4693
0.4756

0.4429
0.4535
0.4625
0.4699
0.4761

0.4441
0.4545
0.4633
0.4706
0.4767

2.0
2.1
2.2
2.3
2.4

0.4772
0.4821
0.4861
0.4893
0.4918

0.4778
0.4826
0.4864
0.4896
0.4920

0.4783
0.4830
0.4868
0.4898
0.4922

0.4788
0.4834
0.4871
0.4901
0.4925

0.4793
0.4838
0.4875
0.4904
0.4927

0.4798
0.4842
0.4878
0.4906
0.4929

0.4803
0.4846
0.4881
0.4909
0.4931

0.4808
0.4850
0.4884
0.4911
0.4932

0.4812
0.4854
0.4887
0.4913
0.4934

0.4817
0.4857
0.4890
0.4916
0.4936

2.5
2.6
2.7
2.8
2.9

0.4938
0.4953
0.4965
0.4974
0.4981

0.4940
0.4955
0.4966
0.4975
0.4982

0.4941
0.4956
0.4967
0.4976
0.4982

0.4943
0.4957
0.4968
0.4977
0.4983

0.4945
0.4959
0.4969
0.4977
0.4984

0.4946
0.4960
0.4970
0.4978
0.4984

0.4948
0.4961
0.4971
0.4979
0.4985

0.4949
0.4962
0.4972
0.4979
0.4985

0.4951
0.4963
0.4973
0.4980
0.4986

0.4952
0.4964
0.4974
0.4981
0.4986

3.0

0.4987

0.4987

0.4987

0.4988

0.4988

0.4989

0.4989

0.4989

0.4990

0.4990

732

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Table of Random Numbers

02711
94873
54921
77640
61545

08182
90935
78680
97636
00835

75997
31684
06635
37397
93251

79866
63952
98689
93379
87203

58095
09865
17306
56454
36759

83319
14491
25170
59818
49197

80295
99518
65928
45827
85967

79741
93394
87709
74164
01704

74599
34691
30533
71666
19634

84379
14985
89736
46977
21898

17147
13748
87455
08999
78804

19519
04742
14813
57409
81333

22497
92460
50373
91185
53809

16857
85801
28037
10200
32471

42426
53444
91182
61411
46034

84822
65626
32786
23392
36306

92598
58710
65261
47797
22498

49186
55406
11173
56377
19239

88247
17173
34376
71635
85428

39967
69776
36408
08601
55721

82173
97594
73881
46071
44492

26921
25168
67176
22912
19686

28472
89178
93504
90326
12495

98958
68190
42636
42453
93135

07960
05043
38233
88108
95185

66124
17407
16154
72064
77799

89731
48201
96451
58601
52441

95069
83917
57925
32357
88272

18625
11413
29667
90610
22024

92405
72920
30859
32921
80631

31864
51574
35350
46943
22665

72170
90692
76132
36502
87226

37722
43339
92925
01172
33304

55794
65689
92124
46045
57975

14636
76539
92634
46991
03985

05148
27909
35681
33804
21566

54505
05467
43690
80006
65796

50113
21727
89136
35542
72915

21119
51141
35599
61056
81466

25228
72949
84138
75666
89205

39437
77082
24544
27503
24590

97957
47784
25649
15558
24545

11838
40098
43370
37738
06435

10433
97962
28007
24849
52758

21564
89845
06779
70722
45685

51570
28392
72402
71859
90151

73558
78187
62632
83736
46516

27495
06112
53956
06016
49644

34533
08169
24709
94397
92686

57808
11261
06978
12529
84870

48155
20226
70178
10169
81084

86226
53752
73444
41465
03701

40359
90648
38790
51935
28598

28723
24362
53626
05711
70013

15364
83314
93780
09799
63794

69125
00014
18629
79077
53169

12609
19207
68766
88159
97054

57171
69413
24371
33437
60303

86857
97016
74639
68519
23259

31702
86290
30782
03040
96196

69202
80561
08199
98883
84568

20777
95787
26703
28220
69286

21727
89426
95128
39358
76054

81511
93325
48599
53720
21615

51887
86412
09333
80161
80883

16175
57479
12584
83371
36797

53746
54194
24374
15181
82845

46516
52153
31232
11131
39139

70339
19197
61782
12219
90900

62727
81877
44032
55920
18172

04269
10538
77843
12034
10524

35173
13124
24955
94636
72829

95745
36099
25900
49455
47641

53893
13140
63843
76362
93315

86022
37706
95029
83532
80875

77722
44562
93859
31062
28090

52498
57179
93634
69903
97728

84193
44693
20205
91186
52560

22448
67877
66294
65768
34937

22571
01549
41218
55949
79548

68935
89450
90696
49006
74591

76632
65665
20451
32047
87025

46984
29190
24211
93086
52368

61772
43709
97310
00112
59416

92786
11172
60446
20470
34417

22651
34481
73530
17136
70557

07086
95977
62865
28255
86746

89754
47535
96574
86328
55809

44143
25658
13829
07293
53628

97687
73898
72226
38809
12000

06315
62386
86848
65643
11084

17012
09184
82122
69480
46534

77103
62092
04028
46598
78957

00968
46617
36959
04501
77353

07235
99419
87827
40403
39578

10728
64230
12813
91408
77868

42189
95034
08627
32343
22970

33292
85481
80699
48130
84349

51487
07857
13345
49303
09184

64443
42510
51695
90689
70603

733

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Student’s t Distribution



–t t Confidence interval



–t 0
Left-tailed test

t

90%

95%

98%

0.10

0.05

0.025

0.01

1
2



–t 0 t Two-tailed test
Confidence Intervals, c

99%

99.9%

80%

Level of Significance for One-Tailed Test, a df ␣

0 t
Right-tailed test

Confidence Intervals, c
80%

1
2

0.005

90%

95%

98%

99%

99.9%

Level of Significance for One-Tailed Test, a
0.0005

df

0.10

Level of Significance for Two-Tailed Test, a

0.05

0.025

0.01

0.005

0.0005

Level of Significance for Two-Tailed Test, a

0.20

0.10

0.05

0.02

0.01

0.001

0.20

0.10

0.05

0.02

0.01

0.001

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

636.619
31.599
12.924
8.610
6.869

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

3.582
3.574
3.566
3.558
3.551

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

5.959
5.408
5.041
4.781
4.587

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

3.544
3.538
3.532
3.526
3.520

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

4.437
4.318
4.221
4.140
4.073

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

3.515
3.510
3.505
3.500
3.496

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

4.015
3.965
3.922
3.883
3.850

51
52
53
54
55

1.298
1.298
1.298
1.297
1.297

1.675
1.675
1.674
1.674
1.673

2.008
2.007
2.006
2.005
2.004

2.402
2.400
2.399
2.397
2.396

2.676
2.674
2.672
2.670
2.668

3.492
3.488
3.484
3.480
3.476

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

3.819
3.792
3.768
3.745
3.725

56
57
58
59
60

1.297
1.297
1.296
1.296
1.296

1.673
1.672
1.672
1.671
1.671

2.003
2.002
2.002
2.001
2.000

2.395
2.394
2.392
2.391
2.390

2.667
2.665
2.663
2.662
2.660

3.473
3.470
3.466
3.463
3.460

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

3.707
3.690
3.674
3.659
3.646

61
62
63
64
65

1.296
1.295
1.295
1.295
1.295

1.670
1.670
1.669
1.669
1.669

2.000
1.999
1.998
1.998
1.997

2.389
2.388
2.387
2.386
2.385

2.659
2.657
2.656
2.655
2.654

3.457
3.454
3.452
3.449
3.447

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

3.633
3.622
3.611
3.601
3.591

66
67
68
69
70

1.295
1.294
1.294
1.294
1.294

1.668
1.668
1.668
1.667
1.667

1.997
1.996
1.995
1.995
1.994

2.384
2.383
2.382
2.382
2.381

2.652
2.651
2.650
2.649
2.648

3.444
3.442
3.439
3.437
3.435
(continued )

734

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Student’s t Distribution (concluded)

Confidence Intervals, c
80%

90%

95%

98%

Confidence Intervals, c
99%

99.9%

80%

Level of Significance for One-Tailed Test, a df 0.10

0.05

0.025

0.01

0.005

90%

95%

98%

99%

99.9%

Level of Significance for One-Tailed Test, a
0.0005

df

0.10

Level of Significance for Two-Tailed Test, a

0.05

0.025

0.01

0.005

0.0005

Level of Significance for Two-Tailed Test, a

0.20

0.10

0.05

0.02

0.01

0.001

71
72
73
74
75

1.294
1.293
1.293
1.293
1.293

1.667
1.666
1.666
1.666
1.665

1.994
1.993
1.993
1.993
1.992

2.380
2.379
2.379
2.378
2.377

2.647
2.646
2.645
2.644
2.643

3.433
3.431
3.429
3.427
3.425

76
77
78
79
80

1.293
1.293
1.292
1.292
1.292

1.665
1.665
1.665
1.664
1.664

1.992
1.991
1.991
1.990
1.990

2.376
2.376
2.375
2.374
2.374

2.642
2.641
2.640
2.640
2.639

3.423
3.421
3.420
3.418
3.416

81
82
83
84
85

1.292
1.292
1.292
1.292
1.292

1.664
1.664
1.663
1.663
1.663

1.990
1.989
1.989
1.989
1.988

2.373
2.373
2.372
2.372
2.371

2.638
2.637
2.636
2.636
2.635

3.415
3.413
3.412
3.410
3.409

86
87
88

1.291
1.291
1.291

1.663
1.663
1.662

1.988
1.988
1.987

2.370
2.370
2.369

2.634
2.634
2.633

3.407
3.406
3.405

0.20

0.10

0.05

0.02

0.01

0.001

89
90

1.291
1.291

1.662
1.662

1.987
1.987

2.369
2.368

2.632
2.632

3.403
3.402

91
92
93
94
95

1.291
1.291
1.291
1.291
1.291

1.662
1.662
1.661
1.661
1.661

1.986
1.986
1.986
1.986
1.985

2.368
2.368
2.367
2.367
2.366

2.631
2.630
2.630
2.629
2.629

3.401
3.399
3.398
3.397
3.396

96
97
98
99
100

1.290
1.290
1.290
1.290
1.290

1.661
1.661
1.661
1.660
1.660

1.985
1.985
1.984
1.984
1.984

2.366
2.365
2.365
2.365
2.364

2.628
2.627
2.627
2.626
2.626

3.395
3.394
3.393
3.392
3.390

120
140
160
180
200
`

1.289
1.288
1.287
1.286
1.286
1.282

1.658
1.656
1.654
1.653
1.653
1.645

1.980
1.977
1.975
1.973
1.972
1.960

2.358
2.353
2.350
2.347
2.345
2.326

2.617
2.611
2.607
2.603
2.601
2.576

3.373
3.361
3.352
3.345
3.340
3.291

735

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Critical Values of the F Distribution (a 5 .05)

.05

0

F

Degrees of Freedom for the Numerator
3

4

5

6

7

8

9

10

12

15

20

24

30

40

161
18.5
10.1
7.71
6.61

200
19.0
9.55
6.94
5.79

216
19.2
9.28
6.59
5.41

225
19.2
9.12
6.39
5.19

230
19.3
9.01
6.26
5.05

234
19.3
8.94
6.16
4.95

237
19.4
8.89
6.09
4.88

239
19.4
8.85
6.04
4.82

241
19.4
8.81
6.00
4.77

242
19.4
8.79
5.96
4.74

244
19.4
8.74
5.91
4.68

246
19.4
8.70
5.86
4.62

248
19.4
8.66
5.80
4.56

249
19.5
8.64
5.77
4.53

250
19.5
8.62
5.75
4.50

251
19.5
8.59
5.72
4.46

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

3.94
3.51
3.22
3.01
2.85

3.87
3.44
3.15
2.94
2.77

3.84
3.41
3.12
2.90
2.74

3.81
3.38
3.08
2.86
2.70

3.77
3.34
3.04
2.83
2.66

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

2.72
2.62
2.53
2.46
2.40

2.65
2.54
2.46
2.39
2.33

2.61
2.51
2.42
2.35
2.29

2.57
2.47
2.38
2.31
2.25

2.53
2.43
2.34
2.27
2.20

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

2.35
2.31
2.27
2.23
2.20

2.28
2.23
2.19
2.16
2.12

2.24
2.19
2.15
2.11
2.08

2.19
2.15
2.11
2.07
2.04

2.15
2.10
2.06
2.03
1.99

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

2.18
2.15
2.13
2.11
2.09

2.10
2.07
2.05
2.03
2.01

2.05
2.03
2.01
1.98
1.96

2.01
1.98
1.96
1.94
1.92

1.96
1.94
1.91
1.89
1.87

30
40
60
120
`

736

2

1
2
3
4
5

Degrees of Freedom for the Denominator

1

4.17
4.08
4.00
3.92
3.84

3.32
3.23
3.15
3.07
3.00

2.92
2.84
2.76
2.68
2.60

2.69
2.61
2.53
2.45
2.37

2.53
2.45
2.37
2.29
2.21

2.42
2.34
2.25
2.18
2.10

2.33
2.25
2.17
2.09
2.01

2.27
2.18
2.10
2.02
1.94

2.21
2.12
2.04
1.96
1.88

2.16
2.08
1.99
1.91
1.83

2.09
2.00
1.92
1.83
1.75

2.01
1.92
1.84
1.75
1.67

1.93
1.84
1.75
1.66
1.57

1.89
1.79
1.70
1.61
1.52

1.84
1.74
1.65
1.55
1.46

1.79
1.69
1.59
1.50
1.39

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Critical Values of the F Distribution (a 5 .01)

.01

0

F
Degrees of Freedom for the Numerator

2

3

4

5

6

7

8

9

10

12

15

20

24

30

40

1
2
3
4
5

Degrees of Freedom for the Denominator

1
4052
98.5
34.1
21.2
16.3

5000
99.0
30.8
18.0
13.3

5403
99.2
29.5
16.7
12.1

5625
99.2
28.7
16.0
11.4

5764
99.3
28.2
15.5
11.0

5859
99.3
27.9
15.2
10.7

5928
99.4
27.7
15.0
10.5

5981
99.4
27.5
14.8
10.3

6022
99.4
27.3
14.7
10.2

6056
99.4
27.2
14.5
10.1

6106
99.4
27.1
14.4
9.89

6157
99.4
26.9
14.2
9.72

6209
99.4
26.7
14.0
9.55

6235
99.5
26.6
13.9
9.47

6261
99.5
26.5
13.8
9.38

6287
99.5
26.4
13.7
9.29

6
7
8
9
10

13.7
12.2
11.3
10.6
10.0

10.9
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

7.56
6.31
5.52
4.96
4.56

7.40
6.16
5.36
4.81
4.41

7.31
6.07
5.28
4.73
4.33

7.23
5.99
5.20
4.65
4.25

7.14
5.91
5.12
4.57
4.17

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

4.25
4.01
3.82
3.66
3.52

4.10
3.86
3.66
3.51
3.37

4.02
3.78
3.59
3.43
3.29

3.94
3.70
3.51
3.35
3.21

3.86
3.62
3.43
3.27
3.13

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
5.18
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

3.41
3.31
3.23
3.15
3.09

3.26
3.16
3.08
3.00
2.94

3.18
3.08
3.00
2.92
2.86

3.10
3.00
2.92
2.84
2.78

3.02
2.92
2.84
2.76
2.69

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

3.03
2.98
2.93
2.89
2.85

2.88
2.83
2.78
2.74
2.70

2.80
2.75
2.70
2.66
2.62

2.72
2.67
2.62
2.58
2.54

2.64
2.58
2.54
2.49
2.45

30
40
60
120
`

7.56
7.31
7.08
6.85
6.63

5.39
5.18
4.98
4.79
4.61

4.51
4.31
4.13
3.95
3.78

4.02
3.83
3.65
3.48
3.32

3.70
3.51
3.34
3.17
3.02

3.47
3.29
3.12
2.96
2.80

3.30
3.12
2.95
2.79
2.64

3.17
2.99
2.82
2.66
2.51

3.07
2.89
2.72
2.56
2.41

2.98
2.80
2.63
2.47
2.32

2.84
2.66
2.50
2.34
2.18

2.70
2.52
2.35
2.19
2.04

2.55
2.37
2.20
2.03
1.88

2.47
2.29
2.12
1.95
1.79

2.39
2.20
2.03
1.86
1.70

2.30
2.11
1.94
1.76
1.59

737

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B.7 Critical Values of Chi-Square

This table contains the values of x2 that correspond to a specific right-tail area and specific number of degrees of freedom.

Example: With 17 df and a .02 area in the upper tail, ␹2 ϭ 30.995
␹2

0
Degrees of
Freedom,
df

0.05

0.02

0.01

2.706
4.605
6.251
7.779
9.236

3.841
5.991
7.815
9.488
11.070

5.412
7.824
9.837
11.668
13.388

6.635
9.210
11.345
13.277
15.086

6
7
8
9
10

10.645
12.017
13.362
14.684
15.987

12.592
14.067
15.507
16.919
18.307

15.033
16.622
18.168
19.679
21.161

16.812
18.475
20.090
21.666
23.209

11
12
13
14
15

17.275
18.549
19.812
21.064
22.307

19.675
21.026
22.362
23.685
24.996

22.618
24.054
25.472
26.873
28.259

24.725
26.217
27.688
29.141
30.578

16
17
18
19
20

23.542
24.769
25.989
27.204
28.412

26.296
27.587
28.869
30.144
31.410

29.633
30.995
32.346
33.687
35.020

32.000
33.409
34.805
36.191
37.566

21
22
23
24
25

29.615
30.813
32.007
33.196
34.382

32.671
33.924
35.172
36.415
37.652

36.343
37.659
38.968
40.270
41.566

38.932
40.289
41.638
42.980
44.314

26
27
28
29
30

738

0.10

1
2
3
4
5

35.563
36.741
37.916
39.087
40.256

38.885
40.113
41.337
42.557
43.773

42.856
44.140
45.419
46.693
47.962

45.642
46.963
48.278
49.588
50.892

Right-Tail Area

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B.8 Wilcoxon T Values

2a
.15

.10

.05

.04

.03

.02

.01

a n .075

.050

.025

.020

.015

.010

.005

4
5
6
7
8

0
1
2
4
7

0
2
3
5

0
2
3

0
1
3

0
2

0
1

0

9
10
11
12
13

9
12
16
19
24

8
10
13
17
21

5
8
10
13
17

5
7
9
12
16

4
6
8
11
14

3
5
7
9
12

1
3
5
7
9

14
15
16
17
18

28
33
39
45
51

25
30
35
41
47

21
25
29
34
40

19
23
28
33
38

18
21
26
30
35

15
19
23
27
32

12
15
19
23
27

19
20
21
22
23

58
65
73
81
89

53
60
67
75
83

46
52
58
65
73

43
50
56
63
70

41
47
53
59
66

37
43
49
55
62

32
37
42
48
54

24
25
26
27
28

98
108
118
128
138

91
100
110
119
130

81
89
98
107
116

78
86
94
103
112

74
82
90
99
108

69
76
84
92
101

61
68
75
83
91

29
30
31
32
33

150
161
173
186
199

140
151
163
175
187

126
137
147
159
170

122
132
143
154
165

117
127
137
148
159

110
120
130
140
151

100
109
118
128
138

34
35
40
50
60

212
226
302
487
718

200
213
286
466
690

182
195
264
434
648

177
189
257
425
636

171
182
249
413
620

162
173
238
397
600

148
159
220
373
567

70
80
90
100

995
1,318
1,688
2,105

960
1,276
1,638
2,045

907
1,211
1,560
1,955

891
1,192
1,537
1,928

872
1,168
1,509
1,894

846
1,136
1,471
1,850

805
1,086
1,410
1,779

739

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Critical Values for the Durbin–Watson d Statistic (a 5 .05)

k51

k52

k53

k54

k55

n

dL,.05

dU,.05

dL,.05

dU,.05

dL,.05

dU,.05

dL,.05

dU,.05

dL,.05

dU,.05

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
40
45
50
55
60
65
70
75
80
85
90
95
100

1.08
1.10
1.13
1.16
1.18
1.20
1.22
1.24
1.26
1.27
1.29
1.30
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.43
1.44
1.48
1.50
1.53
1.55
1.57
1.58
1.60
1.61
1.62
1.63
1.64
1.65

1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.44
1.45
1.45
1.46
1.47
1.48
1.48
1.49
1.50
1.50
1.51
1.51
1.52
1.52
1.53
1.54
1.54
1.54
1.57
1.59
1.60
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.69

0.95
0.98
1.02
1.05
1.08
1.10
1.13
1.15
1.17
1.19
1.21
1.22
1.24
1.26
1.27
1.28
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.43
1.46
1.49
1.51
1.54
1.55
1.57
1.59
1.60
1.61
1.62
1.63

1.54
1.54
1.54
1.53
1.53
1.54
1.54
1.54
1.54
1.55
1.55
1.55
1.56
1.56
1.56
1.57
1.57
1.57
1.58
1.58
1.58
1.59
1.59
1.59
1.60
1.60
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.70
1.70
1.71
1.72

0.82
0.86
0.90
0.93
0.97
1.00
1.03
1.05
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.21
1.23
1.24
1.26
1.27
1.28
1.29
1.31
1.32
1.33
1.34
1.38
1.42
1.45
1.48
1.50
1.52
1.54
1.56
1.57
1.59
1.60
1.61

1.75
1.73
1.71
1.69
1.68
1.68
1.67
1.66
1.66
1.66
1.66
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.66
1.66
1.66
1.66
1.67
1.67
1.68
1.69
1.70
1.70
1.71
1.72
1.72
1.73
1.73
1.74

0.69
0.74
0.78
0.82
0.86
0.90
0.93
0.96
0.99
1.01
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.19
1.21
1.22
1.24
1.25
1.26
1.27
1.29
1.34
1.38
1.41
1.44
1.47
1.49
1.51
1.53
1.55
1.57
1.58
1.59

1.97
1.93
1.90
1.87
1.85
1.83
1.81
1.80
1.79
1.78
1.77
1.76
1.76
1.75
1.74
1.74
1.74
1.73
1.73
1.73
1.73
1.73
1.72
1.72
1.72
1.72
1.72
1.72
1.72
1.73
1.73
1.74
1.74
1.74
1.75
1.75
1.75
1.76

0.56
0.62
0.67
0.71
0.75
0.79
0.83
0.86
0.90
0.93
0.95
0.98
1.01
1.03
1.05
1.07
1.09
1.11
1.13
1.15
1.16
1.18
1.19
1.21
1.22
1.23
1.29
1.34
1.38
1.41
1.44
1.46
1.49
1.51
1.52
1.54
1.56
1.57

2.21
2.15
2.10
2.06
2.02
1.99
1.96
1.94
1.92
1.90
1.89
1.88
1.86
1.85
1.84
1.83
1.83
1.82
1.81
1.81
1.80
1.80
1.80
1.79
1.79
1.79
1.78
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.78
1.78
1.78

SOURCE:

J. Durbin and G. S. Watson, “Testing for Serial Correlation in Least Squares Regression, II,” Biometrika 30 (1951), pp. 159–178. Reproduced by permission of the
Biometrika Trustees.

740

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Critical Values for the Durbin–Watson d Statistic (a 5 .025)

k51

k52

k53

k54

k55

n

dL,.025

dU,.025

dL,.025

dU,.025

dL,.025

dU,.025

dL,.025

dU,.025

dL,.025

dU,.025

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
40
45
50
55
60
65
70
75
80
85
90
95
100

0.95
0.98
1.01
1.03
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.19
1.21
1.22
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.33
1.34
1.35
1.39
1.42
1.45
1.47
1.49
1.51
1.53
1.54
1.56
1.57
1.58
1.59

1.23
1.24
1.25
1.26
1.28
1.28
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.38
1.39
1.40
1.41
1.41
1.42
1.43
1.43
1.44
1.44
1.45
1.48
1.50
1.52
1.54
1.55
1.57
1.58
1.59
1.60
1.61
1.62
1.63

0.83
0.86
0.90
0.93
0.96
0.99
1.01
1.04
1.06
1.08
1.10
1.12
1.13
1.15
1.17
1.18
1.20
1.21
1.22
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.34
1.38
1.41
1.44
1.46
1.48
1.50
1.52
1.53
1.55
1.56
1.57

1.40
1.40
1.40
1.40
1.41
1.41
1.41
1.42
1.42
1.43
1.43
1.44
1.44
1.45
1.45
1.46
1.47
1.47
1.48
1.48
1.48
1.49
1.49
1.50
1.50
1.51
1.53
1.54
1.56
1.57
1.59
1.60
1.61
1.62
1.63
1.64
1.65
1.65

0.71
0.75
0.79
0.82
0.86
0.89
0.92
0.95
0.97
1.00
1.02
1.04
1.06
1.08
1.10
1.12
1.13
1.15
1.16
1.17
1.19
1.20
1.21
1.23
1.24
1.25
1.30
1.34
1.37
1.40
1.43
1.45
1.47
1.49
1.51
1.53
1.54
1.55

1.61
1.59
1.58
1.56
1.55
1.55
1.54
1.54
1.54
1.54
1.54
1.54
1.54
1.54
1.54
1.54
1.55
1.55
1.55
1.55
1.55
1.56
1.56
1.56
1.56
1.57
1.58
1.59
1.60
1.61
1.62
1.63
1.64
1.65
1.65
1.66
1.67
1.67

0.59
0.64
0.68
0.72
0.76
0.79
0.83
0.86
0.89
0.91
0.94
0.96
0.99
1.01
1.03
1.05
1.07
1.08
1.10
1.12
1.13
1.15
1.16
1.17
1.19
1.20
1.25
1.30
1.33
1.37
1.40
1.42
1.45
1.47
1.49
1.50
1.52
1.53

1.84
1.80
1.77
1.74
1.72
1.70
1.69
1.68
1.67
1.66
1.65
1.65
1.64
1.64
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.63
1.62
1.62
1.63
1.63
1.63
1.64
1.64
1.65
1.66
1.66
1.67
1.67
1.68
1.69
1.69
1.70

0.48
0.53
0.57
0.62
0.66
0.70
0.73
0.77
0.80
0.83
0.86
0.88
0.91
0.93
0.96
0.98
1.00
1.02
1.04
1.06
1.07
1.09
1.10
1.12
1.13
1.15
1.21
1.26
1.30
1.33
1.36
1.39
1.42
1.44
1.46
1.48
1.50
1.51

2.09
2.03
1.98
1.93
1.90
1.87
1.84
1.82
1.80
1.79
1.77
1.76
1.75
1.74
1.73
1.73
1.72
1.71
1.71
1.70
1.70
1.70
1.70
1.70
1.69
1.69
1.69
1.69
1.69
1.69
1.69
1.70
1.70
1.70
1.71
1.71
1.71
1.72

SOURCE:

J. Durbin and G. S. Watson, “Testing for Serial Correlation in Least Squares Regression, II,” Biometrika 30 (1951), pp. 159–178. Reproduced by permission of the
Biometrika Trustees.

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B.9C Critical Values for the Durbin–Watson d Statistic (a 5 .01)

k51

k52

k53

k54

k55

n

dL,.01

dU,.01

dL,.01

dU,.01

dL,.01

dU,.01

dL,.01

dU,.01

dL,.01

dU,.01

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
40
45
50
55
60
65
70
75
80
85
90
95
100

0.81
0.84
0.87
0.90
0.93
0.95
0.97
1.00
1.02
1.04
1.05
1.07
1.09
1.10
1.12
1.13
1.15
1.16
1.17
1.18
1.19
1.21
1.22
1.23
1.24
1.25
1.29
1.32
1.36
1.38
1.41
1.43
1.45
1.47
1,48
1.50
1.51
1.52

1.07
1.09
1.10
1.12
1.13
1.15
1.16
1.17
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.32
1.33
1.34
1.34
1.38
1.40
1.43
1.45
1.47
1.49
1.50
1.52
1.53
1.54
1.55
1.56

0.70
0.74
0.77
0.80
0.83
0.86
0.89
0.91
0.94
0.96
0.98
1.00
1.02
1.04
1.05
1.07
1.08
1.10
1.11
1.13
1.14
1.15
1.16
1.18
1.19
1.20
1.24
1.28
1.32
1.35
1.38
1.40
1.42
1.44
1.46
1.47
1.49
1.50

1.25
1.25
1.25
1.26
1.26
1.27
1.27
1.28
1.29
1.30
1.30
1.31
1.32
1.32
1.33
1.34
1.34
1.35
1.36
1.36
1.37
1.38
1.38
1.39
1.39
1.40
1.42
1.45
1.47
1.48
1.50
1.52
1.53
1.54
1.55
1.56
1.57
1.58

0.59
0.63
0.67
0.71
0.74
0.77
0.80
0.83
0.86
0.88
0.90
0.93
0.95
0.97
0.99
1.01
1.02
1.04
1.05
1.07
1.08
1.10
1.11
1.12
1.14
1.15
1.20
1.24
1.28
1.32
1.35
1.37
1.39
1.42
1.43
1.45
1.47
1.48

1.46
1.44
1.43
1.42
1.41
1.41
1.41
1.40
1.40
1.41
1.41
1.41
1.41
1.41
1.42
1.42
1.42
1.43
1.43
1.43
1.44
1.44
1.45
1.45
1.45
1.46
1.48
1.49
1.51
1.52
1.53
1.55
1.56
1.57
1.58
1.59
1.60
1.60

0.49
0.53
0.57
0.61
0.65
0.68
0.72
0.75
0.77
0.80
0.83
0.85
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.01
1.03
1.04
1.06
1.07
1.09
1.10
1.16
1.20
1.25
1.28
1.31
1.34
1.37
1.39
1.41
1.43
1.45
1.46

1.70
1.66
1.63
1.60
1.58
1.57
1.55
1.54
1.53
1.53
1.52
1.52
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.52
1.52
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.60
1.60
1.61
1.62
1.63

0.39
0.44
0.48
0.52
0.56
0.60
0.63
0.66
0.70
0.72
0.75
0.78
0.81
0.83
0.85
0.88
0.90
0.92
0.94
0.95
0.97
0.99
1.00
1.02
1.03
1.05
1.11
1.16
1.21
1.25
1.28
1.31
1.34
1.36
1.39
1.41
1.42
1.44

1.96
1.90
1.85
1.80
1.77
1.74
1.71
1.69
1.67
1.66
1.65
1.64
1.63
1.62
1.61
1.61
1.60
1.60
1.59
1.59
1.59
1.59
1.59
1.58
1.58
1.58
1.58
1.59
1.59
1.60
1.61
1.61
1.62
1.62
1.63
1.64
1.64
1.65

SOURCE:

J. Durbin and G. S. Watson, “Testing for Serial Correlation in Least Squares Regression, II,” Biometrika 30 (1951), pp. 159–178. Reproduced by permission of the
Biometrika Trustees.

742

Lin20522_appb_726-743.indd Page 743 28/10/13 11:55 AM user-f-w-198

/201/MH02018/Lin20522_disk1of1/0078020522/Lin20522_pagefiles

B.10 Factors for Control Charts

Chart for
Averages

Chart for
Ranges

Number of
Items in
Sample,
n

Factors for
Control Limits
A2

d2

D3

D4

2
3
4
5

1.880
1.023
.729
.577

1.128
1.693
2.059
2.326

0
0
0
0

3.267
2.575
2.282
2.115

6
7
8
9
10

.483
.419
.373
.337
.308

2.534
2.704
2.847
2.970
3.078

0
.076
.136
.184
.223

2.004
1.924
1.864
1.816
1.777

11
12
13
14
15

.285
.266
.249
.235
.223

3.173
3.258
3.336
3.407
3.472

.256
.284
.308
.329
.348

1.744
1.716
1.692
1.671
1.652

Factors for
Central Line

Factors for
Control Limits

SOURCE:

Adapted from American Society for Testing and Materials, Manual on
Quality Control of Materials, 1951, Table B2, p. 115. For a more detailed table and explanation, see J. Duncan Acheson, Quality Control and Industrial Statistics,
3d ed. (Homewood, Ill.: Richard D. Irwin, 1974), Table M, p. 927.

743

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