Appendix A. Tables

Table value for z is the probability to the left of z.

Figure . Table value for z is the probability to the left of z.

Table A. Standard Normal Probabilities

z

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

–3.4

.0003

.0003

.0003

.0003

.0003

.0003

.0003

.0003

.0003

.0002

–3.3

.0005

.0005

.0005

.0004

.0004

.0004

.0004

.0004

.0004

.0003

–3.2

.0007

.0007

.0006

.0006

.0006

.0006

.0006

.0005

.0005

.0005

–3.1

.0010

.0009

.0009

.0009

.0008

.0008

.0008

.0008

.0007

.0007

–3.0

.0013

.0013

.0013

.0012

.0012

.0011

.0011

.0011

.0010

.0010

–2.9

.0019

.0018

.0018

.0017

.0016

.0016

.0015

.0015

.0014

.0014

–2.8

.0026

.0025

.0024

.0023

.0023

.0022

.0021

.0021

.0020

.0019

–2.7

.0035

.0034

.0033

.0032

.0031

.0030

.0029

.0028

.0027

.0026

–2.6

.0047

.0045

.0044

.0043

.0041

.0040

.0039

.0038

.0037

.0036

–2.5

.0062

.0060

.0059

.0057

.0055

.0054

.0052

.0051

.0049

.0048

–2.4

.0082

.0080

.0078

.0075

.0073

.0071

.0069

.0068

.0066

.0064

–2.3

.0107

.0104

.0102

.0099

.0096

.0094

.0091

.0089

.0087

.0084

–2.2

.0139

.0136

.0132

.0129

.0125

.0122

.0119

.0116

.0113

.0110

–2.1

.0179

.0174

.0170

.0166

.0162

.0158

.0154

.0150

.0146

.0143

–2.0

.0228

.0222

.0217

.0212

.0207

.0202

.0197

.0192

.0188

.0183

–1.9

.0287

.0281

.0274

.0268

.0262

.0256

.0250

.0244

.0239

.0233

–1.8

.0359

.0351

.0344

.0336

.0329

.0322

.0314

.0307

.0301

.0294

–1.7

.0446

.0436

.0427

.0418

.0409

.0401

.0392

.0384

.0375

.0367

–1.6

.0548

.0537

.0526

.0516

.0505

.0495

.0485

.0475

.0465

.0455

–1.5

.0668

.0655

.0643

.0630

.0618

.0606

.0594

.0582

.0571

.0559

–1.4

.0808

.0793

.0778

.0764

.0749

.0735

.0721

.0708

.0694

.0681

–1.3

.0968

.0951

.0934

.0918

.0901

.0885

.0869

.0853

.0838

.0823

–1.2

.1151

.1131

.1112

.1093

.1075

.1056

.1038

.1020

.1003

.0985

–1.1

.1357

.1335

.1314

.1292

.1271

.1251

.1230

.1210

.1190

.1170

–1.0

.1587

.1562

.1539

.1515

.1492

.1469

.1446

.1423

.1401

.1379

–0.9

.1841

.1814

.1788

.1762

.1736

.1711

.1685

.1660

.1635

.1611

–0.8

.2119

.2090

.2061

.2033

.2005

.1977

.1949

.1922

.1894

.1867

–0.7

.2420

.2389

.2358

.2327

.2296

.2266

.2236

.2206

.2177

.2148

–0.6

.2743

.2709

.2676

.2643

.2611

.2578

.2546

.2514

.2483

.2451

–0.5

.3085

.3050

.3015

.2981

.2946

.2912

.2877

.2843

.2810

.2776

–0.4

.3446

.3409

.3372

.3336

.3300

.3264

.3228

.3192

.3156

.3121

–0.3

.3821

.3783

.3745

.3707

.3669

.3632

.3594

.3557

.3520

.3483

–0.2

.4207

.4168

.4129

.4090

.4052

.4013

.3974

.3936

.3897

.3859

–0.1

.4602

.4562

.4522

.4483

.4443

.4404

.4364

.4325

.4286

.4247

–0.0

.5000

.4960

.4920

.4880

.4840

.4801

.4761

.4721

.4681

.4641

Table value for z is the probability to the left of z.

Figure . Table value for z is the probability to the left of z.

z

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0

.5000

.5040

.5080

.5120

.5160

.5199

.5239

.5279

.5319

.5359

0.1

.5398

.5438

.5478

.5517

.5557

.5596

.5636

.5675

.5714

.5753

0.2

.5793

.5832

.5871

.5910

.5948

.5987

.6026

.6064

.6103

.6141

0.3

.6179

.6217

.6255

.6293

.6331

.6368

.6406

.6443

.6480

.6517

0.4

.6554

.6591

.6628

.6664

.6700

.6736

.6772

.6808

.6844

.6879

0.5

.6915

.6950

.6985

.7019

.7054

.7088

.7123

.7157

.7190

.7224

0.6

.7257

.7291

.7324

.7357

.7389

.7422

.7454

.7486

.7517

.7549

0.7

.7580

.7611

.7642

.7673

.7704

.7734

.7764

.7794

.7823

.7852

0.8

.7881

.7910

.7939

.7967

.7995

.8023

.8051

.8078

.8106

.8133

0.9

.8159

.8186

.8212

.8238

.8264

.8289

.8315

.8340

.8365

.8389

1.0

.8413

.8438

.8461

.8485

.8508

.8531

.8554

.8577

.8599

.8621

1.1

.8643

.8665

.8686

.8708

.8729

.8749

.8770

.8790

.8810

.8830

1.2

.8849

.8869

.8888

.8907

.8925

.8944

.8962

.8980

.8997

.9015

1.3

.9032

.9049

.9066

.9082

.9099

.9115

.9131

.9147

.9162

.9177

1.4

.9192

.9207

.9222

.9236

.9251

.9265

.9279

.9292

.9306

.9319

1.5

.9332

.9345

.9357

.9370

.9382

.9394

.9406

.9418

.9429

.9441

1.6

.9452

.9463

.9474

.9484

.9495

.9505

.9515

.9525

.9535

.9545

1.7

.9554

.9564

.9573

.9582

.9591

.9599

.9608

.9616

.9625

.9633

1.8

.9641

.9649

.9656

.9664

.9671

.9678

.9686

.9693

.9699

.9706

1.9

.9713

.9719

.9726

.9732

.9738

.9744

.9750

.9756

.9761

.9767

2.0

.9772

.9778

.9783

.9788

.9793

.9798

.9803

.9808

.9812

.9817

2.1

.9821

.9826

.9830

.9834

.9838

.9842

.9846

.9850

.9854

.9857

2.2

.9861

.9864

.9868

.9871

.9875

.9878

.9881

.9884

.9887

.9890

2.3

.9893

.9896

.9898

.9901

.9904

.9906

.9909

.9911

.9913

.9916

2.4

.9918

.9920

.9922

.9925

.9927

.9929

.9931

.9932

.9934

.9936

2.5

.9938

.9940

.9941

.9943

.9945

.9946

.9948

.9949

.9951

.9952

2.6

.9953

.9955

.9956

.9957

.9959

.9960

.9961

.9962

.9963

.9964

2.7

.9965

.9966

.9967

.9968

.9969

.9970

.9971

.9972

.9973

.9974

2.8

.9974

.9975

.9976

.9977

.9977

.9978

.9979

.9979

.9980

.9981

2.9

.9981

.9982

.9982

.9983

.9984

.9984

.9985

.9985

.9986

.9986

3.0

.9987

.9987

.9987

.9988

.9988

.9989

.9989

.9989

.9990

.9990

3.1

.9990

.9991

.9991

.9991

.9992

.9992

.9992

.9992

.9993

.9993

3.2

.9993

.9993

.9994

.9994

.9994

.9994

.9994

.9995

.9995

.9995

3.3

.9995

.9995

.9995

.9996

.9996

.9996

.9996

.9996

.9996

.9997

3.4

.9997

.9997

.9997

.9997

.9997

.9997

.9997

.9997

.9997

.9998

Table B. Random Digits

101

19223

95034

05756

28713

96409

12531

42544

82853

102

20826

04475

94351

33439

06175

03307

66233

22350

103

32071

50612

05952

30176

81471

85492

19221

57168

104

21199

86409

30003

05822

51816

41862

34876

76020

105

10455

84585

08091

48657

09879

32897

08853

62247

106

63443

55127

53478

41514

51995

07788

76625

79847

107

82739

57890

20807

47511

81676

55360

94072

02417

108

86824

94361

79090

63600

91936

51541

23963

88545

109

34683

84484

87891

83382

46973

93644

20814

86694

110

87605

13092

97004

12712

89090

34265

57676

08708

111

81486

69487

60513

09297

00412

71238

27649

39950

112

43857

27768

14703

90602

46070

36974

74691

79843

113

30596

91595

16816

21290

02139

39693

15210

67003

114

19762

60189

66177

65805

19248

68403

75439

53656

115

23720

26018

93503

26463

87803

71190

22160

87096

116

53696

38126

21384

38418

16164

69209

75415

76349

117

75230

18842

44098

40784

72352

42232

47847

85306

118

70494

19004

08513

16998

29286

06732

49386

06967

119

37863

76884

98089

18779

03689

77790

33927

95962

120

58433

44928

03929

27859

97915

52224

44127

18994

121

29690

37087

48463

02226

56877

22366

93941

74584

122

21123

86170

53411

30529

97469

24365

49980

95663

123

88890

58862

23433

47347

70805

22045

04481

06427

124

61999

42290

25045

80833

66492

48399

57551

87185

125

73846

21500

27562

03460

21513

50913

39017

46856

126

59321

17125

05234

13443

14507

39680

24430

49485

127

45415

07837

42078

93921

54639

76462

30303

38530

128

20766

40730

22023

50998

13032

37150

85029

55129

129

25942

85230

00761

84894

77771

60839

76961

98370

130

27900

40660

16853

06842

05640

21090

66672

35358

131

51643

57697

43214

90366

19817

43734

98927

15045

132

21683

72667

15479

11720

89142

59861

43415

48417

133

58974

43986

23733

01102

55638

46270

99271

40082

134

17761

08361

01221

72309

17597

48752

73073

03052

135

04460

63771

03141

70392

18954

44344

16167

01776

136

23626

34712

17032

17810

34740

26053

93307

16285

137

32790

17033

55420

97717

61006

64560

22480

07641

138

43736

71972

47283

53324

93486

10687

36572

22854

139

93745

92391

77040

95992

83135

30714

06719

59096

140

47160

09855

48906

85728

84229

80628

03316

57587

141

23867

62845

04461

57908

32363

29866

43069

00888

142

31972

82955

23581

26219

32396

98106

03259

13009

143

69479

02023

42240

58720

78179

51440

83402

34979

144

28722

34202

77501

93305

53698

55189

23017

53861

145

19687

12633

57857

95806

09931

02150

43163

58636

146

22232

22320

40740

78321

65478

77484

33012

69691

Table entry for p is the point t* with probability p to the right.

Figure . Table entry for p is the point t* with probability p to the right.

Table C. t Distribution Critical Values

 

Upper tail probability p

df

.25

.20

.15

.10

.05

.025

.02

.01

.005

.0025

.001

.0005

1

1.000

1.376

1.963

3.078

6.314

12.71

15.89

31.82

63.66

127.3

318.3

636.6

2

.816

1.061

1.386

1.886

2.920

4.303

4.849

6.965

9.925

14.09

22.33

31.60

3

.765

.978

1.250

1.638

2.353

3.182

3.482

4.541

5.841

7.453

10.21

12.92

4

.741

.941

1.190

1.533

2.132

2.776

2.999

3.747

4.604

5.598

7.173

8.610

5

.727

.920

1.156

1.476

2.015

2.571

2.757

3.365

4.032

4.773

5.893

6.869

6

.718

.906

1.134

1.440

1.943

2.447

2.612

3.143

3.707

4.317

5.208

5.959

7

.711

.896

1.119

1.415

1.895

2.365

2.517

2.998

3.499

4.029

4.785

5.408

8

.706

.889

1.108

1.397

1.860

2.306

2.449

2.896

3.355

3.833

4.501

5.041

9

.703

.883

1.100

1.383

1.833

2.262

2.398

2.821

3.250

3.690

4.297

4.781

10

.700

.879

1.093

1.372

1.812

2.228

2.359

2.764

3.169

3.581

4.144

4.587

11

.697

.876

1.088

1.363

1.796

2.201

2.328

2.718

3.106

3.497

4.025

4.437

12

.695

.873

1.083

1.356

1.782

2.179

2.303

2.681

3.055

3.428

3.930

4.318

13

.694

.870

1.079

1.350

1.771

2.160

2.282

2.650

3.012

3.372

3.852

4.221

14

.692

.868

1.076

1.345

1.761

2.145

2.264

2.624

2.977

3.326

3.787

4.140

15

.691

.866

1.074

1.341

1.753

2.131

2.249

2.602

2.947

3.286

3.733

4.073

16

.690

.865

1.071

1.337

1.746

2.120

2.235

2.583

2.921

3.252

3.686

4.015

17

.689

.863

1.069

1.333

1.740

2.110

2.224

2.567

2.898

3.222

3.646

3.965

18

.688

.862

1.067

1.330

1.734

2.101

2.214

2.552

2.878

3.197

3.611

3.922

19

.688

.861

1.066

1.328

1.729

2.093

2.205

2.539

2.861

3.174

3.579

3.883

20

.687

.860

1.064

1.325

1.725

2.086

2.197

2.528

2.845

3.153

3.552

3.850

21

.686

.859

1.063

1.323

1.721

2.080

2.189

2.518

2.831

3.135

3.527

3.819

22

.686

.858

1.061

1.321

1.717

2.074

2.183

2.508

2.819

3.119

3.505

3.792

23

.685

.858

1.060

1.319

1.714

2.069

2.177

2.500

2.807

3.104

3.485

3.768

24

.685

.857

1.059

1.318

1.711

2.064

2.172

2.492

2.797

3.091

3.467

3.745

25

.684

.856

1.058

1.316

1.708

2.060

2.167

2.485

2.787

3.078

3.450

3.725

26

.684

.856

1.058

1.315

1.706

2.056

2.162

2.479

2.779

3.067

3.435

3.707

27

.684

.855

1.057

1.314

1.703

2.052

2.158

2.473

2.771

3.057

3.421

3.690

28

.683

.855

1.056

1.313

1.701

2.048

2.154

2.467

2.763

3.047

3.408

3.674

29

.683

.854

1.055

1.311

1.699

2.045

2.150

2.462

2.756

3.038

3.396

3.659

30

.683

.854

1.055

1.310

1.697

2.042

2.147

2.457

2.750

3.030

3.385

3.646

40

.681

.851

1.050

1.303

1.684

2.021

2.123

2.423

2.704

2.971

3.307

3.551

50

.679

.849

1.047

1.299

1.676

2.009

2.109

2.403

2.678

2.937

3.261

3.496

60

.679

.848

1.045

1.296

1.671

2.000

2.099

2.390

2.660

2.915

3.232

3.460

80

.678

.846

1.043

1.292

1.664

1.990

2.088

2.374

2.639

2.887

3.195

3.416

100

.677

.845

1.042

1.290

1.660

1.984

2.081

2.364

2.626

2.871

3.174

3.390

1000

.675

.842

1.037

1.282

1.646

1.962

2.056

2.330

2.581

2.813

3.098

3.300

∞

.674

.841

1.036

1.282

1.645

1.960

2.054

2.326

2.576

2.807

3.091

3.291

 

50%

60%

70%

80%

90%

95%

96%

98%

99%

99.5%

99.8%

99.9%

Confidence level C

Table entry for p is the value χ2 with probability p to the right.

Figure . Table entry for p is the value χ2 with probability p to the right.

Table D. χ2 Critical Value

 

Tail probability p

df

.25

.20

.15

.10

.05

.025

.02

.01

.005

.0025

.001

1

  1.32

  1.64

  2.07

  2.71

  3.84

  5.02

  5.41

  6.63

  7.88

  9.14

10.83

2

  2.77

  3.22

  3.79

  4.61

  5.99

  7.38

  7.82

  9.21

10.60

11.98

13.82

3

  4.11

  4.64

  5.32

  6.25

  7.81

  9.35

  9.84

11.34

12.84

14.32

16.27

4

  5.39

  5.99

  6.74

  7.78

  9.49

11.14

11.67

13.28

14.86

16.42

18.47

5

  6.63

  7.29

  8.12

  9.24

11.07

12.83

13.39

15.09

16.75

18.39

20.51

6

  7.84

  8.56

  9.45

10.64

12.59

14.45

15.03

16.81

18.55

20.25

22.46

7

  9.04

  9.80

10.75

12.02

14.07

16.01

16.62

18.48

20.28

22.04

24.32

8

10.22

11.03

12.03

13.36

15.51

17.53

18.17

20.09

21.95

23.77

26.12

9

11.39

12.24

13.29

14.68

16.92

19.02

19.68

21.67

23.59

25.46

27.88

10

12.55

13.44

14.53

15.99

18.31

20.48

21.16

23.21

25.19

27.11

29.59

11

13.70

14.63

15.77

17.28

19.68

21.92

22.62

24.72

26.76

28.73

31.26

12

14.85

15.81

16.99

18.55

21.03

23.34

24.05

26.22

28.30

30.32

32.91

13

15.98

16.98

18.20

19.81

22.36

24.74

25.47

27.69

29.82

31.88

34.53

14

17.12

18.15

19.41

21.06

23.68

26.12

26.87

29.14

31.32

33.43

36.12

15

18.25

19.31

20.60

22.31

25.00

27.49

28.26

30.58

32.80

34.95

37.70

16

19.37

20.47

21.79

23.54

26.30

28.85

29.63

32.00

34.27

36.46

39.25

17

20.49

21.61

22.98

24.77

27.59

30.19

31.00

33.41

35.72

37.95

40.79

18

21.60

22.76

24.16

25.99

28.87

31.53

32.35

34.81

37.16

39.42

42.31

19

22.72

23.90

25.33

27.20

30.14

32.85

33.69

36.19

38.58

40.88

43.82

20

23.83

25.04

26.50

28.41

31.41

34.17

35.02

37.57

40.00

42.34

45.31

21

24.93

26.17

27.66

29.62

32.67

35.48

36.34

38.93

41.40

43.78

46.80

22

26.04

27.30

28.82

30.81

33.92

36.78

37.66

40.29

42.80

45.20

48.27

23

27.14

28.43

29.98

32.01

35.17

38.08

38.97

41.64

44.18

46.62

49.73

24

28.24

29.55

31.13

33.20

36.42

39.36

40.27

42.98

45.56

48.03

51.18

25

29.34

30.68

32.28

34.38

37.65

40.65

41.57

44.31

46.93

49.44

52.62

26

30.43

31.79

33.43

35.56

38.89

41.92

42.86

45.64

48.29

50.83

54.05

27

31.53

32.91

34.57

36.74

40.11

43.19

44.14

46.96

49.64

52.22

55.48

28

32.62

34.03

35.71

37.92

41.34

44.46

45.42

48.28

50.99

53.59

56.89

29

33.71

35.14

36.85

39.09

42.56

45.72

46.69

49.59

52.34

54.97

58.30

30

34.80

36.25

37.99

40.26

43.77

46.98

47.96

50.89

53.67

56.33

59.70

40

45.62

47.27

49.24

51.81

55.76

59.34

60.44

63.69

66.77

69.70

73.40

50

56.33

58.16

60.35

63.17

67.50

71.42

72.61

76.15

79.49

82.66

86.66

60

66.98

68.97

71.34

74.40

79.08

83.30

84.58

88.38

91.95

95.34

99.61

80

88.13

90.41

93.11

96.58

101.9

106.6

108.1

112.3

116.3

120.1

124.8

100

109.1

111.7

114.7

118.5

124.3

129.6

131.1

135.8

140.2

144.3

149.4

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