Appendix 1

A1.1 Two-Tailed t-Table

Probability points of the t-distribution with degrees of freedom (df) (Continued)

df

Two-Tail Area Probability

0.2

0.1

0.05

0.01

0.005

0.001

1

3.078

6.314

12.706

63.657

127.321

636.619

2

1.886

2.920

4.303

9.925

14.089

31.599

3

1.638

2.353

3.182

5.841

7.453

12.924

4

1.533

2.132

2.776

4.604

5.598

8.610

5

1.476

2.015

2.571

4.032

4.773

6.869

6

1.440

1.943

2.447

3.707

4.317

5.959

7

1.415

1.895

2.365

3.499

4.029

5.408

8

1.397

1.860

2.306

3.355

3.833

5.041

9

1.383

1.833

2.262

3.250

3.690

4.781

10

1.372

1.812

2.228

3.169

3.581

4.587

11

1.363

1.796

2.201

3.106

3.497

4.437

Continued

12

1.356

1.782

2.179

3.055

3.428

4.318

13

1.350

1.771

2.160

3.012

3.372

4.221

14

1.345

1.761

2.145

2.977

3.326

4.140

15

1.341

1.753

2.131

2.947

3.286

4.073

16

1.337

1.746

2.120

2.921

3.252

4.015

17

1.333

1.740

2.110

2.898

3.222

3.965

18

1.330

1.734

2.101

2.878

3.197

3.922

19

1.328

1.729

2.093

2.861

3.174

3.883

20

1.325

1.725

2.086

2.845

3.153

3.850

21

1.323

1.721

2.080

2.831

3.135

3.819

22

1.321

1.717

2.074

2.819

3.119

3.792

23

1.319

1.714

2.069

2.807

3.104

3.768

24

1.318

1.711

2.064

2.797

3.091

3.745

25

1.316

1.708

2.060

2.787

3.078

3.725

26

1.315

1.706

2.056

2.779

3.067

3.707

27

1.314

1.703

2.052

2.771

3.057

3.690

28

1.313

1.701

2.048

2.763

3.047

3.674

29

1.311

1.699

2.045

2.756

3.038

3.659

30

1.310

1.697

2.042

2.750

3.030

3.646

40

1.303

1.684

2.021

2.704

2.971

3.551

60

1.296

1.671

2.000

2.660

2.915

3.460

120

1.289

1.658

1.980

2.617

2.860

3.373

1000

1.282

1.646

1.962

2.581

2.813

3.300

10000

1.282

1.645

1.960

2.576

2.808

3.291

A1.2 F-Table for 10%

Percentage points of the F-distribution: Upper 10% points (Continued)

dfnum

dfden

1

2

3

4

5

6

7

8

9

10

15

20

1

39.863

49.500

53.593

55.833

57.240

58.204

58.906

59.439

59.858

60.195

61.220

61.740

2

8.526

9.000

9.162

9.243

9.293

9.326

9.349

9.367

9.381

9.392

9.425

9.441

3

5.538

5.462

5.391

5.343

5.309

5.285

5.266

5.252

5.240

5.230

5.200

5.184

4

4.545

4.325

4.191

4.107

4.051

4.010

3.979

3.955

3.936

3.920

3.870

3.844

5

4.060

3.780

3.619

3.520

3.453

3.405

3.368

3.339

3.316

3.297

3.238

3.207

6

3.776

3.463

3.289

3.181

3.108

3.055

3.014

2.983

2.958

2.937

2.871

2.836

7

3.589

3.257

3.074

2.961

2.883

2.827

2.785

2.752

2.725

2.703

2.632

2.595

8

3.458

3.113

2.924

2.806

2.726

2.668

2.624

2.589

2.561

2.538

2.464

2.425

9

3.360

3.006

2.813

2.693

2.611

2.551

2.505

2.469

2.440

2.416

2.340

2.298

10

3.285

2.924

2.728

2.605

2.522

2.461

2.414

2.377

2.347

2.323

2.244

2.201

11

3.225

2.860

2.660

2.536

2.451

2.389

2.342

2.304

2.274

2.248

2.167

2.123

12

3.177

2.807

2.606

2.480

2.394

2.331

2.283

2.245

2.214

2.188

2.105

2.060

13

3.136

2.763

2.560

2.434

2.347

2.283

2.234

2.195

2.164

2.138

2.053

2.007

14

3.102

2.726

2.522

2.395

2.307

2.243

2.193

2.154

2.122

2.095

2.010

1.962

15

3.073

2.695

2.490

2.361

2.273

2.208

2.158

2.119

2.086

2.059

1.972

1.924

16

3.048

2.668

2.462

2.333

2.244

2.178

2.128

2.088

2.055

2.028

1.940

1.891

17

3.026

2.645

2.437

2.308

2.218

2.152

2.102

2.061

2.028

2.001

1.912

1.862

18

3.007

2.624

2.416

2.286

2.196

2.130

2.079

2.038

2.005

1.977

1.887

1.837

19

2.990

2.606

2.397

2.266

2.176

2.109

2.058

2.017

1.984

1.956

1.865

1.814

20

2.975

2.589

2.380

2.249

2.158

2.091

2.040

1.999

1.965

1.937

1.845

1.794

21

2.961

2.575

2.365

2.233

2.142

2.075

2.023

1.982

1.948

1.920

1.827

1.776

22

2.949

2.561

2.351

2.219

2.128

2.060

2.008

1.967

1.933

1.904

1.811

1.759

23

2.937

2.549

2.339

2.207

2.115

2.047

1.995

1.953

1.919

1.890

1.796

1.744

24

2.927

2.538

2.327

2.195

2.103

2.035

1.983

1.941

1.906

1.877

1.783

1.730

25

2.918

2.528

2.317

2.184

2.092

2.024

1.971

1.929

1.895

1.866

1.771

1.718

26

2.909

2.519

2.307

2.174

2.082

2.014

1.961

1.919

1.884

1.855

1.760

1.706

27

2.901

2.511

2.299

2.165

2.073

2.005

1.952

1.909

1.874

1.845

1.749

1.695

28

2.894

2.503

2.291

2.157

2.064

1.996

1.943

1.900

1.865

1.836

1.740

1.685

29

2.887

2.495

2.283

2.149

2.057

1.988

1.935

1.892

1.857

1.827

1.731

1.676

30

2.881

2.489

2.276

2.142

2.049

1.980

1.927

1.884

1.849

1.819

1.722

1.667

40

2.835

2.440

2.226

2.091

1.997

1.927

1.873

1.829

1.793

1.763

1.662

1.605

60

2.791

2.393

2.177

2.041

1.946

1.875

1.819

1.775

1.738

1.707

1.603

1.543

120

2.748

2.347

2.130

1.992

1.896

1.824

1.767

1.722

1.684

1.652

1.545

1.482

100K

2.706

2.303

2.084

1.945

1.847

1.774

1.717

1.670

1.632

1.599

1.487

1.421

K: Multiply this value by 1,000.

A1.3 F-Table for 5%

Percentage points of the F-distribution: Upper 5% points (Continued)

dfnum

dfden

1

2

3

4

5

6

7

8

9

10

15

20

1

161.45

199.50

215.71

224.58

230.16

233.99

236.77

238.88

240.54

241.88

245.95

248.01

2

18.513

19.000

19.164

19.247

19.296

19.330

19.353

19.371

19.385

19.396

19.429

19.446

3

10.128

9.552

9.277

9.117

9.013

8.941

8.887

8.845

8.812

8.786

8.703

8.660

4

7.709

6.944

6.591

6.388

6.256

6.163

6.094

6.041

5.999

5.964

5.858

5.803

5

6.608

5.786

5.409

5.192

5.050

4.950

4.876

4.818

4.772

4.735

4.619

4.558

6

5.987

5.143

4.757

4.534

4.387

4.284

4.207

4.147

4.099

4.060

3.938

3.874

7

5.591

4.737

4.347

4.120

3.972

3.866

3.787

3.726

3.677

3.637

3.511

3.445

8

5.318

4.459

4.066

3.838

3.687

3.581

3.500

3.438

3.388

3.347

3.218

3.150

9

5.117

4.256

3.863

3.633

3.482

3.374

3.293

3.230

3.179

3.137

3.006

2.936

10

4.965

4.103

3.708

3.478

3.326

3.217

3.135

3.072

3.020

2.978

2.845

2.774

11

4.844

3.982

3.587

3.357

3.204

3.095

3.012

2.948

2.896

2.854

2.719

2.646

12

4.747

3.885

3.490

3.259

3.106

2.996

2.913

2.849

2.796

2.753

2.617

2.544

13

4.667

3.806

3.411

3.179

3.025

2.915

2.832

2.767

2.714

2.671

2.533

2.459

14

4.600

3.739

3.344

3.112

2.958

2.848

2.764

2.699

2.646

2.602

2.463

2.388

15

4.543

3.682

3.287

3.056

2.901

2.790

2.707

2.641

2.588

2.544

2.403

2.328

16

4.494

3.634

3.239

3.007

2.852

2.741

2.657

2.591

2.538

2.494

2.352

2.276

17

4.451

3.592

3.197

2.965

2.810

2.699

2.614

2.548

2.494

2.450

2.308

2.230

18

4.414

3.555

3.160

2.928

2.773

2.661

2.577

2.510

2.456

2.412

2.269

2.191

19

4.381

3.522

3.127

2.895

2.740

2.628

2.544

2.477

2.423

2.378

2.234

2.155

20

4.351

3.493

3.098

2.866

2.711

2.599

2.514

2.447

2.393

2.348

2.203

2.124

21

4.325

3.467

3.072

2.840

2.685

2.573

2.488

2.420

2.366

2.321

2.176

2.096

22

4.301

3.443

3.049

2.817

2.661

2.549

2.464

2.397

2.342

2.297

2.151

2.071

23

4.279

3.422

3.028

2.796

2.640

2.528

2.442

2.375

2.320

2.275

2.128

2.048

24

4.260

3.403

3.009

2.776

2.621

2.508

2.423

2.355

2.300

2.255

2.108

2.027

25

4.242

3.385

2.991

2.759

2.603

2.490

2.405

2.337

2.282

2.236

2.089

2.007

26

4.225

3.369

2.975

2.743

2.587

2.474

2.388

2.321

2.265

2.220

2.072

1.990

27

4.210

3.354

2.960

2.728

2.572

2.459

2.373

2.305

2.250

2.204

2.056

1.974

28

4.196

3.340

2.947

2.714

2.558

2.445

2.359

2.291

2.236

2.190

2.041

1.959

29

4.183

3.328

2.934

2.701

2.545

2.432

2.346

2.278

2.223

2.177

2.027

1.945

30

4.171

3.316

2.922

2.690

2.534

2.421

2.334

2.266

2.211

2.165

2.015

1.932

40

4.085

3.232

2.839

2.606

2.449

2.336

2.249

2.180

2.124

2.077

1.924

1.839

60

4.001

3.150

2.758

2.525

2.368

2.254

2.167

2.097

2.040

1.993

1.836

1.748

120

3.920

3.072

2.680

2.447

2.290

2.175

2.087

2.016

1.959

1.910

1.750

1.659

100K

3.842

2.996

2.605

2.372

2.214

2.099

2.010

1.939

1.880

1.831

1.666

1.571

K: Multiply this value by 1,000.

A1.4 F-Table for 1%

Percentage points of the F-distribution: Upper 1% points (Continued)

dfnum

dfden

1

2

3

4

5

6

7

8

9

10

15

20

1

4052.2

4999.5

5403.3

5624.6

5763.6

5859.0

5928.3

5981.1

6022.5

6055.8

6157.3

6208.7

2

98.503

99.000

99.166

99.249

99.299

99.333

99.356

99.374

99.388

99.399

99.433

99.449

3

34.116

30.817

29.457

28.710

28.237

27.911

27.672

27.489

27.345

27.229

26.872

26.690

4

21.198

18.000

16.694

15.977

15.522

15.207

14.976

14.799

14.659

14.546

14.198

14.020

5

16.258

13.274

12.060

11.392

10.967

10.672

10.456

10.289

10.158

10.051

9.722

9.553

6

13.745

10.925

9.780

9.148

8.746

8.466

8.260

8.102

7.976

7.874

7.559

7.396

7

12.246

9.547

8.451

7.847

7.460

7.191

6.993

6.840

6.719

6.620

6.314

6.155

8

11.259

8.649

7.591

7.006

6.632

6.371

6.178

6.029

5.911

5.814

5.515

5.359

9

10.561

8.022

6.992

6.422

6.057

5.802

5.613

5.467

5.351

5.257

4.962

4.808

10

10.044

7.559

6.552

5.994

5.636

5.386

5.200

5.057

4.942

4.849

4.558

4.405

11

9.646

7.206

6.217

5.668

5.316

5.069

4.886

4.744

4.632

4.539

4.251

4.099

12

9.330

6.927

5.953

5.412

5.064

4.821

4.640

4.499

4.388

4.296

4.010

3.858

13

9.074

6.701

5.739

5.205

4.862

4.620

4.441

4.302

4.191

4.100

3.815

3.665

14

8.862

6.515

5.564

5.035

4.695

4.456

4.278

4.140

4.030

3.939

3.656

3.505

15

8.683

6.359

5.417

4.893

4.556

4.318

4.142

4.004

3.895

3.805

3.522

3.372

16

8.531

6.226

5.292

4.773

4.437

4.202

4.026

3.890

3.780

3.691

3.409

3.259

17

8.400

6.112

5.185

4.669

4.336

4.102

3.927

3.791

3.682

3.593

3.312

3.162

18

8.285

6.013

5.092

4.579

4.248

4.015

3.841

3.705

3.597

3.508

3.227

3.077

19

8.185

5.926

5.010

4.500

4.171

3.939

3.765

3.631

3.523

3.434

3.153

3.003

20

8.096

5.849

4.938

4.431

4.103

3.871

3.699

3.564

3.457

3.368

3.088

2.938

21

8.017

5.780

4.874

4.369

4.042

3.812

3.640

3.506

3.398

3.310

3.030

2.880

22

7.945

5.719

4.817

4.313

3.988

3.758

3.587

3.453

3.346

3.258

2.978

2.827

23

7.881

5.664

4.765

4.264

3.939

3.710

3.539

3.406

3.299

3.211

2.931

2.781

24

7.823

5.614

4.718

4.218

3.895

3.667

3.496

3.363

3.256

3.168

2.889

2.738

25

7.770

5.568

4.675

4.177

3.855

3.627

3.457

3.324

3.217

3.129

2.850

2.699

26

7.721

5.526

4.637

4.140

3.818

3.591

3.421

3.288

3.182

3.094

2.815

2.664

27

7.677

5.488

4.601

4.106

3.785

3.558

3.388

3.256

3.149

3.062

2.783

2.632

28

7.636

5.453

4.568

4.074

3.754

3.528

3.358

3.226

3.120

3.032

2.753

2.602

29

7.598

5.420

4.538

4.045

3.725

3.499

3.330

3.198

3.092

3.005

2.726

2.574

30

7.562

5.390

4.510

4.018

3.699

3.473

3.304

3.173

3.067

2.979

2.700

2.549

40

7.314

5.179

4.313

3.828

3.514

3.291

3.124

2.993

2.888

2.801

2.522

2.369

60

7.077

4.977

4.126

3.649

3.339

3.119

2.953

2.823

2.718

2.632

2.352

2.198

120

6.851

4.787

3.949

3.480

3.174

2.956

2.792

2.663

2.559

2.472

2.192

2.035

100K

6.635

4.605

3.782

3.319

3.017

2.802

2.640

2.511

2.408

2.321

2.039

1.878

K: Multiply this value by 1,000.

A1.5 F-Table for 0.1%

Percentage points of the F-distribution: Upper 0.1% points (Continued)

dfnum

dfden

1

2

3

4

5

6

7

8

9

10

15

20

1

405.2K

500.0K

540.4K

562.5K3

576.4K

585.9K

592.9K

598.1K

602.3K92

605.6K

615.8K

620.9K

2

998.50

999.00

999.17

999.25

999.30

999.33

999.36

999.37

999.39

999.40

999.43

999.45

3

167.03

148.50

141.11

137.10

134.58

132.85

131.58

130.62

129.86

129.25

127.37

126.42

4

74.137

61.246

56.177

53.436

51.712

50.525

49.658

48.996

48.475

48.053

46.761

46.100

5

47.181

37.122

33.202

31.085

29.752

28.834

28.163

27.649

27.244

26.917

25.911

25.395

6

35.507

27.000

23.703

21.924

20.803

20.030

19.463

19.030

18.688

18.411

17.559

17.120

7

29.245

21.689

18.772

17.198

16.206

15.521

15.019

14.634

14.330

14.083

13.324

12.932

8

25.415

18.494

15.829

14.392

13.485

12.858

12.398

12.046

11.767

11.540

10.841

10.480

9

22.857

16.387

13.902

12.560

11.714

11.128

10.698

10.368

10.107

9.894

9.238

8.898

10

21.040

14.905

12.553

11.283

10.481

9.926

9.517

9.204

8.956

8.754

8.129

7.804

11

19.687

13.812

11.561

10.346

9.578

9.047

8.655

8.355

8.116

7.922

7.321

7.008

12

18.643

12.974

10.804

9.633

8.892

8.379

8.001

7.710

7.480

7.292

6.709

6.405

13

17.815

12.313

10.209

9.073

8.354

7.856

7.489

7.206

6.982

6.799

6.231

5.934

14

17.143

11.779

9.729

8.622

7.922

7.436

7.077

6.802

6.583

6.404

5.848

5.557

15

16.587

11.339

9.335

8.253

7.567

7.092

6.741

6.471

6.256

6.081

5.535

5.248

16

16.120

10.971

9.006

7.944

7.272

6.805

6.460

6.195

5.984

5.812

5.274

4.992

17

15.722

10.658

8.727

7.683

7.022

6.562

6.223

5.962

5.754

5.584

5.054

4.775

18

15.379

10.390

8.487

7.459

6.808

6.355

6.021

5.763

5.558

5.390

4.866

4.590

19

15.081

10.157

8.280

7.265

6.622

6.175

5.845

5.590

5.388

5.222

4.704

4.430

20

14.819

9.953

8.098

7.096

6.461

6.019

5.692

5.440

5.239

5.075

4.562

4.290

21

14.587

9.772

7.938

6.947

6.318

5.881

5.557

5.308

5.109

4.946

4.437

4.167

22

14.380

9.612

7.796

6.814

6.191

5.758

5.438

5.190

4.993

4.832

4.326

4.058

23

14.195

9.469

7.669

6.696

6.078

5.649

5.331

5.085

4.890

4.730

4.227

3.961

24

14.028

9.339

7.554

6.589

5.977

5.550

5.235

4.991

4.797

4.638

4.139

3.873

25

13.877

9.223

7.451

6.493

5.885

5.462

5.148

4.906

4.713

4.555

4.059

3.794

26

13.739

9.116

7.357

6.406

5.802

5.381

5.070

4.829

4.637

4.480

3.986

3.723

27

13.613

9.019

7.272

6.326

5.726

5.308

4.998

4.759

4.568

4.412

3.920

3.658

28

13.498

8.931

7.193

6.253

5.656

5.241

4.933

4.695

4.505

4.349

3.859

3.598

29

13.391

8.849

7.121

6.186

5.593

5.179

4.873

4.636

4.447

4.292

3.804

3.543

30

13.293

8.773

7.054

6.125

5.534

5.122

4.817

4.581

4.393

4.239

3.753

3.493

40

12.609

8.251

6.595

5.698

5.128

4.731

4.436

4.207

4.024

3.874

3.400

3.145

60

11.973

7.768

6.171

5.307

4.757

4.372

4.086

3.865

3.687

3.541

3.078

2.827

120

11.380

7.321

5.781

4.947

4.416

4.044

3.767

3.552

3.379

3.237

2.783

2.534

100K

10.828

6.908

5.422

4.617

4.103

3.743

3.475

3.266

3.098

2.959

2.513

2.266

K: Multiply entries by 1,000 in first row of F-values, and last value for df.

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