45ENGINEERING PROBABILITY AND STATISTICS

UNIT NORMAL DISTRIBUTION

x f(x) F(x) R(x) 2R(x) W(x)0.00.10.20.30.4

0.50.60.70.80.9

1.01.11.21.31.4

1.51.61.71.81.9

2.02.12.22.32.4

2.52.62.72.82.93.0

Fractiles1.28161.64491.96002.05372.32632.5758

0.39890.39700.39100.38140.3683

0.35210.33320.31230.28970.2661

0.24200.21790.19420.17140.1497

0.12950.11090.09400.07900.0656

0.05400.04400.03550.02830.0224

0.01750.01360.01040.00790.00600.0044

0.17550.10310.05840.04840.02670.0145

0.50000.53980.57930.61790.6554

0.69150.72570.75800.78810.8159

0.84130.86430.88490.90320.9192

0.93320.94520.95540.96410.9713

0.97720.98210.98610.98930.9918

0.99380.99530.99650.99740.99810.9987

0.90000.95000.97500.98000.99000.9950

0.50000.46020.42070.38210.3446

0.30850.27430.24200.21190.1841

0.15870.13570.11510.09680.0808

0.06680.05480.04460.03590.0287

0.02280.01790.01390.01070.0082

0.00620.00470.00350.00260.00190.0013

0.10000.05000.02500.02000.01000.0050

1.00000.92030.84150.76420.6892

0.61710.54850.48390.42370.3681

0.31730.27130.23010.19360.1615

0.13360.10960.08910.07190.0574

0.04550.03570.02780.02140.0164

0.01240.00930.00690.00510.00370.0027

0.20000.10000.05000.04000.02000.0100

0.00000.07970.15850.23580.3108

0.38290.45150.51610.57630.6319

0.68270.72870.76990.80640.8385

0.86640.89040.91090.92810.9426

0.95450.96430.97220.97860.9836

0.98760.99070.99310.99490.99630.9973

0.80000.90000.95000.96000.98000.9900

46 ENGINEERING PROBABILITY AND STATISTICS

STUDENT'S t-DISTRIBUTION

VALUES OFα,n

α

ttα,n

n123456789

1011121314151617181920212223242526272829

∞30

0.103.0781.8861.6381.5331.4761.4401.4151.3971.3831.3721.3631.3561.3501.3451.3411.3371.3331.3301.3281.3251.3231.3211.3191.3181.3161.3151.3141.3131.311

1.2821.310

0.056.3142.9202.3532.1322.0151.9431.8951.8601.8331.8121.7961.7821.7711.7611.7531.7461.7401.7341.7291.7251.7211.7171.7141.7111.7081.7061.7031.7011.699

1.6451.697

0.02512.7064.3033.1822.7762.5712.4472.3652.3062.2622.2282.2012.1792.1602.1452.1312.1202.1102.1012.0932.0862.0802.0742.0692.0642.0602.0562.0522.0482.045

1.9602.042

0.0131.8216.9654.5413.7473.3653.1432.9982.8962.8212.7642.7182.6812.6502.6242.6022.5832.5672.5522.5392.5282.5182.5082.5002.4922.4852.4792.4732.4672.462

2.3262.457

n123456789

1011121314151617181920212223242526272829

∞30

0.151.9631.3861.3501.1901.1561.1341.1191.1081.1001.0931.0881.0831.0791.0761.0741.0711.0691.0671.0661.0641.0631.0611.0601.0591.0581.0581.0571.0561.055

1.0361.055

0.201.3761.0610.9780.9410.9200.9060.8960.8890.8830.8790.8760.8730.8700.8680.8660.8650.8630.8620.8610.8600.8590.8580.8580.8570.8560.8560.8550.8550.854

0.8420.854

0.251.0000.8160.7650.7410.7270.7180.7110.7060.7030.7000.6970.6950.6940.6920.6910.6900.6890.6880.6880.6870.6860.6860.6850.6850.6840.6840.6840.6830.683

0.6740.683

0.00563.6579.9255.8414.6044.0323.7073.4993.3553.2503.1693.1063.0553.0122.9772.9472.9212.8982.8782.8612.8452.8312.8192.8072.7972.7872.7792.7712.7632.756

2.5762.750

48 ENGINEERING PROBABILITY AND STATISTICS

critical values of xcritical values of x2 distribution distributionC

RIT

ICA

L V

ALU

ES O

FX

2 DIS

TRIB

UTI

ON

f (X 2

)

0X

2

X2

α

α,n

Deg

rees

of F

reed

om

.995

.9

90.9

75.9

50.9

00.1

00.0

50.0

25.0

10X

2.0

05

1

2

3

4 5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

40

50

60

70

80

90

100

0.

0000

393

0.

0100

251

0.

0717

212

0.

2069

90

0.41

1740

0.67

5727

0.98

9265

1.34

4419

1.73

4926

2.15

585

2.

6032

1

3.07

382

3.

5650

3

4.07

468

4.

6009

4

5.14

224

5.

6972

4

6.26

481

6.

8439

8

7.43

386

8.

0336

6

8.64

272

9.

2604

2

9.88

623

10

.519

7

11.1

603

11

.807

6

12.4

613

13

.121

1

13.7

867

20

.706

5

27.9

907

35

.534

6

43.2

752

51

.172

0

59.1

963

67

.327

6

0.

0001

571

0.

0201

007

0.

1148

32

0.

2971

10

0.

5543

00

0.

8720

85

1.

2390

43

1.

6464

82

2.

0879

12

2.

5582

1

3.05

347

3.

5705

6

4.10

691

4.

6604

3

5.22

935

5.

8122

1

6.40

776

7.

0149

1

7.63

273

8.

2604

0

8.89

720

9.

5424

9

10.1

9567

10.8

564

11

.524

0

12.1

981

12

.878

6

13.5

648

14

.256

5

14.9

535

22

.164

3

29.7

067

37

.484

8

45.4

418

53

.540

0

61.7

541

70

.064

8

0.00

0982

10.

0506

356

0.21

5795

0.48

4419

0.83

1211

1.23

7347

1.68

987

2.17

973

2.70

039

3.24

697

3.81

575

4.40

379

5.00

874

5.62

872

6.26

214

6.90

766

7.56

418

8.23

075

8.90

655

9.59

083

10.2

8293

10.9

823

11.6

885

12.4

011

13.1

197

13.8

439

14.5

733

15.3

079

16.0

471

16.7

908

24.4

331

32.3

574

40.4

817

48.7

576

57.1

532

65.6

466

74.2

219

0.

0039

321

0.

1025

87

0.

3518

46

0.

7107

21

1.

1454

76

1.

6353

9

2.16

735

2.

7326

4

3.32

511

3.

9403

0

4.57

481

5.

2260

3

5.89

186

6.

5706

3

7.26

094

7.

9616

4

8.67

176

9.

3904

6

10.1

170

10

.850

8

11.5

913

12

.338

0

13.0

905

13

.848

4

14.6

114

15

.379

1

16.1

513

16

.927

9

17.7

083

18

.492

6

26.5

093

34

.764

2

43.1

879

51

.739

3

60.3

915

69

.126

0

77.9

295

0.

0157

908

0.

2107

20

0.

5843

75

1.

0636

23

1.

6103

1

2.20

413

2.

8331

1

3.48

954

4.

1681

6

4.86

518

5.

5777

9

6.30

380

7.

0415

0

7.78

953

8.

5467

5

9.31

223

10

.085

2

10.8

649

11

.650

9

12.4

426

13

.239

6

14.0

415

14

.847

9

15.6

587

16

.473

4

17.2

919

18

.113

8

18.9

392

19

.767

7

20.5

992

29

.050

5

37.6

886

46

.458

9

55.3

290

64

.277

8

73.2

912

82

.358

1

2.

7055

4

4.60

517

6.

2513

9

7.77

944

9.

2363

5

10.6

446

12

.017

0

13.3

616

14

.683

7

15.9

871

17

.275

0

18.5

494

19

.811

9

21.0

642

22

.307

2

23.5

418

24

.769

0

25.9

894

27

.203

6

28.4

120

29

.615

1

30.8

133

32

.006

9

33.1

963

34

.381

6

35.5

631

36

.741

2

37.9

159

39

.087

5

40.2

560

51

.805

0

63.1

671

74

.397

0

85.5

271

96

.578

2 1

07.5

65

118

.498

3.

8414

6

5.99

147

7.

8147

3

9.48

773

11

.070

5

12.5

916

14

.067

1

15.5

073

16

.919

0

18.3

070

19

.675

1

21.0

261

22

.362

1

23.6

848

24

.995

8

26.2

962

27

.587

1

28.8

693

30

.143

5

31.4

104

32

.670

5

33.9

244

35

.172

5

36.4

151

37

.652

5

38.8

852

40

.113

3

41.3

372

42

.556

9

43.7

729

55

.758

5

67.5

048

79

.081

9

90.5

312

101

.879

1

13.1

45

124

.342

5.

0238

9

7.37

776

9.

3484

0

11.1

433

12

.832

5

14.4

494

16

.012

8

17.5

346

19

.022

8

20.4

831

21

.920

0

23.3

367

24

.735

6

26.1

190

27

.488

4

28.8

454

30

.191

0

31.5

264

32

.852

3

34.1

696

35

.478

9

36.7

807

38

.075

7

39.3

641

40

.646

5

41.9

232

43

.194

4

44.4

607

45

.722

2

46.9

792

59

.341

7

71.4

202

83

.297

6

95.0

231

106

.629

1

18.1

36

129

.561

6.

6349

0

9.21

034

11

.344

9

13.2

767

15

.086

3

16.8

119

18

.475

3

20.0

902

21

.666

0

23.2

093

24

.725

0

26.2

170

27

.688

3

29.1

413

30

.577

9

31.9

999

33

.408

7

34.8

053

36

.190

8

37.5

662

38

.932

1

40.2

894

41

.638

4

42.9

798

44

.314

1

45.6

417

46

.963

0

48.2

782

49

.587

9

50.8

922

63

.690

7

76.1

539

88

.379

4 1

00.4

25

112

.329

1

24.1

16

135

.807

7.

8794

4

10.5

966

12

.838

1

14.8

602

16

.749

6

18.5

476

20

.277

7

21.9

550

23

.589

3

25.1

882

26

.756

9

28.2

995

29

.819

4

31.3

193

32

.801

3

34.2

672

35

.718

5

37.1

564

38

.582

2

39.9

968

41

.401

0

42.7

956

44

.181

3

45.5

585

46

.927

8

48.2

899

49

.644

9

50.9

933

52

.335

6

53.6

720

66

.765

9

79.4

900

91

.951

7 1

04.2

15

116

.321

1

28.2

99

140.

169

X2

X2

X2

X2

X2

X2

X2

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, C. M

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, ©19

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88-1

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47ENGINEERING PROBABILITY AND STATISTICS

CR

ITIC

AL

VA

LU

ES

OF

TH

EF

DIS

TR

IBU

TIO

N–

TA

BL

E

For

apa

rtic

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com

bina

tion

of

num

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s of

free

dom

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pres

ents

the

criti

calv

alue

sof F

cor

resp

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toa

spec

ified

uppe

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ea(α

).

Num

erat

ordf

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enom

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ordf

21

23

45

67

89

1012

1520

2430

4060

120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120

161.

418

.51

10.1

37.

716.

615.

995.

595.

325.

124.

964.

844.

754.

674.

604.

544.

494.

454.

414.

384.

354.

324.

304.

284.

264.

244.

234.

214.

204.

184.

174.

084.

003.

923.

84

199.

519

.00

9.55

6.94

5.79

5.14

4.74

4.46

4.26

4.10

3.98

3.89

3.81

3.74

3.68

3.63

3.59

3.55

3.52

3.49

3.47

3.44

3.42

3.40

3.39

3.37

3.35

3.34

3.33

3.32

3.23

3.15

3.07

3.00

215.

719

.16

9.28

6.59

5.41

4.76

4.35

4.07

3.86

3.71

3.59

3.49

3.41

3.34

3.29

3.24

3.20

3.16

3.13

3.10

3.07

3.05

3.03

3.01

2.99

2.98

2.96

2.95

2.93

2.92

2.84

2.76

2.68

2.60

224.

619

.25

9.12

6.39

5.19

4.53

4.12

3.84

3.63

3.48

3.36

3.26

3.18

3.11

3.06

3.01

2.96

2.93

2.90

2.87

2.84

2.82

2.80

2.78

2.76

2.74

2.73

2.71

2.70

2.69

2.61

2.53

2.45

2.37

230.

219

.30

9.01

6.26

5.05

4.39

3.97

3.69

3.48

3.33

3.20

3.11

3.03

2.96

2.90

2.85

2.81

2.77

2.74

2.71

2.68

2.66

2.64

2.62

2.60

2.59

2.57

2.56

2.55

2.53

2.45

2.37

2.29

2.21

234.

019

.33

8.94

6.16

4.95

4.28

3.87

3.58

3.37

3.22

3.09

3.00

2.92

2.85

2.79

2.74

2.70

2.66

2.63

2.60

2.57

2.55

2.53

2.51

2.49

2.47

2.46

2.45

2.43

2.42

2.34

2.25

2.17

2.10

236.

819

.35

8.89

6.09

4.88

4.21

3.79

3.50

3.29

3.14

3.01

2.91

2.83

2.76

2.71

2.66

2.61

2.58

2.54

2.51

2.49

2.46

2.44

2.42

2.40

2.39

2.37

2.36

2.35

2.33

2.25

2.17

2.09

2.01

238.

919

.37

8.85

6.04

4.82

4.15

3.73

3.44

3.23

3.07

2.95

2.85

2.77

2.70

2.64

2.59

2.55

2.51

2.48

2.45

2.42

2.40

2.37

2.36

2.34

2.32

2.31

2.29

2.28

2.27

2.18

2.10

2.02

1.94

240.

519

.38

8.81

6.00

4.77

4.10

3.68

3.39

3.18

3.02

2.90

2.80

2.71

2.65

2.59

2.54

2.49

2.46

2.42

2.39

2.37

2.34

2.32

2.30

2.28

2.27

2.25

2.24

2.22

2.21

2.12

2.04

1.96

1.88

241.

919

.40

8.79

5.96

4.74

4.06

3.64

3.35

3.14

2.98

2.85

2.75

2.67

2.60

2.54

2.49

2.45

2.41

2.38

2.35

2.32

2.30

2.27

2.25

2.24

2.22

2.20

2.19

2.18

2.16

2.08

1.99

1.91

1.83

243.

919

.41

8.74

5.91

4.68

4.00

3.57

3.28

3.07

2.91

2.79

2.69

2.60

2.53

2.48

2.42

2.38

2.34

2.31

2.28

2.25

2.23

2.20

2.18

2.16

2.15

2.13

2.12

2.10

2.09

2.00

1.92

1.83

1.75

245.

919

.43

8.70

5.86

4.62

3.94

3.51

3.22

3.01

2.85

2.72

2.62

2.53

2.46

2.40

2.35

2.31

2.27

2.23

2.20

2.18

2.15

2.13

2.11

2.09

2.07

2.06

2.04

2.03

2.01

1.92

1.84

1.75

1.67

248.

019

.45

8.66

5.80

4.56

3.87

3.44

3.15

2.94

2.77

2.65

2.54

2.46

2.39

2.33

2.28

2.23

2.19

2.16

2.12

2.10

2.07

2.05

2.03

2.01

1.99

1.97

1.96

1.94

1.93

1.84

1.75

1.66

1.57

249.

119

.45

8.64

5.77

4.53

3.84

3.41

3.12

2.90

2.74

2.61

2.51

2.42

2.35

2.29

2.24

2.19

2.15

2.11

2.08

2.05

2.03

2.01

1.98

1.96

1.95

1.93

1.91

1.90

1.89

1.79

1.70

1.61

1.52

250.

119

.46

8.62

5.75

4.50

3.81

3.38

3.08

2.86

2.70

2.57

2.47

2.38

2.31

2.25

2.19

2.15

2.11

2.07

2.04

2.01

1.98

1.96

1.94

1.92

1.90

1.88

1.87

1.85

1.84

1.74

1.65

1.55

1.46

251.

119

.47

8.59

5.72

4.46

3.77

3.34

3.04

2.83

2.66

2.53

2.43

2.34

2.27

2.20

2.15

2.10

2.06

2.03

1.99

1.96

1.94

1.91

1.89

1.87

1.85

1.84

1.82

1.81

1.79

1.69

1.59

1.50

1.39

252.

219

.48

8.57

5.69

4.43

3.74

3.30

3.01

2.79

2.62

2.49

2.38

2.30

2.22

2.16

2.11

2.06

2.02

1.98

1.95

1.92

1.89

1.86

1.84

1.82

1.80

1.79

1.77

1.75

1.74

1.64

1.53

1.43

1.32

253.

319

.49

8.55

5.66

4.40

3.70

3.27

2.97

2.75

2.58

2.45

2.34

2.25

2.18

2.11

2.06

2.01

1.97

1.93

1.90

1.87

1.84

1.81

1.79

1.77

1.75

1.73

1.71

1.70

1.68

1.58

1.47

1.35

1.22

254.

319

.50

8.53

5.63

4.36

3.67

3.23

2.93

2.71

2.54

2.40

2.30

2.21

2.13

2.07

2.01

1.96

1.92

1.88

1.84

1.81

1.78

1.76

1.73

1.71

1.69

1.67

1.65

1.64

1.62

1.51

1.39

1.25

1.00

∞

∞

F(α,

df 1, d

f 2)F

α =

0.05

0