Continuous Distributions Exponential:

Standard Normal:

Confidence Intervals

CONFIDENCE INTERVAL ON A MEAN (σ KNOWN)

where is the critical point corresponding to a tail area of

CONFIDENCE INTERVAL ON A MEAN (σ UNKNOWN)

where is the critical point corresponding to a tail area of

CONFIDENCE INTERVAL ON A DIFFERENCE IN MEANS (σ1 and σ2 KNOWN)

where is the critical point corresponding to a tail area of

CONFIDENCE INTERVAL ON A DIFFERENCE IN MEANS (σ1 = σ2 UNKNOWN)

where Sp is a “pooled” estimator of the unknown standard deviation and is calculated as

But this can only be used if both populations are normally distributed.

CONFIDENCE INTERVAL ON A DIFFERENCE IN MEANS (PAIRED SAMPLES)

But this can only be used if both populations are normally distributed.

CONFIDENCE INTERVAL ON A VARIANCE

But this can only be used if the population is normally distributed.

Hypothesis Testing

HYPOTHESIS TEST ON A MEAN (σ KNOWN)

Ho:

TS: ← uses known σ

Ha: Ha: Ha:

RR: RR: RR:

HYPOTHESIS TEST ON A MEAN (σ UNKNOWN, n LARGE)

Ho:

TS: ← uses s instead of σ

Ha: Ha: Ha:

RR: RR: RR:

HYPOTHESIS TEST ON A MEAN (σ UNKNOWN, n SMALL)

Ho:

TS: ← uses t instead of z

Ha: Ha: Ha:

RR: RR: RR:

DIFFERENCE BETWEEN MEANS (σ KNOWN)

Ho:

TS: ← uses known σ

Ha: Ha: Ha:

RR: RR: RR:

DIFFERENCE BETWEEN MEANS (σ UNKNOWN, n LARGE)

Ho:

TS: ← uses s instead of σ

Ha: Ha: Ha:

RR: RR: RR:

DIFFERENCE BETWEEN MEANS (σ UNKNOWN BUT EQUAL, n SMALL)

Ho:

TS: ← uses t instead of z

Ha: Ha: Ha:

RR: RR: RR:

DIFFERENCE BETWEEN MEANS (σ UNKNOWN AND UNEQUAL)

Ho:

TS:

⇐ round down!

Ha: Ha: Ha:

RR: RR: RR:

DIFFERENCE BETWEEN MEANS (PAIRED SAMPLES)

Ho:

TS:

Ha: Ha: Ha:

RR: RR: RR:

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

x x x x−xx−x

f

46 ENGINEERING PROBABILITY AND STATISTICS

STUDENT'S t-DISTRIBUTION

VALUES OF

α,n

α

t

tα,n

n

12345

6789

10

1112131415

1617181920

2122232425

26272829

∞30

0.10

3.0781.8861.6381.5331.476

1.4401.4151.3971.3831.372

1.3631.3561.3501.3451.341

1.3371.3331.3301.3281.325

1.3231.3211.3191.3181.316

1.3151.3141.3131.311

1.2821.310

0.05

6.3142.9202.3532.1322.015

1.9431.8951.8601.8331.812

1.7961.7821.7711.7611.753

1.7461.7401.7341.7291.725

1.7211.7171.7141.7111.708

1.7061.7031.7011.699

1.6451.697

0.025

12.7064.3033.1822.7762.571

2.4472.3652.3062.2622.228

2.2012.1792.1602.1452.131

2.1202.1102.1012.0932.086

2.0802.0742.0692.0642.060

2.0562.0522.0482.045

1.9602.042

0.01

31.8216.9654.5413.7473.365

3.1432.9982.8962.8212.764

2.7182.6812.6502.6242.602

2.5832.5672.5522.5392.528

2.5182.5082.5002.4922.485

2.4792.4732.4672.462

2.3262.457

n

12345

6789

10

1112131415

1617181920

2122232425

26272829

∞30

0.15

1.9631.3861.3501.1901.156

1.1341.1191.1081.1001.093

1.0881.0831.0791.0761.074

1.0711.0691.0671.0661.064

1.0631.0611.0601.0591.058

1.0581.0571.0561.055

1.0361.055

0.20

1.3761.0610.9780.9410.920

0.9060.8960.8890.8830.879

0.8760.8730.8700.8680.866

0.8650.8630.8620.8610.860

0.8590.8580.8580.8570.856

0.8560.8550.8550.854

0.8420.854

0.25

1.0000.8160.7650.7410.727

0.7180.7110.7060.7030.700

0.6970.6950.6940.6920.691

0.6900.6890.6880.6880.687

0.6860.6860.6850.6850.684

0.6840.6840.6830.683

0.6740.683

0.005

63.6579.9255.8414.6044.032

3.7073.4993.3553.2503.169

3.1063.0553.0122.9772.947

2.9212.8982.8782.8612.845

2.8312.8192.8072.7972.787

2.7792.7712.7632.756

2.5762.750

47ENGINEERING PROBABILITY AND STATISTICS

CR

ITIC

AL

VA

LU

ES

OF

TH

EF

DIS

TR

IBU

TIO

N–

TA

BL

E

For

ap

art

icu

lar

com

bin

ati

on

of

nu

mer

ato

ra

nd

den

om

ina

tor

deg

rees

of

free

dom

,en

try

rep

rese

nts

the

crit

ical

va

lues

of

F c

orr

esp

on

din

g

toa

spec

ifie

du

pp

erta

il a

rea

(α).

Nu

mer

ato

rdf

1D

enom

ina

tor

df2

12

34

56

78

91

01

21

52

02

43

04

06

01

20

1 2 3 4 5 6 7 8 91

0 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

.61

5.99

5.59

5.32

5.12

4.9

6

4.84

4.75

4.67

4.60

4.5

4

4.49

4.45

4.41

4.38

4.3

5

4.32

4.30

4.28

4.26

4.2

4

4.23

4.21

4.20

4.18

4.1

7

4.08

4.00

3.92

3.84

199.

519

.00

9.55

6.94

5.7

9

5.14

4.74

4.46

4.26

4.1

0

3.98

3.89

3.81

3.74

3.6

8

3.63

3.59

3.55

3.52

3.4

9

3.47

3.44

3.42

3.40

3.3

9

3.37

3.35

3.34

3.33

3.3

2

3.23

3.15

3.07

3.00

215.

719

.16

9.28

6.59

5.4

1

4.76

4.35

4.07

3.86

3.7

1

3.59

3.49

3.41

3.34

3.2

9

3.24

3.20

3.16

3.13

3.1

0

3.07

3.05

3.03

3.01

2.9

9

2.98

2.96

2.95

2.93

2.9

2

2.84

2.76

2.68

2.60

224.

619

.25

9.12

6.39

5.1

9

4.53

4.12

3.84

3.63

3.4

8

3.36

3.26

3.18

3.11

3.0

6

3.01

2.96

2.93

2.90

2.8

7

2.84

2.82

2.80

2.78

2.7

6

2.74

2.73

2.71

2.70

2.6

9

2.61

2.53

2.45

2.37

230.

219

.30

9.01

6.26

5.0

5

4.39

3.97

3.69

3.48

3.3

3

3.20

3.11

3.03

2.96

2.9

0

2.85

2.81

2.77

2.74

2.7

1

2.68

2.66

2.64

2.62

2.6

0

2.59

2.57

2.56

2.55

2.5

3

2.45

2.37

2.29

2.21

234.

019

.33

8.94

6.16

4.9

5

4.28

3.87

3 .58

3.37

3.2

2

3.09

3.00

2.92

2.85

2.7

9

2.74

2.70

2.66

2.63

2.6

0

2.57

2.55

2.53

2.51

2.4

9

2.47

2.46

2.45

2.43

2.4

2

2.34

2.25

2.17

2.10

236.

819

.35

8.89

6.09

4.8

8

4.21

3.79

3.50

3.29

3.1

4

3.01

2.91

2.83

2.76

2.7

1

2.66

2.61

2.58

2.54

2.5

1

2.49

2.46

2.44

2.42

2.4

0

2.39

2.37

2.36

2.35

2.3

3

2.25

2.17

2.09

2.01

238.

919

.37

8.85

6.04

4.8

2

4.15

3.73

3.44

3.23

3.0

7

2.95

2.85

2.77

2.70

2.6

4

2.59

2.55

2.51

2.48

2.4

5

2.42

2.40

2.37

2.36

2.3

4

2.32

2.31

2.29

2.28

2.2

7

2.18

2.10

2.02

1.94

240.

519

.38

8.81

6.00

4.7

7

4.10

3.68

3.39

3.18

3.0

2

2.90

2.80

2.71

2.65

2.5

9

2.54

2.49

2.46

2.42

2.3

9

2.37

2.34

2.32

2.30

2.2

8

2.27

2.25

2.24

2.22

2.2

1

2.12

2.04

1.96

1.88

241.

919

.40

8.79

5.96

4.7

4

4.06

3.64

3.35

3.14

2.9

8

2.85

2.75

2.67

2.60

2.5

4

2.49

2.45

2.41

2.38

2.3

5

2.32

2.30

2.27

2.25

2.2

4

2.22

2.20

2.19

2.18

2.1

6

2.08

1.99

1.91

1.83

243.

919

.41

8.74

5.91

4.6

8

4.00

3.57

3.28

3.07

2.9

1

2.79

2.69

2.60

2.53

2.4

8

2.42

2.38

2.34

2.31

2.2

8

2.25

2.23

2.20

2.18

2.1

6

2.15

2.13

2.12

2.10

2.0

9

2.00

1.92

1 .83

1.75

245.

919

.43

8.70

5.86

4.6

2

3.94

3.51

3.22

3.01

2.8

5

2.72

2.62

2.53

2.46

2.4

0

2.35

2.31

2.27

2.23

2.2

0

2.18

2.15

2.13

2.11

2.0

9

2.07

2.06

2.04

2.03

2.0

1

1.92

1.84

1.75

1.67

248.

019

.45

8.66

5.80

4.5

6

3.87

3.44

3.15

2.94

2.7

7

2.65

2.54

2.46

2.39

2.3

3

2.28

2.23

2.19

2.16

2.1

2

2.10

2.07

2.05

2.03

2.0

1

1.99

1.97

1.96

1.94

1.9

3

1.84

1.75

1.66

1.57

249.

119

.45

8.64

5.77

4.5

3

3.84

3.41

3.12

2.90

2.7

4

2.61

2.51

2.42

2.35

2.2

9

2.24

2.19

2.15

2.11

2.0

8

2.05

2.03

2.01

1.98

1.9

6

1.95

1.93

1.91

1.90

1.8

9

1.79

1.70

1.61

1.52

250.

119

.46

8.62

5.75

4.5

0

3.81

3.38

3.08

2.86

2.7

0

2.57

2.47

2.38

2.31

2.2

5

2.19

2.15

2.11

2.07

2.0

4

2.01

1.98

1.96

1.94

1.9

2

1.90

1.88

1.87

1.85

1.8

4

1.74

1.65

1.55

1.46

251.

119

.47

8.59

5.72

4.4

6

3.77

3.34

3.04

2.83

2.6

6

2.53

2.43

2.34

2.27

2.2

0

2.15

2.10

2.06

2.03

1.9

9

1.96

1.94

1.91

1.89

1.8

7

1.85

1.84

1.82

1.81

1.7

9

1.69

1.59

1.50

1.39

252.

219

.48

8.57

5.69

4.4

3

3.74

3.30

3.01

2.79

2.6

2

2.49

2.38

2.30

2.22

2.1

6

2.11

2.06

2.02

1.98

1.9

5

1.92

1.89

1.86

1.84

1.8

2

1.80

1.79

1.77

1.75

1.7

4

1.64

1.53

1.43

1.32

253.

319

.49

8.55

5.66

4.4

0

3.70

3.27

2.97

2.75

2.5

8

2.45

2.34

2.25

2.18

2.1

1

2.06

2.01

1.97

1.93

1.9

0

1.87

1.84

1.81

1.79

1.7

7

1.75

1.73

1.71

1.70

1.6

8

1.58

1.47

1.35

1.22

254.

319

.50

8.53

5.63

4.3

6

3.67

3.23

2.93

2.71

2.5

4

2.40

2.30

2.21

2.13

2.0

7

2.01

1.96

1.92

1.88

1.8

4

1.81

1.78

1.76

1.73

1.7

1

1.69

1.67

1.65

1.64

1.6

2

1.51

1.39

1.25

1.00

∞

∞

F(α,

df 1, d

f 2)F

α =

0.0

5

0

48 ENGINEERING PROBABILITY AND STATISTICS

critical values of x2 distributionC

RIT

ICA

L V

AL

UE

S O

FX

2 D

IST

RIB

UT

ION

f (X 2

)

0X

2

X2

α

α,n

Deg

rees

of

Fre

edom

.9

95

.990

.975

.950

.900

.100

.050

.025

.010

X2

.005

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

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99

140.

169

X2

X2

X2

X2

X2

X2

X2

X2

X2

Sour

ce: T

hom

pson

, C. M

., “T

able

s of

the

Perc

enta

ge P

oint

s of

the

X2 –Dis

trib

utio

n,”

Biom

etri

ka, ©

1941

, 32,

188

-189

. Rep

rodu

ced

by p

erm

issi

on o

f th

e Bi

omet

rika

Tru

stee

s.