I. Descriptive Statistics

= - Z*on

U=

bt=

sr=

sp= (q - l) + (nz -L)

b0 + brr

\{q-E)(u.i-il)l@5-D2

bs=!-bfi

,=#>("#)(T)

b, = rfu-'S,

sbl = trc;a

f1\rn--r)z

(nt - 1)s!+ (n2 -lsl

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II. Probability

P(AU B) -- P(A) + P(B) - P@n B)

P@lD=W

E(X) = tL, =\rnpu

var(X) = o2, = \t"n - lt")z pt

If X has a binomial distribution with parameters n and p, then:

P(x =n=(T)nr\-d"-k

IIa = 1W

o, = J71p(1fi

ITh=P

n^=.M--p I n

If r is the mean of a random sample of size n from an infinite population with mean pand standard deviation o, then:

Fz=V

6=_+* Jn

III. Inferential Statistics

Standardized test statistic: , :tutit:it , l*t."t:t, , ,,snnoaro oevlauon or stansuc

Confidence intervat statistic t (critical value) . (standard deviation of statistic)

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Single-Sample

Two-Sample

Difference of

sample means

Special case when61=62

Difference of

sample proportions

Special case whenPr=Pz

,[N:6r+

m$- il + p20- p2)

chi-square test statistic - 5' (observ-e-L:-e-:Pected)2

H expecrco

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Table entry

for a is the

probability

lying below a.

Table A

-3.4

-3.3_.L

'-3.1

-3.0

.0003

.0005

.0007

.0009

.0013

.0003

.0005

.0006

.0009

.0013

.0080 .0078

.0104 .0102

.0136 .0t32

.0774 .0170

.0222 .02t7

.0281 .0274

.0003 .0003 .0003

.0004 .0004 .0004

.0006 .0006 .0006

.0009 .0008 .0008

.0012 .00t2 .0011

.0075 .0073

.0099 .0096

.0129 .0125

.0166 .0162

.02t2 .0207

.0268 .0262

.0003

.0004

.0006

.0008

.0011

.0071 .0069

.0094 .0091

.0L22 .0119

.0158 .0154

.0202 .0197

.0256 .0250

.0003

.0004

.0005

.0007

.0010

.0002

.0003

.0005

.0007

.0010

.0838 .0823

.1003 .0985

.1190 .tr70

.1401 .1379

.1635 .1611

Standard normal probabilities

2 .00 .01 .02 .03 .04 .05

Probability

I

z

.06 .07 ,08 .09

.0003

.0005

.0007

.0010

.0013

.0082

.0003

.0004

.000s

.0008

.0011

-2.4qt

-2.1

-2.0

-1.9

-1.3

-L.2

-1.1

-1.0

-0.9

.0107

.0139

.0179

.0228

.0287

.0068 .0066 .0064

.0089 .0087 .0084

.0116 .0113 .0110

.0150 .0146 .0143

.0t92 .0188 .0183

.0244 .0239 .0233

_0968 .0951

.1151 .1131

.1357 .1335

.1587 .1562

.1841 .1814

.0901 .0885

.1075 .1056

.L27r .L25t

.t492 .1469

.1736 .L7LL

.0869 .0853

.1038 .L020

.1230 .L210

.t446 .1423

.1685 .1660

.0934

.ttt2

.1314

.1539

.1788

.0ei8

.1093

.1292

.1515

.1762

-0.3

-0.2

-0.1

-0.0

.382r .3783 .3745

.4207 .4168 .4129

.4602 .4562 .4522

.s000 .4960 .4920

.3707 .3669 .3632 .3594

.4090 .4052 .4013 .3974

.4483 .4443 .4404 .4364

.4880 .4840 .4801 .476r

.3557 .3520 .3483

.3936 .3897 .3859

.4325 .4286 .4247

.472t .4681 .4641

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Thble entryfor a is theprobabilitylying below e.

Table A (Conti,nued,)

z

Probabilitv

I

z

Standard normal probabilities

.02 .03 .04 .05 .08 .09.07.06

0.0

0.1

0.2

0.3

0.4

1.0

1.1

t.2

1.3

1.4

1.5

1.6

,,2.3

2.4OE

2.6

.5000 .5040

.5398 .5438

.5793 .5832

.6t79 .6217

.6554 .6591

.5160

.5557

.5948

.6331

.6700

.5199

.5596

.5987

.6368

.6736

.8508 .8531

.8729 .8749

.8925 .8944

.9099 .9115

.925I .9265

.9382 .9394

.9495 .9505

.5239 .5279

.5636 .5675

.6026 .6064

.6406 .6443

.6772 .6808

.8554 .8577

.8770 .8790

.8962 .8980

,9131 .9147

.9279 .9292

.9406 .9418

.9515 .9525

.5080

.5478

.5871

.6255

.6628

.5120

.5517

.5910

.6293

.6664

.5319

.57t4

.6103

.6480

.6844

.5359

.5753

.6141

.6517

.6879

.8413 .8438

.8643 .8665

.8849 .8869

.9032 .9049

.9t92 .9207

.9332 .9345

.9452 .9463

.8461 .8485

.8686 .8708

.8888 .8907

.9066 .9082

.9222 .9236

.9357 .9370

.9474 .9484

.8599 .862t

.8810 .8830

.8997 .9015

.9162 .9177

.9306 .9319

.9429 .9441

.9535 .9545

a,

3.4

.9861 .9864

.9893 .9896

.9918 .9920

.9938 .9940

.9953 .9955

.9868 .9871

.9898 .9901

.9922 .9925

.9941 .9943

.9956 .9957

.9875 .9878

.9904 .9906

.9927 .9929

.9945 .9946

.99s9 .9960

.9881 .9884

.9909 _9911

.9931 .9932

.9948 .9949

.9961 .9962

.9887 .9890

.9913 .9916

.9934 .9936

.9951 .9952

.9963 .9964

.9995 .9995 .9995

.9996 .9996 .9997

.9997 .9997 .9998

.9993 .9993 .9994

.9995 .9995 .9995

.9997 .9997 .9997

.9994 .9994 .9994 .9994

.9996 .9996 .9996 .9996

.9997 .9997 .9997 .9997

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Table entry forpand C is the pointt* with probabilityp lying above itand probability Clying between

-t* and t*.

Table B

?.€i-:$

2l22

23

24

25

40

50

60

80

100

1000

Probabilityp

I

t*t distribution critical values

Tail

df .25 .20 .15 .10 .05 .025

It3

4

5

.002s .001 .0005

1.000 1.376 1.963 3.078 6.314 t2.7t 15.89 31.82 63.66

.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 31.60

.765 .978 Lzffi 1.638 2.353 3.182 3.482 4.54t 5.841 7.453 10.21 L2.92

.74t .941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610

.727 .920 1.156 L.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869

.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437

.873 r.083 1.356 1.782 2.179 2.303 2.681 3.0s5 3.428 3.930 4.318

.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221

.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140

.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.M3

1.063 1.323 1.72t 2.080 2.189 2.518 2.831 3.135 3.527 3.819

1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792

1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768

1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745

1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725

1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551

t.047 1.299 t.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496

1.045 1.296 t.67t 2.000 2.099 2.390 2.660 2.915 3.232 3.460

1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416

1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.t74 3.390

1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300

1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291

70% 95% 96% 98% 99% 99.5% 99.8% 99.9%

Confidence level C

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.697

.695

.694

.692

.691

.859

.858

.858

.857

.856

.686

.686

.685

.685

.684

.851

.849

.848

.846

.845

.842

.841

60%

.681

.679

.679

.678

.677

.675

.674

50%

18

Thble entryforp is thepoint (22) withprobabilityplying above it.

Table C

1

t

J

4

5

1.32

2.77

4.11

5.39

6.63

11

12

13

14

l5

.25 .20 .15dT

Probabilityp

I

(x2)

,fzcritical values

Tail probabilityp

.10 .05 .02s

l.6t3.22

4.U

5.99

7.29

24.93 26.17

26.04 27.30

27.14 28.43

28.24 29.55

n.u 30.68

45.62 47.27

56.33 58.16

66.98 68.97

88.13 90.41

109.1 111.7

3.84 5.02

5.99 7.38

7.81 9.35

9.49 11.14

11.07 12.83

6.63 7.88

9.2r 10.60

11.34 12.84

13.28 14.86

15.09 16.75

9.14 10.83

11.98 13.82

14.32 16.27

t6.42 18.47

18.39 20.51

2.07

319

5.32

6.74

8.r2

2.71

4.61

6.25

7.78

9.24

5.41

7.82

9.84

11.67

13.39

13.70 14.63 t5.77

14.85 15.81 16.99

15.98 16.98 18.20

17.12 18.15 19.41

18.25 19.31 20.60

17.28 19.68

18.55 21.03

19.81 22.36

21.06 23.68

22.31 25.00

29.62

30.81

32.01

33.20

34.38

32.67

33.92

35.17

36.42

37.65

2r.92 22.62

23.34 24.05

24.74 25.47

26.12 26.87

27.49 28.26

26.76 28.n T.26

28.30 30.32 32.91

29.82 31.88 34.53

31.32 33.43 36.12

32.80 34.95 37.70

41.40 43.78 46.80

42.80 45.20 48.27

44.18 46.62 49:73

45.56 48.03 51.18

46.93 49.44 52.62

24.72

26.22

27.69

29.14

30.58

21

,t

23

24

25

27.66

28.82

29.98

31.13

32.28

35.48

36.78

38.08

39.36

40.65

36.34 38.93

37.66 40.29

38.97 At.M

40.27 42.98

4t.57 44.31

40

50

60

80

100

49.24

60.35

71.U

93.11

114.7

51.81 55.76

63.17 67.50

74.40 79.08

96.58 101.9

118.5 124.3

59.34 60.44

71.42 72.61

83.30 84.58

106.6 108.1

t29.6 131.1

63.69 66.77

76.15 79.49

88.38 91.95

112.3 116.3

135.8 140.2

69.70 73.40

82.66 86.66

95.34 99.61

120.1 124.8

144.3 149.4

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