Statistics Formulas

Formulas and an d Tables Tables

for Essentials of Statistics, by Mario F. Triola ©2002 by Addison-Wesley. Ch. 2: Descriptive Statistics x x

s s s

Sx n

x2E,m,x1E

Mean

f . x S

S f

Å Å 3 Å

Ch. 6: Confidence Intervals (one population)

> ! n

where E 5 za 2

S (x 2 x) 2

Standard deviation

n21

n ( Sx 2 ) 2 ( Sx ) 2 n (n 2 1)

4 3

Standard deviation (shortcut)

n S f ( f . x2 ) 2 S f ( f . x )

4

2

n ( n 2 1)

Standard deviation (frequency table)

or E 5 ta 2

0

P (A ) 5 1 2 P (A) Rule of complements n! Permutations (no elements alike) nPr 5 (n 2 r) ! n! Permutations (n ( n1 alike, ...) n1! n2! . . . n k! n! Combinations nCr 5 ( n 2 r ) ! r!

Ch. 4: Probability Distributions . P(x ) Mean (prob. dist.) 2 2 Standard deviation (prob. dist.)   [x . P(x)]   n! . px . qn  x Binomial probability P(x)  (n  x)! x! Mean (binomial)   n . p 2 Variance (binomial)   n . p . q

( n 2 1 ) s2 xR2

n5 n5

Standard deviation (binomial)

xx x or Standard score z s  Central limit theorem  x   

n

2

,s ,

( n 2 1 ) s2 x L2

Variance

Central limit theorem (Standard error)

B> R 3 >4 3 >4 za 2s

2

E za 2 2 . 0.25 E2 pˆ ˆ qˆ za 2 2 ˆ

Mean Proportion

ˆ and ˆ q are known) Proportion ( p

E2

Ch. 7: Test Statistics (one population) z5 t5 z5

> ! > !

x2m

Mean—one population ( known or n  30)

s n x2m s

Mean—one population ( unknown and n  30)

n

ˆ pˆ 2 p

Proportion—one population

Å

pq

x2 5

n ( n 2 1 ) s2

s

2

Standard deviation or variance— one population

Ch. 8: Test Statistics (two populations) z5

( x1 2 x2 ) 2 ( m1 2 m2 )

Å > !

s 12

s 22

n1 1 n2

Ch. 5: Normal Distribution

  x

Proportion pˆ ˆ ˆ qˆ

> Å n

where E 5 za 2

 x



(s unknown and n  30)

pˆ  E  p  pˆ  E

n5

if A, P (A or B) 5 P (A) 1 P (B) if A, B are mutually exclusive P (A or B) 5 P (A) 1 P (B) 2 P (A and B) A, B are not mutually exclusive if A, if if A, A, B are independent P (A and B) 5 P (A) . P (B) if if A, P (A and B) 5 P (A) . P (B A) if A, B are dependent

n . p . q

s

(s known or n  30)

Ch. 6: Sample Size Determination

Ch. 3: Probability

 

s

> ! n

Mean (frequency table)

variance  s2



Mean

t5

z5

d 2 md sd

n

Two means—independent (1, 2 known or n1  30 and n2  30)

Two means—matched pairs (df  n  1)

pˆ 1 2 ˆ pˆ 2 ) 2 ( p1 2 p2 ) ( ˆ

Å

pq pq n1 1 n2

Two proportions

Formulas and Tables

for Essentials of Statistics, by Mario F. Triola ©2002 by Addison-Wesley. Ch. 8: Confidence Intervals (two populations) ( x1 2 x2 ) 2 E , ( m1 2 m2 ) , ( x1 2 x2 ) 1 E

> Å n

s 12

where E 5 za 2

1

1

s 22

Ch. 10: Multinomial and Contingency Tables (Indep.)

‹

n2

2

x 5 g x2 5 g

( s1, s2 known or n1 . 30 and n2 . 30 ) s > ! n d

(df  n  1)

( pˆ 1 2 pˆ 2 ) 2 E , ( p1 2 p2 ) , ( pˆ 1 2 pˆ 2 ) 1 E

> Å pnq 1 pnq ˆ 1 ˆ 1

where E 5 za 2

1

b1 5

"

r2 5 se 5

"

nSxy 2 ( Sx ) ( Sy ) n ( Sy 2 ) 2 ( Sy ) 2

nSxy 2 ( Sx ) ( Sy ) n ( Sx 2 ) 2 ( Sx ) 2

ˆ y 5 b0 1 b1x

( Sy ) ( Sx 2 ) 2 ( Sx ) ( Sxy ) n ( Sx 2 ) 2 ( Sx ) 2

Estimated eq. of regression line

explained variation total variation

Å

y) 2 S ( y 2 ˆ n22

or

Å

Sy2 2 b0Sy 2 b1Sxy n22

yˆ  E  y  ˆ yE where E  t 2s e 

rs 5 1 2

a

6Sd2 n ( n2 2 1 )

Contingency table [df  (r  1)(c  1)]

E

(row total) (column total) (grand total)

1



1 n(x 0  x )2  n n( x 2)  ( x )2

Rank correlation

critical value for n . 30:

Ch. 10: One-Way Analysis of a Variance F5

F5

2

n ( Sx 2 ) 2 ( Sx ) 2

b0 5 y 2 b1x or b0 5

E

(O 2 E) 2

where E 5

ˆ 2 ˆ 2

Ch. 9: Linear Correlation/Regression Correlation r 5

Multinomial (df  k  1)

(Matched Pairs)

d 2 E , md , d 1 E

where E 5 ta 2

(O 2 E) 2

! b 6z n21

2 ns 2 x

sp2

k samples each of size n (num. df  k  1; den. df  k(n  1))

MS(treatment)

←

MS(error)

←

MS(treatment) 5 MS(error) 5

df  k  1 df  N  k

SS(treatment) k21

SS(error) N2k

MS(total) 5

SS(total) N21

SS(treatment) 5 n1 ( x1 2 x ) 2 1 . . . 1 nk ( xk 2 x ) 2 SS(error) 5 ( n1 2 1 ) s21 1 . . . 1 ( nk 2 1 ) s 2k SS(total) 5 S ( x 2 x ) 2 SS(total) 5 SS(treatment) 1 SS(error)

HYPOTHESIS TESTING 1. 2. 3. 4.

5. 6. 7. 8.

Identify the specific claim or hypothesis to be tested and put it in symbolic form. Give the symbolic form that must be true when the original claim is false. Of the two symbolic expressions obtained so far, let the null hypothesis H 0 be the one that contains the condition of equality; H 1 is the other statement. Select the significance level a based on the seriousness of a type I error. Make a small if the consequences of rejecting a true H 0 are severe. The values of 0.05 and 0.01 are very common. Identify the statistic that is relevant to this test, and identify its sampling distribution. Determine the test statistic and either the P-value or the critical values, and the critical region. Draw a graph. Reject H 0: Test statistic is in the critical region or P-value # a. Fail to reject H 0: Test statistic is not in the critical region or P-value . a. Restate this previous conclusion in simple, nontechnical terms.

FINDING P-VALUES Start

What type of test ? Two-tailed

Left-tailed

Left

P -value

area to the left of the test statistic 

P -value

P -value



Is the test statistic to the right or left of center ?

twice

the area to the left of the test statistic P -value is

twice this area. m

Test statistic

Right-tailed

Right

P -value

twice

the area to the right of the test statistic P -value is

P -value

area to the right of the test statistic 

P -value

twice this area. m

m

Test statistic



Test statistic

m

Test statistic

z

0

TABLE A-2

Standard Normal ( z) Distribution

z

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0 0.1 0.2 0.3 0.4

.0000 .0398 .0793 .1179 .1554

.0040 .0438 .0832 .1217 .1591

.0080 .0478 .0871 .1255 .1628

.0120 .0517 .0910 .1293 .1664

.0160 .0557 .0948 .1331 .1700

.0199 .0596 .0987 .1368 .1736

.0239 .0636 .1026 .1406 .1772

.0279 .0675 .1064 .1443 .1808

.0319 .0714 .1103 .1480 .1844

.0359 .0753 .1141 .1517 .1879

0.5 0.6 0.7 0.8 0.9

.1915 .2257 .2580 .2881 .3159

.1950 .2291 .2611 .2910 .3186

.1985 .2324 .2642 .2939 .3212

.2019 .2357 .2673 .2967 .3238

.2054 .2389 .2704 .2995 .3264

.2088 .2422 .2734 .3023 .3289

.2123 .2454 .2764 .3051 .3315

.2157 .2486 .2794 .3078 .3340

.2190 .2517 .2823 .3106 .3365

.2224 .2549 .2852 .3133 .3389

1.0 1.1 1.2 1.3 1.4

.3413 .3643 .3849 .4032 .4192

.3438 .3665 .3869 .4049 .4207

.3461 .3686 .3888 .4066 .4222

.3485 .3708 .3907 .4082 .4236

.3508 .3729 .3925 .4099 .4251

.3531 .3749 .3944 .4115 .4265

.3554 .3770 .3962 .4131 .4279

.3577 .3790 .3980 .4147 .4292

.3599 .3810 .3997 .4162 .4306

.3621 .3830 .4015 .4177 .4319

1.5 1.6 1.7 1.8 1.9

.4332 .4452 .4554 .4641 .4713

.4345 .4463 .4564 .4649 .4719

.4357 .4474 .4573 .4656 .4726

.4370 .4484 .4582 .4664 .4732

.4382 .4495 .4591 .4671 .4738

.4394 .4505 .4599 .4678 .4744

.4406 .4515 .4608 .4686 .4750

.4418 .4525 .4616 .4693 .4756

.4429 .4535 .4625 .4699 .4761

.4441 .4545 .4633 .4706 .4767

2.0 2.1 2.2 2.3 2.4

.4772 .4821 .4861 .4893 .4918

.4778 .4826 .4864 .4896 .4920

.4783 .4830 .4868 .4898 .4922

.4788 .4834 .4871 .4901 .4925

.4793 .4838 .4875 .4904 .4927

.4798 .4842 .4878 .4906 .4929

.4803 .4846 .4881 .4909 .4931

.4808 .4850 .4884 .4911 .4932

.4812 .4854 .4887 .4913 .4934

.4817 .4857 .4890 .4916 .4936

2.5 2.6 2.7 2.8 2.9

.4938 .4953 .4965 .4974 .4981

.4940 .4955 .4966 .4975 .4982

.4941 .4956 .4967 .4976 .4982

.4943 .4957 .4968 .4977 .4983

.4945 .4959 .4969 .4977 .4984

.4946 .4960 .4970 .4978 .4984

.4948 .4961 .4971 .4979 .4985

.4949 .4962 .4972 .4979 .4985

.4951 .4963 .4973 .4980 .4986

.4952 .4964 .4974 .4981 .4986

3.0 3.10 and higher

.4987

.4987

.4987

.4988

.4988

.4989

.4989

.4989

.4990

.4990

∗

∗

.4999

NOTE: For values of z above 3.09, use 0.4999 for the area. *Use these common values that result from interpolation: z score

Area

1.645

0.4500

2.575

0.4950

From Frederick C. Mosteller and Robert E. K. Rourke, Sturdy Statistics, 1973, Addison-Wesley Publishing Co., Reading, MA. Reprinted with permission of Frederick Mosteller.

Student t distribution Right tail

Left tail

a

a

Critical t score (negative) TABLE A-3

Two tails

a

/2

a

/2

Critical t scoreCritical t scoreCritical t score (positive) (negative) (positive)

t Distribution

a Degrees of Freedom

.005 (one tail) .01 (two tails)






1 2 3 4 5

63.657 9.925 5.841 4.604 4.032

31.821 6.965 4.541 3.747 3.365

12.706 4.303 3.182 2.776 2.571

6.314 2.920 2.353 2.132 2.015

3.078 1.886 1.638 1.533 1.476

1.000 .816 .765 .741 .727

6 7 8 9 10

3.707 3.500 3.355 3.250 3.169

3.143 2.998 2.896 2.821 2.764

2.447 2.365 2.306 2.262 2.228

1.943 1.895 1.860 1.833 1.812

1.440 1.415 1.397 1.383 1.372

.718 .711 .706 .703 .700

11 12 13 14 15

3.106 3.054 3.012 2.977 2.947

2.718 2.681 2.650 2.625 2.602

2.201 2.179 2.160 2.145 2.132

1.796 1.782 1.771 1.761 1.753

1.363 1.356 1.350 1.345 1.341

.697 .696 .694 .692 .691

16 17 18 19 20

2.921 2.898 2.878 2.861 2.845

2.584 2.567 2.552 2.540 2.528

2.120 2.110 2.101 2.093 2.086

1.746 1.740 1.734 1.729 1.725

1.337 1.333 1.330 1.328 1.325

.690 .689 .688 .688 .687

21 22 23 24 25

2.831 2.819 2.807 2.797 2.787

2.518 2.508 2.500 2.492 2.485

2.080 2.074 2.069 2.064 2.060

1.721 1.717 1.714 1.711 1.708

1.323 1.321 1.320 1.318 1.316

.686 .686 .685 .685 .684

26 27 28 29 Large (z)

2.779 2.771 2.763 2.756 2.575

2.479 2.473 2.467 2.462 2.326

2.056 2.052 2.048 2.045 1.960

1.706 1.703 1.701 1.699 1.645

1.315 1.314 1.313 1.311 1.282

.684 .684 .683 .683 .675

Formulas and Tables

for Essentials of Statistics, by Mario F. Triola ©2002 by Addison-Wesley. TABLE A-4

Chi-Square ( x2 ) Distribution Area to the Right of the Critical Value

Degrees of Freedom

0.995

0.99

0.975

0.95

0.90

0.10

0.05

0.025

0.01

0.005

1 2 3 4 5

— 0.010 0.072 0.207 0.412

— 0.020 0.115 0.297 0.554

0.001 0.051 0.216 0.484 0.831

0.004 0.103 0.352 0.711 1.145

0.016 0.211 0.584 1.064 1.610

2.706 4.605 6.251 7.779 9.236

3.841 5.991 7.815 9.488 11.071

5.024 7.378 9.348 11.143 12.833

6.635 9.210 11.345 13.277 15.086

7.879 10.597 12.838 14.860 16.750

6 7 8 9 10

0.676 0.989 1.344 1.735 2.156

0.872 1.239 1.646 2.088 2.558

1.237 1.690 2.180 2.700 3.247

1.635 2.167 2.733 3.325 3.940

2.204 2.833 3.490 4.168 4.865

10.645 12.017 13.362 14.684 15.987

12.592 14.067 15.507 16.919 18.307

14.449 16.013 17.535 19.023 20.483

16.812 18.475 20.090 21.666 23.209

18.548 20.278 21.955 23.589 25.188

11 12 13 14 15

2.603 3.074 3.565 4.075 4.601

3.053 3.571 4.107 4.660 5.229

3.816 4.404 5.009 5.629 6.262

4.575 5.226 5.892 6.571 7.261

5.578 6.304 7.042 7.790 8.547

17.275 18.549 19.812 21.064 22.307

19.675 21.026 22.362 23.685 24.996

21.920 23.337 24.736 26.119 27.488

24.725 26.217 27.688 29.141 30.578

26.757 28.299 29.819 31.319 32.801

16 17 18 19 20

5.142 5.697 6.265 6.844 7.434

5.812 6.408 7.015 7.633 8.260

6.908 7.564 8.231 8.907 9.591

7.962 8.672 9.390 10.117 10.851

9.312 10.085 10.865 11.651 12.443

23.542 24.769 25.989 27.204 28.412

26.296 27.587 28.869 30.144 31.410

28.845 30.191 31.526 32.852 34.170

32.000 33.409 34.805 36.191 37.566

34.267 35.718 37.156 38.582 39.997

21 22 23 24 25

8.034 8.643 9.260 9.886 10.520

8.897 9.542 10.196 10.856 11.524

10.283 10.982 11.689 12.401 13.120

11.591 12.338 13.091 13.848 14.611

13.240 14.042 14.848 15.659 16.473

29.615 30.813 32.007 33.196 34.382

32.671 33.924 35.172 36.415 37.652

35.479 36.781 38.076 39.364 40.646

38.932 40.289 41.638 42.980 44.314

41.401 42.796 44.181 45.559 46.928

26 27 28 29 30

11.160 11.808 12.461 13.121 13.787

12.198 12.879 13.565 14.257 14.954

13.844 14.573 15.308 16.047 16.791

15.379 16.151 16.928 17.708 18.493

17.292 18.114 18.939 19.768 20.599

35.563 36.741 37.916 39.087 40.256

38.885 40.113 41.337 42.557 43.773

41.923 43.194 44.461 45.722 46.979

45.642 46.963 48.278 49.588 50.892

48.290 49.645 50.993 52.336 53.672

40 50 60 70 80 90 100

20.707 27.991 35.534 43.275 51.172 59.196 67.328

22.164 29.707 37.485 45.442 53.540 61.754 70.065

24.433 32.357 40.482 48.758 57.153 65.647 74.222

26.509 34.764 43.188 51.739 60.391 69.126 77.929

29.051 37.689 46.459 55.329 64.278 73.291 82.358

51.805 63.167 74.397 85.527 96.578 107.565 118.498

55.758 67.505 79.082 90.531 101.879 113.145 124.342

59.342 71.420 83.298 95.023 106.629 118.136 129.561

63.691 76.154 88.379 100.425 112.329 124.116 135.807

66.766 79.490 91.952 104.215 116.321 128.299 140.169

From Donald B. Owen, Handbook of Statistical Tables, ©1962 Addison-Wesley Publishing Co., Reading, MA. Reprinted with permission of the publisher.

Formulas and Tables

for Essentials of Statistics, by Mario F. Triola ©2002 by Addison-Wesley.

Linear Correlation (Section 9-2)

Rank Correlation (Section 9-5)

TABLE A-5

TABLE A-6

Critical Values of the Pearson Correlation Coefficient r

Critical Values of Spearman’s Rank Correlation Coefficient r s

n

a



.05

a



.01

4 5 6 7

.950 .878 .811 .754

.999 .959 .917 .875

8 9 10 11

.707 .666 .632 .602

.834 .798 .765 .735

12 13 14 15

.576 .553 .532 .514

.708 .684 .661 .641

16 17 18 19

.497 .482 .468 .456

.623 .606 .590 .575

20 25 30 35

.444 .396 .361 .335

.561 .505 .463 .430

40 45 50 60

.312 .294 .279 .254

.402 .378 .361 .330

70 80 90 100

.236 .220 .207 .196

.305 .286 .269 .256

NOTE: To test H 0: r 0 against H 1: r  0, reject H 0 if the absolute value of r is greater than the critical value in the table. 

n

a



0.05

a



0.01

5 6 7 8 9 10

— .886 .786 .738 .683 .648

— — — .881 .833 .794

11 12 13 14 15

.623 .591 .566 .545 .525

.818 .780 .745 .716 .689

16 17 18 19 20

.507 .490 .476 .462 .450

.666 .645 .625 .608 .591

21 22 23 24 25

.438 .428 .418 .409 .400

.576 .562 .549 .537 .526

26 27 28 29 30

.392 .385 .377 .370 .364

.515 .505 .496 .487 .478

HYPOTHESIS TEST: WORDING OF FINAL CONCLUSION Start Wording of final conclusion

Does the original claim contain the condition of equality?

Yes (Original claim contains equality and becomes H 0 )

No (Original claim does not contain equality and becomes H 1)

Do you reject H0?

Yes (Reject H 0 )

No (Fail to reject H 0 )

Do you reject H0?

“There is sufficient evidence to warrant rejection of the claim that . . . (original claim).”

(This is the only case in which the original claim is rejected.)

“There is not sufficient evidence to warrant rejection of the claim that . . . (original claim).”

(This is the “The sample data only case in Yes support the claim  which the (Reject H 0 ) that . . . (original claim).” original claim is supported.)

No (Fail to reject H 0 )

“There is not sufficient sample evidence to  support the claim that . . . (original claim).”

Statistics Formulas

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