For Elementary Statistics, Eighth Edition, by Mario F. Triola ©2001 by ...

Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola ?2001 by Addison Wesley Longman Publishing Company, Inc.

Ch. 2: Descriptive Statistics

x

Sx n

Mean

Sf . x

x

Mean (frequency table)

Sf

S(x 2 x)2

s? n21

Standard deviation

n(Sx2) 2 (Sx)2 Standard deviation

s ? n(n 2 1)

(shortcut)

n3S(f . x2) 4 2 3S(f . x) 42 Standard deviation

s?

n(n 2 1)

(frequency table)

variance s2

Ch. 3: Probability

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)

if A, B are not mutually exclusive P(A and B) 5 P(A) . P(B) if A, B are independent P(A and B) 5 P(A) . P(B 0A) if A, B are dependent P(A) 5 1 2 P(A) Rule of complements

n! nPr 5 (n 2 r)! Permutations (no elements alike)

n! n1! n2! . . . nk! Permutations (n1 alike, ...)

n! nCr 5 (n 2 r)! r! Combinations

Ch. 4: Probability Distributions

x . P(x) Mean (prob. dist.)

[x2 . P(x)] 2 Standard deviation (prob. dist.)

P(x)

(n

n! x)!

x!

.

px

.

qn x

Binomial probability

n.p

Mean (binomial)

2 n . p . q

Variance (binomial)

n . p . q

P(x)

x . e x!

Standard deviation (binomial)

Poisson Distribution where e 2.71828

Ch. 5: Normal Distribution

z

x

s

x

or

x

Standard score

x Central limit theorem

x

n

Central limit theorem (Standard error)

Ch. 6: Confidence Intervals (one population)

x 2 E , m , x 1 E Mean

s where E 5 za>2 !n

( known or n 30)

s or E 5 ta>2 !n

( unknown and n 30)

p^ E p p^ E Proportion p^ q^

where E 5 za>2? n

(n 2 1)s2

(n 2 1)s2

xR2

, s2 ,

x

2 L

Variance

Ch. 6: Sample Size Determination

za>2s 2 n 5 B R Mean

E

3za>242 . 0.25

n5 E2

Proportion

3za>242p^ q^

n 5 E2

Proportion (p^ and q^ are known)

Ch. 8: Confidence Intervals (two populations)

d 2 E , md , d 1 E (Matched Pairs)

where

E

5

ta>2

sd !n

(df n 1)

(x1 2 x2) 2 E , (m1 2 m2) , (x1 2 x2) 1 E (Indep.)

where

E

5

za>2?

s

2 1

n1

1

s

2 2

n2

<

(s1, s2 known or n1 . 30 and n2 . 30)

< E

5

ta>2?

s21 n1

1

s22 n2

(df smaller of n1 1, n2 1)

(unequal population variances and n1 30 or n2 30)

< E

5

ta>2?

sp2 n1

1

sp2 n2

(df 5 n1 1 n2 2 2)

sp2

5

(n1 2 1)s21 (n1 2 1)

1 1

(n2 (n2

2 2

1 ) s22 1)

(equal population variances and n1 30 or n2 30) (p^ 1 2 p^ 2) 2 E , (p1 2 p2) , (p^ 1 2 p^ 2) 1 E

where

E

5

za>2?

p^ 1q^ 1 n1

1

p^ 2q^ 2 n2

Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola ?2001 by Addison Wesley Longman Publishing Company, Inc.

Ch. 7: Test Statistics (one population)

x 2 m Mean--one population

z5 s> !n

( known or n 30)

x 2 m Mean--one population

t5 s> !n

( unknown and n 30)

p^ 2 p

z5

Proportion--one population

pq

?n

(n 2 1)s2

x2 5

s2

Standard deviation or variance-- one population

Ch. 8: Test Statistics (two populations)

z

5

(x1

2 x2) 2 (m1 2

s

2 1

? n1

1

s

2 2

n2

m2)

Two means--independent (1, 2 known or n1 30 and n2 30)

t 5 d 2 md sd> !n

Two means--matched pairs (df n 1)

z 5 (p^ 1 2 p^ 2) 2 (p1 2 p2) pq pq

? n1 1 n2

Two proportions

F

5

s21 s22

Standard deviation or variance--

two

populations

(where

s

2 1

s

22)

t

5

(x1

2

x2) 2 (m1 s21 1 s22

2

m2)

? n1 n2

df smaller of n1 1, n2 1

Two means--independent; unequal variances

(and n1 30 or n2 30)

t 5 (x1 2 x2) 2 (m1 2 m2)

sp2 ? n1

1

sp2 n2

(df n1 n2 2)

where

sp2

5

(n1

2 1)s21 1 (n2 2 n1 1 n2 2 2

1 ) s22

Two means--independent; equal variances (and n1 30 or n2 30)

Ch. 10: Multinomial and Contingency Tables

(O 2 E)2 x2 5 g

E

Multinomial (df k 1)

(O 2 E)2 x2 5 g

E

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

(row total) (column total)

where E 5

(grand total)

Ch. 9: Linear Correlation/Regression

nSxy 2 (Sx) (Sy) Correlation r 5

"n(Sx2) 2 (Sx)2"n(Sy2) 2 (Sy)2

nSxy 2 (Sx) (Sy) b1 5 n(Sx2) 2 (Sx)2

(Sy) (Sx2) 2 (Sx) (Sxy)

b0 5 y 2 b1x or b0 5

n(Sx2) 2 (Sx)2

y^ 5 b0 1 b1x Estimated eq. of regression line

explained variation r2 5

total variation

se

5

S(y ?n

2 2

y^ )2 2

or

Sy2 ?

2

b0Sy n2

2 2

b1Sxy

y^ E y y^ E

where E t2se

1

1 n

n(x0 n(x2)

x)2 (x)2

Ch. 11: One-Way Analysis of a Variance

F

5

ns2x2 sp2

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

MS ( treatment ) F5

MS ( error )

df k 1 df N k

SS ( treatment )

MS(treatment) 5

k21

SS ( error ) MS(error) 5 N 2 k

SS ( total ) MS(total) 5 N 2 1

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)s2k SS(total) 5 S(x 2 x)2

SS(total) 5 SS(treatment) 1 SS(error)

Ch. 11: Two-Way Analysis of Variance

MS ( interaction ) Interaction: F 5

MS ( error ) MS(row factor) Row Factor: F 5 MS(error)

MS(column factor) Column Factor: F 5

MS ( error )

<

<

Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola ?2001 by Addison Wesley Longman Publishing Company, Inc.

Ch. 13: Nonparametric Tests

(x 1 0.5) 2 (n>2)

z5

Sign test for n 25

!n>2

z5

T 2 n(n 1 1)>4 Wilcoxon signed ranks n(n 1 1) (2n 1 1) (matched pairs and n 30)

?

24

z

5

R2 mR

5

R2

n1(n1 1 n2 1 1) 2

sR

n1n2(n1 1 n2 1 1)

?

12

Wilcoxon rank-sum (two independent samples)

H5

12

a R21 1 R22 1 . . . 1 R2k b 2 3(N 1 1)

N(N 1 1) n1 n2

nk

Kruskal-Wallis (chi-square df k 1)

6Sd2 rs 5 1 2 n(n2 2 1) Rank correlation acritical value for n . 30: 6 z b

!n 2 1

z 5 G 2 mG 5 sG

G 2 a 2n1n2 1 1b n1 1 n2

(2n1n2) (2n1n2 2 n1 2 n2)

? (n1 1 n2)2(n1 1 n2 2 1)

Runs test for n 20

Ch. 12: Control Charts

R chart: Plot sample ranges UCL: D4R Centerline: R LCL: D3R

x chart: Plot sample means UCL: x 1 A2R Centerline: x LCL: x 2 A2R

p chart: Plot sample proportions pq

UCL: p 1 3 ? n Centerline: p

pq LCL: p 2 3 ? n

TABLE A-6 Critical Values of the Pearson Correlation Coefficient r

n

.05

.01

4

.950

.999

5

.878

.959

6

.811

.917

7

.754

.875

8

.707

.834

9

.666

.798

10

.632

.765

11

.602

.735

12

.576

.708

13

.553

.684

14

.532

.661

15

.514

.641

16

.497

.623

17

.482

.606

18

.468

.590

19

.456

.575

20

.444

.561

25

.396

.505

30

.361

.463

35

.335

.430

40

.312

.402

45

.294

.378

50

.279

.361

60

.254

.330

70

.236

.305

80

.220

.286

90

.207

.269

100

.196

.256

NOTE: To test H0: 0 against H1: 0, reject H0 if the absolute value of r is greater than the critical value in the table.

Control Chart Constants

Subgroup Size n

2 3 4 5 6 7

A2

1.880 1.023 0.729 0.577 0.483 0.419

D3

0.000 0.000 0.000 0.000 0.000 0.076

D4

3.267 2.574 2.282 2.114 2.004 1.924

HYPOTHESIS TESTING

1. Identify the specific claim or hypothesis to be tested and put it in symbolic form. 2. Give the symbolic form that must be true when the original claim is false. 3. Of the two symbolic expressions obtained so far, let the null hypothesis H0 be the one that

contains the condition of equality; H1 is the other statement. 4. Select the significance level based on the seriousness of a type I error. Make small if

the consequences of rejecting a true H0 are severe. The values of 0.05 and 0.01 are very common. 5. Identify the statistic that is relevant to this test, and identify its sampling distribution. 6. Determine the test statistic and either the P-value or the critical values, and the critical region. Draw a graph. 7. Reject H0: Test statistic is in the critical region or P-value # a. Fail to reject H0: Test statistic is not in the critical region or P-value . a. 8. Restate this previous conclusion in simple, nontechnical terms.

FINDING P-VALUES

Start

Left -tailed

What type of test

?

Two-tailed

Right -tailed

Is Left the test statistic Right

to the right or left of center ?

P-value area to the left of the test statistic

P - value

P-value twice the area to the left of the test statistic

P-value is twice this area.

m Test statistic

m Test statistic

P-value twice the area to the right of the test statistic

P-value is twice this area.

P-value area to the right of the test statistic

P -value

m Test statistic

m Test statistic

0z

TABLE A-2 Standard Normal (z) Distribution

z

0.0 0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9

1.0 1.1 1.2 1.3 1.4

1.5 1.6 1.7 1.8 1.9

2.0 2.1 2.2 2.3 2.4

2.5 2.6 2.7 2.8 2.9

3.0 3.10 and higher

.00

.0000 .0398 .0793 .1179 .1554

.1915 .2257 .2580 .2881 .3159

.3413 .3643 .3849 .4032 .4192

.4332 .4452 .4554 .4641 .4713

.4772 .4821 .4861 .4893 .4918

.4938 .4953 .4965 .4974 .4981

.4987

.4999

.01

.0040 .0438 .0832 .1217 .1591

.1950 .2291 .2611 .2910 .3186

.3438 .3665 .3869 .4049 .4207

.4345 .4463 .4564 .4649 .4719

.4778 .4826 .4864 .4896 .4920

.4940 .4955 .4966 .4975 .4982

.4987

.02

.0080 .0478 .0871 .1255 .1628

.1985 .2324 .2642 .2939 .3212

.3461 .3686 .3888 .4066 .4222

.4357 .4474 .4573 .4656 .4726

.4783 .4830 .4868 .4898 .4922

.4941 .4956 .4967 .4976 .4982

.4987

.03

.0120 .0517 .0910 .1293 .1664

.2019 .2357 .2673 .2967 .3238

.3485 .3708 .3907 .4082 .4236

.4370 .4484 .4582 .4664 .4732

.4788 .4834 .4871 .4901 .4925

.4943 .4957 .4968 .4977 .4983

.4988

.04

.05

.0160 .0557 .0948 .1331 .1700

.0199 .0596 .0987 .1368 .1736

.2054 .2389 .2704 .2995 .3264

.2088 .2422 .2734 .3023 .3289

.3508 .3729 .3925 .4099 .4251

.3531 .3749 .3944 .4115 .4265

.4382 .4394

.4495 .4505

.4591 .4599

.4671 .4678

.4738 .4744

.4793 .4838 .4875 .4904 .4927

.4798 .4842 .4878 .4906 .4929

.4945 .4959 .4969 .4977 .4984

.4946 .4960 .4970 .4978 .4984

.4988 .4989

.06

.0239 .0636 .1026 .1406 .1772

.2123 .2454 .2764 .3051 .3315

.3554 .3770 .3962 .4131 .4279

.4406 .4515 .4608 .4686 .4750

.4803 .4846 .4881 .4909 .4931

.4948 .4961 .4971 .4979 .4985

.4989

.07

.08

.0279 .0675 .1064 .1443 .1808

.0319 .0714 .1103 .1480 .1844

.2157 .2486 .2794 .3078 .3340

.2190 .2517 .2823 .3106 .3365

.3577 .3790 .3980 .4147 .4292

.3599 .3810 .3997 .4162 .4306

.4418 .4525 .4616 .4693 .4756

.4429 .4535 .4625 .4699 .4761

.4808 .4850 .4884 .4911 .4932

.4812 .4854 .4887 .4913 .4934

.4949 .4951

.4962 .4963 .4972 .4973 .4979 .4980 .4985 .4986

.4989 .4990

.09

.0359 .0753 .1141 .1517 .1879

.2224 .2549 .2852 .3133 .3389

.3621 .3830 .4015 .4177 .4319

.4441 .4545 .4633 .4706 .4767

.4817 .4857 .4890 .4916 .4936

.4952 .4964 .4974 .4981 .4986

.4990

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.

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