Formulas and Tables by Mario F. Triola Copyright 2010 Pearson ...

Formulas and Tables by Mario F. Triola Copyright 2010 Pearson Education, Inc.

Ch. 3: Descriptive Statistics

x ?x Mean n

?f # x

x

Mean (frequency table)

?f

?1x - x22

sB n - 1

Standard deviation

n 1?x 22 - 1?x22 Standard deviation

s B n 1n - 12

(shortcut)

n 3 ?1 f # x224 - 3 ?1 f # x242 Standard deviation

sB

n 1n - 12

(frequency table)

variance s 2

Ch. 4: Probability

P 1A or B2 = P 1A2 + P 1B2 if A, B are mutually exclusive P 1A or B2 = P 1A2 + P 1B2 - P 1A and B2

if A, B are not mutually exclusive

P 1A and B2 = P 1A2 # P 1B2 if A, B are independent

P 1A and B2 = P 1A2 # P1B A2 if A, B are dependent

P 1A2 = 1 - P 1A2 Rule of complements n!

nPr = 1n - r2! Permutations (no elements alike) n!

n 1! n 2! . . . n k! Permutations (n1 alike, ? ) n!

nCr = 1n - r2! r ! Combinations

Ch. 5: Probability Distributions

m = ?x # P 1x2 Mean (prob. dist.)

# s = 2?3x2 P 1x24 - m2 Standard deviation (prob. dist.)

# # P 1x2

=

1n

n! - x2! x !

px qn-x

Binomial probability

m = n #p

Mean (binomial)

s2 = n # p # q

Variance (binomial)

s = 2n # p # q

Standard deviation (binomial)

# mx e -m

P 1x2 = x!

Poisson distribution where e 2.71828

Ch. 6: Normal Distribution

x-x x-m

z=

or

Standard score

s

s

mx = m Central limit theorem

s Central limit theorem

sx = 2n

(Standard error)

Ch. 7: Confidence Intervals (one population)

^p E p ^p E Proportion

pNqN where E = z a>2B n

x - E 6 m 6 x + E Mean

s where E = z a>2 1n (s known)

s or E = ta>2 1n

(s unknown)

1n - 12s2

1n - 12s2

xR2

6 s2 6

x

2 L

Variance

Ch. 7: Sample Size Determination

3z a>242 . 0.25

n= E2

Proportion

3z a>242pNqN

n=

E2

Proportion (^p and q^ are known)

2

z a>2s

n = B R Mean

E

Ch. 9: Confidence Intervals (two populations)

1pN 1 - pN 22 - E 6 1p1 - p22 6 1pN 1 - pN 22 + E

pN 1qN 1 pN 2qN 2 where E = z a>2B n 1 + n 2

1x 1 - x 22 - E 6 1m1 - m22 6 1x 1 - x 22 + E (Indep.)

< where E

=

t a>2B ns 211

+

s

2 2

n2

(df smaller of n1 1, n2 1)

(s1 and s2 unknown and not assumed equal)

< sp2

sp2

E = ta>2Bn 1 + n 2

1df = n 1 + n 2 - 22

sp2

=

1n 1

-

12s

2 1

1n 1 - 12

+ +

1n 2 1n 2

-

12s

2 2

12

(s1 and s2 unknown but assumed equal)

< E

=

z

s 12 a>2B n 1

+

s22 n2

(s1, s2 known)

d - E 6 md 6 d + E (Matched pairs)

where E

=

t

a>2

sd 1n

(df n 1)

Formulas and Tables by Mario F. Triola Copyright 2010 Pearson Education, Inc.

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Ch. 8: Test Statistics (one population)

pN - p

z=

Proportion--one population

pq

Bn

x - m Mean--one population

z= s> 1n

( known)

x - m Mean--one population

t= s> 1n

( unknown)

1n - 12s2

x2 =

s2

Standard deviation or variance-- one population

Ch. 9: Test Statistics (two populations)

1pN 1 - pN 22 - 1p1 - p22 z=

< pq pq

Bn1 + n2

Two proportions p = x1 + x2 n1 + n2

1x 1 - x 22 - 1m1 - m22

t=

s

2 1

s

2 2

+

Bn1 n2

df smaller of n1 1, n2 1

Two means--independent; s1 and s2 unknown, and not assumed equal.

1x 1 - x 22 - 1m1 - m22

t=

< sp2

sp2

(df n1 n2 2)

+ Bn1 n2

sp2

=

1n 1

-

12s

2 1

n1 +

+ 1n 2 n2 - 2

12s

2 2

Two means--independent; s1 and s2 unknown, but assumed equal.

1x 1 - x 22 - 1m1 - m22

z=

s

2 1

s22

Bn1 + n2

Two means--independent; 1, 2 known.

t = d - md sd> 1n

Two means--matched pairs (df n 1)

F

=

s

2 1

s

2 2

Standard deviation or variance--

two

populations

(where

s

2 1

s

22)

Ch. 11: Goodness-of-Fit and Contingency Tables

1O - E22 Goodness-of-fit

x2 = g E

(df k 1)

1O - E22 Contingency table

x2 = g E

[df (r 1)(c 1)]

1row total21column total2

where E =

1grand total2

1b - c - 122 McNemar's test for matched

x2 =

b+c

pairs (df 1)

Ch. 10: Linear Correlation/Regression

n?xy - 1?x21?y2 Correlation r = 2n1?x 22 - 1?x22 2n1?y22 - 1?y22

a AzxzyB

or r = n-1

where z x = z score for x z y = z score for y

Slope:

n?xy - 1?x21?y2 b1 = n 1?x 22 - 1?x22

sy or b1 = r sx

y-Intercept:

1?y21?x 22 - 1?x21?xy2

b0 = y - b1x or b0 =

n 1?x 22 - 1?x22

yN = b0 + b1x Estimated eq. of regression line

r 2 = explained variation total variation

se

=

?1y - yN22 B n-2

?y2 or B

-

b0?y n-2

b1?xy

yN - E 6 y 6 yN + E Prediction interval

1

n1x 0 - x22

where E = ta>2se B1 + n + n1?x 22 - 1?x22

Ch. 12: One-Way Analysis of Variance

Procedure for testing H0: m1 = m2 = m3 = ? 1. Use software or calculator to obtain results. 2. Identify the P-value. 3. Form conclusion:

If P-value a, reject the null hypothesis of equal means.

If P-value a, fail to reject the null hypothesis of equal means.

Ch. 12: Two-Way Analysis of Variance

Procedure:

1. Use software or a calculator to obtain results. 2. Test H0: There is no interaction between the row factor and

column factor. 3. Stop if H0 from Step 2 is rejected.

If H0 from Step 2 is not rejected (so there does not appear to be an interaction effect), proceed with these two tests:

Test for effects from the row factor. Test for effects from the column factor.

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Formulas and Tables by Mario F. Triola Copyright 2010 Pearson Education, Inc.

Ch. 13: Nonparametric Tests

1x + 0.52 - 1n>22

z=

Sign test for n 25

1n>2

z=

T - n 1n + 12>4 Wilcoxon signed ranks n 1n + 1212n + 12 (matched pairs and n 30)

B

24

n 11n 1 + n 2 + 12

z = R - mR = R -

2

sR

n 1n 21n 1 + n 2 + 12

B

12

Wilcoxon rank-sum (two independent samples)

H

=

12 N1N +

a

R

2 1

12 n 1

+

R

2 2

n2

+

. . .

+

R

2 k

b

nk

-

31N

+

12

Kruskal-Wallis (chi-square df k 1)

6?d 2 rs = 1 - n1n2 - 12 Rank correlation

acritical value for n 7 30: ; z b 1n - 1

z = G - mG = sG

G - a 2n 1n 2 + 1 b n1 + n2

Runs test

12n 1n 2212n 1n 2 - n 1 - n 22 for n 20

B 1n 1 + n 2221n 1 + n 2 - 12

Ch. 14: Control Charts

R chart: Plot sample ranges

UCL: D4R Centerline: R

LCL: D3R x chart: Plot sample means

UCL:xx + A2R Centerline: xx LCL: xx - A2R

p chart: Plot sample proportions pq

UCL: p + 3 B n

Centerline: p pq

LCL: p - 3 B n

TABLE A-6 Critical Values of the Pearson Correlation

Coefficient r

n

a = .05

4

.950

5

.878

6

.811

7

.754

8

.707

9

.666

10

.632

11

.602

12

.576

13

.553

14

.532

15

.514

16

.497

17

.482

18

.468

19

.456

20

.444

25

.396

30

.361

35

.335

40

.312

45

.294

50

.279

60

.254

70

.236

80

.220

90

.207

100

.196

a = .01

.990 .959 .917 .875 .834 .798 .765 .735 .708 .684 .661 .641 .623 .606 .590 .575 .561 .505 .463 .430 .402 .378 .361 .330 .305 .286 .269 .256

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

Control Chart Constants

Subgroup Size n

A2

D3

2

1.880 0.000

3

1.023 0.000

4

0.729 0.000

5

0.577 0.000

6

0.483 0.000

7

0.419 0.076

D4

3.267 2.574 2.282 2.114 2.004 1.924

General considerations ? Context of the data ? Source of the data ? Sampling method ? Measures of center ? Measures of variation ? Nature of distribution ? Outliers ? Changes over time ? Conclusions ? Practical implications

FINDING P-VALUES

HYPOTHESIS TEST: WORDING OF FINAL CONCLUSION

Inferences about M: choosing between t and normal distributions

t distribution:

s not known and normally distributed population

or s not known and n 30

Normal distribution: or

s known and normally distributed population s known and n 30

Nonparametric method or bootstrapping: Population not normally distributed and n 30

NEGATIVE z Scores

z

0

TABLE A-2 Standard Normal (z) Distribution: Cumulative Area from the LEFT

z

- 3.50 and lower - 3.4 - 3.3 - 3.2 - 3.1 - 3.0 - 2.9 - 2.8 - 2.7 - 2.6 - 2.5 - 2.4 - 2.3 - 2.2 - 2.1 - 2.0 - 1.9 - 1.8 - 1.7 - 1.6 - 1.5 - 1.4 - 1.3 - 1.2 - 1.1 - 1.0 - 0.9 - 0.8 - 0.7 - 0.6 - 0.5 - 0.4 - 0.3 - 0.2 - 0.1 - 0.0

.00

.0001 .0003 .0005 .0007 .0010 .0013 .0019 .0026 .0035 .0047 .0062 .0082 .0107 .0139 .0179 .0228 .0287 .0359 .0446 .0548 .0668 .0808 .0968 .1151 .1357 .1587 .1841 .2119 .2420 .2743 .3085 .3446 .3821 .4207 .4602 .5000

.01

.02

.03

.04

.05

.06

.07

.08

.0003 .0005 .0007 .0009 .0013 .0018 .0025 .0034 .0045 .0060 .0080 .0104 .0136 .0174 .0222 .0281 .0351 .0436 .0537 .0655 .0793 .0951 .1131 .1335 .1562 .1814 .2090 .2389 .2709 .3050 .3409 .3783 .4168 .4562

.4960

.0003 .0005 .0006 .0009 .0013 .0018 .0024 .0033 .0044 .0059 .0078 .0102 .0132 .0170 .0217 .0274 .0344 .0427 .0526 .0643 .0778 .0934 .1112 .1314 .1539 .1788 .2061 .2358 .2676 .3015 .3372 .3745 .4129 .4522

.4920

.0003 .0004 .0006 .0009 .0012 .0017 .0023 .0032 .0043 .0057 .0075 .0099 .0129 .0166 .0212 .0268 .0336 .0418 .0516 .0630 .0764 .0918 .1093 .1292 .1515 .1762 .2033 .2327 .2643 .2981 .3336 .3707 .4090 .4483

.4880

.0003 .0004

.0003 .0004

.0006 .0006

.0008 .0012 .0016

.0008 .0011 .0016

.0023 .0022

.0031 .0041 .0055

.0030 .0040 .0054

.0073 .0071

.0096 .0125 .0162

.0094 .0122 .0158

.0207

.0202

.0262 .0329 .0409 .0505 .0618 .0749 .0901

.0256 .0322 .0401

* .0495

.0606 .0735 .0885

.1075

.1056

.1271 .1492 .1736

.1251 .1469 .1711

.2005 .1977

.2296 .2611 .2946

.2266 .2578 .2912

.3300 .3264

.3669 .4052 .4443

.3632 .4013 .4404

.4840 .4801

.0003 .0004 .0006 .0008 .0011 .0015 .0021 .0029 .0039 .0052 .0069 .0091 .0119 .0154 .0197 .0250 .0314 .0392 .0485 .0594 .0721 .0869 .1038 .1230 .1446 .1685 .1949 .2236 .2546 .2877 .3228 .3594 .3974 .4364

.4761

.0003 .0004

.0003 .0004

.0005 .0005

.0008 .0011 .0015

.0007 .0010 .0014

.0021

.0020

.0028 .0038 .0051 .0068

.0027 .0037

* .0049

.0066

.0089 .0116 .0150

.0087 .0113 .0146

.0192

.0188

.0244 .0307 .0384

.0239 .0301 .0375

.0475

.0465

.0582 .0708 .0853

.0571 .0694 .0838

.1020

.1003

.1210 .1423 .1660

.1190 .1401 .1635

.1922

.1894

.2206 .2514 .2843

.2177 .2483 .2810

.3192

.3156

.3557 .3936 .4325

.3520 .3897 .4286

.4721

.4681

NOTE: For values of z below - 3.49, use 0.0001 for the area. *Use these common values that result from interpolation:

z score Area - 1.645 0.0500 - 2.575 0.0050

.09

.0002 .0003 .0005 .0007 .0010 .0014 .0019 .0026 .0036 .0048 .0064 .0084 .0110 .0143 .0183 .0233 .0294 .0367 .0455 .0559 .0681 .0823 .0985 .1170 .1379 .1611 .1867 .2148 .2451 .2776 .3121 .3483 .3859 .4247 .4641

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