For Elementary Statistics, Tenth Edition, by Mario F ...

Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola

Copyright 2006 Pearson Education, Inc.

Ch. 3: Descriptive Statistics

x Sx Mean n

Sf . x

x

Mean (frequency table)

Sf

S(x 2 x)2

s

Standard deviation

? n21

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. 4: 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. 5: Probability Distributions

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

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

P(x)

n!

. px . qnx Binomial probability

(n x)! x!

n.p

Mean (binomial)

2 n . p . q

Variance (binomial)

n . p . q x . e

P(x) x!

Standard deviation (binomial)

Poisson Distribution where e 2.71828

Ch. 6: Normal Distribution

z

x

s

x

or

x

Standard score

x Central limit theorem

x

n

Central limit theorem (Standard error)

Ch. 7: Confidence Intervals (one population)

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

where E 5 za>2? n

x 2 E , m , x 1 E Mean

s where E 5 za>2 !n (s known )

or

E

5

ta>2

s !n

(s unknown)

(n 2 1)s2

(n 2 1)s2

xR2

, s2 ,

x

2 L

Variance

Ch. 7: Sample Size Determination

3za>242 . 0.25

n5

Proportion

E2

3za>242p^ q^

n 5 E2

Proportion (p^ and q^ are known)

za>2s 2 n 5 B R Mean

E

Ch. 9: Confidence Intervals (two populations)

(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

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

< where

E

5

ta>2?

s21 n1

1

s22 n2

(df smaller of n1 1, n2 1)

(s1 and s2 unknown and not assumed equal)

< 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)

(s1 and s2 unknown but assumed equal)

< E

5

za>2?

s

2 1

n1

1

s

2 2

n2

(s1, s2 known)

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

where

E

5

ta>2

sd !n

(df n 1)

Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.

<

Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola

Copyright 2006 Pearson Education, Inc.

Ch. 8: Test Statistics (one population)

p^ 2 p

z5

Proportion--one population

pq

?n

x 2 m Mean--one population

z5 s> !n

( known)

x 2 m Mean--one population

t5 s> !n

( unknown)

(n 2 1)s2

x2 5

s2

Standard deviation or variance-- one population

Ch. 9: Test Statistics (two populations)

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

? n1 1 n2

Two proportions

t

5

(x1

2

x2) 2 (m1 2

s21 ? n1

1

s22 n2

m2)

df smaller of n1 1, n2 1

Two means--independent; s1 and s2 unknown, and

not assumed equal.

< t

5

(x1

2 x2) 2 (m1 2 sp2 1 sp2

? n1 n2

m2)

(df n1 n2 2)

sp2

5

(n1

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

1 ) s22

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

z

5

(x1

2 x2) 2 (m1 2

s 12 ? n1

1

s

2 2

n2

m2)

Two means--independent; 1, 2 known.

t 5 d 2 md sd> !n

Two means--matched pairs (df n 1)

F

5

s21 s22

Standard deviation or variance--

two

populations

(where

s

2 1

s

22)

Ch. 11: 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)

( 0 b 2 c 0 2 1)2 McNemar's test

x2 5

b1c

for matched pairs

(df 1)

Ch. 10: 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 Prediction interval

where E t2se

1

1 n

n(x0 n(x2)

x)2 (x)2

Ch. 12: One-Way Analysis of a Variance

Procedure for testing H0: m1 5 m2 5 m3 5 c

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 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 1 is rejected.

If H0 from Step 1 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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Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.

Formulas and Tables for Elementary Statistics, Tenth Edition, by Mario F. Triola

Copyright 2006 Pearson Education, 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. 14: 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

a .05

a .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: r 0 against H1: r 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

Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.

FINDING P-VALUES

Start

Left -tailed Left

What type of test

?

Two-tailed

Right-tailed

Is the test statistic to the right or left of

center ?

Right

P-value area to the left of the

test statistic

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

P - value

P-value is twice this area.

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

P-value area to the right of the

test statistic

P-value is twice this area.

P-value

Test statistic

Test statistic

Test statistic

Test statistic

Start HYPOTHESIS TEST: WORDING OF FINAL CONCLUSION

Wording of final conclusion

Does the original claim contain

the condition of equality ?

Yes

(Original claim contains equality)

Do you reject

H0?

Yes (Reject H0 )

"There is sufficient evidence to warrant rejection of the claim

that . . . (original claim)."

No (Original claim does not contain equality and becomes H1)

No (Fail to reject H0 )

"There is not sufficient evidence to warrant rejection of the claim

that . . . (original claim)."

Do you reject

H0?

Yes

"The sample data

support the claim

(Reject H0 ) that . . . (original claim)."

No (Fail to reject H0 )

"There is not sufficient sample evidence to support the claim

that . . . (original claim)."

(This is the only case in which the original claim

is rejected.)

(This is the only case in which the original claim

is supported.)

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 distrubted and n 30

Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.

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

.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

.02

.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

.03

.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

.04

.05

.0003 .0003 .0004 .0004 .0006 .0006 .0008 .0008 .0012 .0011 .0016 .0016 .0023 .0022 .0031 .0030 .0041 .0040 .0055 .0054 .0073 .0071 .0096 .0094 .0125 .0122 .0162 .0158 .0207 .0202 .0262 .0256 .0329 .0322 .0409 .0401 .0505 * .0495 .0618 .0606 .0749 .0735 .0901 .0885 .1075 .1056 .1271 .1251 .1492 .1469 .1736 .1711 .2005 .1977 .2296 .2266 .2611 .2578 .2946 .2912 .3300 .3264 .3669 .3632 .4052 .4013 .4443 .4404 .4840 .4801

.06

.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

.07

.08

.0003 .0003 .0004 .0004 .0005 .0005 .0008 .0007 .0011 .0010 .0015 .0014 .0021 .0020 .0028 .0027 .0038 .0037 .0051 * .0049 .0068 .0066 .0089 .0087 .0116 .0113 .0150 .0146 .0192 .0188 .0244 .0239 .0307 .0301 .0384 .0375 .0475 .0465 .0582 .0571 .0708 .0694 .0853 .0838 .1020 .1003 .1210 .1190 .1423 .1401 .1660 .1635 .1922 .1894 .2206 .2177 .2514 .2483 .2843 .2810 .3192 .3156 .3557 .3520 .3936 .3897 .4325 .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

Copyright 2007 Pearson Education, publishing as Pearson Addison-Wesley.

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