1

Schaum’s Outline

Probability and Statistics

Chapter 9

Examples by Steve Brochu

Mark Thomas

Analysis of Variance

9-2

Outline Chapter 9

t test versus F test

Analysis of variance

Test differences of means across groups

Variation within groups

Variation between groups

Consider (Variation between)/ (Variation within)

Explanatory Power of Regression

(Variation explained/Variation unexplained)

9-3

Analysis of Variance – F testt tests

inferences on one parameter

unknown variances, small sample

F tests

Analysis of variance

difference of means

often groups > 2

across models

Do variables in regression model explain y

Which model is better

9-4

Analysis of Variance – F test

Uranium Mines

j different sized mines

do costs differ for the j different sized mines

(j = 1,. . .,a a=3)

1 = small

2 = medium

3 = large

sample 15 mines, 5 (k) in each category

sample k mines in each category k = 1,5

Cost per ton

44 = + ejk ei ~ N(0, 2)

9-5

Analysis of Variance Uranium Mine Cost

tons produced Cost Per Ton

j k 25000 100 110 120 130 140 50000 100 105 110 115 120

100000 95 98 100 102 105

Xjk = jejk

Ho: 1 = 2= 3

H1: not all equal

9-6

Variation Within Groups

Vw = jk(Xjk- Xj.)2

tons produced

j Cost Per Ton (k) xj.= 5k=1xkj

25000 100 110 120 130 140 120

50000 100 105 110 115 120 110

100000 95 98 100 102 105 100

Within group variation

(100-120)2 ( ) 0 100 400

(100-110)2 25 0 25 100

( 95- 100)2 4 0 4 25 Vw = 1308

9-7Distribution of Variation Within

Groups

(Xjk- j)2/2 ~ 21

jk(Xjk- j)2~ 2T = 2

ab

Vw = jk(Xjk- Xj.)2/ 2 = 2T-a = 2

ab-a

9-8

Variation Between Groups

Vb = jk (Xj.- X)2 =bj (Xj.- X)2

tons produced

j Cost Per Ton (k) Xj.

25000 100 110 120 130 140 120

50000 100 105 110 115 120 110

100000 95 98 100 102 105 100 x 110

5(100-110)2 + (110-110)2 + ( )2 = 1000

9-9Distribution of Variation

Between GroupsTotal Variation

V = jk(Xjk- X)2 = jk(Xjk- Xj.)2 + jk(Xj. - X)2

Vw + Vb

V = jk(Xjk- X)2 = jk(Xjk- Xj.)2 + jk(Xj. - X)2

2 2 2

If all s the same then

T-1 = T-a + ?

? ~ T-1 - T-a = T-1-(T-a) = a-1

Vb/ 2~ a-1

9-10

AOV Hypothesis Tests

2df1

df1 ~ Fdf1,df2

2df2

df2

Under null hypothesis

Ho: 1 = 2= 3

H1: not all equal

Vb/(a-1) = ŝb2 = 500/(3-1) = 4.587

Vw/(ab-a) ŝw2 1308/(15-3)

Critical F2, 12 = 3.89

9-11

AOV with Unequal Number of Observations

Vb = jk (Xj.- X)2 =jnj (Xj.- X)2

Vw = jk(Xjk- Xj.)2

Fa-1,T-a = vb/a-1

vw/T-a

9-12

Fit of Whole Regression

y = xxkxk+ e

R2 = 1 – êi2 / y'i

2

y'i2 = ŷ -x)2 + êi

2

Ho: = =k = 0

H1: k not all equal to zero

Total SS = Explained SS + Error SS

Under null hypothesis

Total SS/2 ~ T-1

9-13

Fit of Whole Regression

Explained SS = Total SS - Error SS

2 2 2

Under null ~ T-1 - ~ T-K

Explained SS ~T-1-(T-K) = K-1

2

Under null

Explained SS/2

K-1 ~ FK-1, T-K

Error SS/ 2

T-K

9-14

Fit of Whole Regression

Under null

Explained SS/2

K-1 ~ FK-1, T-K

Error SS/ 2

T-K

9-15

Analysis of Variance – Summary

Differences between t and F testing

Analysis of Variance (ANOVA)

Tests for equivalence of multiple means (μ1 = μ2 …)

Utilizes identity that: Total SS = Explained SS + Error SS

Compares variation between groups to variation within groups using F test

Test statistic is:

Need Modification if unequal observations each group

9-155

anawithin

betw Fs

s ,12

2

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