Schaum s outline probability and statistics chapter 9 examples by steve brochu mark thomas
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Analysis of Variance. Schaum’s Outline Probability and Statistics Chapter 9 Examples by Steve Brochu Mark Thomas. Outline Chapter 9. t test versus F test Analysis of variance Test differences of means across groups Variation within groups Variation between groups

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Schaum’s Outline Probability and Statistics Chapter 9 Examples by Steve Brochu Mark Thomas

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Schaum s outline probability and statistics chapter 9 examples by steve brochu mark thomas

Analysis of Variance

Schaum’s Outline

Probability and Statistics

Chapter 9

Examples by Steve Brochu

Mark Thomas


Outline chapter 9

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)


Analysis of variance f test

Analysis of Variance – F test

  • t 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


Analysis of variance f test1

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)


Analysis of variance uranium mine cost

Analysis of Variance Uranium Mine Cost

Xjk =j + ejk

Ho: 1 = 2= 3

H1: not all equal


Variation within groups

Variation Within Groups

  • Vw = jk(Xjk- Xj.)2

  • tons produced

  • jCost Per Ton (k) xj.= 5k=1xkj

  • 25000 100 110 120 130 140120

  • 50000 100 105 110 115 120110

  • 100000 95 98 100 102 105100

  • Within group variation

  • (100-120)2 ( ) 0 100 400

  • (100-110)2 25 0 25 100

  • ( 95- 100)2 4 0 4 25Vw =1308


Distribution of variation within groups

Distribution of Variation Within Groups

  • (Xjk- j)2/2 ~ 21

  • jk(Xjk- j)2~ 2T = 2ab

  • Vw = jk(Xjk- Xj.)2/ 2= 2T-a = 2ab-a


Variation between groups

Variation Between Groups

  • Vb = jk (Xj.- X)2 =bj (Xj.- X)2

  • tons produced

  • j Cost Per Ton (k)Xj.

  • 25000 100 110120130140120

  • 50000 100 105110115120110

  • 100000 95 98100102105100 x 110

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


Distribution of variation between groups

Distribution of Variation Between Groups

  • Total 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


Aov hypothesis tests

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


Aov with unequal number of observations

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


Fit of whole regression

Fit of Whole Regression

  • y = 1 + 2x2 +3x3 + . . . kxk+ e

  • R2 = 1 – Sêi2 /Sy'i2

  • Sy'i2 =S( ŷ -x)2 + Sêi2

  • Ho:2 = 3 = . . . = k = 0

  • H1: 2, 3,, .. k not all equal to zero

  • Total SS = Explained SS + Error SS

  • Under null hypothesis

  • Total SS/2 ~ T-1


Fit of whole regression1

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


Fit of whole regression2

Fit of Whole Regression

  • Under null

  • Explained SS/2

  • K-1 ~ FK-1, T-K

  • Error SS/ 2

  • T-K


Analysis of variance summary

Analysis of Variance – Summary

9-155

  • 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


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