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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

• 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)

• 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

• 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)

Xjk =j + ejk

Ho: 1 = 2= 3

H1: not all equal

• 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

• (Xjk- j)2/2 ~ 21

• jk(Xjk- j)2~ 2T = 2ab

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

• Vb = jk (Xj.- X)2 =bj (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

• 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

• 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

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

• Vw = jk(Xjk- Xj.)2

• Fa-1,T-a = vb/a-1

• vw/T-a

• 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

• 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

• Under null

• Explained SS/2

• K-1 ~ FK-1, T-K

• Error SS/ 2

• T-K

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