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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

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

- 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 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)

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

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

- 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

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

- 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

- 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

- 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 Regression

- Explained SS = Total SS - Error SS
- 22 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 Regression

- Under null
- Explained SS/2
- K-1 ~ FK-1, T-K
- Error SS/ 2
- T-K

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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