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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. 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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  1. Analysis of Variance Schaum’s Outline Probability and Statistics Chapter 9 Examples by Steve Brochu Mark Thomas

  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)

  3. 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

  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)

  5. Analysis of Variance Uranium Mine Cost Xjk =j + ejk Ho: 1 = 2= 3 H1: not all equal

  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

  7. 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

  8. Variation Between Groups • 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

  9. 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

  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

  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

  12. 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

  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

  14. Fit of Whole Regression • Under null • Explained SS/2 • K-1 ~ FK-1, T-K • Error SS/ 2 • T-K

  15. 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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