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PPT16 ANOVA - McGill University Advanced Business...

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McGill University Advanced Business Statistics MGSC-272
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Fall’07 Analysis of Variance Read: Business Statistics (A Second Course) , Custom Edition for McGill University Chapter 10
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ANOVA ANOVA is a statistical test of significance for the equality of several (2 or more) population (treatment) means. Assumptions: 1. All populations are normally distributed. 2. All populations have the same variance: σ 1 2 = σ 2 2 = σ 3 2 = ........ = σ k 2 = σ 2 . 3. Independent random samples.
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Notation Y ij = the i th observation from the j th treatment. = the mean of the j th sample. T j = the total of the j th sample. n j = the size of the j th sample. n T = n j = the total number of observations. p = the number of treatments. = the overall (grand) mean. μ j = the mean of the j th treatment. σ = the common standard deviation for all treatments. j Y Y
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ANOVA Table Source of Sum of Variation Squares df Mean Square F* Treatment SSTR p - 1 MSTR = SSTR/(p - 1) MSTR/MSE Error SSE n T – p MSE = SSE/(n T - p) Total SSTO n T - 1
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ANOVA Test of Hypothesis H 0 : μ 1 = μ 2 = μ 3 = ........ = μ p ( All the population means are equal). H 1 : Not all μ j are equal (At least one of the means is different). TS: F* = MSTR/MSE (or MSB/MSW). CV: F(1 - α ; p - 1; n T - p ). DR: Conclude H 0 if F * CV and Conclude H 1 if F * > CV.
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The Formulas 2 SSTO = ( - ) ij Y Y ∑∑ ( 29 2 SSTR j j n Y Y = - 2 SSE = ( - ) = SSTO - SSTR ij j Y Y ∑∑
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Partition Theorem SSTO = SSTR + SSE df TO = df TR + df E
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One-way ANOVA A Completely Randomized Design A completely randomized design to compare p treatment means is one in which the treatments are randomly assigned to the experimental units, or in which independent random samples are drawn from each of the p populations.
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EXAMPLE A student society has undertaken a survey of the cost of a “night out” in three different cities: Montreal, Toronto and Ottawa. The results of the survey are as follows (in dollars): Montreal Toronto Ottawa 16 19 23 17 26 19 22 24 24 14 24 19 27 20
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Summary Statistics Sample sizes : n 1 = 6 n 2 = 5 n 3 = 3 Means: 1 18 Y = 2 24 Y = 3 22 Y = n T = 14; Y = 294 / 14 = 21
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Calculating Sums of Squares SSTR = = 102. SSE = = (16 - 18)2 + (17 - 18)2 + (22 - 18)2 + (14 - 18)2 + (19 - 18)2 + (20 - 18)2 + (19 - 24)2 + (26 - 24)2 + (24 - 24)2 + (24 - 24)2 + (27 - 24)2 + (23 - 22)2 + (19 - 22)2 + (24 - 22)2 = 94. SSTO = SSTR + SSE = 102 + 94 = 196. 2 2 2 2 ( ) 6 (18 21) 5 (24 21) 3 (22 21) j j n Y Y - = - + - + - 2 ( ) ij j Y Y - ∑∑
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ANOVA Table Source of Variation Sum of Squares df Mean Square F* Treatment 102 2 51.00 5.97 Error 94 11 8.56 Total 196 13 Note: E(MSE) = σ 2 MSE s 2 .
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The test of hypothesis H 0 : μ 1 = μ 2 = μ 3 (All three population means are equal). H 1 : Not all μ j are equal (At least one of the means is different). TS: F* = 51.00/8.56 = 5.97 CV: F .05 ; (2, 11) =3.98 DR: Do not reject H 0 if F* 3.88, Reject H0 if F* > 3.98. Conclusion: Reject H 0 i.e. the three means are not equal
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Anova in Minitab One-way ANOVA: Cost versus City Source DF SS MS F P City 2 102.00 51.00 5.97 0.018 Error 11 94.00 8.55 Total 13 196.00 S = 2.923 R-Sq = 52.04% R-Sq(adj) = 43.32% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---------+---------+---------+---------+ Montreal 6 18.000 2.898 (--------*--------) Ottawa 3 22.000 2.646 (-----------*------------) Toronto 5 24.000 3.082 (---------*---------) ---------+---------+---------+---------+ 18.0 21.0 24.0 27.0 Pooled StDev = 2.923
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ANOVA: The Regression Approach City 1 City 2 City 3 Montreal Toronto Ottawa 16 19 23 17 26 19 22 24 24 14 24 19 27 20
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The Regression Model Y = β 0 + β 1 X 1 + β 2 X 2 + ε 1 2 1 if City = Toronto 0 if not 1 if City = Ottawa 0 if not X X
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