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Analysis of Variance

Analysis of Variance. A procedure for testing the hypothesis that three or more population means are equal. H 0 : µ 1 = µ 2 = µ 3 = . . . µ k H 1 : At least one mean is different. ANOVA methods require the F-distribution.

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Analysis of Variance

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  1. Analysis of Variance A procedure for testing the hypothesis that three or more population means are equal. H0: µ1 = µ2 = µ3 = . . . µk H1: At least one mean is different

  2. ANOVA methods require the F-distribution 1.The F-distribution is not symmetric; it is skewed to the right. 2. The values of F can be 0 or positive, they cannot be negative. 3. There is a different F-distribution for each pair of degrees of freedom for the numerator and denominator.

  3. F - distribution Not symmetric (skewed to the right)

  4. One-Way ANOVA Assumptions 1. The populations have normal distributions. 2. The populations have the same variance . 3. The samples are simple random samples. 4. The samples are independent of each other. 5. The different samples are from populations that are categorized in only one way.

  5. Relationships Among Components of ANOVA

  6. Fundamental Concept Estimate the common value of 2 using 1. The variance between samples (also called variation due to treatment) is an estimate of the common population variance that is based on the variability among the sample means. 2. The variance within samples (also called variation due to error) is an estimate of the common population variance based on the sample variances.

  7. Comparing Two Groups Data: Observations on two groups Procedure: 1) Calculate the means of and Sum of Squares for both groups individually. 2) Combine the groups and repeat Step (1).

  8. Comparing Two Groups Procedure: 3) Calculate deviation^2 for each group mean and the Combined mean. 4) Multiply (3) x #observations in each group. 5) Sum the Squares,

  9. Comparing Two Groups Procedure: 6) Ensure that Σ ESS(A) + ESS(B) + RSS = TSS 7) Calculate the F-stat 8) Compare with F-critical numerator = k-1 denominator = N-k

  10. RSS / k-1 F = Σ ESS / N-k Test Statistic for One-Way ANOVA A large F test statistic is evidence against equal population means.

  11. variance between samples F = variance within samples Test Statistic for One-Way ANOVA A large F test statistic is evidence against equal population means.

  12. http://www.uwsp.edu/psych/stat/12/anova-1w.htm

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