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Chapter 10

Chapter 10. Hypothesis Testing III (ANOVA). Chapter Outline. Introduction The Logic of the Analysis of Variance The Computation of ANOVA Computational Shortcut A Computational Example. Chapter Outline. A Test of Significance for ANOVA

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Chapter 10

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  1. Chapter 10 Hypothesis Testing III (ANOVA)

  2. Chapter Outline • Introduction • The Logic of the Analysis of Variance • The Computation of ANOVA • Computational Shortcut • A Computational Example

  3. Chapter Outline • A Test of Significance for ANOVA • An Additional Example for Computing and Testing the Analysis of Variance • The Limitations of the Test • Interpreting Statistics: Does Sexual Activity Vary by Marital Status?

  4. In This Presentation • The basic logic of ANOVA • A sample problem applying ANOVA • The Five Step Model

  5. Basic Logic • ANOVA can be used in situations where the researcher is interested in the differences in sample means across three or more categories.

  6. Basic Logic • Examples: • How do Protestants, Catholics and Jews vary in terms of number of children? • How do Republicans, Democrats, and Independents vary in terms of income? • How do older, middle-aged, and younger people vary in terms of frequency of church attendance?

  7. Basic Logic • ANOVA asks “are the differences between the sample means so large that we can conclude that the populations represented by the samples are different?” • The H0 is that the population means are the same: • H0: μ1= μ2= μ3 = … = μk

  8. Basic Logic • If the H0 is true, the sample means should be about the same value. • If the H0 is false, there should be substantial differences between categories, combined with relatively little difference within categories. • The sample standard deviations should be low in value.

  9. Basic Logic • If the H0 is true, there will be little difference between sample means. • If the H0 is false, there will be big difference between sample means combined with small values for s.

  10. Basic Logic • The larger the differences between the sample means, the more likely the H0is false.-- especially when there is little difference within categories. • When we reject the H0, we are saying there are differences between the populations represented by the sample.

  11. Steps in Com putation of ANOVA • Find SST by Formula 10.10. • Find SSB by Formula 10.4. • Find SSW by subtraction (Formula 10.11).

  12. Steps in Computation of ANOVA • Calculate the degrees of freedom (Formulas 0.5 and 10.6). • Construct the mean square estimates by dividing SSB and SSW by their degrees of freedom. (Formulas 10.7 and 10.8). • Find F ratio by Formula 10.9.

  13. Example of Computation of ANOVA • Problem 10.6 • Does voter turnout vary by type of election? Data are presented for local, state, and national elections.

  14. Example of Computation of ANOVA

  15. Example of Computation of ANOVA • The difference in the means suggests that turnout does vary by type of election. • Turnout seems to increase as the scope of the election increases. • Are these differences significant?

  16. Example of Computation of ANOVA • Use Formula 10.10 to find SST. • Use Formula 10.4 to find SSB • Find SSW by subtraction • SSW = SST – SSB • SSW = 10,612.13 - 3,342.99 • SSW= 7269.14 • Use Formulas 10.5 and 10.6 to calculate degrees of freedom.

  17. Example of Computation of ANOVA • Use Formulas 10.7 and 10.8 to find the Mean Square Estimates: • MSW = SSW/dfw • MSW =7269.14/33 • MSW = 220.28 • MSB = SSB/dfb • MSB = 3342.99/2 • MSB = 1671.50

  18. Example of Computation of ANOVA • Find the F ratio by Formula 10.9: • F = MSB/MSW • F = 1671.95/220.28 • F = 7.59

  19. Step 1 Make Assumptions and Meet Test Requirements • Independent Random Samples • LOM is I-R • The dependent variable (e.g., voter turnout) should be I-R to justify computation of the mean. ANOVA is often used with ordinal variables with wide ranges. • Populations are normally distributed. • Population variances are equal.

  20. Step 2 State the Null Hypothesis • H0: μ1 = μ2= μ3 • The H0 states that the population means are the same. • H1: At least one population mean is different. • If we reject the H0, the test does not specify which population mean is different from the others.

  21. Step 3 Select the S.D. and Determine the C.R. • Sampling Distribution = F distribution • Alpha = 0.05 • dfw = (N – k) = 33 • dfb = k – 1 = 2 • F(critical) = 3.32 • The exact dfw (33) is not in the table but dfw = 30 and dfw = 40 are. Choose the largerF ratio as F critical.

  22. Step 4 Calculate the Test Statistic • F (obtained) = 7.59

  23. Step 5 Making a Decision and Interpreting the Test Results • F (obtained) = 7.59 • F (critical) = 3.32 • The test statistic is in the critical region. Reject the H0. • Voter turnout varies significantly by type of election.

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