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Chapter 10 Hypothesis Testing Using Analysis of Variance (ANOVA)

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

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Outline The basic logic of ANOVA A sample problem applying ANOVA The Five Step Model

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Logic (cont.) Examples: How do Protestants, Catholics and Jews vary in terms of number of children? How do Liberals, Conservatives, and NDP supporters vary in terms of income? How do older, middle-aged, and younger people vary in terms of frequency of church attendance?

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Logic (cont.) 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 H 0 is that the population means are the same: H 0: μ 1 = μ 2 = μ 3 = … = μ k

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Logic (cont.) If the H 0 is true, the sample means should be about the same value. If the H 0 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.

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Logic (cont.) If the H 0 is true, there will be little difference between sample means. If the H 0 is false, there will be big difference between sample means combined with small values for s.

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Logic (cont.) The larger the differences between the sample means, the more likely the H 0 is false.-- especially when there is little difference within categories. When we reject the H 0, we are saying there are differences between the populations represented by the sample. The ANOVA test uses the F-statistic and the F- distribution (Appendix D). Table uses degrees of freedom for the between number dfb = k-1 and the within number dfw = n – k.

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Steps in Computation of ANOVA 1.Find SST (Formula 10.10 or 9.10): 2. Find SSB (Formula 10.4 or 9.4): 3. Find SSW by subtraction (Formula 10.11 or 9.11):

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Steps in Computation of ANOVA 4. Calculate the degrees of freedom (see 10.5,10.6 or 9.5, 9.6): dfb = k-1 and dfw = n – k. 5. Construct the mean square estimates by dividing SSB and SSW by their degrees of freedom (10.7,10.8 or 9.7, 9.8): MS w = SSW / dfw MS b = SSB / dfb 6. Find F ratio by Formula 10.9: F = MS b / MS w

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Example: 1 st ed. p. 293, #10.6, 2 nd p. 263 9.6 Does voter turnout vary by type of election? Calculate the data for 3 types of elections and make a table: Grand mean: MunicipalProvincialFederal ∑X i 441559723 ∑X 2 20,21327,60745,253 Group Means 36.7546.5860.25

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Example (cont.) 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?

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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.

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Step 2: State the Null Hypothesis H 0 : μ 1 = μ 2 = μ 3 The H 0 states that the population means are the same. H 1 : At least one population mean is different. If we reject the H 0, the test does not specify which population mean is different from the others.

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Step 3: Select the Sampling Distribution and Determine the Critical Region 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 larger F ratio as F critical.

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Step 4: Calculate the Test Statistic 1.SST: 2. SSB = 3. SSW by subtraction: SSW = SST – SSB = 10,612.13 - 3,342.99 = 7269.14

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Step 4: Calculate the Test Statistic (cont.) Calculate degrees of freedom: dfw = n-k = 33 and dfb = k-1 = 2 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

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Step 4: Calculate the Test Statistic (cont.) Find the F ratio by Formula 10.9 (9.9): F = MSB/MSW F = 1671.95/220.28 F = 7.59

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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 H 0. Voter turnout varies significantly by type of election. Now… Work with a partner and try p. 293, 8e p. 273 #10.5 (2 nd ed # 9.5)

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