0673chapter12 - Chapter 12 Analysis of Variance 12-2...

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Chapter 12 Analysis of Variance 12-2 One-Way ANOVA 1. a. One-way analysis of variance is appropriate for these data because they represent three or more populations categorized by a single characteristic that distinguishes the populations from each other. The distinguishing characteristic in this case is epoch. b. One-way analysis of variance tests the equality of two or more population means by analyzing sample variances. It finds a difference in the population means if the variance between the sample means is larger than can be expected considering the variance within the samples. 2. One test is better than three tests because a single test allows the conclusion to be made at the stated level of significance. Suppose, for example, all tests are conducted at the α = 0.05 level of significance: for a single test, P(no type I error) = 1– α = 0.95; for three independent tests, P(no type I error) = (1– α ) 3 = (0.95) 3 = 0.857 – and so P(type I error) = 0.143 instead of 0.05. Simply put, multiple tests increase the likelihood of chance differences between the samples causing rejection of H o when there are actually no differences between the populations. 3. We should reject the hypothesis that the three epochs have the same mean skull breadth. There is sufficient evidence to conclude that at least one of the means is different from the others. 4. No. Rejection of the hypothesis that all means are the same implies only that at least one of the means is not the same as the others. The hypothesis that all the means are the same is rejected when the variation between the means is larger than it is expected to be were all the means equal, but there is nothing in the mathematics of the test statistic that can identify which individual contributions to the variation between means are significant. NOTE: When testing the hypothesis that three or more groups have the same mean, the test statistic is F, where F is the ratio is the ratio of the variance between the groups to the variance within the groups as defined in the text. This manual generally uses the generic notation F = 22 Bp s/ s . As in previous chapters, the superscripts and subscripts (the numerator df and denominator df) may be used to identify which F distribution to look up in the tables. 5. H o : μ 1 = μ 2 = μ 3 H 1 : at least one μ i is different α = 0.05 and df num = 2, df den = 33 C.V. F = F α = F 0.05 = 3.3158 calculations: F = s = 669.0011/70.6481 = 9.4695 [TI-83/84+] P-value = P( 2 33 F >9.4695) = 0.0006 [TI-83/84+] conclusion: Reject Ho; there is sufficient evidence to reject the claim that μ 1 = μ 2 = μ 3 . There is sufficient evidence to reject the claim that the three books have the same mean Flesch Reading Ease score. F 1 3.3158 2 33 0.05
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418 CHAPTER 12 Analysis of Variance 6. H o : μ 1 = μ 2 = μ 3 H 1 : at least one μ i is different α = 0.05 and df num = 2, df den = 33 C.V. F = F α = F 0.05 = 3.3158 calculations: F = 22 Bp s/ s = 133.3/34.1 = 3.91 [Minitab] P-value = P( 2 33 F >3.91) = 0.030 [Minitab] conclusion: Reject Ho; there is sufficient evidence to reject the claim that μ 1 = μ 2 = μ 3 . There is sufficient evidence to reject the claim that the three books have the same mean numbers of words per sentence.
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0673chapter12 - Chapter 12 Analysis of Variance 12-2...

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