12-1 One-Way ANOVA 613 PART 1 Basics of One-Way Analysis of Variance When testing for equality of three or more population means, use the method of one-way analysis of variance. DEFINITION One-way analysis of variance (ANOVA) is a method of testing the equality of three or more population means by analyzing sample variances. One-way analysis of variance is used with data categorized with onefactor (or treatment), so there is one characteristic used to separate the sample data into the different categories. One-Way Analysis of Variance for Testing Equality of Three or More Population Means Objective Use samples from three or more different populations to test a claim that the populations all have the same mean. Requirements 1. The populations have distributions that are approximately normal. This is a loose requirement, because the method works well unless a population has a distribution that is very far from normal. If a population does have a distribution that is far from normal, use the Kruskal-Wallis test described in Section 13-5. 2. The populations have the same variance s 2 (or standard deviation s). This is a loose requirement, because the method works well unless the population variances differ by large amounts. Statistician George E. P. Box showed that as long as the sample sizes are equal (or nearly equal), the largest variance can be up to nine times the smallest variance and the results of ANOVA will continue to be essentially reliable. 3. The samples are simple random samples of quantitative data. 4. The samples are independent of each other. (The samples are not matched or paired in any way.) 5. The different samples are from populations that are categorized in only one way. Procedure for Testing H0: M1 = M2 = M3 = P= Mk KEY ELEMENTS 1. Use technology to obtain results that include the test statistic and P-value. 2. Identify the P-value from the display. (The ANOVA test is right-tailed because only large values of the test statistic cause us to reject equality of the population means.) 3. Form a conclusion based on these criteria that use the significance level a: • Reject: If the P@value … a, reject the null hypothesis of equal means and conclude that at least one of the population means is different from the others. • Fail to Reject: If the P@value 7 a, fail to reject the null hypothesis of equal means. The term treatment is used because early applications of analysis of variance involved agricultural experiments in which different plots of farmland were treated with different fertilizers, seed types, insecticides, and so on. Table 12-1 uses the one “treatment” (or factor) of size category of the automobile. That factor of size has four different categories: small, midsize, large, and SUV.
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