Analysis of Variance 10.4 558 CHAPTER 10 Chi-Square Tests and the F-Distribution What You Should Learn How to use one-way analysis of variance to test claims involving three or more means An introduction to two-way analysis of variance One-Way ANOVA Two-Way ANOVA One-Way ANOVA Suppose a medical researcher is analyzing the effectiveness of three types of pain relievers and wants to determine whether there is a difference in the mean lengths of time it takes the three medications to provide relief. To determine whether such a difference exists, the researcher can use the F@distribution together with a technique called analysis of variance. Because one independent variable is being studied, the process is called one-way analysis of variance. One-way analysis of variance is a hypothesis-testing technique that is used to compare the means of three or more populations. Analysis of variance is usually abbreviated as ANOVA. DEFINITION To begin a one-way analysis of variance test, you should first state the null and alternative hypotheses. For a one-way ANOVA test, the null and alternative hypotheses are always similar to these statements. H0: m1 = m2 = m3 = c= mk (All population means are equal.) Ha: At least one mean is different from the others. When you reject the null hypothesis in a one-way ANOVA test, you can conclude that at least one of the means is different from the others. Without performing more statistical tests, however, you cannot determine which of the means is different. Before performing a one-way ANOVA test, you must check that these conditions are satisfied. 1. Each sample must be randomly selected from a normal, or approximately normal, population. 2. The samples must be independent of each other. 3. Each population must have the same variance. The test statistic for a one-way ANOVA test is the ratio of two variances: the variance between samples and the variance within samples. Test statistic = Variance between samples Variance within samples 1. The variance between samples measures the differences related to the treatment given to each sample. This variance, sometimes called the mean square between, is denoted by MSB. 2. The variance within samples measures the differences related to entries within the same sample and is usually due to sampling error. This variance, sometimes called the mean square within, is denoted by MSW.
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