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Unformatted text preview: testing the equality of the means “analysis of VARIANCE”? Answer: We are going to compare two estimators of the common population variance, 2 . MS Groups (Mean Square between the Groups): estimator of 2 if the null hypothesis H0 is true, else
tends to be too big. MSE (Mean Square Within or due to Error): a good (unbiased estimator of 2 These two estimates are used to form the F statistic: F Variation among sample means
Natural variation within groups MS Groups
. MSE If this F ratio is too BIG we would reject the null hypothesis. 170 The Logic behind the ANOVA F‐Test Look at the plots below. For each Scenario, we have plotted data obtained by taking independent random samples of size 10 from three populations. For Scenarios A and B, the three populations each had a normal distribution and the population means were 60, 65, and 70, respectively. So the population means are indeed not all equal. In Scenario A, the population standard deviations were all equal to 1.5. In Scenario B, the population standard deviations were all equal to 3. So in each case the assumption that the populations have equal standard deviations is met. Scenario A Samples from 3 populations whose means are different. Variability within each population is small. Difference between sample means more readily seen. F statistic somewhat big. Scenario B Samples from 3 populations whose means are different. Variability within each population is larger. Difference between sample means not readily seen. F statistic smaller. Which of the above two scenarios do you think would provide more evidence that at least one of the population means is different from the others? Scenario A or Scenario B? 171 Below is a final set of plots for three independent random samples of size 10 each taken from a population with a normal model with a population mean of 65 and population standard deviation of 1.5. So in Scenario C, the population means are indeed all equal––that is, the null hypothesis tested in one‐way ANOVA is true. Notice th...
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