SECTION 10.4 Analysis of Variance 559 To perform a one-way ANOVA test, these conditions must be met. 1. Each of the k samples, k Ú 3, 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. If these conditions are met, then the sampling distribution for the test is approximated by the F@distribution. The test statistic is F = MSB MSW . The degrees of freedom are d.f.N = k - 1 Degrees of freedom for numerator and d.f.D = N - k Degrees of freedom for denominator where k is the number of samples and N is the sum of the sample sizes. One-Way Analysis of Variance Test If there is little or no difference between the means, then MSB will be approximately equal to MSW and the test statistic will be approximately 1. Values of F close to 1 suggest that you should fail to reject the null hypothesis. However, if one of the means differs significantly from the others, then MSB will be greater than MSW and the test statistic will be greater than 1. Values of F significantly greater than 1 suggest that you should reject the null hypothesis. So, all one-way ANOVA tests are right-tailed tests. That is, if the test statistic is greater than the critical value, then H0 will be rejected. Finding the Test Statistic for a One-Way ANOVA Test In Words In Symbols 1. Find the mean and variance of xi = Σx n , s2 i = Σ1x - xi2 2 n - 1 each sample. 2. Find the mean of all entries in x = Σx N all samples. 3. Find the sum of squares between SSB = Σni(xi - x)2 the samples. 4. Find the sum of squares within SSW = Σ1ni - 12s 2 i the samples. 5. Find the variance between the MSB = SSB d.f.N = Σni(xi - x)2 k - 1 samples. 6. Find the variance within the MSW = SSW d.f.D = Σ1ni - 12s 2 i N - k samples. 7. Find the test statistic. F = MSB MSW GUIDELINES Note that in Step 1 of the guidelines above, you are summing the values from just one sample. In Step 2, you are summing the values from all of the samples. The sums SSB and SSW are explained on the next page. Study Tip The notations ni , xi , and si 2 represent the sample size, mean, and variance of the ith sample, respectively. Also, note that x is sometimes called the grand mean.
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