SECTION 10.4 Analysis of Variance 563 Compact Midsize Large 12 21 18 23 23 17 17 19 15 20 14 17 25 14 20 18 21 17 24 26 17 27 18 13 29 25 17 16 31 27 24 22 24 21 25 21 Using Technology to Perform a One-Way ANOVA Test A researcher believes that for city driving, the fuel economy of compact, midsize, and large cars are the same. The gas mileages (in miles per gallon) for city driving for several randomly selected cars from each category are shown in the table at the left. Assume that the populations are normally distributed, the samples are independent, and the population variances are equal. At a = 0.05, can you reject the claim that the mean gas mileages for city driving are the same for the three categories? Use technology to test the claim. (Source: Fueleconomy.gov) SOLUTION Here are the null and alternative hypotheses. H0: m1 = m2 = m3 (Claim) Ha: At least one mean is different from the others. The results obtained by performing the test using Excel are shown below. From the results, you can see that P ≈ 0.02. Because P 6 a, you reject the null hypothesis. EXCEL 13 15 160.6897 608.3103 769 D E F C B A 1 2 3 4 5 6 7 Anova: Single factor SUMMARY compact midsize large 8 9 10 11 12 15 16 ANOVA Groups Source of Variation Count 291 313 2 33 35 22.38462 20.86667 80.34487 18.43364 28.75641 16.69524 8 134 16.75 4.214286 4.358599 0.020906 Sum Average Variance SS df MS F P-value Between Groups 13 Within Groups 14 Total Interpretation There is enough evidence at the 5% level of significance to reject the claim that the mean gas mileages for city driving are the same. TRY IT YOURSELF 2 The data shown in the table represent the GPAs of randomly selected freshmen, sophomores, juniors, and seniors. At a = 0.05, can you conclude that there is a difference in the means of the GPAs? Assume that the populations of GPAs are normally distributed and that the population variances are equal. Use technology to test the claim. Freshmen 2.34 2.38 3.31 2.39 3.40 2.70 2.34 Sophomores 3.26 2.22 3.26 3.29 2.95 3.01 3.13 3.59 2.84 3.00 Juniors 2.80 2.60 2.49 2.83 2.34 3.23 3.49 3.03 2.87 Seniors 3.31 2.35 3.27 2.86 2.78 2.75 3.05 3.31 Answer: Page A43 EXAMPLE 2 Tech Tip Here are instructions for performing a one-way analysis of variance test on a TI-84 Plus. Begin by storing the data in L1, L2, and so on. STAT Choose the TESTS menu. H: ANOVA( Then enter L1, L2, and so on, separated by commas.
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