SECTION 10.3 Comparing Two Variances 553 Normal solution Treated solution n = 25 s2 = 180 n = 20 s2 = 56 Performing a Two-Sample F@Test A restaurant manager is designing a system that is intended to decrease the variance of the time customers wait before their meals are served. Under the old system, a random sample of 10 customers had a variance of 400. Under the new system, a random sample of 21 customers had a variance of 256. At a = 0.10, is there enough evidence to convince the manager to switch to the new system? Assume both populations are normally distributed. SOLUTION Because 400 7 256, s2 1 = 400 and s 2 2 = 256. Therefore, s 2 1 and s 2 1 represent the sample and population variances for the old system, respectively. With the claim “the variance of the waiting times under the new system is less than the variance of the waiting times under the old system,” the null and alternative hypotheses are H0: s 2 1 … s 2 2 and Ha: s 2 1 7 s 2 2. (Claim) Note that the test is a right-tailed test with a = 0.10, and the degrees of freedom are d.f.N = n1 - 1 = 10 - 1 = 9 and d.f.D = n2 - 1 = 21 - 1 = 20. So, the critical value is F0 = 1.96 and the rejection region is F 7 1.96. The test statistic is F = s2 1 s2 2 = 400 256 ≈ 1.56. The figure shows the location of the rejection region and the test statistic F. Because F is not in the rejection region, you fail to reject the null hypothesis. 1 2 3 4 F F0 = 1.96 F≈ 1.56 α= 0.10 Interpretation There is not enough evidence at the 10% level of significance to convince the manager to switch to the new system. TRY IT YOURSELF 3 A medical researcher claims that a specially treated intravenous solution decreases the variance of the time required for nutrients to enter the bloodstream. Independent samples from each type of solution are randomly selected, and the results are shown in the table at the left. At a = 0.01, is there enough evidence to support the researcher’s claim? Assume the populations are normally distributed. Answer: Page A43 EXAMPLE 3 Picturing the World Does location have an effect on the variance of real estate selling prices? A random sample of selling prices (in thousands of dollars) of existing homes sold in the California counties of Los Angeles and San Diego is shown in the table. (Adapted from National Association of Realtors and Move, Inc.) Los Angeles San Diego 443 639 347 369 495 649 595 665 595 699 649 779 250 425 449 595 580 649 369 549 Assuming each population of selling prices is normally distributed, is it possible to use a two-sample F@test to compare the population variances?
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