554 CHAPTER 10 Chi-Square Tests and the F -Distribution Stock A Stock B n2 = 30 s2 = 3.5 n1 = 31 s1 = 5.7 Location A Location B n = 16 s = 0.95 n = 22 s = 0.78 Using Technology for a Two-Sample F@Test You want to purchase stock in a company and are deciding between two different stocks. Because a stock’s risk can be associated with the standard deviation of its daily closing prices, you randomly select samples of the daily closing prices for each stock to obtain the results shown at the left. At a = 0.05, can you conclude that one of the two stocks is a riskier investment? Assume the stock closing prices are normally distributed. SOLUTION Because 5.72 7 3.52, s2 1 = 5.7 2 and s2 2 = 3.5 2. Therefore, s2 1 and s 2 1 represent the sample and population variances for Stock B, respectively. With the claim “one of the two stocks is a riskier investment,” the null and alternative hypotheses are H0: s 2 1 = s 2 2 and Ha: s 2 1 ≠ s 2 2. (Claim) Note that the test is a two-tailed test with 1 2a = 1 210.052 = 0.025, and the degrees of freedom are d.f.N = n1 - 1 = 31 - 1 = 30 and d.f.D = n2 - 1 = 30 - 1 = 29. So, the critical value is F0 = 2.09 and the rejection region is F 7 2.09. To perform a two-sample F@test using a TI-84 Plus, begin with the STAT keystroke. Choose the TESTS menu and select E:2–SampFTest. Then set up the two-sample F@test as shown in the first screen below. Because you are entering the descriptive statistics, select the Stats input option. When entering the original data, select the Data input option. The other displays below show the results of selecting Calculate or Draw. The test statistic F ≈ 2.65 is in the rejection region, so you reject the null hypothesis. Interpretation There is enough evidence at the 5% level of significance to support the claim that one of the two stocks is a riskier investment. TRY IT YOURSELF 4 A biologist claims that the pH levels of the soil in two geographic locations have equal standard deviations. Independent samples from each location are randomly selected, and the results are shown at the left. At a = 0.01, is there enough evidence to reject the biologist’s claim? Assume the pH levels are normally distributed. Answer: Page A43 You can also use a P-value to perform a two-sample F-test. For instance, in Example 4, note that the TI-84 Plus displays P = .0102172459. Because P 6 a, you reject the null hypothesis. EXAMPLE 4 TI-84 PLUS 2-SampFTest Inpt: Data Stats Sx1:5.7 n1:31 Sx2:3.5 n2:30 s1:≠s2 <s2 >s2 Calculate Draw TI-84 PLUS 2-SampFTest s1≠s2 F=2.652244898 p=.0102172459 Sx1=5.7 Sx2=3.5 È n1=31 TI-84 PLUS F=2.6522 p=0.0102
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