450 CHAPTER 8 Hypothesis Testing with Two Samples Sample Statistics for Post-Traumatic Stress Disorder Treatment Medication Received medication Received placebo n1 = 42 x1 = 28 np 1 ≈ 0.6667 n2 = 37 x2 = 12 np 2 ≈ 0.3243 A Two-Sample z-Test for the Difference Between Proportions A medical research team conducted a study to test the effect of a medication to treat patients with post-traumatic stress disorder (PTSD). At the end of the study, the researchers found that of the 42 randomly selected subjects who took the medication, 28 no longer met the diagnostic criteria for PTSD. Of the 37 randomly selected subjects who took a placebo, 12 no longer met the diagnostic criteria for PTSD. At a = 0.01, can you support the claim that the reduction rate in the diagnostic criteria for PTSD is higher for those who took the medication than for those who took the placebo? (Source: Nature Medicine) SOLUTION The samples are random and independent. Also, the weighted estimate of p1 and p2 is p = x1 + x2 n1 + n2 = 28 + 12 42 + 37 = 40 79 ≈ 0.5063 and the value of q is q = 1 - p ≈ 1 - 0.5063 = 0.4937. Because n1p ≈ 4210.50632, n1q ≈ 4210.49372, n2p ≈ 3710.50632, and n2q ≈ 3710.49372 are at least 5, you can use a two-sample z@test. The claim is “the reduction rate in the diagnostic criteria for PTSD is higher for those who took the medication than for those who took the placebo.” So, the null and alternative hypotheses are H0: p1 … p2 and Ha: p1 7 p2 (claim). Because the test is right-tailed and the level of significance is a = 0.01, the critical value is z0 = 2.33. The rejection region is z 7 2.33. The standardized test statistic is z = 1 np 1 - np 22 - 1p1 - p22 B p qa 1 n1 + 1 n2b ≈ 10.6667 - 0.32432 - 0 B 10.5063210.49372a 1 42 + 1 37b ≈ 3.04. The figure below shows the location of the rejection region and the standardized test statistic z. Because z is in the rejection region, you reject the null hypothesis. 1 − = 0.99 z ≈ 3.04 α α= 0.01 z 0 −1 −3 −2 −4 1 2 3 4 z0 = 2.33 Interpretation There is enough evidence at the 1% level of significance to support the claim that the reduction rate in the diagnostic criteria for PTSD is higher for those who took the medication than for those who took the placebo. TRY IT YOURSELF 2 Consider the results of the study discussed on page 417. At a = 0.05, can you reject the claim that the proportion of people who use yoga who are from the western region of the United States is greater than or equal to the proportion of people who do not use yoga who are from the western region of the United States? Answer: Page A41 Study Tip To find p1 n and p 2 n use np 1 = x1 n1 and np 2 = x2 n2 . EXAMPLE 2
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