LESSON 9.2 - TESTING A CLAIM ABOUT A
DIFFERENCE BETWEEN TWO PrOPORTIONS
DOES TAKING ASPIRIN HELP PREVENT HEART ATTACKS?
The Physicians' Health Study I was a medical experiment that helped answer this question. The subjects in this experiment were 22,071 male physicians. About half \( (11,037) \) of these subjects took an aspirin tablet every other day and the remaining subjects \( (11,034) \) took a dummy pill that looked and tasted like the aspirin but had no active ingredient. After several years, 239 of the control group but only 139 of the aspirin group had suffered heart attacks.
1. Does this study provide convincing evidence that aspirin helps prevent heart attacks for healthy male physicians like those in this study? Justify your answer.
2. Based on your conclusion in Question 1, is it possible you made a Type I error or a Type II error? Explain.
3. Should you generalize the result in Question 1 to all healthy males? Why or why not?
Statistics and Probability with Applications \( 3 \mathrm{e} \)
Answer
We should not generalize the result to all healthy males because the sample in the study was specific to male physicians, and other factors or lifestyles could play a role in the effectiveness of aspirin for different groups of healthy males.
Step 1 ：Let \(p_1\) be the proportion of heart attacks in the aspirin group and \(p_2\) be the proportion in the control group. We want to test the null hypothesis \(H_0: p_1 = p_2\) against the alternative hypothesis \(H_a: p_1 < p_2\).
Step 2 ：First, find the sample proportions: \(\hat{p}_1 = \frac{139}{11,037} = 0.0126\) and \(\hat{p}_2 = \frac{239}{11,034} = 0.0217\). Then, we compute the pooled proportion: \(\hat{p} = \frac{139 + 239}{11,037 + 11,034} = 0.0172\)
Step 3 ：Next, calculate the test statistic: \(Z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\frac{\hat{p}(1 - \hat{p})}{11,037} + \frac{\hat{p}(1 - \hat{p})}{11,034}}} = \frac{0.0126 - 0.0217}{\sqrt{\frac{0.0172(0.9828)}{11,037} + \frac{0.0172(0.9828)}{11,034}}} = -4.644\)
Step 4 ：Using a significance level of 0.05 and looking up the test statistic value in the standard normal table, we find a p-value of 0.0002, which is less than 0.05, so we reject the null hypothesis \(H_0\).
Step 5 ：Since we rejected the null hypothesis, there is a possibility of making a Type I error, which is concluding that aspirin helps prevent heart attacks when it does not.
Step 6 ：We should not generalize the result to all healthy males because the sample in the study was specific to male physicians, and other factors or lifestyles could play a role in the effectiveness of aspirin for different groups of healthy males.
