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.