In a​ randomized, placebo-controlled​ study, 1027 healthy women 45 years of age or older were randomly divided into two groups. The women in group 1 received 100 mg of a drug every other​ day; the women in group 2 received a placebo every other day. The women were monitored for 10 years to determine if they experienced a cardiovascular event​ (such as a heart attack or​ stroke). Of the 509 in the drug​ group, 11 experienced a heart attack. Of the 518 women in the placebo​ group, 16 experienced a heart attack. Determine the test statistic

To determine the test statistic for comparing the proportions of heart attacks between the two groups in the study, we can use a two-proportion z-test. The test statistic for a two-proportion z-test is given by:

\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{p(1 - p) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]

Where:
- \(\hat{p}_1\) is the sample proportion of heart attacks in group 1 (drug group).
- \(\hat{p}_2\) is the sample proportion of heart attacks in group 2 (placebo group).
- \(n_1\) is the sample size of group 1.
- \(n_2\) is the sample size of group 2.
- \(p\) is the pooled sample proportion.

First, let's calculate the sample proportions:

\[ \hat{p}_1 = \frac{11}{509} \]
\[ \hat{p}_2 = \frac{16}{518} \]

Next, we calculate the pooled sample proportion \( p \):

\[ p = \frac{11 + 16}{509 + 518} = \frac{27}{1027} \]

Now, we calculate the test statistic:

\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{p(1 - p) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]

Plugging in the values:

\[ \hat{p}_1 = \frac{11}{509} \approx 0.0216 \]
\[ \hat{p}_2 = \frac{16}{518} \approx 0.0309 \]
\[ p = \frac{27}{1027} \approx 0.0263 \]

Then:

\[ z = \frac{0.0216 - 0.0309}{\sqrt{0.0263(1 - 0.0263) \left(\frac{1}{509} + \frac{1}{518}\right)}} \]

\[ z = \frac{-0.0093}{\sqrt{0.0263 \times 0.9737 \left(\frac{1}{509} + \frac{1}{518}\right)}} \]

\[ z = \frac{-0.0093}{\sqrt{0.0256 \left(\frac{1}{509} + \frac{1}{518}\right)}} \]

\[ z = \frac{-0.0093}{\sqrt{0.0256 \left(0.001964 + 0.001931\right)}} \]

\[ z = \frac{-0.0093}{\sqrt{0.0256 \times 0.003895}} \]

\[ z = \frac{-0.0093}{\sqrt{0.00009968}} \]

\[ z = \frac{-0.0093}{0.009984} \]

\[ z \approx -0.9317 \]

The test statistic \( z \) is approximately \(-0.9317\).

Would you like more details or have any questions about this calculation?

Here are 8 related questions to expand on this information:
1. What is the null hypothesis for the two-proportion z-test in this context?
2. What is the alternative hypothesis for this test?
3. How do you determine the critical value for the z-test at a given significance level?
4. How can you interpret the p-value obtained from the z-test?
5. What assumptions must be satisfied for the two-proportion z-test to be valid?
6. How would you conduct a power analysis for this study?
7. Can you explain the concept of the pooled proportion in hypothesis testing?
8. How would the test statistic change if the number of heart attacks in the drug group was higher?

**Tip:** Always verify that the sample sizes are large enough to meet the normality assumption for the two-proportion z-test. This typically requires that both \( n_1\hat{p}_1 \), \( n_1(1 - \hat{p}_1) \), \( n_2\hat{p}_2 \), and \( n_2(1 - \hat{p}_2) \) are at least 5.

Learn how to calculate the two-proportion z-test to compare heart attack rates between a drug and placebo group in a randomized study. Detailed step-by-step solution includes sample proportions, pooled proportion, and calculation of test statistic. Suitable for advanced undergraduate students and researchers.

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