Problem
Hypothesis Test for the Difference in Two Proportions You wish to test the following claim \( \left(H_{d}\right) \) at a significance level of \( \alpha=0.02 \). \[ \begin{array}{l} H_{o}: p_{1}=p_{2} \\ H_{a}: p_{1} \neq p_{2} \end{array} \] You obtain 208 successes in a sample of size \( n_{1}=291 \) from the first population. in a sample of size \( n_{2}=263 \) from the second population. What is the test statistic for this sample? (Report answer accurate to three decima test statistic \( = \) What is the \( p \)-value for this sample? (Report answer accurate to four decimal places \( \mathrm{p} \)-value \( = \) The \( p \)-value is... less than (or equal to) \( \alpha \) greater than \( \alpha \)
Solution
Step 1 ：1. Calculate \(\hat{p_c}\) and \(\hat{p_1}\) and \(\hat{p_2}\): \[\hat{p_c} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{208}{291 + 263}\, \hat{p_1} = \frac{208}{291}\, \hat{p_2} = \frac{208 - 208}{263} \]
Step 2 ：2. Calculate the standard error: \[SE = \sqrt{\frac{\hat{p_c} (1 - \hat{p_c})}{n_1} + \frac{\hat{p_c} (1 - \hat{p_c})}{n_2}} \]
Step 3 ：3. Calculate the test statistic: \[Z = \frac{(\hat{p_1} - \hat{p_2}) - 0}{SE} \]
Step 4 ：4. Find the \(p\)-value by using the standard normal table with Z
Step 5 ：5. Compare the \(p\)-value with the significance level \(\alpha = 0.02\)
