You wish to test the following claim \( \left(H_{a}\right) \) at a significance level of \( \alpha=0.05 \) \[ \begin{array}{l} H_{o}: p_{1}=p_{2} \\ H_{a}: p_{1} \neq p_{2} \end{array} \] You obtain \( 45.2 \% \) successes in a sample of size \( n_{1}=531 \) from the first population. You obta successes in a sample of size \( n_{2}=657 \) from the second population. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic \( = \) What is the p-value for this sample? (Report answer accurate to four decimal places.) \( \mathrm{p}-\mathrm{value}= \) The \( p \)-value is... less than (or equal to) \( \alpha \) greater than \( \alpha \) \( 0^{6} \) This test statistic leads to a decision to... CHAPTER 15 SU....docx E 1 Type here to search esc ₹ 1 [* 5
Solution
Step 1 ：\( X_{1} = 0.452n_{1} = 240 \)
Step 2 ：\( X_{2} = n_{2} - n_{1}X_{1} = 657 - 240 = 417 \)
Step 3 ：\( \hat{p} = \frac{X_{1} + X_{2}}{n_{1} + n_{2}} = \frac{240 + 417}{531 + 657} = 0.5040 \)
Step 4 ：\( \hat{p}_{1} = \frac{X_{1}}{n_{1}} = \frac{240}{531} = 0.4520 \)
Step 5 ：\( \hat{p}_{2} = \frac{X_{2}}{n_{2}} = \frac{417}{657} = 0.6347 \)
Step 6 ：\( \text{test statistic} = \frac{\hat{p}_{1} - \hat{p}_{2}}{\sqrt{\frac{\hat{p}(1 - \hat{p})}{n_{1}} + \frac{\hat{p}(1 - \hat{p})}{n_{2}}}} = \frac{0.4520 - 0.6347}{\sqrt{\frac{0.5040(1 - 0.5040)}{531} + \frac{0.5040(1 - 0.5040)}{657}}} = -6.363 \)
Step 7 ：\( \text{p-value} = 2 * \text{P}(Z \leq -6.363) = 2 * (1 - \text{P}(Z \leq 6.363)) \approx 2 * 0 = 0.0000 \)
Step 8 ：\text{The p-value is less than } \( \alpha \)
Step 9 ：\text{This test statistic leads to a decision to reject the null hypothesis.}
