Avery Goodrich 10/31/23 5:26 PM This quiz: 9 point(s) isite Question 4 of 9 possible This question: 1 point(s) possible Submit quiz Check here for instructional material to complete this problem. Evaluate $E=z^{*} \sqrt{\frac{\hat{p}_{1} \cdot \hat{q}_{1}}{n_{1}}+\frac{\hat{p}_{2} \cdot \hat{q}_{2}}{n_{2}}}$ for $z^{*}=2.575, x_{1}=44, x_{2}=94, n_{1}=50, n_{2}=107, \hat{p}_{1}=\frac{x_{1}}{n_{1}}, \hat{q}_{1}=1-\hat{p}_{1}$. $\hat{p}_{2}=\frac{x_{2}}{n_{2}}$, and $\hat{q}_{2}=1-\hat{p}_{2}$. $E=\square$ (Round to four decimal places as needed.)
Solution
Step 1 ：Given the values of $z^{*}=2.575, x_{1}=44, x_{2}=94, n_{1}=50, n_{2}=107$, we first need to calculate the values of $\hat{p}_{1}, \hat{q}_{1}, \hat{p}_{2},$ and $\hat{q}_{2}$.
Step 2 ：We calculate $\hat{p}_{1}$ as $\frac{x_{1}}{n_{1}}$ which gives $\hat{p}_{1} = \frac{44}{50} = 0.88$.
Step 3 ：Next, we calculate $\hat{q}_{1}$ as $1-\hat{p}_{1}$ which gives $\hat{q}_{1} = 1 - 0.88 = 0.12$.
Step 4 ：We calculate $\hat{p}_{2}$ as $\frac{x_{2}}{n_{2}}$ which gives $\hat{p}_{2} = \frac{94}{107} = 0.8785$ (rounded to four decimal places).
Step 5 ：Next, we calculate $\hat{q}_{2}$ as $1-\hat{p}_{2}$ which gives $\hat{q}_{2} = 1 - 0.8785 = 0.1215$ (rounded to four decimal places).
Step 6 ：Now, we substitute these values into the expression for $E$ and calculate the result. $E=z^{*} \sqrt{\frac{\hat{p}_{1} \cdot \hat{q}_{1}}{n_{1}}+\frac{\hat{p}_{2} \cdot \hat{q}_{2}}{n_{2}}}$
Step 7 ：Substituting the values, we get $E=2.575 \sqrt{\frac{0.88 \cdot 0.12}{50}+\frac{0.8785 \cdot 0.1215}{107}} = 0.1436$ (rounded to four decimal places).
Step 8 ：Final Answer: The value of $E$ is $\boxed{0.1436}$ (rounded to four decimal places as needed).
