A research center conducted a survey on family-leave practices and attitudes. Respondents were asked to complete this sentence: "When a family member has a serious health condition, caregiver responsibilities fall..." with choices being "mainly on women," mainly on men," or "on both men and women equally." The percentages for each response are shown in the accompanying table. For these age groups, the responses fell only into the two categories shown in the table. Assume a sample size of 1300 for each age group. Can we conclude that there is a difference in the proportion of people aged $30-49$ and aged $50-64$ who feel the primary caregiver responsibility falls on women? Use a significance level of 0.01 .
\begin{tabular}{lcc}
Age & \begin{tabular}{c}
Fall mainly \\
on women
\end{tabular} & \begin{tabular}{c}
Fall equally on \\
men and women
\end{tabular} \\
$30-49$ & $55 \%$ & $45 \%$ \\
$50-64$ & $57 \%$ & $43 \%$
\end{tabular}
Consider the first sample to be the $30-49$ group, the second sample to be the $50-64$ group, and the number of successes to be the number of people who feel the primary caregiver responsibility falls on women. What are the null and alternative hypotheses for the hypothesis test?
A.
\[
\begin{array}{l}
H_{0}: p_{1} \neq p_{2} \\
H_{a}: p_{1}=p_{2}
\end{array}
\]
D. H.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{a}: p_{1}> p_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: p_{1}> p_{2} \\
H_{a}: p_{1}=p_{2}
\end{array}
\]
E.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{a}: p_{1} \neq p_{2}
\end{array}
\]
C.
\[
\begin{array}{l}
H_{0}: p_{1}< p_{2} \\
H_{a}: p_{1}=p_{2}
\end{array}
\]
F.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{a}: p_{1}< p_{2}
\end{array}
\]
Identify the test statistic.
\[
z=\square
\]
(Round to two decimal places as needed.)
Answer
So, the test statistic is \(\boxed{z = -1.15}\) (rounded to two decimal places).
Step 1 ：Define the null hypothesis (H0) and the alternative hypothesis (Ha).
Step 2 ：\(H0: p1 = p2\) (There is no difference in the proportion of people aged 30-49 and aged 50-64 who feel the primary caregiver responsibility falls on women.)
Step 3 ：\(Ha: p1 \neq p2\) (There is a difference in the proportion of people aged 30-49 and aged 50-64 who feel the primary caregiver responsibility falls on women.)
Step 4 ：The correct answer is E.
Step 5 ：Perform a two-proportion z-test to identify the test statistic. The formula for the test statistic in a two-proportion z-test is \(z = \frac{p1 - p2}{\sqrt{p * (1 - p) * ((1/n1) + (1/n2))}}\), where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and p is the pooled sample proportion.
Step 6 ：Given that \(n1 = n2 = 1300\), \(p1 = 0.55\), and \(p2 = 0.57\), calculate p as follows: \(p = \frac{n1*p1 + n2*p2}{n1 + n2} = \frac{1300*0.55 + 1300*0.57}{1300 + 1300} = 0.56\).
Step 7 ：Substitute these values into the formula to get: \(z = \frac{0.55 - 0.57}{\sqrt{0.56 * (1 - 0.56) * ((1/1300) + (1/1300))}} = \frac{-0.02}{0.01732} = -1.15\).
Step 8 ：So, the test statistic is \(\boxed{z = -1.15}\) (rounded to two decimal places).
