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).