In a survey of 200 females who recently completed high school, $76 \%$ were enrolled in college. In a survey of 150 males who recently completed high school, $70 \%$ were enrolled in college. At $\alpha=0.06$, can you reject the claim that there is no difference in the proportion of college enrollees between the two groups? Assume the random samples are independent. Complete parts (a) through (e).
(a) Identify the claim and state $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$.
The claim is "the proportion of female college enrollees is the same as the proportion of male college enrollees."
Let $p_{1}$ represent the population proportion for female college enrollees and $p_{2}$ represent the population proportion for male college enrollees. State $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$.
Choose the correct answer below.
A.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{a}: p_{1} \neq p_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: p_{1}p_{2}
\end{array}
\]
C.
\[
\begin{array}{l}
H_{0}: p_{1} \geq p_{2} \\
H_{a}: p_{1}p_{2} \\
H_{a}: p_{1} \leq p_{2}
\end{array}
\]
(b) Find the critical value(s) and identify the rejection region(s).
The critical value(s) is(are) $\square$.
(Use a comma to separate answers as needed. Type an integer or a decimal. Round to two decimal places as needed.)
Solution
Step 1 ：The claim is 'the proportion of female college enrollees is the same as the proportion of male college enrollees.' Let \(p_{1}\) represent the population proportion for female college enrollees and \(p_{2}\) represent the population proportion for male college enrollees. The null hypothesis \(H_{0}\) and the alternative hypothesis \(H_{a}\) are as follows: \[H_{0}: p_{1}=p_{2}\] \[H_{a}: p_{1} \neq p_{2}\]
Step 2 ：The critical value is a point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis. If the absolute value of your test statistic is greater than the critical value, you can declare statistical significance and reject the null hypothesis. The rejection region is the range of values for which we can reject the null hypothesis.
Step 3 ：To find the critical value, we can use the z-score formula for a two-tailed test, which is \(Z = ±Zα/2\), where α is the significance level. In this case, α = 0.06, so α/2 = 0.03. We can look up this value in the z-table to find the critical value.
Step 4 ：The critical value is approximately \(\boxed{1.88}\). The rejection regions are \(Z < -1.88\) and \(Z > 1.88\).
