In a survey of 240 males ages 20 to $24,38 \%$ were neither in school nor working. In a survey of 200 females ages 20 to $24,43 \%$ were neither in school nor working. These samples are random and independent. At $\alpha=0.05$, can you support the claim that the proportion of males ages 20 to 24 who were neither in school nor working is less than the proportion of females ages 20 to 24 who were neither in school nor working? Complete parts (a) through (e) below.
(a) Identify the claim and state $\mathrm{H}_{0}$ and $\mathrm{H}_{\mathrm{a}}$.
The claim is "the proportion of males ages 20 to 24 who were neither in school nor working is less than the proportion of females ages 20 to 24 who were neither in school nor working."
Let $p_{1}$ and $p_{2}$ be the two population proportions for males and females, respectively. State $H_{0}$ and $H_{a}$.
Choose the correct answer below.
A.
\[
\begin{array}{l}
H_{0}: p_{1}>p_{2} \\
H_{a}: p_{1} \leq p_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: p_{1} \neq p_{2} \\
H_{a}: p_{1}=p_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: p_{1}p_{2}
\end{array}
\]
Solution
Step 1 ：The claim is 'the proportion of males ages 20 to 24 who were neither in school nor working is less than the proportion of females ages 20 to 24 who were neither in school nor working.'
Step 2 ：Let \(p_{1}\) and \(p_{2}\) be the two population proportions for males and females, respectively.
Step 3 ：The null hypothesis, denoted by \(H_{0}\), is usually the hypothesis that sample observations result purely from chance.
Step 4 ：The alternative hypothesis, denoted by \(H_{a}\), is the hypothesis that sample observations are influenced by some non-random cause.
Step 5 ：In this case, the claim is that the proportion of males ages 20 to 24 who were neither in school nor working is less than the proportion of females ages 20 to 24 who were neither in school nor working. This claim is the alternative hypothesis.
Step 6 ：The null hypothesis is the opposite of the claim, which is that the proportion of males ages 20 to 24 who were neither in school nor working is greater than or equal to the proportion of females ages 20 to 24 who were neither in school nor working.
Step 7 ：So, the correct answer is: \(H_{0}: p_{1} \geq p_{2}\) and \(H_{a}: p_{1} < p_{2}\)
Step 8 ：Final Answer: \(\boxed{H_{0}: p_{1} \geq p_{2}, H_{a}: p_{1} < p_{2}}\)
