In a random sample of males, it was found that 22 write with their left hands and 222 do not. In a random sample of females, it was found that 64 write with their left hands and 464 do not. Use a 0.01 significance level to test the claim that the rate of left-handedness among males is less than that among females. Complete parts (a) through (c) below.
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of males and the second sample to be the sample of females. What are the null and alternative hypotheses for the hypothesis test?
A.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}< p_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}> p_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
E.
\[
\begin{array}{l}
H_{0}: p_{1} \neq p_{2} \\
H_{1}: p_{1}=p_{2}
\end{array}
\]
c.
\[
\begin{array}{l}
H_{0}: p_{1} \geq p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
F.
\[
\begin{array}{l}
H_{0}: p_{1} \leq p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
Identify the test statistic.
\[
\mathrm{z}=
\]
(Round to two decimal places as needed.)

Step 1 ：We are given two samples, one of males and one of females. In the male sample, 22 are left-handed and 222 are not. In the female sample, 64 are left-handed and 464 are not. We are asked to test the claim that the rate of left-handedness among males is less than that among females at a 0.01 significance level.

Step 2 ：To test this claim, we will perform a hypothesis test for two proportions. The null hypothesis \(H_{0}\) is that the proportions of left-handedness among males and females are equal, i.e., \(p_{1}=p_{2}\). The alternative hypothesis \(H_{1}\) is that the proportion of left-handedness among males is less than that among females, i.e., \(p_{1}

Step 3 ：The test statistic for this problem is a z-score, which can be calculated using the formula: \[z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}\] where \(\hat{p}_1\) and \(\hat{p}_2\) are the sample proportions of left-handedness among males and females respectively, \(\hat{p}\) is the pooled sample proportion, and \(n_1\) and \(n_2\) are the sample sizes for males and females respectively.

Step 4 ：Substituting the given values into the formula, we get: \[n1 = 244, n2 = 528, \hat{p}_1 = 0.09016393442622951, \hat{p}_2 = 0.12121212121212122, \hat{p} = 0.11139896373056994\]

Step 5 ：Calculating the z-score, we get: \[z = -1.2748100646672456\]

Step 6 ：Rounding to two decimal places, the test statistic is \(\boxed{-1.27}\).