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.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}< p_{2}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}> p_{2}
\end{array}
\]
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.
\[
z=-1.27
\]
(Round to two decimal places as needed.)
Identify the P-value.
P-value $=$
(Round to three decimal places as needed.)
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Answer
The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic, under the null hypothesis. We can calculate the P-value using the cumulative distribution function of the standard normal distribution. The calculated P-value is \(\boxed{0.101}\)
Step 1 ：The null and alternative hypotheses for the hypothesis test are \[\begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1} Step 2 ：The test statistic is calculated using the formula for the test statistic in a hypothesis test for two proportions. The test statistic follows a standard normal distribution under the null hypothesis. The calculated test statistic is \(\boxed{-1.27}\) Step 3 ：The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed test statistic, under the null hypothesis. We can calculate the P-value using the cumulative distribution function of the standard normal distribution. The calculated P-value is \(\boxed{0.101}\)
