Since an instant replay system for tennis was introduced at a major tournament, men challenged 1405 referee calls, with the result that 426 of the calls were overturned. Women challenged 758 referee calls, and 212 of the calls were overturned. Use a 0.01 significance level to test the claim that men and women have equal success in challenging calls. Complete parts (a) through (c) below.
a. Test the claim using a hypothesis test.
Consider the first sample to be the sample of male tennis players who challenged referee calls and the second sample to be the sample of female tennis players who challenged referee calls. 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.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
c.
\[
\begin{array}{l}
H_{0}: p_{1} \neq p_{2} \\
H_{1}: p_{1}=p_{2}
\end{array}
\]
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D.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}< p_{2}
\end{array}
\]
E.
\[
\begin{array}{l}
H_{0}: p_{1} \geq p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
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.)
Answer
\(\boxed{H_{0}: p_{1}=p_{2}, H_{1}: p_{1} \neq p_{2}, z = 1.14}\)
Step 1 ：State the null hypothesis and the alternative hypothesis. The null hypothesis is that the proportion of successful challenges for men (\(p_1\)) is equal to the proportion of successful challenges for women (\(p_2\)). The alternative hypothesis is that the proportions are not equal. So, we have: \[\begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1} \neq p_{2} \end{array}\]
Step 2 ：Calculate the pooled sample proportion (\(p\)), which is the total number of successes divided by the total sample size. The formula is: \[p = \frac{x_1 + x_2}{n_1 + n_2}\]
Step 3 ：Calculate the test statistic (\(z\)). The formula for the z-score is: \[z = \frac{p_1 - p_2}{\sqrt{\left(\frac{p(1 - p)}{n_1}\right) + \left(\frac{p(1 - p)}{n_2}\right)}}\]
Step 4 ：Substitute the values into the formulas. We get \(p = 0.29496070272769304\) and \(z = 1.1444143324302583\)
Step 5 ：Round the test statistic to two decimal places. We get \(z = 1.14\)
Step 6 ：\(\boxed{H_{0}: p_{1}=p_{2}, H_{1}: p_{1} \neq p_{2}, z = 1.14}\)
