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.
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.
\[
\begin{array}{l}
H_{0}: p_{1}=p_{2} \\
H_{1}: p_{1}> p_{2}
\end{array}
\]
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. $H_{0}: p_{1} \geq p_{2}$
$H_{1}: p_{1} \neq p_{2}$
c.
\[
\begin{array}{l}
H_{0}: p_{1} \neq p_{2} \\
H_{1}: p_{1}=p_{2}
\end{array}
\]
F.
\[
\begin{array}{l}
H_{0}: p_{1} \leq p_{2} \\
H_{1}: p_{1} \neq p_{2}
\end{array}
\]
7
Question 4 (1/1)
Question $8(0.50 / 1)$
Identify the test statistic.
\[
z=1.14
\]
(Round to two decimal places as needed.)
Identify the P-value.
P-value $=$
(Round to three decimal places as needed.)
example
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Step 1 ：Define the null and alternative hypotheses. The null hypothesis is that the proportion of successful challenges for men (p1) is equal to the proportion of successful challenges for women (p2). The alternative hypothesis is that these proportions are not equal. Therefore, the hypotheses are: \[\begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1} \neq p_{2} \end{array}\]

Step 2 ：Calculate the test statistic and the P-value. The test statistic for a hypothesis test for the difference between two proportions is a z-score, which is 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, \(\hat{p}\) is the pooled sample proportion, and \(n_1\) and \(n_2\) are the sample sizes.

Step 3 ：Calculate the sample proportions: \(p1 = 0.303202846975089\), \(p2 = 0.2796833773087071\), and the pooled sample proportion: \(p = 0.29496070272769304\).

Step 4 ：Calculate the standard error: \(se = 0.020551533653407082\).

Step 5 ：Calculate the z-score: \(z = 1.1444143324302583\).

Step 6 ：Calculate the P-value: \(p_value = 0.252451847093357\).

Step 7 ：The calculated z-score is approximately 1.14, which matches the given test statistic. The calculated P-value is approximately 0.252, which is much larger than the significance level of 0.01. This means that we do not have enough evidence to reject the null hypothesis that the proportions of successful challenges for men and women are equal.

Step 8 ：Final Answer: The null and alternative hypotheses for the hypothesis test are: \[\begin{array}{l} H_{0}: p_{1}=p_{2} \\ H_{1}: p_{1} \neq p_{2} \end{array}\] The test statistic is \(z \approx 1.14\), and the P-value is \(P-value \approx 0.252\).