Since an instant replay system for tennis was introduced at a major tournament, men challenged 1432 referee calls, with the result that 422 of the calls were overturned. Women challenged 758 referee calls, and 225 of the calls were overturned. Use a 0.05 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?
O A. Но: Р1 ≥ Рг
Hy: P1 7P2
• D. Ho: P1 = P2
11: P1 7P2
Identify the test statistic.
Z =
• B. Ho: P1 = P2
H1: P1 > P2
ФЕ. Но: P1 = Р2
H1: P1 # P2
(Round to two decimal places as needed.)

To test the claim that men and women have equal success rates in challenging calls, we'll perform a hypothesis test for the difference between two population proportions.

### Part (a): Define the Hypotheses

Given the information provided:

1. **Null Hypothesis (H0)**: The success rates for men and women are equal.
2. **Alternative Hypothesis (H1)**: The success rates for men and women are not equal.

Therefore, the correct answer is:
- **Option D**: \( H_0: P_1 = P_2 \) and \( H_1: P_1 \neq P_2 \)

### Part (b): Calculate the Test Statistic

#### Step 1: Define the Proportions
1. For men:
   - Total challenges: \( n_1 = 1432 \)
   - Successful challenges (overturned calls): \( x_1 = 422 \)
   - Proportion of successful challenges: \( \hat{p}_1 = \frac{x_1}{n_1} = \frac{422}{1432} \approx 0.2947 \)

2. For women:
   - Total challenges: \( n_2 = 758 \)
   - Successful challenges (overturned calls): \( x_2 = 225 \)
   - Proportion of successful challenges: \( \hat{p}_2 = \frac{x_2}{n_2} = \frac{225}{758} \approx 0.2963 \)

#### Step 2: Calculate the Pooled Proportion
The pooled proportion \( \hat{p} \) is calculated as follows:
\[
\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{422 + 225}{1432 + 758} = \frac{647}{2190} \approx 0.2950
\]

#### Step 3: Calculate the Standard Error
The standard error \( SE \) for the difference between two proportions is given by:
\[
SE = \sqrt{\hat{p} \left( 1 - \hat{p} \right) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}
\]
Substituting the values:
\[
SE = \sqrt{0.2950 \times (1 - 0.2950) \times \left( \frac{1}{1432} + \frac{1}{758} \right)} \approx 0.0262
\]

#### Step 4: Calculate the Z-Statistic
The Z-statistic for the difference in proportions is calculated by:
\[
Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.2947 - 0.2963}{0.0262} \approx -0.06
\]

### Part (c): Conclusion of the Hypothesis Test

Using a significance level of \( \alpha = 0.05 \):
- Since this is a two-tailed test, we look at the critical values for \( Z \) at \( \pm 1.96 \).
- The calculated \( Z \)-value of \( -0.06 \) is within the range \( -1.96 < Z < 1.96 \), so we do not reject the null hypothesis.

### Conclusion
There is not enough evidence at the 0.05 significance level to conclude that there is a difference in success rates between men and women in challenging calls.

---

Do you have questions, or would you like further details?

#### Follow-Up Questions:
1. How would the result change if we used a one-tailed test instead?
2. What is the practical interpretation of failing to reject the null hypothesis in this context?
3. How would a larger sample size impact the significance of the test?
4. Could the result change if the success rates for both groups were higher overall?
5. What are other potential factors that could influence the success rate in challenging calls?

**Tip:** When comparing proportions, always check if the sample sizes are large enough for reliable results.

Since an instant replay system for tennis was introduced at a major tournament, men challenged 1432 referee calls, with the result that 422 of the calls were overturned. Women challenged 758 referee calls, and 225 of the calls were overturned. Use a 0.05 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?
O A. Но: Р1 ≥ Рг
Hy: P1 7P2
• D. Ho: P1 = P2
11: P1 7P2
Identify the test statistic.
Z =
• B. Ho: P1 = P2
H1: P1 > P2
ФЕ. Но: P1 = Р2
H1: P1 # P2
(Round to two decimal places as needed.)

Analyze the success rates of men and women in challenging referee calls at a major tennis tournament using a hypothesis test for the difference in two population proportions. This problem involves calculating a Z-statistic and comparing it against a 0.05 significance level. Includes step-by-step calculations of the pooled proportion, standard error, and interpretation of the test result.

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