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To solve this hypothesis testing question, let’s go through each part step-by-step.

### Part (a): Identify the Hypotheses
The question states that we want to test whether men and women have **equal success** in challenging calls.

- \( p_1 \): Proportion of successful challenges for men
- \( p_2 \): Proportion of successful challenges for women

Since we're testing for **equality** of proportions, the correct hypotheses are:
- **Null Hypothesis \( H_0 \)**: \( p_1 = p_2 \)
- **Alternative Hypothesis \( H_1 \)**: \( p_1 \neq p_2 \)

Therefore, the correct answer for part (a) is:
- **Option E**: \( H_0: p_1 = p_2 \) and \( H_1: p_1 \neq p_2 \)

### Part (b): Test Statistic
The test statistic for comparing two proportions is given by:

\[
Z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}}
\]

where:
- \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions for men and women, respectively.
- \( \hat{p} \) is the combined proportion, calculated as:

\[
\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}
\]

#### Given Values:
- Men: \( x_1 = 432 \) successful challenges out of \( n_1 = 1432 \).
- Women: \( x_2 = 75 \) successful challenges out of \( n_2 = 250 \).

1. **Calculate \( \hat{p}_1 \) and \( \hat{p}_2 \):**

   \[
   \hat{p}_1 = \frac{432}{1432} \approx 0.3017
   \]
   \[
   \hat{p}_2 = \frac{75}{250} = 0.30
   \]

2. **Calculate \( \hat{p} \):**

   \[
   \hat{p} = \frac{432 + 75}{1432 + 250} = \frac{507}{1682} \approx 0.3014
   \]

3. **Calculate the Standard Error:**

   \[
   \text{SE} = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} = \sqrt{0.3014 \times 0.6986 \left( \frac{1}{1432} + \frac{1}{250} \right)}
   \]

   This gives approximately \( \text{SE} \approx 0.0308 \).

4. **Calculate \( Z \):**

   \[
   Z = \frac{0.3017 - 0.30}{0.0308} \approx 0.10
   \]

So, the test statistic is:
- \( Z = 0.10 \) (rounded to two decimal places).

### Part (c): P-Value
For a two-tailed test with \( Z = 0.10 \), we can use a Z-table or calculator.

- **P-value** for \( Z = 0.10 \) in a two-tailed test is approximately **0.920** (rounded to three decimal places).

### Summary of Answers
1. **Hypotheses**: Option E: \( H_0: p_1 = p_2 \), \( H_1: p_1 \neq p_2 \)
2. **Test Statistic**: \( Z = 0.10 \)
3. **P-Value**: 0.920

Would you like a deeper explanation of any part of this solution?

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### Follow-Up Questions

1. What does a high P-value indicate in hypothesis testing?
2. How would the test change if we were only testing if \( p_1 > p_2 \)?
3. Why is it necessary to pool proportions when calculating the test statistic?
4. How would you interpret the result if the significance level was 0.01?
5. What are some possible limitations of using hypothesis tests on proportions?

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### Quick Tip
When performing a two-tailed test, remember that you need to double the one-tail area from the Z-table to find the total P-value.

Consider the sample of male and female tennis players who challenged referee calls. Using the data given, conduct a hypothesis test to see if men and women have equal success rates in challenging calls. Use a 0.05 significance level.

Perform a hypothesis test to determine if male and female tennis players have equal success in challenging referee calls, using a two-proportion Z-test at a 0.05 significance level. Learn step-by-step calculations, including test statistics and p-value interpretation.

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