Since an instant replay system for tennis was introduced at a major tournament, men challenged 1,416 referee calls, with the result that 426 of the calls were overturned. Women challenged 755 referee calls, and 225 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) and (b) below.
(a) Test the claim using a hypothesis test.
[tex]\[ P\text{-value} = 0.891 \][/tex] (Round to three decimal places as needed.)
What is the conclusion based on the hypothesis test?
The [tex]\(P\)[/tex]-value is greater than the significance level of [tex]\(\alpha = 0.01\)[/tex], so fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that women and men have equal success in challenging calls.
(b) Test the claim by constructing an appropriate confidence interval.
The [tex]\(99\%\)[/tex] confidence interval is [tex]\(\square \ \textgreater \ \left( p_1 - p_2 \right) \ \textless \ \square\)[/tex].
Let's follow the steps to test the claim and constructing the appropriate confidence interval.
### a. Hypothesis Test
First, we need to establish the null and alternative hypotheses:
- Null Hypothesis [tex]\(H_0\)[/tex]: Men and women have equal success rates in challenging referee calls. Mathematically, this means: [tex]\( p_1 - p_2 = 0 \)[/tex] - Alternative Hypothesis [tex]\(H_a\)[/tex]: Men and women do not have equal success rates in challenging referee calls. Mathematically, this means: [tex]\( p_1 - p_2 \neq 0 \)[/tex]
Given data: - Men challenged 1416 referee calls with 426 overturned. - Women challenged 755 referee calls with 225 overturned.
From these data, we have already calculated the following: - Men’s success rate [tex]\( p_1 = \frac{426}{1416} = 0.301 \)[/tex] - Women’s success rate [tex]\( p_2 = \frac{225}{755} = 0.298 \)[/tex]
Since the [tex]\( P \)[/tex]-value (0.891) is greater than the significance level [tex]\( \alpha = 0.01 \)[/tex]: - We fail to reject the null hypothesis.
Thus, there is not sufficient evidence to warrant rejection of the claim that men and women have equal success in challenging calls.
### b. Confidence Interval
Next, we construct a 99% confidence interval for the difference in proportions [tex]\( \left( p_1 - p_2 \right) \)[/tex]:
The 99% confidence interval for the difference between the two proportions was calculated as: - Lower bound: [tex]\( -0.050 \)[/tex] - Upper bound: [tex]\( 0.056 \)[/tex]
This interval is: [tex]\[ \left( -0.050, 0.056 \right) \][/tex]
### Final Answer for Confidence Interval:
The 99% confidence interval is [tex]\( -0.050 < (p_1 - p_2) < 0.056 \)[/tex].
### Summary
- Hypothesis Test Conclusion: We fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that men and women have equal success in challenging calls. - 99% Confidence Interval: [tex]\( -0.050 < (p_1 - p_2) < 0.056 \)[/tex]
