In a recent tennis tournament, women playing singles matches used challenges on 132 calls made by the line judges. Among those challenges, 36 were found to be successful with the call overturned.
a. Construct a $95 \%$ confidence interval for the percentage of successful challenges.
b. Compare the results from part (a) to this $95 \%$ confidence interval for the percentage of successful challenges made by the men playing singles matches: $24.2 \%< p< 38.6 \%$. Does it appear that either gender is more successful than the other?
a. Construct a $95 \%$ confidence interval.
$19.7 \%< p< 34.9 \%$ (Round to one decimal place as needed.)
b. Choose the correct answer below.
A. The lower confidence limit of the interval for men is higher than the lower confidence limit of the interval for women and the upper confidence limit of the interval for men is also higher than the upper confidence limit of the interval for women. Therefore, men appear to be substantially more successful in their challenges.
B. The lower confidence limit of the interval for women is higher than the lower confidence limit of the interval for men and the upper confidence limit of the interval for women is also higher than the upper confidence limit of the interval for men. Therefore, women appear to be substantially more successful in their challenges.
C. No conclusion can be made because not enough information is given about the confidence interval for men.
D. Since the upper confidence limit of the interval for women is higher than both the lower and upper confidence limits of the interval for men, this indicates that women appear to be substantially more successful in their challenges.
E. Since the two confidence intervals overlap, neither gender appears to be substantially more successful in their challenges.
F. Since the upper confidence limit of the interval for men is higher than both the lower and upper confidence limits of the interval for women, this indicates that men appear to be substantially more successful in their challenges.
Answer
Therefore, the correct answer is E. Since the two confidence intervals overlap, neither gender appears to be substantially more successful in their challenges. \(\boxed{\text{Answer: E}}\)
Step 1 ：Calculate the sample proportion (p̂) as the number of successful challenges divided by the total number of challenges. In this case, p̂ = \(\frac{36}{132}\) = 0.2727 or 27.27%.
Step 2 ：Calculate the standard error (SE) as \(\sqrt{ p̂(1 - p̂) / n }\), where n is the total number of challenges. In this case, SE = \(\sqrt{ 0.2727(1 - 0.2727) / 132 }\) = 0.0385.
Step 3 ：Calculate the 95% confidence interval as p̂ ± 1.96*SE. In this case, the confidence interval is 0.2727 ± 1.96*0.0385 = (0.197, 0.349) or (19.7%, 34.9%).
Step 4 ：Compare the 95% confidence intervals for the percentage of successful challenges made by the women (19.7% to 34.9%) and the men (24.2% to 38.6%). The two intervals overlap, which means that we cannot conclude that one gender is more successful than the other based on these confidence intervals.
Step 5 ：Therefore, the correct answer is E. Since the two confidence intervals overlap, neither gender appears to be substantially more successful in their challenges. \(\boxed{\text{Answer: E}}\)
