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 \%
Solution
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. \(\boxed{\text{Answer: E}}\)
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Solution
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. \(\boxed{\text{Answer: E}}\)
