n recent tennis tournament, women playing singles matches used challenges on 135 calls made by the line judges. Among those challenges, 36 were found to be successful with the call overtu Construct a \( 95 \% \) confidence interval for the percentage of successful challenges o. Compare the results from part (a) to this \( 95 \% \) confidence interval for the percentage of successful challenges made by the men playing singles matches: \( 22.1 \%

Ask by Cervantes Stuart. in the United States

Mar 15,2025

To construct a \( 95\% \) confidence interval for the percentage of successful challenges, we start by calculating the sample proportion, which is given by: \[ \hat{p} = \frac{x}{n} = \frac{36}{135} \approx 0.2667 \] Next, we calculate the standard error (SE) for the sample proportion: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.2667(1 - 0.2667)}{135}} \approx 0.0415 \] Now, we find the critical value (z*) for a \( 95\% \) confidence level, which is approximately \( 1.96 \). Using this, we can construct the confidence interval: \[ \text{Margin of Error (ME)} = z^* \cdot SE = 1.96 \cdot 0.0415 \approx 0.0815 \] Finally, we calculate the confidence interval: \[ \hat{p} - ME < p < \hat{p} + ME \] \[ 0.2667 - 0.0815 < p < 0.2667 + 0.0815 \] \[ 0.1852 < p < 0.3482 \] Rounding to one decimal place, the \( 95\% \) confidence interval for the percentage of successful challenges is approximately: \( 18.5\% < p < 34.8\% \). Now, comparing this interval with the men's \( 95\% \) confidence interval of \( 22.1\% < p < 38.8\% \): - The intervals overlap, bounding both men's and women's successful challenge rates between 22.1% and 34.8%. This suggests that while a higher proportion of successful challenges can be found in men's matches (up to 38.8%), women also have a notable range of successful challenges (up to 34.8%). Overall, it indicates no strong evidence that one gender is significantly better than the other based solely on these intervals.