In a recent tennis toumament, 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 ca 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: $$ 22.1 \%

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In a recent tennis toumament, 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 ca 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: $$ 22.1 \%< $$ p < $$ 38.8 \% $$. Does it appe more successful than the other?

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**a. Constructing a 95% Confidence Interval for the Percentage of Successful Challenges**

Given:
- Number of challenges, $$ n = 135 $$
- Number of successful challenges, $$ x = 36 $$

The sample proportion is
$$
\hat{p} = \frac{36}{135} \approx 0.2667.
$$

The standard error (SE) for the proportion is calculated as
$$
SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.2667(1-0.2667)}{135}}.
$$
Calculating the numerator:
$$
0.2667 \times 0.7333 \approx 0.1956.
$$
Thus,
$$
SE \approx \sqrt{\frac{0.1956}{135}} \approx \sqrt{0.001448} \approx 0.03806.
$$

For a $$95\%$$ confidence interval, we use the critical value $$ z = 1.96 $$. The margin of error (ME) is
$$
ME = z \times SE \approx 1.96 \times 0.03806 \approx 0.0746.
$$

The confidence interval for the proportion is then:
$$
\hat{p} \pm ME \approx 0.2667 \pm 0.0746.
$$
Calculating the bounds:
$$
\text{Lower bound} \approx 0.2667 - 0.0746 \approx 0.1921,
$$
$$
\text{Upper bound} \approx 0.2667 + 0.0746 \approx 0.3413.
$$

Expressed in percentages, the $$95\%$$ confidence interval for the successful challenges is approximately:
$$
19.2\% \text{ to } 34.1\%.
$$

---

**b. Comparison with Men’s Confidence Interval**

The $$95\%$$ confidence interval for successful challenges for men playing singles is given as:
$$
22.1\% \text{ to } 38.8\%.
$$

Comparison:
- Women's interval: approximately $$19.2\%$$ to $$34.1\%$$.
- Men's interval: $$22.1\%$$ to $$38.8\%$$.

Observations:
- The intervals overlap between approximately $$22.1\%$$ and $$34.1\%$$.
- Although the men's interval is shifted slightly upward (with a higher upper bound), both intervals share a common area which indicates that there is no statistically significant evidence that one group is more successful than the other based solely on these intervals.

Thus, based on the overlapping confidence intervals, it does not appear that the women were more successful (or less successful) than the men in making successful challenges.
**a. Confidence Interval for Women's Successful Challenges** - **Sample Proportion ($$ \hat{p} $$)**: Approximately 26.67% - **Standard Error (SE)**: Approximately 3.81% - **Margin of Error (ME)**: Approximately 7.46% - **95% Confidence Interval**: 19.2% to 34.1% **b. Comparison with Men’s Confidence Interval** - **Men’s Interval**: 22.1% to 38.8% - **Overlap**: The intervals overlap between 22.1% and 34.1%, indicating no significant difference in success rates between women and men. **Conclusion**: Based on the overlapping confidence intervals, there is no evidence that women were more successful than men in making successful challenges.

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