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 confidence interval for the percentage of successful challenges
o. Compare the results from part (a) to this confidence interval for the percentage of successful challenges made by the men playing singles matches: . Does it appear that
more successful than the other?
a. Construct a confidence interval.
(Round to one decimal place as needed)

Question

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 confidence interval for the percentage of successful challenges
o. Compare the results from part (a) to this confidence interval for the percentage of successful challenges made by the men playing singles matches: . Does it appear that
more successful than the other?
a. Construct a confidence interval.
(Round to one decimal place as needed)

UpStudy AI · Accepted Answer

Let the sample proportion be  
$$
\hat{p}=\frac{36}{135}\approx 0.2667.
$$

The standard error (SE) is given by  
$$
\text{SE}=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.2667\cdot0.7333}{135}}=\sqrt{0.001448}\approx 0.0381.
$$

For a $$ 95\% $$ confidence interval, we use the $$ z $$-value $$ z^* = 1.96 $$. Thus, the margin of error (ME) is  
$$
\text{ME}=1.96\times0.0381\approx 0.0746.
$$

The confidence interval is then  
$$
\hat{p}\pm \text{ME}=0.2667\pm0.0746,
$$
which gives  
$$
(0.2667-0.0746,\;0.2667+0.0746)=(0.1921,0.3413).
$$

Expressed as percentages and rounded to one decimal place, the $$ 95\% $$ confidence interval for the percentage of successful challenges among women is  
$$
\boxed{19.2\% < p < 34.1\%}.
$$

Comparing with the $$ 95\% $$ confidence interval for the men’s matches, $$22.1\% < p < 38.8\%$$, we see that the intervals overlap (from about $$22.1\%$$ to $$34.1\%$$). This overlap suggests that there is no conclusive evidence that the success rate of challenges is different between the two groups.
The $$ 95\% $$ confidence interval for the percentage of successful challenges among women is $$ 19.2\% < p < 34.1\% $$. Comparing this to the men's interval of $$ 22.1\% < p < 38.8\% $$, the intervals overlap, indicating that there is no clear evidence that one group has a higher success rate than the other.

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