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Math  /  Data & Statistics

QuestionIn a recent tennis tournament, women playing singles matches used challenges on 138 calls made by the line judges. Among those challenges, 37 were found to be successful with the call overturned. a. Construct a 99%99 \% confidence interval for the percentage of successful challenges. b. Compare the results from part (a) to this 99%99 \% confidence interval for the percentage of successful challenges made by the men playing singles matches: 20.7%<p<40.8%20.7 \%<p<40.8 \%. Does it appear that either gender is more successful than the other? a. Construct a 99\% confidence interval. □\square \% < p < □\square \% (Round to one decimal place as needed.)

Studdy Solution

STEP 1

1. The sample size is 138 challenges.
2. The number of successful challenges is 37.
3. We are constructing a 99% confidence interval for the proportion of successful challenges.
4. The confidence interval formula for a proportion is used: p^±zp^(1−p^)n\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

STEP 2

1. Calculate the sample proportion of successful challenges.
2. Determine the critical value for a 99% confidence interval.
3. Calculate the standard error.
4. Construct the confidence interval.
5. Compare the confidence interval with the given interval for men.

STEP 3

Calculate the sample proportion p^\hat{p}:
p^=37138\hat{p} = \frac{37}{138}
p^≈0.2681\hat{p} \approx 0.2681

STEP 4

Determine the critical value zz for a 99% confidence interval. For a 99% confidence level, z≈2.576z \approx 2.576.

STEP 5

Calculate the standard error (SE):
SE=p^(1−p^)n=0.2681×(1−0.2681)138SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.2681 \times (1 - 0.2681)}{138}}
SE≈0.0389SE \approx 0.0389

STEP 6

Construct the confidence interval:
p^±z×SE=0.2681±2.576×0.0389\hat{p} \pm z \times SE = 0.2681 \pm 2.576 \times 0.0389
Calculate the margin of error:
ME=2.576×0.0389≈0.1002ME = 2.576 \times 0.0389 \approx 0.1002
Confidence interval:
0.2681−0.1002<p<0.2681+0.10020.2681 - 0.1002 < p < 0.2681 + 0.1002
0.1679<p<0.36830.1679 < p < 0.3683
Convert to percentage:
16.8%<p<36.8%16.8\% < p < 36.8\%

STEP 7

Compare the confidence interval for women 16.8%<p<36.8%16.8\% < p < 36.8\% with the given interval for men 20.7%<p<40.8%20.7\% < p < 40.8\%.
Both intervals overlap, indicating that neither gender appears to be significantly more successful than the other.
The 99% confidence interval for the percentage of successful challenges by women is:
16.8%<p<36.8%\boxed{16.8\% < p < 36.8\%}

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