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)
Construct a
o. Compare the results from part (a) to this
more successful than the other?
a. Construct a
Upstudy AI Solution
Tutor-Verified Answer
Answer
The
confidence interval for the percentage of successful challenges among women is
. Comparing this to the men’s interval of
, the intervals overlap, indicating that there is no clear evidence that one group has a higher success rate than the other.
Solution
Let the sample proportion be
The standard error (SE) is given by
For a
confidence interval, we use the
-value
. Thus, the margin of error (ME) is
The confidence interval is then
which gives
Expressed as percentages and rounded to one decimal place, the
confidence interval for the percentage of successful challenges among women is
Comparing with the
confidence interval for the men’s matches,
, we see that the intervals overlap (from about
to
). This overlap suggests that there is no conclusive evidence that the success rate of challenges is different between the two groups.
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Beyond the Answer
To construct a
confidence interval for the percentage of successful challenges, we start by calculating the sample proportion, which is given by:
Next, we calculate the standard error (SE) for the sample proportion:
Now, we find the critical value (z*) for a
confidence level, which is approximately
.
Using this, we can construct the confidence interval:
Finally, we calculate the confidence interval:
Rounding to one decimal place, the
confidence interval for the percentage of successful challenges is approximately:
Now, comparing this interval with the men’s
confidence interval of
:
- 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.