The authors of a paper were interested in estimating the mean time that patients who broke a hip had to
wait for surgery before and after the opening of a new hospital facility. They reported that for a
representative sample of 405 patients seen at this hospital before the new facility was opened, the mean
wait time for surgery was 31.4 hours and the standard deviation of wait times was 27.0 hours. For a
representative sample of 224 patients seen after the new facility was opened, the mean wait time for
surgery was 28.2 hours and the standard deviation of wait times was 16.7 hours.
Construct a 95% confidence interval for the mean wait time (in hours) of patients seen before the
new facility was opened.
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To construct a 95% confidence interval for the mean wait time of patients seen before the new facility was opened, we can use the formula:
Confidence Interval = (mean - margin of error, mean + margin of error)
First, we need to calculate the margin of error, which can be found using the formula:
Margin of Error = (z-value) * (standard deviation / √n)
where z-value is the critical value associated with the desired confidence level (95%), standard deviation is the standard deviation of the sample (27.0), and n is the sample size (405).
Using a standard normal distribution table or calculator, the z-value associated with a 95% confidence level is approximately 1.96.
Now let's calculate the margin of error:
Margin of Error = (1.96) * (27.0 / √405) = 3.061
The mean wait time for patients seen before the new facility was opened is 31.4 hours.
To construct the confidence interval, we can plug in the values:
Confidence Interval = (31.4 - 3.061, 31.4 + 3.061)
Confidence Interval = (28.339, 34.461)
Therefore, the 95% confidence interval for the mean wait time of patients seen before the new facility was opened is (28.339, 34.461) hours.
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