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 after the new facility was opened. (Enter your answer using interval notation. Round your numerical values to three decimal places.)
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To construct a 95% confidence interval for the mean wait time of patients seen after the new facility was opened, we can use the formula:
Confidence Interval = sample mean ± margin of error
The margin of error can be calculated as:
Margin of error = critical value * standard error
First, we need to find the critical value corresponding to a 95% confidence level. Since we have a large sample size, we can use the z-table. For a 95% confidence level, the critical value is approximately 1.96.
Next, we need to calculate the standard error:
Standard error = standard deviation / √(sample size)
For the sample of patients seen after the new facility was opened:
Mean wait time = 28.2 hours,
Standard deviation = 16.7 hours,
Sample size = 224.
Standard error = 16.7 / √(224)
Now, we can calculate the margin of error:
Margin of error = 1.96 * (16.7 / √224)
Finally, we can construct the confidence interval:
Confidence Interval = 28.2 ± (1.96 * (16.7 / √224))
Calculating the expression:
Confidence Interval = 28.2 ± (1.96 * 1.114)
Confidence Interval ≈ 28.2 ± 2.181
Calculating the lower and upper bounds of the confidence interval:
Lower bound = 28.2 - 2.181 ≈ 25.019
Upper bound = 28.2 + 2.181 ≈ 31.381
Therefore, the 95% confidence interval for the mean wait time of patients seen after the new facility was opened is approximately [25.019, 31.381].
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