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. (Enter your answer using interval notation. Round your numerical values to three decimal places.)
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To construct a confidence interval for the mean wait time of patients seen before the new facility was opened, we can use the formula:
Confidence interval = sample mean ± (critical value) * (standard deviation/square root of sample size)
First, let's calculate the sample mean, standard deviation, and sample size for the patients seen before the new facility was opened:
Sample mean before = 31.4 hours
Standard deviation before = 27.0 hours
Sample size before = 405
Next, we need to find the critical value for a 95% confidence interval. Since the sample size is large (n > 30), we can use the Z-distribution. The critical value for a 95% confidence interval is approximately 1.96.
Now, we can plug in the values into the formula:
Confidence interval before = 31.4 ± (1.96) * (27.0/√405)
Calculating this:
Confidence interval before = 31.4 ± (1.96) * (27.0/√405)
Confidence interval before = 31.4 ± 3.540
Confidence interval before = (27.86, 34.94)
Therefore, the 95% confidence interval for the mean wait time of patients seen before the new facility was opened is (27.86, 34.94) hours.
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