Final answer:
The 95% confidence interval for the emergency room wait time, calculated using the Student's t-distribution, is (24.01, 25.99) after rounding. The closest provided option to our calculated confidence interval is b) (24.15, 25.85).
Step-by-step explanation:
The student is asking for a 95% confidence interval estimate for the population mean wait time in an emergency room using the Student's t-distribution. With a sample mean («x») of 25 minutes and a sample standard deviation (s) of 2 minutes from 18 patients, we can calculate this confidence interval.
Since the population standard deviation is unknown, we use the Student's t-distribution rather than the normal distribution.
To find the confidence interval, we need to determine the critical value of t for 17 degrees of freedom (n - 1) at the 95% confidence level. We use the sample standard deviation and the sample size to calculate the standard error, and then find the margin of error by multiplying the standard error with the critical value of t.
The confidence interval is found by adding and subtracting this margin of error from the sample mean. Assuming a critical value of 2.110 (from a t-distribution table or calculator), the confidence interval is calculated as follows:
- Standard Error (SE) = s / (sqrt(n)) = 2 / sqrt(18) = 0.471
- Margin of Error (ME) = t * SE = 2.110 * 0.471 = 0.994
- Lower limit = «x» - ME = 25 - 0.994 = 24.01
- Upper limit = «x» + ME = 25 + 0.994 = 25.99
We then round these to two decimal places to get the confidence interval: (24.01, 25.99).
The correct answer found does not exactly match any option provided by the student. However, the closest option, considering rounding principles, to our calculation is option b) (24.15, 25.85).