You wish to estimate the population mean emergency room wait time (in minutes) at UVA hospital. From a random sample of 1,000 emergency room visits, you find a 95% confidence interval to be (24,36). Suppose you instead wish to calculate a 90% confidence interval for the population for the population mean using the same sample data. Which of the following intervals could be the 90% confidence interval?

Question

a. (23,37)
b. (24, 36)
c. (24, 34)
d. (25, 35)

Answer 1

The formula for a confidence interval is:

CI = sample mean ± margin of error

Since the sample mean is not provided in the question, we cannot calculate the margin of error directly. However, we can make some observations to determine which interval could be the 90% confidence interval.

1. The margin of error will decrease as we decrease the confidence level from 95% to 90%. This means that the range of the interval will become narrower.

2. The 90% confidence interval will be entirely contained within the 95% confidence interval.

Based on these observations, we can eliminate options a and d immediately, as they have wider ranges than the 95% confidence interval.

Now, we need to compare options b and c to the 95% confidence interval.

Option b (24, 36) is the same as the 95% confidence interval. Therefore, it cannot be the 90% confidence interval, as it does not have a narrower range.

Option c (24, 34) has a narrower range than the 95% confidence interval. It is possible for the 90% confidence interval to have a narrower range, so option c could be the 90% confidence interval.

Therefore, the answer is c. (24, 34)

Answer 2

why would option d be eliminated if the range is in the 95% confidence interval and a smaller ranger?

Answer 3

I apologize for the confusion in my previous response. You are correct, option d should not be eliminated based on the range of the interval alone. Option d is still a valid possibility for the 90% confidence interval since it has a narrower range and is entirely contained within the 95% confidence interval. Therefore, both options b (24, 36) and d (25, 35) could be the 90% confidence interval. I apologize for any confusion caused by my previous incorrect response.