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?
a. (23,37)
b. (24, 36)
c. (24, 34)
d. (25, 35)
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)
why would option d be eliminated if the range is in the 95% confidence interval and a smaller ranger?
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.
To calculate a confidence interval for a population mean, you need to know the sample mean, the standard deviation or standard error, and the desired level of confidence.
In this case, the sample data provides you with a 95% confidence interval for the population mean emergency room wait time at UVA hospital. The interval is (24,36), but this interval corresponds to a 95% confidence level.
To calculate a 90% confidence interval using the same sample data, you need to find the margin of error. The margin of error is given by the formula:
Margin of Error = Critical Value * Standard Error
The critical value depends on the desired level of confidence. For a 90% confidence level, the critical value is smaller than the critical value for a 95% confidence level. Therefore, the margin of error for a 90% confidence interval will be smaller than the margin of error for a 95% confidence interval.
To select the correct 90% confidence interval from the options given, you need to choose the interval that has a smaller range while still including the sample mean. Let's consider each option:
a. (23,37): This interval has a larger range than the original interval and includes values outside the original interval. Therefore, it cannot be the correct 90% confidence interval.
b. (24,36): This is the original 95% confidence interval, but we need to calculate a 90% confidence interval. Since the margin of error for a 90% confidence interval is smaller, this interval is too wide to be the correct 90% confidence interval.
c. (24,34): This interval has a smaller range than the original interval, and it is entirely contained within the original interval. Therefore, it is a plausible 90% confidence interval.
d. (25,35): This interval also has a smaller range than the original interval, but it does not include the sample mean. Therefore, it cannot be the correct 90% confidence interval.
Based on this analysis, the correct 90% confidence interval is option c, (24, 34).