Final answer:
To calculate the 99.5% confidence interval for a given sample, use the z-score for the confidence level, find the standard error, and then calculate the margin of error (EBM). Subtract and add the EBM to the sample mean to construct the interval.
Step-by-step explanation:
To find the 99.5% confidence interval for a sample mean, we first need to understand that the confidence interval provides a range in which we expect the true population mean to lie, given the sample statistics. For a sample size of 66 with a mean of 42.5 and a standard deviation of 7.8, we can calculate the margin of error (EBM) using the z-score corresponding to the 99.5% confidence level.
Firstly, we find the z-score that corresponds to the 99.5% confidence level, which is typically 2.576 (this value can be found in z-score tables or by using a statistical software). Then, we calculate the standard error of the mean, which is the standard deviation divided by the square root of the sample size. The formula for the standard error (SE) is SE = σ / √(n), where σ is the standard deviation and n is the sample size.
Subsequently, we compute the margin of error (EBM) using the formula EBM = z * SE. Once we have the EBM, we construct the confidence interval by subtracting and adding the EBM from the sample mean: CI = (x-bar - EBM, x-bar + EBM).
Applying these calculations to our given problem:
- SE = 7.8 / √(66)
- EBM = 2.576 * SE
- CI = (42.5 - EBM, 42.5 + EBM)
After calculating, we can express the 99.5% confidence interval using the calculated EBM value.