When the population standard deviation is known, the confidence interval for the mean uses the

When the population standard deviation is known, the confidence interval for the mean uses the

• A. Z-distribution
• B. T-distribution
• C. Chi-square distribution
• D. F-distribution

Answer: (A)

When the population standard deviation is known, the confidence interval for the mean uses the Z-distribution because the sampling distribution of the sample mean is Normal with a fixed standard error sigma divided by the square root of n. This lets us apply the standard normal critical values to form the interval around the sample mean. For a chosen confidence level, the interval is xbar ± z_(alpha/2) * sigma / sqrt(n); for example, a 95% interval uses about 1.96 as the critical value. If sigma isn’t known, we estimate it with the sample standard deviation and switch to the t-distribution with n−1 degrees of freedom to reflect extra uncertainty in the estimate. As n grows, the t-distribution approaches the Z distribution. The chi-square and F distributions are used for variances and variances ratios, not for constructing a confidence interval for the mean, which is why they aren’t used here.

When the population standard deviation is known, the confidence interval for the mean uses the Z-distribution because the sampling distribution of the sample mean is Normal with a fixed standard error sigma divided by the square root of n. This lets us apply the standard normal critical values to form the interval around the sample mean. For a chosen confidence level, the interval is xbar ± z_(alpha/2) * sigma / sqrt(n); for example, a 95% interval uses about 1.96 as the critical value. If sigma isn’t known, we estimate it with the sample standard deviation and switch to the t-distribution with n−1 degrees of freedom to reflect extra uncertainty in the estimate. As n grows, the t-distribution approaches the Z distribution. The chi-square and F distributions are used for variances and variances ratios, not for constructing a confidence interval for the mean, which is why they aren’t used here.