Which formula provides a confidence interval for the population mean when σ is known?

Which formula provides a confidence interval for the population mean when σ is known?

• A. x̄ ± zα/2 (σ / √n)
• B. x̄ ± tα/2 (s / √n)
• C. x̄ ± zα/2 (s / √n)
• D. x̄ ± tα/2 (σ / √n)

Answer: (A)

When the population standard deviation is known, the sampling distribution of the sample mean is normal with standard error equal to σ/√n. That lets us use the standard normal critical value z_{α/2} to form a confidence interval around the sample mean. The interval becomes x̄ ± z_{α/2} (σ/√n). Using z reflects the exact knowledge of σ and the normality of the sampling distribution, giving the correct width for the interval. If you replace σ with the sample standard deviation s, you’re no longer using the known parameter, so the appropriate approach changes to the t distribution with degrees of freedom n−1, leading to x̄ ± t_{α/2, n−1} (s/√n). That’s more conservative and applies when σ is unknown. Using s in place of σ while still using the z value would misstate the uncertainty.

When the population standard deviation is known, the sampling distribution of the sample mean is normal with standard error equal to σ/√n. That lets us use the standard normal critical value z_{α/2} to form a confidence interval around the sample mean. The interval becomes x̄ ± z_{α/2} (σ/√n). Using z reflects the exact knowledge of σ and the normality of the sampling distribution, giving the correct width for the interval.

If you replace σ with the sample standard deviation s, you’re no longer using the known parameter, so the appropriate approach changes to the t distribution with degrees of freedom n−1, leading to x̄ ± t_{α/2, n−1} (s/√n). That’s more conservative and applies when σ is unknown. Using s in place of σ while still using the z value would misstate the uncertainty.