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

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

• A. x̄ ± zα/2 (σ / √n)
• B. x̄ ± tα/2 (s / √n)
• C. x̄ ± tα/2, df = n−1 (s / √n)
• D. x̄ ± zα/2 (s / √n)

Answer: (C)

When sigma is unknown, you estimate the variability of the sample mean using the sample standard deviation s, which makes the sampling distribution of x̄ heavier-tailed than a normal distribution. The appropriate interval uses the t distribution and includes the degrees of freedom n−1 to reflect this estimation. The confidence interval is x̄ ± t_{α/2, n−1} · (s/√n). This accounts for the extra uncertainty from estimating sigma with s. Using a z critical value would be appropriate only if sigma were known (or for very large samples where t and z are nearly the same). Substituting s into a z-based formula ignores that extra variability, which is why that approach is not correct for smaller samples.

When sigma is unknown, you estimate the variability of the sample mean using the sample standard deviation s, which makes the sampling distribution of x̄ heavier-tailed than a normal distribution. The appropriate interval uses the t distribution and includes the degrees of freedom n−1 to reflect this estimation. The confidence interval is x̄ ± t_{α/2, n−1} · (s/√n).

This accounts for the extra uncertainty from estimating sigma with s. Using a z critical value would be appropriate only if sigma were known (or for very large samples where t and z are nearly the same). Substituting s into a z-based formula ignores that extra variability, which is why that approach is not correct for smaller samples.