Which statement is true about confidence intervals when the population standard deviation is unknown?

Which statement is true about confidence intervals when the population standard deviation is unknown?

• A. Always use the z-distribution regardless of sigma.
• B. Use the t-distribution only when the population is known.
• C. Use the t-distribution when the population standard deviation is unknown.
• D. Use the t-distribution when the sample size is extremely large.

Answer: (C)

When the population standard deviation is unknown, confidence intervals for the mean are built using the t-distribution because the variability is estimated from the sample data. Instead of dividing by the true sigma, you use the sample standard deviation s, which adds extra uncertainty. The resulting t-statistic has heavier tails and uses n−1 degrees of freedom, so the interval width reflects that extra variability: X̄ ± t_{α/2, n−1} · (s/√n). As n grows large, the t-distribution looks more like the standard normal, but the standard practice remains to use the t distribution whenever sigma is unknown, especially for smaller samples. This is why the statement about using the t-distribution when sigma is unknown is true.

When the population standard deviation is unknown, confidence intervals for the mean are built using the t-distribution because the variability is estimated from the sample data. Instead of dividing by the true sigma, you use the sample standard deviation s, which adds extra uncertainty. The resulting t-statistic has heavier tails and uses n−1 degrees of freedom, so the interval width reflects that extra variability: X̄ ± t_{α/2, n−1} · (s/√n). As n grows large, the t-distribution looks more like the standard normal, but the standard practice remains to use the t distribution whenever sigma is unknown, especially for smaller samples. This is why the statement about using the t-distribution when sigma is unknown is true.