When should the t-distribution be used instead of the normal distribution for confidence intervals?

When should the t-distribution be used instead of the normal distribution for confidence intervals?

• A. When the population standard deviation is known.
• B. When the sample size is very large.
• C. When the data are categorical.
• D. When the population standard deviation is unknown.

Answer: (D)

The main idea is that you use the t-distribution for confidence intervals when the population standard deviation is unknown and you must estimate variability from the sample. When sigma is unknown, you replace it with the sample standard deviation s, so the standard error becomes s/√n and the critical value comes from a t distribution with n−1 degrees of freedom. The resulting interval is X̄ ± t_{α/2, n−1} (s/√n). The t distribution has heavier tails than the normal, which accounts for the extra uncertainty from estimating sigma with s, especially in small samples, producing wider intervals to maintain the intended confidence level. As n grows large, the t distribution approaches the normal, and the two methods give similar results. If sigma were known, you would use the normal distribution instead. Categorical data or other scenarios aren’t about estimating a mean with an unknown standard deviation.

The main idea is that you use the t-distribution for confidence intervals when the population standard deviation is unknown and you must estimate variability from the sample. When sigma is unknown, you replace it with the sample standard deviation s, so the standard error becomes s/√n and the critical value comes from a t distribution with n−1 degrees of freedom. The resulting interval is X̄ ± t_{α/2, n−1} (s/√n). The t distribution has heavier tails than the normal, which accounts for the extra uncertainty from estimating sigma with s, especially in small samples, producing wider intervals to maintain the intended confidence level. As n grows large, the t distribution approaches the normal, and the two methods give similar results. If sigma were known, you would use the normal distribution instead. Categorical data or other scenarios aren’t about estimating a mean with an unknown standard deviation.