When the population standard deviation is unknown and the sample size is moderate to large, the t-distribution should be used.

When the population standard deviation is unknown and the sample size is moderate to large, the t-distribution should be used.

• A. True
• B. False
• C. Only for very small samples
• D. Never

Answer: (A)

When the population standard deviation is unknown, you must rely on the sample standard deviation to estimate variability, and the t distribution is designed to handle that extra uncertainty. The t distribution has heavier tails than the normal distribution, which accounts for the fact that we’re estimating sigma from the data. This is true for any sample size, but as the sample gets larger, the t distribution becomes almost indistinguishable from the normal distribution, so using it remains valid and conservative for moderate to large samples. If the population standard deviation were known, you would use the normal distribution instead. Saying the statement is false, or that the t distribution is needed only for very small samples or never, contradicts the standard approach to inference about the mean when sigma is unknown.

When the population standard deviation is unknown, you must rely on the sample standard deviation to estimate variability, and the t distribution is designed to handle that extra uncertainty. The t distribution has heavier tails than the normal distribution, which accounts for the fact that we’re estimating sigma from the data. This is true for any sample size, but as the sample gets larger, the t distribution becomes almost indistinguishable from the normal distribution, so using it remains valid and conservative for moderate to large samples. If the population standard deviation were known, you would use the normal distribution instead. Saying the statement is false, or that the t distribution is needed only for very small samples or never, contradicts the standard approach to inference about the mean when sigma is unknown.