Under which scenario is a t-test appropriate for inference about a population mean?

Under which scenario is a t-test appropriate for inference about a population mean?

• A. Small samples or unknown population variance; use t-distribution.
• B. Large samples with known population variance; use z-distribution.
• C. Data are categorical and count-based.
• D. The sample size is exactly 100 and the population variance is known.

Answer: (A)

A t-test is used when you want to infer the population mean from a sample but you don’t know the population’s variance. You estimate the variance from the sample, and that extra uncertainty is captured by the t-distribution, which has heavier tails than the normal distribution. This makes the t-test more reliable for small samples because it appropriately widens the confidence intervals and t-statistic tails to reflect the unknown variance. If the population variance were known and the sample size is large, the normal (z) approach becomes appropriate since the uncertainty from estimating variance is removed and the distribution approximates normal more closely. Categorical data aren’t analyzed with a t-test, and even a large sample with known variance would typically use the z-test rather than a t-test.

A t-test is used when you want to infer the population mean from a sample but you don’t know the population’s variance. You estimate the variance from the sample, and that extra uncertainty is captured by the t-distribution, which has heavier tails than the normal distribution. This makes the t-test more reliable for small samples because it appropriately widens the confidence intervals and t-statistic tails to reflect the unknown variance. If the population variance were known and the sample size is large, the normal (z) approach becomes appropriate since the uncertainty from estimating variance is removed and the distribution approximates normal more closely. Categorical data aren’t analyzed with a t-test, and even a large sample with known variance would typically use the z-test rather than a t-test.