Which scenario is not appropriate for performing a t-test to compare means?

Which scenario is not appropriate for performing a t-test to compare means?

• A. Two independent small samples with unknown variance.
• B. A single sample mean with unknown variance.
• C. Large sample with known variance.
• D. Paired design with small sample.

Answer: (C)

T-tests are used to compare means when the population variance is unknown and must be estimated from the data. In a large-sample situation where the variance is known, you don’t estimate spread from the sample—so the test statistic follows a normal distribution with the known variance. That is why a z-test is the appropriate choice here, not a t-test. The other scenarios fit the t-test framework: two independent small samples with unknown variance use a two-sample t-test; a single sample mean with unknown variance uses a one-sample t-test; and a paired design with small samples uses a paired t-test on the differences, which also relies on estimating the variance from the data. So, knowing the variance changes the appropriate method from t to z.

T-tests are used to compare means when the population variance is unknown and must be estimated from the data. In a large-sample situation where the variance is known, you don’t estimate spread from the sample—so the test statistic follows a normal distribution with the known variance. That is why a z-test is the appropriate choice here, not a t-test. The other scenarios fit the t-test framework: two independent small samples with unknown variance use a two-sample t-test; a single sample mean with unknown variance uses a one-sample t-test; and a paired design with small samples uses a paired t-test on the differences, which also relies on estimating the variance from the data. So, knowing the variance changes the appropriate method from t to z.