The t-statistic is a test statistic with a known distribution. Which distribution is associated with the t-statistic?

• A. Chi-square distribution
• B. F-distribution
• C. Normal distribution
• D. t-distribution

Answer: (D)

The t-statistic is tied to the t-distribution because it arises when you estimate the population standard deviation from the sample. Using the sample standard deviation introduces extra uncertainty, and the t-distribution adjusts for that by having heavier tails than the normal distribution. The exact shape depends on the degrees of freedom, which for a one-sample mean test are typically n − 1. As the sample size grows, the t-distribution becomes increasingly similar to the standard normal distribution, which is why large samples often use z instead of t.

The other distributions aren’t the appropriate reference for the t-statistic. The chi-square distribution is associated with sums of squared standardized deviations and is used in variance-related tests. The F-distribution is a ratio of two scaled chi-square variables and appears in ANOVA and regression contexts. The normal distribution describes the distribution you’d use if the population variance were known or in large samples where the z-statistic applies.