A t-statistic is used to test hypotheses about what, under what condition?

A t-statistic is used to test hypotheses about what, under what condition?

• A. A t-statistic is used to test hypotheses about an unknown population mean when the population standard deviation is unknown.
• B. Differences in variances between groups.
• C. Proportions in a binomial distribution.
• D. The median of a skewed distribution.

Answer: (A)

Testing hypotheses about a population mean when the population standard deviation is unknown is what the t-statistic is for. Since sigma isn’t known, you estimate variability with the sample standard deviation, and the test statistic follows a t distribution with n minus 1 degrees of freedom. This distribution has heavier tails than the normal, especially with small samples, so the test accounts for the extra uncertainty. If the population standard deviation were known, you’d use a z-test instead, which relies on the standard normal distribution. The other options don’t fit because they involve different questions or statistics: testing variances between groups uses an F-test (or related variance-equality tests), testing proportions in a binomial setup uses a z-test for proportions or a chi-square test, and assessing a median in a skewed distribution is typically handled by nonparametric tests, not the t-statistic.

Testing hypotheses about a population mean when the population standard deviation is unknown is what the t-statistic is for. Since sigma isn’t known, you estimate variability with the sample standard deviation, and the test statistic follows a t distribution with n minus 1 degrees of freedom. This distribution has heavier tails than the normal, especially with small samples, so the test accounts for the extra uncertainty. If the population standard deviation were known, you’d use a z-test instead, which relies on the standard normal distribution.

The other options don’t fit because they involve different questions or statistics: testing variances between groups uses an F-test (or related variance-equality tests), testing proportions in a binomial setup uses a z-test for proportions or a chi-square test, and assessing a median in a skewed distribution is typically handled by nonparametric tests, not the t-statistic.