Which statistic is used to test hypotheses about an unknown population mean when the population standard deviation is unknown?

Which statistic is used to test hypotheses about an unknown population mean when the population standard deviation is unknown?

• A. T-statistic
• B. Z-statistic
• C. Chi-square statistic
• D. F-statistic

Answer: (A)

When you’re testing a population mean but the population standard deviation is unknown, you rely on the t-statistic. Since you don’t know the true spread, you estimate it with the sample standard deviation, s, and assess how far the sample mean is from the hypothesized mean in units of s/√n. This gives t = (X̄ − μ0) / (s/√n), which follows a t distribution with n−1 degrees of freedom. The t distribution has heavier tails to reflect the extra uncertainty from estimating sigma, and as the sample size grows it looks more and more like the normal distribution. The Z statistic would be used only if the population standard deviation were known (or in very large samples where that knowledge effectively applies); chi-square and F statistics are used for variance-related tests or ANOVA, not for testing a single mean with unknown sigma.

When you’re testing a population mean but the population standard deviation is unknown, you rely on the t-statistic. Since you don’t know the true spread, you estimate it with the sample standard deviation, s, and assess how far the sample mean is from the hypothesized mean in units of s/√n. This gives t = (X̄ − μ0) / (s/√n), which follows a t distribution with n−1 degrees of freedom. The t distribution has heavier tails to reflect the extra uncertainty from estimating sigma, and as the sample size grows it looks more and more like the normal distribution. The Z statistic would be used only if the population standard deviation were known (or in very large samples where that knowledge effectively applies); chi-square and F statistics are used for variance-related tests or ANOVA, not for testing a single mean with unknown sigma.