Which test is used to compare a sample mean to a hypothesized population mean when the population standard deviation is unknown?

Which test is used to compare a sample mean to a hypothesized population mean when the population standard deviation is unknown?

• A. Z-test
• B. Paired t-test
• C. One-sample t-test
• D. Chi-square test

Answer: (C)

When you have a single sample and you want to compare its mean to a hypothesized population mean while the population standard deviation is unknown, you use the one-sample t-test. This test accounts for the extra uncertainty from estimating the population spread with the sample standard deviation, s. You compute the t statistic as t = (X̄ − μ0) / (s / √n), and compare it to the t distribution with n − 1 degrees of freedom. Because s introduces extra variability, the t distribution is broader than the normal distribution, especially with small samples, making the test more conservative. A Z-test would be used only if the population standard deviation were known (or the sample size was so large that s could be treated as known). The paired t-test is for related or repeated measures on the same subjects, not a single sample versus a fixed mean. The chi-square test is for categorical data or for testing variances/grequencies, not for comparing a mean to a hypothesized value.

When you have a single sample and you want to compare its mean to a hypothesized population mean while the population standard deviation is unknown, you use the one-sample t-test. This test accounts for the extra uncertainty from estimating the population spread with the sample standard deviation, s. You compute the t statistic as t = (X̄ − μ0) / (s / √n), and compare it to the t distribution with n − 1 degrees of freedom. Because s introduces extra variability, the t distribution is broader than the normal distribution, especially with small samples, making the test more conservative. A Z-test would be used only if the population standard deviation were known (or the sample size was so large that s could be treated as known). The paired t-test is for related or repeated measures on the same subjects, not a single sample versus a fixed mean. The chi-square test is for categorical data or for testing variances/grequencies, not for comparing a mean to a hypothesized value.