t-test or z-test for a sample of 12 with unknown sigma?

Question

You want to test a hypothesis about a population mean. You have a sample of 12 observations, the population standard deviation is unknown, and the data look roughly normal.

Which test statistic should you use, t or z, and why?

Accepted Answer

Use a t-test. The choice hinges on whether the population standard deviation \sigma is known: - If \sigma is known, standardize with \sigma/\sqrt{n} and use the z-test. - If \sigma is unknown, you must estimate it from the sample with s, which adds extra uncertainty. That extra uncertainty is captured by the t-distribution, whose heavier tails give slightly wider critical values. Here \sigma is unknown and the sample is small (n = 12), so the t-test with n - 1 = 11 degrees of freedom is correct. The roughly-normal data supports the t-test's normality assumption, which matters most at small sa…

t-test or z-test for a sample of 12 with unknown sigma?

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