Z and T-tests

Z-tests and t-tests are used to compare sample means against a known or hypothesized population mean. A z-test is used when the sample size is large (typically over 30) and the population standard deviation is known. A t-test is used for smaller sample sizes or when the population standard deviation is unknown. The t-distribution accounts for greater uncertainty and has heavier tails compared to the normal distribution.

Use a z-test if the population variance is known or if the sample is very large (n > 200), because the t-distribution approximates the normal distribution at large sample sizes. For comparing proportions, a z-test is generally used because the distribution of proportions is better approximated by a binomial distribution, which can be modeled by a normal distribution via CLT.