When is a z-test appropriate?

When is a z-test appropriate?

• A. Large samples or known population variance; test statistic follows z.
• B. When the population distribution is non-normal, use z.
• C. Small samples with unknown variance; test statistic follows t.
• D. Only for proportions.

Answer: (A)

A z-test is appropriate when you know the population standard deviation (or have a very large sample so the sampling distribution of the mean is essentially normal with a known standard error). The test statistic is z = (X̄ − μ0) / (σ/√n), and under the null this follows a standard normal distribution. This is why large samples or known variance make the z-test the right choice: you can standardize the gap between the sample mean and the hypothesized mean using the true spread of the population. If the population variance isn’t known and you have a small sample, you don’t rely on the z distribution because you’re uncertain about σ; the appropriate approach is the t-test, which uses a t distribution with n−1 degrees of freedom to account for that extra uncertainty. Using the z distribution for non-normal populations isn’t generally valid with small samples, and the idea that z tests are “only for proportions” isn’t correct—the z test can be used for means (and, with large samples, for proportions too).

A z-test is appropriate when you know the population standard deviation (or have a very large sample so the sampling distribution of the mean is essentially normal with a known standard error). The test statistic is z = (X̄ − μ0) / (σ/√n), and under the null this follows a standard normal distribution. This is why large samples or known variance make the z-test the right choice: you can standardize the gap between the sample mean and the hypothesized mean using the true spread of the population.

If the population variance isn’t known and you have a small sample, you don’t rely on the z distribution because you’re uncertain about σ; the appropriate approach is the t-test, which uses a t distribution with n−1 degrees of freedom to account for that extra uncertainty. Using the z distribution for non-normal populations isn’t generally valid with small samples, and the idea that z tests are “only for proportions” isn’t correct—the z test can be used for means (and, with large samples, for proportions too).