Ztest

A z-test, or z-test for short, is a statistical hypothesis test that uses the standard normal distribution to assess whether a population parameter equals a specified value. It is most appropriate when the sampling distribution of the statistic is normal and the population standard deviation is known, or when sample sizes are large enough for the normal approximation to be valid. The test is commonly applied to testing a population mean or a population proportion.

Common forms include the one-sample z-test for a mean with known sigma, and the z-test for a population proportion. The corresponding test statistics are, for a mean: z = (X̄ − μ0) / (σ / √n); and for a proportion: z = (p̂ − p0) / √[p0(1 − p0) / n]. Under the null hypothesis, these statistics are approximately standard normal, allowing the calculation of p-values and confidence statements. In two-sample settings, z-tests can be used when the variances are known or when sample sizes are large enough for the large-sample normal approximation.

Assumptions typically include random or representative sampling, independence of observations, and, for the mean test, a known population standard deviation or a sufficiently large sample size. For the proportion test, the normal approximation is valid when np0 and n(1 − p0) are both at least about 5. If the population variance is unknown or the sample size is small, a t-test is usually preferred. Z-tests are widely used in quality control, survey analysis, and large-sample inferential settings.