ztests

zTests is a family of statistical hypothesis tests that use the standard normal distribution to assess claims about population parameters. They are typically applied when the population standard deviation is known or when sample sizes are large enough for the normal approximation to be reliable. The term commonly encompasses one-sample z-tests for a mean, two-sample z-tests for comparing means, and z-tests for proportions.

In a mean z-test with known sigma, the test statistic is z = (Xbar - mu0) / (sigma / sqrt(n)). For a one-proportion z-test, z = (phat - p0) / sqrt(p0(1-p0)/n). Two-sample z-tests extend these formulas to differences in means or proportions and may use pooled or unpooled variance estimates depending on variance equality assumptions. Some formulations also address paired data under a normal-approximation framework.

zTests are widely used in quality control, manufacturing, clinical trials, and online experiments (A/B testing), where large samples or known variances are common. They are implemented in many statistics libraries and software packages as ztest or similar functions, often alongside t-tests and nonparametric alternatives to enable quick inference and reporting of p-values and confidence intervals.

Limitations include reliance on normal approximation, which may be inappropriate for small samples or unknown variances. Z-tests are sensitive to departures from normality and to outliers. In such cases, or when sample sizes are small, t-tests, exact binomial tests, or nonparametric methods are preferred.