Z-test

thumb| A '''Z-test' is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-test tests the mean of a distribution. For each significance level in the confidence interval, the Z''-test has a single critical value (for example, 1.96 for 5% two-tailed), which makes it more convenient than the Student's t-test whose critical values are defined by the sample size (through the corresponding degrees of freedom). Both the Z-test and Student's t-test have similarities in that they both help determine the significance of a set of data. However, the Z-test requires knowing the population deviation, which is sometimes difficult to determine, making the t-test more convenient.

==Applicability== Because of the central limit theorem, many test statistics are approximately normally distributed for large samples. Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance is known. If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n T that is approximately normally distributed under the null hypothesis is as follows: Estimate the expected value μ of T under the null hypothesis and obtain an estimate s of the standard deviation of T. Determine the properties of T: one-tailed or two-tailed. For null hypothesis '''H0: μ ≥ μ0 vs alternative hypothesis H1: μ 0, it is lower/left-tailed (one-tailed). For null hypothesis H0: μ ≤ μ0 vs alternative hypothesis H1: μ > μ0, it is upper/right-tailed (one-tailed). For null hypothesis H0: μ = μ0 vs alternative hypothesis H1: μ ≠ μ0', it is two-tailed. Calculate the standard score: Z = \frac{\bar{T} - \mu_0}{s}