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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).

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 < 30), the Student's t-test may be more appropriate.

How to perform a Z test when T is a statistic that is approximately normally distributed under the null hypothesis is as follows:

First, estimate the expected value μ of T under the null hypothesis, and obtain an estimate s of the standard deviation of T.

Second, determine the properties of T : one tailed or two 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 lower/left-tailed (one tailed).

For Null hypothesis H0: μ=μ0 vs alternative hypothesis H1: μ≠μ0 , it is two-tailed.

Third, calculate the standard score :

${\displaystyle Z={\frac {({\bar {X}}-\mu _{0})}{s}}}$ ,

which one-tailed and two-tailed p-values can be calculated as Φ(Z) (for upper/right-tailed tests), Φ(−Z)(for lower/left-tailed tests) and 2Φ(−|Z|) (for two-tailed tests) where Φ is the standard normal cumulative distribution function.

## Use in location testing

1. The term "Z-test" is often used to refer specifically to the one-sample location test comparing the mean of a set of measurements to a given constant when the sample variance is known. For example, if the observed data X1, ..., Xn are (i) independent, (ii) have a common mean μ, and (iii) have a common variance σ2, then the sample average X has mean μ and variance