A Z-test is a type of parametric statistical test utilized for comparing two populations, which have large population sizes and the same variance values, to determine if their population means differ. It is a common test used for hypothesis testing, along with the Student’s t-test. A Z-test is an approximation which is based from the probability distribution of the test statistic.

A basic Z-test, using standard units, corresponds to the following formula:

Z = Xbar – μ_{0} /σ

Variables:

μ_{0 }= mean of the population

Xbar = mean of the samples

σ = standard deviation

A Z-test is used often for large populations in which the variance is known. For populations where the population is small and the variance is unknown, a t-test is more appropriate. Furthermore, histograms are often constructed before computing a Z-test in order to verify that the sample being used comes from a normally distributed population.

When conducting a Z-test, the first step is to state the null and alternative hypotheses. After this, a decision needs to be made on whether the test is a right-tailed test, a left-tailed test or a 2-tailed test. This is dependent upon the structure of the null hypothesis.

Additionally, Z-tests have a single critical value for different significance levels. Typically the significance value required is stated initially. The following displays the critical regions for testing the population means using a Z-test^{1}:

μ < μ_{0} - Reject the null if: Z < -z_{α}

μ > μ_{0} - Reject the null if: Z > z_{α}

μ does not equal μ_{0} - Reject the null if Z < -z_{α/2 }or Z > z_{α/2}

References:

1. Miller, I. and Freund, J.E. (2011). *Probability and Statistics for Engineers, 8th Edition. *Boston, MA.