Z-test - Use in Location Testing

Use in Location Testing

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. If the observed data X₁, ..., X_n are (i) uncorrelated, (ii) have a common mean μ, and (iii) have a common variance σ2, then the sample average X has mean μ and variance σ2 / n. If our null hypothesis is that the mean value of the population is a given number μ₀, we can use X −μ₀ as a test-statistic, rejecting the null hypothesis if X −μ₀ is large.

To calculate the standardized statistic Z = (X − μ₀) / s, we need to either know or have an approximate value for σ2, from which we can calculate s2 = σ2 / n. In some applications, σ2 is known, but this is uncommon. If the sample size is moderate or large, we can substitute the sample variance for σ2, giving a plug-in test. The resulting test will not be an exact Z-test since the uncertainty in the sample variance is not accounted for — however, it will be a good approximation unless the sample size is small. A t-test can be used to account for the uncertainty in the sample variance when the sample size is small and the data are exactly normal. There is no universal constant at which the sample size is generally considered large enough to justify use of the plug-in test. Typical rules of thumb range from 20 to 50 samples. For larger sample sizes, the t-test procedure gives almost identical p-values as the Z-test procedure.

Other location tests that can be performed as Z-tests are the two-sample location test and the paired difference test.