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# Z-tests

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# Z-tests

In earlier sections of this Handbook we have described the Normal- or z-transform and the Normal distribution. As such these topics describe how the Normal distribution, and the unit Normal in particular, may be used as a model of the population from which particular samples have been obtained. The process of statistical inference in such cases is one of checking that the Normal distribution model is a realistic assumption, and that the sample is a random sample from such a population, and then inferring the behavior of the population on the basis of the sample. Z-tests are essentially of this form, but typically refer to random samples of 30+ values, and generally involve evaluating whether the sample mean value, as an estimate of the population mean, differs from a particular value, a, by a large margin, or whether the difference appears to be not significant. Similar tests can be applied to the difference between two means and for tests on proportions. In all cases the process essentially involves producing a z-transform of the data, where the denominator is the estimated standard error (since we are testing mean values and proportions rather than source data items). For the reasons described above it is common to test for Normality before conducting  such tests. For small samples, typically of less than 30-50 values, the small sample approximation to the Normal, known as the t-distribution should be used, in the manner described as t-tests.