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# Tests on the Mean

This chapter contains statistical tests on the mean of a normal population. Frequent questions are if the mean equals a specific value (mostly the null) or if two populations have the same mean or differ by a specific value. Depending on the sampling strategy and on knowledge of the data generation process, the assumptions of known or unknown variances must be distinguished. In most situations the variance is unknown–probably the reason why neither SAS nor R provides procedures to calculate tests for the case of known variances. However, rare situations exist where the variance of the underlying Gaussian distribution is known. We provide some code that demonstrates how these–so-called z-tests–can be calculated. If the variance has to be estimated from the sample, the test statistic distribution changes from standard normal to Student's t-distribution. Here the degrees of freedom vary depending on the specific test problem, for example, in the two population case on whether the variances are assumed to be equal or not. SAS by the procedure proc ttest and R by the function t.test provided convenient ways to calculate these tests. For k-sample tests (F-test) please refer to Chapter 17 which covers ANOVA tests.

# 2.1 One-sample tests

In this section we deal with the question, if the mean of a normal population differs from a predefined value. Whether the variance of the underlying Gaussian distribution is known or not determines the use of the z-test or the ...

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