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The term classical tests we are using in this topic is not a widely recognized terminology, but is a useful umbrella under which a number of well-known statistical tests can be placed. In this topic we include many of the standard tests described in Chapter 8 'Classical Tests' of Crawley's (2007) excellent textbook on R [CRA1], although some topics such as correlation analysis and contingency table analysis we describe in separate topics in this Handbook.
The tests we describe relate to the mean values and variances of samples, and the overall distribution of such samples. Typically we are looking to see whether a sample mean provides evidence that the population mean has a particular value, such as 0 or 5.5, and within what the range of values the population mean might be expected to be found. For larger samples and/or where the population variance is known, this involves the use of z-tests, whilst for smaller samples, t-tests are used. These tests are essentially the same, with both types being built on the assumption that the sample data are drawn randomly from a population whose form is a Normal distribution. To check that a sample is indeed drawn from a particular distribution, such as the Normal, goodness of fit tests are carried out before other forms of analysis. Classical tests that do not require such strict assumptions are summarized in the section Non-parametric analysis.
We also cover a number of tests that relate to the estimates of variances in one or more samples. These include F-tests, Barlett's M-test, and a number of similar tests for the homogeneity of variance among Normally distributed observations.