RE : "F and T tests"
One should be careful in simply referring to "F and T tests". There is no such thing as a uniquely identified/determined "F test" nor a uniquely identified/determined "T test". In both cases one is simply identifying the distribution of a test statistic.
The F distribution is used in connection with one form of a test of a hypothesis of the form
H0: mu1 = mu2 =..... = mn
H1: at least one of the mui's is differentwhere mu1,..., mun are the means. But the degrees of freedom can be different depending on exactly how the populations are identified and how the experiment is conducted. Note that there is an underlying assumption of equal variances.
The F distribution can also be used to test the equality of variances but the test statistic is quite different. In this case the null hypothesis could be two sided , one sided (and a choice of two different sides).
The F distribution can also be used in connection with tests pertaining to the coefficients in a regression model, again the test statistic is quite different.
A t distributed statistic occurs in many quite different instances, again the actual test statistic will vary significantly and even for the same statistic the hypotheses may vary, e.g., one sided vs two sided.
For example a t distributed statistic arises when testing the equality of two means (at least two different cases, one where the variances are assumed equal but unknown and one uses a pooled estimate of variance, a second one where the variances are not assumed equal. In both cases the variances are assumed unknown. Normality is a critical underlying assumption in both cases.
Contrast this with a paired sample "t test". The hypotheses are quite different, the test statistic is computed in a very different way and the underlying assumptions are quite different, e.g., there is only one variance not two. The degrees of freedom are computed differently.
A t distributed statistic can arise when testing whether a correlation coefficient is zero or not zero ( different formula for the degrees of freedom and very different test statistic), hypothesis can be two sided or one sided.
The output of a typical statistics software package, when fitting a regression model, will give "p values" for each of the coefficients, these are commonly obtained from "t tests" although some software now incorporates the use of bootstrap methods.
As always with testing of hypotheses one should be aware that there are two kinds of possible errors but depending on which hypothesis is "accepted" only one of the two errors is possible. E.g., if you accept the null hypothesis you can't make a Type I error.
Equality of parameters such as means, variances is not sufficient to claim identical populations, i.e., that the samples came from the same populations.
Donald Myers
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