A function of the ratio of two independent random variables, each of which has a chi-square distribution and is divided by its number of degrees of freedom. 

What is an f-distribution?

1. The hypothesis that the means of a given set of normally distributed populations, all having the same standard deviation, are equal. This is perhaps the best-known F-test, and plays an important role in the analysis of variance (ANOVA).
2.  Lack-of-fit sum of squares: The hypothesis that a proposed regression model fits the data well. 
3.  The hypothesis that a data set in a regression analysis follows the simpler of two proposed linear models that are nested within each other.

Common examples of f-tests

Method for adjusting significance levels in a linear regression analysis to account for multiple comparisons.
Scheffé's method is a single-step multiple comparison procedure which applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method.
It works on similar principles as the Working–Hotelling procedure for estimating mean responses in regression, which applies to the set of all possible factor levels.

Scheffe's Method

Used for the purpose of comparing statistical models
Test statistic must have an f-distribution under the null hypothesis.
Example of application: finding the model that best fits the population from which the data were sampled.

University of Michigan - Ann Arbor

Method of fitting regression models in which the choice of predictive variables is carried out by an automatic procedure - adding or subtracting variables at each step depending on the results of predetermined tests 
Examples of tests that could be used:
1. F-tests
2. t-tests
3. R Squared

Stepwise Model Selection

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

F-test

Statistical measure of how close the data are to the fitted regression line.
r - correlation coefficient
r squared - coefficient of determination


R Squared

The $$t$$-test is the most frequently used null hypothesis test for evaluating the statistical relationship between two means. There are three primary variations of this procedure—the one-sample $$t$$-test, the dependent-samples $$t$$-test, and the independent-samples $$t$$-test—each designed to accommodate slightly different research designs.

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