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. 

University of Michigan - Ann Arbor

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.

F-test

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

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.

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