# F-Test

We start by introducing to the F-test. For learning more, access the F-test assignment help and F-test homework help sections.

## An overview of F-Test

Any test statistic which has an F-distribution in the null hypothesis is considered as the F-test. Most commonly used to compare statistical models, the f-distribution has to be structured around a data set, in order to identify the model which best fits the population from which the data was sampled. The name ‘F-test’ was coined by George W. Snedecor. It was in the honour of Sir Ronald A. Fisher. Initially, this statistic was developed as the variance ratio in the 1920s.

### Examples of F-test

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Below is a list of common examples of the use of F-tests.

- A hypothesis which states that the means of a given set of normally distributed populations, all with the same standard deviation, are equal. This is the most commonly used instance of ‘F-test’ and plays a crucial part in the analysis of variance or ANOVA.
- The hypothesis that a proposed regression model fits the data set
- A hypothesis when the data of a regression analysis, replicates in the simpler of the two proposed linear models.

In addition, there are some statistical procedures, like Scheffé’s method for multiple comparisons linear models adjustments, also use F-tests. The F-test is designed to verify the hypothesis that the two population variances are equal. This test compares the ratio of the two terms to make this observation. Hence, if the variances are equal, then the ratio of the two terms will be 1.

### Discuss the relation between F-test and Chi-square test

An F-distribution is formed by the ratio of the two independent chi-square variables. These two terms are divided by their respective degrees of freedom. Since F is formed by chi-square, F-distribution has picked on many of the chi-square properties. These properties are listed below

- The F-values are greater than 0 ie they are non-negative
- F-distribution is a non-symmetric distribution
- The mean is approximately equal to 1
- There are two different degrees of freedom, one is for the numerator, and the other for the denominator.
- There are different F distributions. One for each pair of degrees of freedom.

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### F-test: Observation and results of F-Tests

For the hypothesis, we assume that the null hypothesis states the ratio of variances is equal to 1. This ratio of sample variances will be used as the test statistic. If the null hypothesis is false, then we reject the hypothesis that the ratio is equal to 1.

There are different F-tables and each has a different level of significance. It is necessary to find the correct level of significance. The critical value can be found out by looking up the numerator degrees of freedom and the denominator degrees of freedom.

In a right tailed test, it can be observed that all of the tables only give one level of significance. This is based on the assumption that the F distribution is asymmetric, and there are no negative values. It is not possible to take the right critical value with the opposite sign to find the left critical value.

In order to determine the left critical value, we have to reverse the degrees of freedom and then look up the right critical value. The reciprocal of this value is considered.

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