An F-test is conducted on the basis of F statistic. F statistic is defined as the ratio between the two independent chi square variates divided by their respective degree of freedom. F-test follows Snedecor’s F- distribution.
The F-test has some applications in statistical theory. This document will discuss the applications of F-test in detail.
F-test can be used to test the equality of two population variances. Suppose a researcher wants to test whether or not two independent samples have been drawn from normal populations with the same variability. In this case, the researcher uses the F-test to do this study. The F-test can also be used to know whether there is any homogeneity between the two independent estimates of the population variance.
A practical example can show the above case in which the F-test is applied. Suppose two sets of pumpkins were grown under two different experimental conditions. Then, a random sample of size 9 and 11 were taken from the two different conditions. Those samples indicate that the standard deviations of their weights are 0.6 and 0.8 respectively. On making an assumption that the distribution of their weights is normal, the researcher conducts an F-test to test the hypothesis of whether or not the true variances are equal.
F-test can be used to test the significance of an observed multiple correlation coefficient. F-test can also be used to test the significance of an observed sample correlation ratio. The term sample correlation ratio is a measure of relationship between the statistical dispersion between the categories within the sample and the sample as a whole. Its significance is tested by the researcher using the F-test. F-test can also be used to test for the linearity in the regression model.
The most popular usage of F-test is that of Analysis of Variance (ANOVA) which plays a very important and fundamental role in Design of Experiments in Agricultural Statistics. In analysis of variance (ANOVA), F-test is carried out to test the equality of several means.
There is a relationship between t and F distributions, as in the F-test. This relationship states that if a statistic t follows Student’s t distribution with ‘n’ degrees of freedom, then the square of this statistic will follow Snedecor’s F distribution, as in F-test with 1 and n degrees of freedom.
There is also a relationship between F-test and chi square distribution. This relationship states that if the degree of freedom and the second chi square variate goes to infinity, then the F distribution (as in F-test) will follow the chi square distribution.
Due to such relationships, the F-test has many properties, much like the chi square test. The F-values in F-test are all non negative. The F-distribution (as in F-test) is always non symmetrically distributed. The mean in F-distribution (as in F-test) is approximately one. There are two independent degrees of freedom in F distribution, one is the numerator and the other is the denominator. There are many different F distributions (as in F-test), one for every pair of degrees of freedom.
F-test is a parameteric test that helps the researcher draw an inference about the data that is being drawn from a particular population. F-test is called a parameteric test because of the presence of parameters in the F- test. These parameters in F-test are mean and variance. The mode of F-test, i.e. the value that is most frequently in a data set, is always less than unity. According to the Karl Pearson’s coefficient of skewness, F-test is highly positively skewed. The probability distribution of F increases steadily before reaching the peak, and then again it starts decreasing in order to become tangential at infinity. Thus, we can say that the axis of F is asymptote to the right tail.
