F -DISTRIBUTION AND ITS APPLICATIONS
F-statistic is the ratio of two sums of the squares of deviations of observations from respective means. The sampling distribution of the statistic is F-distribution.
Let X and Y be two independent χ2 random variates with m and n degrees of freedom respectively.
Then F = is said to follow F-distribution with (m, n) degrees of freedom. This F-distribution is named after the famous statistician R.A. Fisher (1890 to 1962).
Let (X1, X2 , …, Xm) and (Y1, Y2, …, Yn) be two independent random samples drawn from N(μX, σX2) and N(μY, σY2) populations respectively.
(1) Hence, F-Statistic is defined as
(2) F-Statistic is also defined as the ratio of two mean square errors.
The following are some of the important applications where the sampling distribution of the respective statistic under H0 is F–distribution.
(i) Testing the equality of variances of two normal populations. [Using (1)]
(ii) Testing the equality of means of k (>2) normal populations. [Using (2)]
(iii) Carrying out analysis of variance for two-way classified data. [Using (2)]