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In F (n1, n2) distribution if we let n2 à∞, then χ2 = n1F follows χ2 – distribution with n1 degrees of freedom.

Proof:

We have

Which is the probability density function of chi square distribution with n1 degrees of freedom.

12.5.5 F test for testing the significance of an observed multiple correlation coefficient

If R is the observed multiple correlation coefficient of a variate with k other variates in a random sample of size n from a (k+1) variate population, then Prof. R.A. Fisher proved that under the null hypothesis (H0
) that the multiple correlation coefficient in the population is zero, the statistic

f test for significance

Conforms to F – distribution with (k, n-k-1) degrees of freedom.

12.5.6 F test for significance of an observed sample correlation ratio

Under the null hypothesis that population correlation ratio is zero, the test statistic is

observed sample correlation

Where N is the size of the sample (from a bivariate normal population) arranged in h arrays.

12.5.7. F test for testing the linearity of regression

For a sample of size N arranged in h arrays, from a bi variate normal population, the test statistic for testing the hypothesis of linearity of regression is

f test for testing

12.5.8. F – test for equality of several means

This test is carried out by the technique of analysis of variance, which plays a very important and fundamental role in Design of Experiments in Agricultural Statistics.

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