Module 5. Test of significance

Lesson 20

F- TEST AND ITS APPLICATIONS

20.1 Introduction

A large number of research experiments are conducted to examine the effect of various factors on the production and quality attributes of milk and milk products. F-test is used either for testing the hypothesis about the equality of two population variances or the equality of two or more population means. The equality of two population means was dealt with t-test. Besides a t-test, we can also apply F-test for testing equality of two population means. Sir Ronald A. Fisher defined a statistic Z which is based upon ratio of two sample variances. In this lesson we will consider the distribution of ratio of two sample variances which was worked out by G.W. Snedecor.

20.2  F-Statistic

Let X_1i (i=1,2,…,n₁)  be a random sample of size n₁from the first population with variance σ₁² and X_2j (j=1,2,…,n₂) be another independent random sample of size n₂ from the second normal  population with variance σ₂². The F- statistic is defined as the ratio of estimates of two variances as given below:

where, S₁² > S₂² and are unbiased estimates of population variances which are given by:

It follows Snedecor’s F- distribution with (n₁-1, n₂-1) d.f. i.e., F~F (n₁ – 1, n₂ – 1). Further, if X is a χ²-variate with n₁ d.f. and Y is another independent χ²-variate with n₂ d.f., then F-statistic is defined as: i.e. F-statistic is the ratio of two independent Chi-square variates divided by their respective degrees of freedom. This statistic follows G.W. Snedecor F distribution with (n₁,n₂) d.f. The sampling distribution of F-statistic does not involve any population parameter and depends only on the degrees of freedom n₁and n₂.

20.3  Application of F- Distribution

F-distribution has a number of applications in statistics, some of which are given below

a) F-test for equality of population variances

b) F test for testing equality of several population  means

20.3.1  F-test for equality of population variances

Suppose we are interested to find if two normal populations have same variance. Let X_1i (i=1,2,…,n₁)  be a random sample of size n₁from the first population with variance σ₁² and X_2j (j=1,2,…,n₂) be another independent random sample of size n₂ from the second normal  population with variance σ₂². The two samples are independent of each other.

Under the null hypothesis H₀: σ₁² = σ₂² = σ² i.e., population variances are equal. In other words the two independent estimates of the common population variance do not differ. The test statistic F = (S₁²/ S₂²) ~ F(n₁ – 1, n₂ – 1) (S₁² > S₂²)

Where S₁² and S₂² are unbiased estimates of the common population variance σ² and are given by

It follows Snedecor’s F- distribution with (n₁-1, n₂-1) d.f. i.e., F~F (n₁ – 1, n₂ – 1). Since F-test is based on the ratio of two variances it is also known as Variance Ratio Test. If calculated value of F is greater than tabulated value at α% level of significance H₀ is rejected at α percent level of significance which implies that the two independent estimates of population variances are heterogeneous. On the other hand if calculated value of F is less than tabulated F then H₀ is not rejected at α percent level of significance suggesting that the estimates of population variances are homogenous.

Remarks:

· Since the available tables of the significant values of F are for the right –tail test, i.e., against an alternative σ₁² > σ₂², in numerical problems we will take greater of the variance S₁² or S₂² in the numerator and adjust for the degrees of freedom i.e., degree of freedom of the large variance must be taken in the numerator while computing F.

· In numerical problems, usually sample variance s² is given from which S² can be obtained on using the relation n s² = (n-1) S².

The procedure is illustrated by following examples

Example 1: In a sample of 8 observations, the sum of squared deviations of items from the mean was 94.5. In another sample of 10 observations, the value was found to be 101.7. Test whether the difference is significant at 5% level of significance.

Solution:  Let us take the hypothesis that there is no difference in the variances of two samples i.e., H₀: σ₁² = σ₂², with an alternative H₀: σ₁² ≠ σ₂². we are given

Applying F test

Tabulated value of F (7, 9) d.f. at 5% level of significance is 3.29.

Calculated value of F is less than the tabulated value, it is not significant. Hence H₀ may be accepted and conclude that the difference in the variances of two samples is not significant at 5% level of significance which implies that the two population variances are homogenous or the sample have been drawn from the population having same variances.

Example 2: Two random samples drawn from normal population are :

Sample I	20	16	26	27	23	22	18	24	25	19
Sample II	27	33	42	35	32	34	38	28	41	43	30	37

Obtain estimates of the variances of two populations and test whether two populations have same variances.

Solution: Let us take the hypothesis that the two populations have the same variance i.e.,

H₀: σ₁² = σ₂² = σ₀². With an alternative H₀: σ₁² ≠ σ₂².

Table 20.1

Sample I X1	22		Sample II X2
20	-2	4	27	-8.00	64
16	-6	36	33	-2.00	4
26	4	16	42	7.00	49
27	5	25	35	0.00	0
23	1	1	32	-3.00	9
22	0	0	34	-1.00	1
18	-4	16	38	3.00	9
24	2	4	28	-7.00	49
25	3	9	41	6.00	36
19	-3	9	43	8.00	64
			30	-5.00	25
			37	2.00	4
220		120	420		314

Applying F test

Tabulated value of F(11,9) d.f. at 5% level of significance is 3.10.

Calculated value of F is less than the tabulated value, it is not significant. Hence H₀ may be accepted and conclude that the difference in the variances of two samples is not significant at 5% level of significance which implies that the two populations have the same variance.

20.3.2  F test for equality of several population means

This test is widely used in the technique of analysis of variance which plays a very important and fundamental role in Design of experiments which is discussed in details in next module.