F-distribution [Test for equality of variances]

F-distribution [Test for equality of variances]

[Snedecor's F-distribution]

 

1.  The F-distribution

Suppose that two independent normal populations are of interest, when the population means and variances, say µ₁, µ₂ , σ²₁ and σ²₂, are unknown. We wish to test hypothesis about the equality of the two variances, say, H₀ : σ²₁ = σ²₂ Assume that two random samples of size n₂ from population 2 are available, and let S²₁ and S²₂ be the sample variances. We wish to test the hypothesis.

H₀ : σ²₁ = σ²₂

H₁ : σ²₁ ≠ σ²₂

The development of a test procedure for these hypothesis requires a new probability distribution called F distribution.

If S²₁ and S²₂ are the variances of two samples of sizes n1 and n, respectively, the estimates of the population variance based on these samples are respectively

S²₁  = n₁ S²₁  / n₁ – 1 and S²₂  = n₂ S²₂  / n₂ – 1

The quantities v₁ = n₁ -1 and v₂  = n₂ -1  are called the degrees of freedom of these estimates. We want to test if these estimates S²₁ and S²₂ are significantly different or if the samples may be regarded as drawn from the same population or from two populations with same variance σ².

If S²₁ = S²₂, then F = 1. Hence, our object is to find how far any observed value of F differs from unity, consistent with our assumption of the equality of the population variances.

The area of the curve y f(F) to the right of Fo gives the probability that F > F₀. Here, F₀ is the critical value of F. If F > F₀  the difference of variances is significant. If F < F₀, the difference is not significant.

The critical values of F, say F₀, is got from F-table corresponding to degrees of freedom (v₁, v₂) at a level of significance.

If F < F₀, critical value, the difference is not significant and if F > F₀, the difference is significant.

Note. F > 0 always.

Applications :

(i) . -F-test is used to test (i) whether two independent samples have been drawn from the normal populations with the same variance σ², or

(ii) whether the two independent estimates of the population variance are homogeneous or not.

 

2.  Properties of the F-distribution

1. The probability curve of the F-distribution is roughly sketched below.

2. The square of the t-variate with n degrees of freedom follows a F-distribution with l and n degrees of freedom.

3. The mean of the F-distribution is V₂ / V₂ -2 (V₂ > 2)

4. The variance of the F-distribution is

Note :

1. We should always make F > 1. This is done by taking the larger of the two estimates of σ² as σ²₁ and by assuming that the corresponding degree of freedom as V₁.

2. To test if two small samples have been drawn from the same normal population, it is not enough to test if their means differ significantly or not, because in this test we assumed that the two samples came from the same population or from populations with equal variance. So, before applying the t-test for the significance of the difference of the sample means, we should satisfy ourselves about the equality of the population variances by F-test.

 

Example 1.5.1

A group of 10 rats fed on diet A and another group of 8 rats fed on diet B, recorded the following increase in weight.

Find if the variances are significantly different. [A.U N/D 2011] [A.U A/M 2015 R-13] [A.U N/D 2013 R-13]

Solution:

Given: n₁ = 10, n₂ = 8

Here Cal F < table F

i.e., 1.03 < 3.29

We conclude that the two samples have come from populations with equal variances.

 

Example 1.5.2

Two independent samples of sizes 9 and 7 from a normal population had the following values of the variables.

Do the estimates of the population variance differ significantly at 5% level? [A.U M/J 2014] [A.U A/M 2017 R-13]

Solution :

Given: n₁ = 9, n₂ = 7

Here Cal F < table F

i.e., 1. 3968 < 3.58

We conclude that the difference is not significant.

 

Example 1.5.3

In one sample of 8 observations, the sum of the squares of deviations of the sample values from the sample mean was 84.4 and in the other sample of 10 observations was 102.6. Test whether this difference is significant at 5% level, given that the 5 percent of F for v1 = 7 and v2 = 9 degrees of freedom is 3.29.

Solution: Given: n₁ = 8, n₂ = 10

Here

Cal F < Table F

i.e., 1.06 < 3.29

We conclude that the difference is not significant.

 

Example 1.5.4

In one sample of 10 observations, the sum of the squares of the deviations of the sample values from the sample mean was 120 and in another sample of 12 observations it was 314. Test whether this difference is significant at 5% level of significance.

Solution: Given: n₁ = 10, n₂ = 12

Here

Cal F <  Table F

i.e., 2.14 < 3.11

We conclude that the samples might have come from two populations having the same variance.

 

Example 1.5.5

Two random samples give the following results

Test whether the samples could have come from the same normal population. [A.U M/J 2006, M/J 2012] [A.U N/D 2016 R-13]

Solution:

 A normal population has two parameters namely the mean μ and the variance σ² . If we want to test the samples from the same normal population, we have to test

(i) the equality of population variances (using F-test)

(ii) the equality of population means (using t-test)

Since t-test σ²₁ = σ²₂ assumes we shall first apply F-test and then t-test.

(i) F - test

Here Cal F < Table F

i.e., 1.02 < 2.90

[Note: If F-test failed, then t-test should not be used]

We conclude that the samples might have come from two populations having the same variance.

 

Example 1.5.6

Two samples of sizes 9 and 8 give the sum of the squares of deviations from their respective means equal to 160 and 91 respectively. Can they be regarded as drawn from the same normal population?

Solution:

Given: n₁ = 9, n₂ = 8

Here Cal F < Table F

i.e., 1.54 < 3.73

We conclude that the samples might have come from two populations having the same variance.

 

Example 1.5.7

An instructor has two classes A and B, in a particular subject, class A has 16 students while class B has 25 students. On the same examination, although these was no significant difference in mean grade class A has standard deviation of 9, while class B had a standard deviation level of 12. Can we conclude at the 0.01 level of significance that the variability of class B is greater than that of class A. [A.U N/D 2020, A/M 2021 R-17]

Solution :

Given:

n₁ = 16, s₁ = 9

n₂ = 25, s₂  = 12

6. Conclusion:

Here, Cal F < Table F

i.e., 1.736 < 3.325

So, we accept H₀ at 1% level of significance.

 

EXERCISES 1.5

1. Two random samples drawn from two normal populations are :

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

[Ans. They have same variances]

2. Two random samples of sizes 8 and 7 had the following values of the variables.

Do the estimates of population variance differ significantly?

[Ans. Not significant] [A.U Tvli M/J 2011] [A.U A/M 2015 R-8]

3. Two random samples drawn from two normal populations have the variable values as below:

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

4. In two groups of 10 persons each, the increases in weight due to two different diets in the same period were in kgs.

Find whether the variances are significantly different.

[Ans. Not significant]

5. The nicotine contents in two samples (random) of tobacco are given below:

Can you say that the two samples came from the same population?

[Ans. F = 4.07, variances are equal

|t| = 1.92, not significant; came from same population]

6. The time taken by labourers in performing a job by two different ways are given below:

Test for the difference of variances of the populations. [A.U N/D 2012]

[Ans. Not significant]

7. The two random samples reveal the following data:

Test whether the samples come from the same normal population.

 [Ans. F = 1.05; |t| = 9.73]

8. Two random samples reveal the following data. Test for the variance equality of the populations.

9. The following data relate to a random sample of government employees woled an onlsy aldertry in two states of the Indian Union.

Test whether the samples come from the same normal population.

 [Ans. F 1.05; t = 9.73]

10. Two samples were drawn independently from two normal populations.

The data are :

Test whether the population variances are equal.

[Ans. F=1.154]

11. The random samples were drawn from two normal populations and the following results were obtained.

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

[Ans. σ²₁  = σ²₂ ; F = 1.935, Not significant]

12. Two samples are drawn from the two normal populations. From the following data test whether the two samples have the same variance at 5% level.

 [Ans. σ²₁ = σ²₂ ; F = 1.467]

13. Two methods of performing a certain operation are compared. The following data are obtained.

Test the hypothesis that the two variances are equal at 5% level of significance.