Testing Hypothesis about Homogeneity of Two Samples Lecture 40

 

Testing Hypothesis about Homogeneity of Two Samples

                 (Brandt-Snedecor Test)

Lecture 40

The goodness of fit test is used to test that the sample data came from a theoretical distribution, and the test of independence is to determine if the two variable classifications are independent. The goodness of fit and test of independence are not enough to portray a real picture. This can be improved to determine if the samples came from the same population or if both the populations are identical for a categorical variable. To determine if the samples came from the same population or the two populations have the same distribution, another test known as the test for homogeneity can be used. To calculate the test statistic for a test for homogeneity, the data will be displayed in a 2 x k contingency table.

To test the hypothesis that the two samples come from the same population of a single categorical variable or the samples are homogeneous. The Brandt-Snedecor test can be used to test the null hypothesis that two samples came from the same population.

Suppose we select two independent samples of size n from a population, and we wish to test the null hypothesis, whether the two samples are homogenous. The values of both samples presented below,

	1	2	….	i	….	n	Total
Sample I	a1	a2	….	ai	….	an	A
Sample II	b1	b2	….	bi	….	bn	B
Total	c1	c2	….	ci	….	cn	N

To test H0: the two samples are homogenous, the Brandt Snedecore test is given by:

Testing Procedure:

i. State the null & alternative hypothesis

H0: The two samples came from the same population.

Vs.

H1: The two samples did not come from the same population.

ii. The significance level: α

iii. The test statistic: Brandt Snedcore Test

vi. Critical Region:

Reject H0 when χ² ≥ χ²α(k-1)

v. Computation:

vi. Remarks.

Example 9.22: A random sample of 50 men and another sample of 50 women were asked about their educational backgrounds in a particular neighbourhood. They were divided into three categories classified as intermediate, associate diploma, and BS honour. The results are arranged in the table below:

	Intermediate	Associate Diploma	Bs Honor	Total
Male	13	25	12	50
Female	23	20	7	50
Total	36	45	19	100

Test whether the male and female are homogenous in respect of educational levels at a 0.05 significance level.

Solution:

i. State the null & alternative hypothesis

H0: The male and female are homogenous.

Vs.

H1: The male and female are not homogenous.

ii. The significance level: α = 0.05

iii. The test statistic: Brandt Snedcore Test

vi. Critical Region:

Reject H0 when χ² ≥ χ²0.05(2) = 5.99

v. Computation:

vi. Remarks: The computed chi-square value falls in the rejection region; the sample data does not provide sufficient evidence to reject the null hypothesis. Thus, it is concluded that the male and female are homogenous in respect of educational levels.

Example 9.23: Voters were surveyed before and after a recent earthquake to find out which of the three candidates they intended to vote for in the upcoming municipal council election. Has the situation changed since the earthquake? The survey’s results are displayed in the following table.

	A	B	C
Before	167	128	135
After	214	197	225

Solution:

i. State the null & alternative hypothesis

H0: The voters intention is not to change before and after the earthquake.

Vs.

H1: The voters intention is change before and after the earthquake.

ii. The significance level: α = 0.05

iii. The test statistic: Brandt Snedcore Test

vi. Critical Region:

Reject H0 when χ² ≥ χ²0.05(2) = 5.99

v. Computation:

	A	B	C	Total
Before	167	128	135	430
After	214	197	225	636
Total	381	325	360	1066

vi. Remarks: The computed chi-square value falls in the rejection region; the sample data does not provide sufficient evidence to reject the null hypothesis. Thus, it is concluded that the before and after the intention of voters are homogenous.

• Read More: Introduction to t distribution