Chapter 11: The Chi-Square Distribution
11.4 Test for Homogeneity
Learning Objectives
By the end of this section, the student should be able to:
- calculate the test statistic for a test for homogeneity
- test for homogeneity, and draw a conclusion about whether two populations have the same distribution
The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to draw a conclusion about whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence.
Note
The expected value for each cell needs to be at least five in order for you to use this test.
Hypotheses
H0: The distributions of the two populations are the same.
Ha: The distributions of the two populations are not the same.
Test Statistic: Use a [latex]{\chi }^{2}[/latex] test statistic. It is computed in the same way as the test for independence.
Degrees of Freedom (df)df = number of columns – 1
Requirements: All values in the table must be greater than or equal to five.
Common Uses: Comparing two populations. For example: men vs. women, before vs. after, east vs. west. The variable is categorical with more than two possible response values.
Example
Do male and female college students have the same distribution of living arrangements? Use a level of significance of 0.05. Suppose that 250 randomly selected male college students and 300 randomly selected female college students were asked about their living arrangements: dormitory, apartment, with parents, other. The results are shown in [link]. Do male and female college students have the same distribution of living arrangements?
| Dormitory | Apartment | With Parents | Other | |
| Males | 72 | 84 | 49 | 45 |
| Females | 91 | 86 | 88 | 35 |
Solution
H0: The distribution of living arrangements for male college students is the same as the distribution of living arrangements for female college students.
Ha: The distribution of living arrangements for male college students is not the same as the distribution of living arrangements for female college students.
Degrees of Freedom (df):
df = number of columns – 1 = 4 – 1 = 3
Distribution for the test:[latex]{\chi }_{3}^{2}[/latex]
Calculate the test statistic: χ2 = 10.1287 (calculator or computer)
Probability statement: p-value = P(χ2 >10.1287) = 0.0175
Press the MATRX key and arrow over to EDIT. Press 1:[A]. Press 2 ENTER 4 ENTER. Enter the table values by row. Press ENTER after each. Press 2nd QUIT. Press STAT and arrow over to TESTS. Arrow down to C:χ2-TEST. Press ENTER. You should see Observed:[A] and Expected:[B]. Arrow down to Calculate. Press ENTER. The test statistic is 10.1287 and the p-value = 0.0175. Do the procedure a second time but arrow down to Draw instead of calculate.
Compare α and t