Chi-Square Test of Homogeneity

Learning Outcomes

  • Conduct a chi-square test of homogeneity and interpret the conclusion in context

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 X2 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 1

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 the table below. Do male and female college students have the same distribution of living arrangements?

Distribution of Living Arrangements for College Males and College Females
Dormitory Apartment With Parents Other
Males 72 84 49 45
Females 91 86 88 35

try it 1

Do families and singles have the same distribution of cars? Use a level of significance of 0.05. Suppose that 100 randomly selected families and 200 randomly selected singles were asked what type of car they drove: sport, sedan, hatchback, truck, van/SUV. The results are shown in the table. Do families and singles have the same distribution of cars? Test at a level of significance of 0.05.

Sport Sedan Hatchback Truck Van/SUV
Family 5 15 35 17 28
Single 45 65 37 46 7

Example 2

Both before and after a recent earthquake, surveys were conducted asking voters which of the three candidates they planned on voting for in the upcoming city council election. Has there been a change since the earthquake? Use a level of significance of 0.05. The table below shows the results of the survey. Has there been a change in the distribution of voter preferences since the earthquake?

Perez Chung Stevens
Before 167 128 135
After 214 197 225

try it 2

Ivy League schools receive many applications, but only some can be accepted. At the schools listed in the table, two types of applications are accepted: regular and early decision.

Application Type Accepted Brown Columbia Cornell Dartmouth Penn Yale
Regular 2,115 1,792 5,306 1,734 2,685 1,245
Early Decision 577 627 1,228 444 1,195 761

We want to know if the number of regular applications accepted follows the same distribution as the number of early applications accepted. State the null and alternative hypotheses, the degrees of freedom and the test statistic, sketch the graph of the p-value, and draw a conclusion about the test of homogeneity.