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 inside each cell needs to be at least five in order for you to use this test.
Hypotheses
H_{0}: The distributions of the two populations are the same.
H_{a}: The distributions of the two populations are not the same.
Test StatisticUse a ${\chi}^{2}$ test statistic. It is computed in the same way as the test for independence.
Degrees of Freedom (df)df = number of columns - 1
RequirementsAll values in the table must be greater than or equal to five.
Common UsesComparing 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 11.11
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 Table 11.18. 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 |
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 Table 11.19. 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 |
Ivy League schools receive many applications, but only some can be accepted. At the schools listed in Table 11.20, 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 χ^{2} distribution and show the critical value and the calculated value of the test statistic, and draw a conclusion about the test of homogeneity.