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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.
The expected value for each cell needs to be at least five in order for you to use this test.
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 |
H
0 : The distribution of living arrangements for male college students is the same as the distribution of living arrangements for female college students.
H
a : 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:
Calculate the test statistic:
χ
2 = 10.1287 (calculator or computer)
Probability statement:
p -value =
P (
χ
2 >10.1287) = 0.0175
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 the
p -value: Since no
α is given, assume
α = 0.05.
p -value = 0.0175.
α >
p -value.
Make a decision: Since
α >
p -value, reject
H
0 . This means that the distributions are not the same.
Conclusion: At a 5% level of significance, from the data, there is sufficient evidence to conclude that the distributions of living arrangements for male and female college students are not the same.
Notice that the conclusion is only that the distributions are not the same. We cannot use the test for homogeneity to draw any conclusions about how they differ.
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