Summary: The Test for Homogeneity is used to make 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 Goodness of Fit test can be used to decide whether a population fits a given distribution, but the Goodness of Fit test will not suffice to compare whether two populations follow the same unknown distribution. A different test, called the Test for Homogeneity, can be used to make 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.
Use a
df = number of columns - 1
All values in the table must be greater than or equal to 5.
Comparing two populations. For example: men versus women, before vs. after, east vs. west. The variable is categorical with more than two possible response values.
Do male and female college students have the same distribution of living conditions? 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 conditions: Dormitory, Apartment, With Parents, Other. The results are shown in the table below.
Dormitory | Apartment | With Parents | Other | |
Males | 72 | 84 | 49 | 45 |
Females | 91 | 86 | 88 | 35 |
Do male and female college students have the same distribution of living conditions?
Degrees of Freedom (df):
df = number of columns - 1 = 4 - 1 = 3
Distribution for the test:
Calculate the test statistic:
Probability statement:
TI-83+ and TI-84 calculator: 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 Draw
instead of
calculate
.
Compare
Make a decision: Since
Conclusion: At a 5% level of significance, from the data, there is sufficient evidence
to
conclude that the distributions of living conditions 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 make any conclusions about how they differ.
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.
Perez | Chung | Stevens | |
Before | 167 | 128 | 135 |
After | 214 | 197 | 225 |
Has there been a change in the distribution of voter preferences since the earthquake?
Degrees of Freedom (df):
df = number of columns - 1 = 3 - 1 = 2
Distribution for the test:
Calculate the test statistic:
Probability statement:
TI-83+ and TI-84 calculator: Press the MATRX
key and arrow over to
EDIT
. Press 1:[A]
. Press 2 ENTER 3 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 3.2603 and the Draw
instead of
calculate
.
Compare
Make a decision: Since
Conclusion: At a 5% level of significance, from the data, there is insufficient evidence
to
conclude that the distribution of voter preferences was not the same before and after the earthquake.
** Contributed by Dr. Larry Green
"Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"