11.5 Test for homogeneity

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.

Hypotheses

Test statistic

Degrees of freedom ( df )

Requirements

Common uses

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?

Distribution of living arragements for college males and college females

	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: ${\chi }_3^2$

Calculate the test statistic: ${\chi }_c^2$ = 10.129

The graph of the Chi-square shows the distribution and marks the critical value with three degrees of freedom at 95% level of confidence, α = 0.05, 7.815. The graph also marks the calculated χ ² test statistic of 10.129. Comparing the test statistic with the critical value, as we have done with all other hypothesis tests, we reach the conclusion.

Make a decision: Because the calculated test statistic is in the tail we cannot accept 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.

Try it

Ivy League schools receive many applications, but only some can be accepted. At the schools listed in [link] , 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.

H ₀ : The distribution of regular applications accepted is the same as the distribution of early applications accepted.

H _a : The distribution of regular applications accepted is not the same as the distribution of early applications accepted.
df = 5
χ ² test statistic = 430.06

Press the MATRX key and arrow over to EDIT . Press 1:[A] . Press 3 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 430.06 and the p -value = 9.80E-91. Do the procedure a second time but arrow down to Draw instead of calculate .

References

Data from the Insurance Institute for Highway Safety, 2013. Available online at www.iihs.org/iihs/ratings (accessed May 24, 2013).

“Energy use (kg of oil equivalent per capita).” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/EG.USE.PCAP.KG.OE/countries (accessed May 24, 2013).

“Parent and Family Involvement Survey of 2007 National Household Education Survey Program (NHES),” U.S. Department of Education, National Center for Education Statistics. Available online at http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2009030 (accessed May 24, 2013).

“Parent and Family Involvement Survey of 2007 National Household Education Survey Program (NHES),” U.S. Department of Education, National Center for Education Statistics. Available online at http://nces.ed.gov/pubs2009/2009030_sup.pdf (accessed May 24, 2013).

Chapter review

To assess whether two data sets are derived from the same distribution—which need not be known, you can apply the test for homogeneity that uses the chi-square distribution. The null hypothesis for this test states that the populations of the two data sets come from the same distribution. The test compares the observed values against the expected values if the two populations followed the same distribution. The test is right-tailed. Each observation or cell category must have an expected value of at least five.

Formula review

${\displaystyle \sum }_{i\cdot j}\frac{{(O-E)}^2}{E}$ Homogeneity test statistic where: O = observed values
E = expected values
i = number of rows in data contingency table
j = number of columns in data contingency table

df = ( i −1)( j −1) Degrees of freedom

Use the following information to answer the next five exercises: Do private practice doctors and hospital doctors have the same distribution of working hours? Suppose that a sample of 100 private practice doctors and 150 hospital doctors are selected at random and asked about the number of hours a week they work. The results are shown in [link] .