10.1 CHI-SQUARE: TESTS OF INDEPENDENCE AND HOMOGENEITYPart 2: Tests of Homogeneity

Tests of Homogeneity

Homogeneous: of the same structure or composed of similar

parts

When we perform a test of homogeneity, we use a contingency table and the chi-square distribution to determine whether several populations share the same proportions of distinct categories.

A test of homogeneity tests the claim that different populations share the same proportions of specified characteristics.

Tests of Independence VS Tests of Homogeneity

The computational process for conducting the tests are the same.

Two main differences:1. Sampling Method

For tests of independence, we use one random sample and observe how the sample members are distributed among distinct categories.

For tests of homogeneity, we take random samples from two or more populations and see how many members of each population are distributed over distinct categories.

2. Hypotheses For tests of independence

Variable A and variable B are independent Variable A and variable B are not independent

For tests of homogeneity All population proportions are equal At least two population proportions are different

How to Test for Homogeneity of Populations

1. State the hypotheses and identify the level of significance

All population proportions are equal At least two population proportions are different

2. Construct a contingency table (the rows represent different populations and the columns represent the characteristics). Compute the expected frequency for each cell.

Note: the sample size must be “sufficiently large” meaning that for each cell,

3. Compute the sample test statistic(using the Ti-83/84)

4. Find the P-Value (using the Ti-83/84)5. Conclude the Test

Compare the P-value to

6. Interpret your results

Note: when we reject the null hypothesis, we do not know which proportions differ among the populations, we only know that the populations differ in some of the proportions sharing a characteristic.

Example: Test for Homogeneity

Tim is doing a research project involving pet preferences among students at his college. He took random samples of 300 female and 250 male students. Each sample member responded to the survey question “If you could own only one pet, what kind would you choose?” The possible responses were: “dog,” “cat,” “other pet,” “no pet.” The results of the study follow.

Does the same proportion of males as females prefer each type of pet? Use a 1% level of significance. We’ll answer this question in several steps

Pet Preference