Question

Explain the difference between situations that would lead to a chi-square test for homogeneity and those that would lead to a chi-square test for independence.

Short answer

The difference primarily lies in the purpose and setup of the experiment. Chi-square test for homogeneity is used when you're comparing the same variable across different groups within a population, while Chi-square test for independence is used when you're observing two categorical variables within the same population to see if they are related.

Step-by-step solution

Define Chi-square test for Homogeneity

Firstly, the chi-square test for homogeneity is used when the goal is to compare the distribution of outcomes in different groups. These groups are usually a subset of the same population and each group is surveyed only once. Consider a scenario where three flavours of a soda (cherry, lemon, and vanilla) are tested in three different cities to see if the preference is the same in all cities. This is a case for the chi-square test for homogeneity as it is testing whether different groups (cities) follow the same distribution (soda preference)

Define Chi-square test for Independence

Now, the chi-square test for independence is used when you have two categorical variables from the same population. You want to see if there is a significant relationship between those two variables. Consider a scenario where a researcher wants to test whether gender is independent of career choice. This involves a single population (all people) but looks at two distinct factors (gender and career choice). Thus, this is a case for using a chi-square test for independence as it checks if two factors (career choice and gender) are independent.

Summary of Differences

In summary, your use of chi-square test for homogeneity versus chi-square test for independence is based on whether you're comparing different categorical variables from the same group (independence) or comparing the same variable from different groups (homogeneity). Make sure to appropriately define your populations and groups when choosing a test.