Question

a doctor wanted to study the effect of four different treatments on mental health. a group of 100 adults experiencing depression volunteered for the study. the doctor randomly assigned one - fourth of them to each of four groups. group 1 followed a specific exercise plan, group 2 followed a specific diet plan, group 3 followed an exercise and diet plan, and group 4 did not follow any plan. after 4 weeks, the doctor contacted each participant and asked them if they felt any better. the results are displayed in the table.

the doctor would like to know if these data provide convincing evidence that the distribution of responses differs across the treatment groups in the population of all patients like these. what is the appropriate inference procedure?
chi - square test for goodness of fit because the data come from one random sample
chi - square test for homogeneity because the data come from independent treatment groups
chi - square test for association/independence because the data come from one random sample
chi - square test for association/independence because the data come from independent treatment groups

Explanation:

• Chi - square test for goodness of fit: Compares observed frequencies to expected frequencies for a single categorical variable. Here, we have multiple groups (treatment groups), so this is not appropriate.
• Chi - square test for association/independence: Tests if two categorical variables (e.g., treatment and response) are related. But it is for a single sample. Here, we have independent treatment groups.
• Chi - square test for homogeneity: Used to test if the distribution of a categorical variable (response: yes/no) is the same across multiple independent groups (treatment groups). The data come from independent treatment groups (exercise only, diet only, both, neither), and we want to check if the distribution of responses (feel better or not) differs across these groups.

Answer:

chi - square test for homogeneity because the data come from independent treatment groups