**Lesson 10.3 – Testing the Relationship between Two Categorical Variables**

- State hypotheses for a test about the relationship between two categorical variables.
- Calculate expected counts for a test about the relationship between two categorical variables.
- Calculate the test statistic for a test about the relationship between two categorical variables.

**Activity: What was your favorite toy as a child?**

We love revisiting contexts whenever possible. It provides continuity and allows us to make connections within the content. It also promotes the use of multiple approaches and flexible thinking. We used this question and context in lesson 2.1 with segmented bar graphs and in chapter 4 with conditional probability. Just as we did in those lessons, we’re looking for an association. This type of test is called the Chi-Square test for association.

Since the Large counts condition tells us the expected counts have to be greater than 5, we need a fairly large sample size. Since we already collected this data in the fall for all of our classes, we combined it all and used it today. Begin by filling out the center cells for the class. Let the students find the column and table totals. Discuss what we are trying to show with the test and work that into a discussion of the hypotheses. The hypotheses are:

**Null Hypothesis:** There is not an association.

**Alternative Hypothesis:** There is an association.

Now we need to calculate the expected counts. There is a formula for this, but we want students to understand where the formula comes from so we are going to work through an example with them using a Think Aloud. Begin by filling in the column and row totals for the expected counts table. Leave the center cells blank for now. We want to think proportionately about how many students we expect in each cell. Let’s work through one example, Male and Barbies.

There are 49 males total out of the 124 intro stats students, or 39.5%. Forty students chose Barbies. So if gender does not impact which toy a student chooses, then 39.5% of the people who chose Barbies should be male, or 0.395 x 40 = 15.8 males. This is how we calculate the expected counts. We can generalize this to come up with the formula for calculating expected counts.

Let students complete the rest of the expected counts and then calculate the Chi-Square test statistic.