Example

Gender and Body Image

What is your perception of your own body? Do you feel that you are overweight, underweight, or about right? A random sample of 1,200 U.S. college students answered this question as part of a larger survey. The following table shows part of the responses.

Notice that the sample is a random sample from a single population: U.S. college students. Notice also that we collected data on two categorical variables for each student: gender and body image. This is the type of situation that is appropriate for a chi-square test of independence.

Step 1: State the hypotheses.

Here are two equivalent ways we can state the hypotheses for a test of independence.

• H₀: There is no relationship between gender and body image for U.S. college students.
• H_a: There is a relationship between gender and body image for U.S. college students.

We could also state the hypotheses like this:

• H₀: Gender and body image are independent in the population of U.S. college students.
• H_a: Gender and body image are dependent in the population of U.S. college students.

Step 2: Collect and analyze the data.

We summarized the data for this sample in a two-way table:

To investigate the relationship between gender and body image, we compare the percentage of males and females who gave each body image response. In Relationships in Categorical Data with Intro to Probability, we called these conditional percentages.

Comment: In this situation, gender is the explanatory variable. Body image is the response variable. We compare the distribution of the response variable (body image) for the groups defined by the explanatory variable (males and females). This means that explanatory category totals are the denominator of each fraction. In this case, we use male and female totals in the denominators.

We graphed these conditional percentages using ribbon charts for a visual comparison. In this sample, we can see that a larger percentage of females (73.7%) than males (67%) perceive their body weight to be “about right.” A larger percentage of females (21.4%) than males (16.4%) also perceive themselves to be overweight. Males are as likely to say they are overweight (16.4%) as underweight (16.6%), but females as much less likely to perceive themselves to be underweight (4.9%).

What do these conditional percentages have to do with independence?

Recall the definition of independence from Probability and Probability Distribution. Two events, A and B, are independent if the probability of A is the same as the probability of A when B has already occurred. We write this statement as P(A) = P(A | B). In this context, if gender and body image are independent variables, then gender does not affect the probability that a student will give a specific answer to the body image question. We use relative frequencies from the sample to represent these probabilities.

For example, we would expect the following probabilities to be the same if gender and body image are independent variables.

• P(about right) = 855/1,200 = 0.713.
• P(about right | female) = 560/760 = 0.737, which is the conditional percent shown in the ribbon chart for females.
• P(about right | male) = 295/440 = 0.67, also a conditional percent shown in the ribbon chart for males.

Obviously, when we compare the conditional percentages for males and females in this sample, we see that they differ. But do they differ enough to conclude that variables are dependent? Or could these results have come from a population where gender and body image are independent? In which case, the differences are due to chance fluctuation that happens in random sampling. We need to conduct a test of independence to find out.