Module 11: Chi-Square Tests
Test of Independence (2 of 3)
Test of Independence (2 of 3)
Learning outcomes
- Conduct a chi-square test of independence. Interpret the conclusion in context.
Here we continue our chi-square test of independence for the variables gender and body image in the population of U.S. college students.
Example
Gender and Body Image Continued
Step 1: State the hypotheses.
Here are our hypotheses from the previous page:
- H0: There is no relationship between gender and body image for U.S. college students. (The variables are independent.)
- Ha: There is a relationship between gender and body image for U.S. college students. (The variables are dependent.)
Step 2: Collect and analyze the data.
If the variables are independent, the percentage of males and females with a given response will be the same or at least close. Previously, we determined that in our sample, there are differences in the percentage of males and females who answer “about right,” “overweight,” or “underweight.”
We need to determine if these differences are typical in random samples from a population where gender and body image are independent. Perhaps the differences we see in this sample are just fluctuations expected in random sampling. Or perhaps these differences are too large to be explained by chance. We will not know until we complete the hypothesis test.
Step 3: Assess the evidence.
We need to determine the expected values and the chi-square test statistic so that we can find the P-value.
Calculating Expected Values for a Test of Independence
Expected counts always describe what we expect to see in a sample if the null hypothesis is true. In this situation, if gender and body image are independent, then we expect the probability that a student answers “about right” in the sample to be the same probability that a male (or a female) student answers “about right” (similarly for “overweight” or “underweight” responses).
Here are the calculations of expected counts for the response “about right”:
- Probability that a student will answer “about right”: P(about right) = (855/1,200) = 0.7125
- Expected count of females in the sample who will answer “about right”: 0.7125(760) = 541.5
- Expected count of males in the sample who will answer “about right”: 0.7125(440) = 313.5
Here are the calculations of expected counts for the response “overweight”:
- Probability that a student will answer “overweight”: P(overweight) = (235/1,200) = 0.1958
- Expected count of females in the sample who will answer “overweight”: 0.1958(760) = 148.8
- Expected count of males in the sample who will answer “overweight”: 0.1958(440) = 86.2
Try It
Try It
Example
More on Gender and Body Image
Checking Conditions
The conditions for use of the chi-square distribution are the same as we learned previously:
- The sample is random.
- All of the expected counts are 5 or greater.
Since the data meets the conditions, we can proceed with calculating χ2 test statistic.
Calculating the Chi-Square T