http://www.ck12.org Chapter 10. Chi-Square
10.2 Test of Independence
Learning Objectives
- Understand how to draw and calculate appropriate data from tables needed to run a Chi-Square test.
- Run a Test of Independence to determine whether two variables are independent or not.
- Use a Test of Homogeneity to examine the proportions of a variable attributed to different populations.
Introduction
As mentioned in the previous lesson, the Chi-Square test can be used to (1) estimate how closely an observed
distribution matches an expected distribution(Goodness-of-Fit test)or (2) estimating whether two random variables
are independent of one another (theTest of Independence). In this lesson, we will examine the Test of Independence
in greater detail.
The Chi-Square Test of Independence is used toassess if two factors are related.This test is often used in social
science research to determine if factors are independent of each other. For example, we would use this test to
determine relationships between voting patterns and race, income and gender, and behavior and education.
In general, when running the Test of Independence, we ask “Is VariableXindependentof VariableY?” It is
important to note that this test does not testhowthe variables are related, just simply whether or not they are
independent of one another. For example, we can test if income and gender are independent, the Test of Independence
cannot help us assess how one category might affect the other.
Drawing and Calculating Data from Tables
As mentioned in the previous lesson, tables help us frame our hypotheses and solve problems. Often, we use tables
to list the variables and observation patterns that will help us to run the Chi-Square test. For example, we could use
a table to record the answers to phone surveys or observed behavior patterns.
Example:We would use acontingency tableto record the data when analyzing whether women are more likely
to vote for a Republican or Democratic candidate when compared to men. Specifically, we want to know if voting
patterns are independent of gender. Hypothetical data for 76 females and 62 males is in the contingency table below.
TABLE10.6: Frequency of California Citizens voting for a Republican or Democratic Candidate
Democratic Republican Total
Female 48 28 76
Male 36 26 62
Total 84 54 138Similar to the Chi-Square Goodness-of-Fit test, the Chi-Square Test of Independence is a comparison of the differ-
ence between the observed andexpected values.However, in this test we need to calculate the expected value using