Putting It Together: Chi-Square Tests

Let’s Summarize

In this module, Chi-Square Tests, we discussed three different hypothesis tests using the chi-square test statistic:

• Goodness-of-fit for a one-way table
• Test of independence for a two-way table
• Test of homogeneity for a two-way table

Goodness-of-Fit test for a One-Way Table

• In a goodness-of-fit test, we consider one population and one categorical variable.
• The goodness-of-fit test expands the z-test for a population proportion that we learned in Inference for One Proportion by looking at the distribution of proportions for all categories defined by the categorical variable.
• The goodness-of-fit test determines whether a set of categorical data comes from a claimed distribution. The null hypothesis is that the proportion in each category in the population has a specific distribution. The alternative hypothesis says that the proportions in the population are not distributed as stated in the null hypothesis.
• To test our hypotheses, we select a random sample from the population and gather data for one categorical variable.

Test of Independence for a Two-Way Table

• In the test of independence, we consider one population and two categorical variables.
• In Probability and Probability Distribution, we learned that two events are independent if P(A|B) = P(A), but we did not pay attention to variability in the sample. With the chi-square test of independence, we have a method for deciding whether our observed P(A|B) is “too far” from our observed P(A) to infer independence in the population.
• The null hypothesis says the two variables are independent (or not associated). The alternative hypothesis says the two variables are dependent (or associated).
• To test our hypotheses, we select a single random sample and gather data for two different categorical variables.

Test of Homogeneity for a Two-Way Table

• In the test of homogeneity we consider two or more populations (or two or more subgroups of a population) and a single categorical variable.
• The test of homogeneity expands on the test for a difference in two population proportions that we learned in Inference for Two Proportions by comparing the distribution of the categorical variable across multiple groups or populations.
• The null hypothesis says that the distribution of proportions for all categories is the same in each group or population. The alternative hypothesis says that the distributions differ.
• To test our hypotheses, we select a random sample from each population or subgroup independently. We gather data for one categorical variable.

Chi-Square Test Statistic and Distribution

For all chi-square tests, the chi-square test statistic χ² is the same. It measures how far the observed data are from the null hypothesis by comparing observed counts and expected counts. Expected counts are the counts we expect to see if the null hypothesis is true.

[latex]X^2 = \Sigma \frac {(observed-expected)^2} {expected}[/latex]

The chi-square model is a family of curves that depend on degrees of freedom. For a one-way table the degrees of freedom equals (r – 1). For a two-way table, the degrees of freedom equals (r – 1)(c – 1). All chi-square curves are skewed to the right with a mean equal to the degrees of freedom.

A chi-square model is a good fit for the distribution of the chi-square test statistic only if the following conditions are met:

• The sample is randomly selected.
• All expected counts are 5 or greater.

If these conditions are met, we use the chi-square distribution to find the P-value. We use the same logic that we have used in all hypothesis tests to draw a conclusion based on the P-value. If the P-value is at least as small as the significance level, we reject the null hypothesis and accept the alternative hypothesis. The P-value is the likelihood that results from random samples have a χ² value equal to or greater than that calculated from the data if the null hypothesis is true.