3.15 Carrying Out a Chi-Square Test for Homogeneity or Independence

Calculate expected counts for two-way tables of categorical data.

• The expected counts (under the null hypothesis) in a particular cell of a two-way table of categorical data can be calculated using the formula row total column totalexpected count total table= .

Calculate the appropriate test statistic and p-value for a chisquare test for homogeneity or independence.

• The appropriate test statistic for a chi-square test for homogeneity or independence is the chi-square statistic 2 2 Observed Count Expec t Expected Cou ted Count nχ − , where the sum is taken over all cells of the two-way table. The chi-square statistics have a chi-square distribution with degrees of freedom equal to number of rows −⋅1 number of columns −1 when the null hypothesis is true.
• The p-value for a chi-square test for independence or homogeneity is found from a chi-square distribution using a table or technology.

Interpret the p-value for the chi-square test for homogeneity or independence.

• The p-value is the probability of obtaining a test statistic as extreme or more extreme than the test statistic that was observed (i.e., in the direction of the alternative hypothesis) given that the null hypothesis is true. An interpretation of the p-value for the chi-square test for homogeneity or independence should include a statement that the p-value is computed by assuming the null hypothesis is true in context.

Justify a claim about the population(s) based on the results of a chi-square test for homogeneity or independence.

• A formal decision in a hypothesis test explicitly compares the p-value to the significance level, . If the p-value , then reject the null hypothesis for the appropriate chi-square test. If the p-value , then fail to reject the null hypothesis.
• The results of a chi-square test for homogeneity or independence can serve as the statistical reasoning to support the answer to an investigative question about the population that was sampled (independence) or the populations that were sampled (homogeneity).
• A conclusion for a chi-square test for homogeneity or independence is stated in context consistent with, and in terms of, the alternative hypothesis using non-definitive language. The conclusion should contain a reference to the population(s).