Chi-Square Tests of Independence for Categorical Data

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What is a chi-square test of independence?

A hypothesis test that uses categorical data in a two-way table to assess whether two variables are associated or independent in the population.

How do you calculate an expected cell count?

Use $E=\frac{(\text{row total})(\text{column total})}{\text{grand total}}$. It is the count anticipated in that cell if the variables are independent.

What does the chi-square statistic measure?

It combines the discrepancies between observed and expected counts across all cells: $\chi^2=\sum (O-E)^2/E$. Larger values indicate greater overall disagreement with independence.

How do you find degrees of freedom for an independence test?

For a table with $r$ rows and $c$ columns, use $df=(r-1)(c-1)$. For a two-row, four-column table, the degrees of freedom are 3.

What does failing to reject the null hypothesis mean?

The sample does not provide sufficient evidence of an association at the chosen significance level. It does not prove that the variables are independent.

Quiz questions

1. A contingency table has two rows and four columns. What are the degrees of freedom for a chi-square test of independence?Answer: 3, because $(2-1)(4-1)=3$.
2. Which expression calculates the contribution of one cell to the chi-square statistic?Answer: $(O-E)^2/E$
3. Why can a chi-square test statistic not be negative?Answer: Each squared difference between an observed and expected count is nonnegative.
4. According to the lab guidance, what minimum expected count should each cell have?Answer: At least five.
5. A chi-square test gives $p=0.03$ with significance level $\alpha=0.05$. What is the appropriate decision?Answer: Reject $H_0$; the data provide statistically significant evidence of an association.