Chi-square tests practice questions (AP Statistics)

Sample questions

Key terms

The definitions and formulas AP Statistics expects for chi-square tests, typeset as they appear on the paper.

Chi-square statistic

The sum over all cells of the squared difference between observed and expected counts, divided by the expected count. Large values are evidence against the null hypothesis; the test is always right-tailed.

$$\chi^2 = \sum \frac{(O - E)^2}{E}$$

Goodness-of-fit test

Tests whether the distribution of one categorical variable matches hypothesised proportions. Expected count for a category is $n$ times its hypothesised proportion; degrees of freedom are the number of categories minus 1.

$$E_i = n\,p_i, \qquad \text{df} = k - 1$$

Test for homogeneity

Compares the distribution of a categorical variable across two or more populations or treatment groups, from separate samples. Same statistic and expected counts as independence; the difference is the design.

Test for independence

Tests whether two categorical variables are associated in one population, using a single sample classified two ways.

$$E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}, \qquad \text{df} = (r - 1)(c - 1)$$

Conditions for chi-square procedures

Random sample or random assignment, 10 percent condition when sampling without replacement, and all expected counts at least 5. Check the expected counts, not the observed ones.

Follow-up analysis

After rejecting $H_0$, identify the cells that contribute most to the statistic and say whether their observed counts were above or below expected, in context.

Frequently asked questions

Other AP Statistics topics

• Exploring one-variable data
• Exploring two-variable data
• Collecting data
• Probability, random variables and distributions
• Sampling distributions
• Inference for proportions
• Inference for means
• Inference for slopes

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