Statistics · Unit 8: Inference for Categorical Data: Chi-Square · 14 min read · Updated 2026-05-11
Selecting an Inference Procedure — AP Statistics
AP Statistics · Unit 8: Inference for Categorical Data: Chi-Square · 14 min read
1. Overview of the Inference Selection Skill★★☆☆☆⏱ 3 min
This AP Statistics skill requires you to match a given research question, study design, and type of categorical data to the correct inference procedure, rather than just calculating a test statistic or p-value. Per the AP CED, this topic contributes 2-5% of multiple-choice points and 1-2 points on nearly every chi-square-focused FRQ, with points awarded solely for correct selection.
Unlike calculation-focused problems, this topic tests conceptual understanding of how study design and research question drive inference choice, a core competency the AP exam prioritizes. Many students lose easy points here by mixing up the three chi-square procedures, so mastering this selection step is critical for full credit.
2. Identifying a Chi-Square Goodness-of-Fit Test★★☆☆☆⏱ 3 min
The null hypothesis for a GOF test always specifies hypothesized proportions for each category: $H_0: p_1 = p_{1,0}, p_2 = p_{2,0}, ..., p_k = p_{k,0}$, where $k$ is the number of categories. The alternative hypothesis is that at least one $p_i$ does not equal the hypothesized value.
Only one group/sample
One categorical variable
A specific hypothesized distribution or set of proportions is provided
Common contexts: testing claimed ratios, die fairness, expected demographic distributions
Exam tip: If the problem gives you a pre-specified set of proportions or a ratio to test against, it is a goodness-of-fit test 99% of the time on the AP exam.
3. Identifying a Chi-Square Test for Homogeneity★★★☆☆⏱ 3 min
A common source of confusion: tests for homogeneity produce two-way contingency tables, just like tests for independence, but the sampling design is the key difference. For homogeneity, you sample separately from each pre-defined group, so group sizes are fixed before data collection.
The research question for homogeneity is always: *Does the distribution of [response variable] differ across [multiple groups]?* The null hypothesis is that the distribution of the response variable is the same for all groups, and the alternative is that at least one group has a different distribution.
Exam tip: If the problem explicitly states it took separate random samples from each of multiple groups and wants to compare distributions, it is always a test for homogeneity.
4. Identifying a Chi-Square Test for Independence★★★☆☆⏱ 3 min
The null hypothesis is that the two variables are independent in the population; the alternative is that they are dependent (associated). Like the test for homogeneity, this uses a two-way contingency table, but the sampling design differs: for independence, you take one random sample, no group totals are fixed in advance, both row and column totals are random.
One random sample from the population
Two categorical variables measured on each individual
Research question asks if there is an association or relationship between the two variables
Exam tip: If the research question asks 'is there an association between' two categorical variables, it is always a test for independence.
5. Concept Check★★★☆☆⏱ 5 min
Common Pitfalls
Why: Both produce the same test statistic calculation, so students assume they are interchangeable, but AP grading requires matching procedure to study design
Why: Students see 'distribution' and default to goodness-of-fit regardless of the number of groups
Why: Students see proportions and default to z-procedures, which are only for one or two proportions
Why: Any 2x2 table can use either procedure, but AP questions expect the procedure matching the research question
Why: Students only remember to check conditions for calculation, not when justifying procedure selection