Skills Focus: Selecting an Inference Procedure — AP Statistics Study Guide

For: AP Statistics candidates sitting AP Statistics.

Covers: Chi-square goodness-of-fit test, chi-square test for homogeneity, chi-square test for independence, parameter identification, conditions checking, and matching study design to the correct inference procedure for categorical data.

You should already know: Basic chi-square test calculation mechanics. Types of study design, including independent samples and single random samples. How to write null and alternative hypotheses for inference.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Statistics style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Skills Focus: Selecting an Inference Procedure?

This 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 Statistics Course and Exam Description (CED), this topic is embedded within Unit 8: Inference for Categorical Data: Chi-Square, and it contributes directly to 2-5% of multiple-choice points and 1-2 points on nearly every chi-square focused free response question. This skill appears on both MCQ (where you select the correct procedure from four options) and FRQ (where you are asked to identify the appropriate inference procedure as the first step of a multi-part question, 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 or confusing them with other proportion inference methods, so mastering this selection step is critical for full credit.

2. Identifying a Chi-Square Goodness-of-Fit Test

The first chi-square procedure you need to recognize is the chi-square goodness-of-fit (GOF) test. A GOF test is appropriate when you have exactly one categorical variable measured on a single random sample from a population, and your goal is to test whether the distribution of the variable in the population matches a specific hypothesized distribution. In other words, GOF answers the question: “Does the observed distribution of our variable fit the distribution we expected under the null hypothesis?”

The null hypothesis for a GOF test always takes the form: $H_0:p_1=p_{1,0},p_2=p_{2,0},...,p_k=p_{k,0}$, where $k$ is the number of categories of the variable, and $p_{i,0}$ is the hypothesized proportion for category $i$ given in the problem. The alternative hypothesis is always: $H_a:$ At least one $p_i$ does not equal the hypothesized value.

Key clues that indicate GOF in a problem: (1) only one group/sample, (2) one categorical variable, (3) a specific hypothesized distribution or set of proportions is given in the problem. Common contexts include testing if a ratio (like Mendelian genetics, die fairness, or demographic distribution) matches a claimed value.

Worked Example

A plant biologist expects that the ratio of purple-flowered to white-flowered to blue-flowered offspring from a cross will be 2:1:1. She collects a random sample of 100 offspring from the cross and counts how many plants have each flower color. She wants to test whether her expected ratio is correct. What is the appropriate inference procedure?

Solution:

1. First, count the number of samples and variables: we have one random sample of 100 offspring, and one categorical variable (flower color, 3 levels).
2. Identify the research goal: the biologist wants to test if the observed distribution of flower colors matches her hypothesized 2:1:1 ratio, which gives specific hypothesized proportions ($0.5,0.25,0.25$) for each category.
3. There is no comparison of multiple groups, and no test of association between two variables.
4. Therefore, the appropriate procedure is a chi-square goodness-of-fit test.

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

A chi-square test for homogeneity is used when you have two or more independent groups (populations), and you measure the same single categorical variable on each group to test whether the distribution of that variable is the same (homogeneous) across all groups. Unlike goodness-of-fit, you have multiple separate samples here: you fix the size of each group in advance when designing the study, then count how many observations fall into each category of the response variable.

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; the alternative is that at least one group has a different distribution.

A common point of confusion: tests for homogeneity result in a two-way contingency table, same as tests for independence, but the sampling design is different, which is the key to distinguishing them. Key clues: separate random samples from each group, goal of comparing distributions across groups.

Worked Example

A researcher wants to test whether the distribution of vaccine acceptance (fully accepting, hesitant, refusing) differs between three groups of adults: rural, suburban, and urban residents. The researcher selects independent random samples of 200 adults from each of the three residence types, then records each adult’s vaccine acceptance category. What inference procedure is appropriate?

Solution:

1. Count groups and variables: we have three independent groups (rural, suburban, urban), with sample sizes fixed in advance at 200 per group. We measure one categorical response variable: vaccine acceptance, with three levels.
2. Research goal: test whether the distribution of vaccine acceptance is the same (homogeneous) across the three residence groups.
3. We are not testing association between two variables from one single sample, and we are not fitting to a pre-specified distribution, so it cannot be GOF or independence.
4. Therefore, the appropriate procedure is a chi-square test for homogeneity.

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

The third chi-square procedure is the chi-square test for independence, used when you have one single random sample from a population, and you measure two different categorical variables on each individual in the sample. The research question asks whether the two variables are independent (no association) in the population, or dependent (associated).

The null hypothesis for a test for independence is that the two variables are independent in the population; the alternative is that they are dependent (associated). Like the test for homogeneity, this procedure uses a two-way contingency table to organize counts, so it is easy to mix up the two. The key difference is sampling design: for independence, you take one random sample, and count both variables for every individual. No group totals are fixed in advance; both row and column totals are random, based on the sample. For homogeneity, by contrast, you fix the size of each group (row or column) in advance when sampling.

Common clues that point to a test for independence: (1) one sample, (2) two categorical variables measured on each unit, (3) research question asks if there is an association or relationship between the two variables.

Worked Example

A sociologist collects a random sample of 500 working adults in a large city. She records two variables for each adult: highest level of education completed (high school, bachelor’s, graduate degree) and whether they report being satisfied with their job (satisfied, dissatisfied). She wants to test whether job satisfaction is associated with education level. What is the appropriate inference procedure?

Solution:

1. Count samples and variables: there is one random sample of 500 working adults, with two categorical variables measured on each individual (education level, job satisfaction).
2. No group sizes were fixed in advance: the number of adults with each education level is a random result of sampling, not set by the researcher.
3. The research question asks whether there is an association between the two variables, which matches the goal of a test for independence.
4. Therefore, the appropriate procedure is a chi-square test for independence.

Exam tip: If the research question asks “is there an association between” two categorical variables, it is always a test for independence.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Calling a test for homogeneity a test for independence, or vice versa, because both use two-way tables. Why: Both produce the same test statistic calculation, so students assume they are interchangeable, but AP grading requires matching procedure to study design. Correct move: Always check sampling design first: separate samples from fixed groups = homogeneity; one sample with two variables measured = independence.
• Wrong move: Using a chi-square goodness-of-fit test for two or more groups. Why: Students see "distribution" and default to goodness-of-fit regardless of number of groups. Correct move: Count the number of independent samples before selecting: one sample = GOF, more than one = homogeneity.
• Wrong move: Calling a one-variable test with three or more categories a set of multiple z-tests for proportions. Why: Students see proportions and default to z-procedures, which are only for one or two proportions. Correct move: If you have one variable with more than two categories and a hypothesized distribution, use chi-square GOF instead of multiple z-tests.
• Wrong move: Using a chi-square test for independence when a two-sample z-test for difference in proportions is requested. Why: Any 2x2 table can use either procedure, but AP questions expect the procedure matching the research question. Correct move: If you have exactly two independent groups and the question asks for a difference in proportions, use the two-proportion z-test.
• Wrong move: Forgetting to mention the expected counts condition when justifying a chi-square procedure. Why: Students only remember to check conditions for calculation, not when justifying procedure selection. Correct move: Always state the expected counts condition (all expected counts ≥ 5) as part of your justification for any chi-square procedure on FRQ.

6. Practice Questions (AP Statistics Style)

Question 1 (Multiple Choice)

A researcher studying diet wants to test whether the distribution of preferred protein source (beef, poultry, fish, plant-based) is the same for people following three different diet plans: low-carb, low-fat, and intermittent fasting. The researcher selects 150 people at random from each diet plan and records their preferred protein source. Which of the following is the appropriate inference procedure? A) Chi-square goodness-of-fit test B) Chi-square test for homogeneity C) Chi-square test for independence D) Two-sample z-test for a difference in proportions

Worked Solution: We first confirm the study design: three separate independent groups with fixed sample sizes (150 per diet plan), and one categorical response variable (preferred protein source). The research question asks whether the distribution of the response variable is the same across all three groups. This matches the criteria for a chi-square test for homogeneity. It is not goodness-of-fit because there are multiple groups, not one. It is not independence because we sampled each group separately instead of taking one sample and measuring two variables. It is not a two-sample z-test because we have three groups, not two. The correct answer is B.

Question 2 (Free Response)

A community garden association wants to study the traits of tomato plants grown in their garden. (a) A horticulturist claims that the height of mature tomato plants (short, medium, tall) follows the distribution 25% short, 50% medium, 25% tall. The association collects a random sample of 80 tomato plants to test this claim. Identify the appropriate inference procedure. Do not carry out the test. (b) The association wants to study whether there is an association between type of fertilizer (organic, synthetic) and tomato flavor rating (poor, average, excellent). They collect a random sample of 100 tomato plants grown in the garden, and record both variables for each plant. Identify the appropriate inference procedure. Do not carry out the test. (c) A researcher claims that the proportion of tomato plants that get blight is the same for plants grown in full sun, partial sun, and full shade. The association selects separate random samples of 40 plants from each light condition, and counts how many get blight in each group. Identify the appropriate inference procedure. Do not carry out the test.

Worked Solution: (a) We have one random sample of tomato plants, one categorical variable (height), and we test whether the observed distribution matches a pre-specified hypothesized distribution. The appropriate procedure is chi-square goodness-of-fit test. (b) We have one random sample of tomato plants, two categorical variables (fertilizer type, flavor rating) measured on each plant, and the research question asks for an association between the two variables. The appropriate procedure is chi-square test for independence. (c) We have three independent groups (full sun, partial sun, full shade) with separate random samples from each group, and the research question asks whether the distribution of blight status is the same across all groups. The appropriate procedure is chi-square test for homogeneity.

Question 3 (Application / Real-World Style)

A credit card company wants to test whether the distribution of missed payments (0, 1, 2, 3 or more) is different for customers in three age groups: 18-25, 26-40, 41+. The company takes a random sample of 200 customers from each age group, and records the number of missed payments per customer grouped into the four categories. Identify the correct inference procedure, and explain why the other two chi-square procedures are not appropriate.

Worked Solution: The appropriate procedure is the chi-square test for homogeneity. First, the company took independent random samples of 200 customers from each of the three age groups, so the size of each group is fixed in advance. The goal is to test whether the distribution of missed payments (one categorical response variable) is the same across the three age groups. This is not a goodness-of-fit test because goodness-of-fit requires only one sample and a pre-specified hypothesized distribution, which we do not have here. It is not a test for independence because a test for independence requires one single sample of all customers with two variables (age group and missed payments) measured on each individual, rather than separate samples from each age group with fixed sizes. The chi-square test for homogeneity will allow the company to test whether missed payment distribution differs by age group.

7. Quick Reference Cheatsheet

8. What's Next

Mastering selection of chi-square inference procedures is the critical prerequisite for carrying out full chi-square inference, from checking conditions to calculating test statistics, interpreting p-values, and drawing contextual conclusions. Next, you will practice full inference workflows for each of the three chi-square procedures, where you will apply the selection skill you learned here before moving to calculation and conclusion. Without correctly identifying the right procedure first, you will lose all core points for any inference FRQ, even if your calculation is correct for the wrong procedure. This skill also builds on your earlier understanding of inference selection for z and t procedures, and prepares you for the cumulative FRQ on the AP exam that requires selecting an inference procedure from any unit.

• Chi-Square Goodness-of-Fit Tests
• Chi-Square Tests for Homogeneity and Independence
• Inference for One Proportion
• Inference for Two Proportions