AP Statistics · Lesson 8 of 8 · Unit 4

Free-response Question 3: choosing and completing an inference procedure

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Free-response Question 3 is worth 10 points (12.5% of your total AP exam score) and always tests statistical inference from Unit 3 or Unit 4. Because the question is scored analytically (each component has its own point, earned independently unless the scoring guideline says otherwise), mastering the exact steps and required phrasing helps you earn as many points as possible.

By the end you’ll be able to

Determine the appropriate inference procedure by name (one- or two-sample z for proportions, chi-square for homogeneity or independence, one-sample, matched pairs or two-sample t for means) from the type of variable, the number of samples or groups and the goal (test or interval).
Identify and classify the variables, and define the parameter or parameters and the population or populations in context.
Verify each condition (randomization, the 10% condition when relevant, and normality, sample data or expected counts) explicitly with the numbers from the problem.

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Core ideas (continued)
Worked examples
Exam technique
Common mistakes
Quick recap
Check your understanding (3 questions)

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What the exam asks

Question 3 presents a real-world research scenario and asks you to complete either a hypothesis test or a confidence interval across labeled parts (A, B, C, etc., with subparts i and ii). Every step of the inference process is broken out into specific prompts: identifying variables, naming the procedure, defining parameters or stating hypotheses, checking conditions, calculating the interval or test statistic and p-value, and interpreting or justifying a conclusion.

Core ideas

1. Choosing and naming the exact procedure

First, classify the response variable (categorical $\rightarrow$ proportions/counts; quantitative $\rightarrow$ means), count the samples/treatments, and determine the goal (estimating a parameter $\rightarrow$ interval; testing a claim $\rightarrow$ test).

Variable Type	Samples / Groups	Official Procedure Name (Interval or Test)
Categorical (binary)	1 random sample	One-sample $z$ -interval (or $z$ -test) for a population proportion
Categorical (binary)	2 independent samples / groups	Two-sample $z$ -interval (or $z$ -test) for a difference between two population proportions
Categorical ( $\ge 2$ categories)

2. Defining parameters and stating hypotheses

Always define parameters as population values (in an experiment, the true value for each treatment), naming the population (or treatment group) in context.

Proportions ( $p$ ): “Let $p$ be the population proportion of all [population units] that [have the characteristic].”
Means ( $\mu$ ): “Let $\mu$ be the population mean [variable with units] for all [population units].” For matched pairs, define $\mu_d$ and state the (e.g., ).

3. Verifying conditions with numbers

Never write a bare checklist. Every condition must be tied to the scenario with explicit numbers:

Randomization: Quote the problem (e.g., “two independent random samples of 50 trees and 60 trees” or “40 volunteers were randomly assigned to the two treatments”).
10% condition (sampling without replacement only): Show that each sample size $n$ is at most 10% of its population size $N$ (e.g., “ $50 \le 10\%$ of all oak trees in the state”). Do not check this for randomized experiments.
Large counts / Sample data / Expected counts:
- 1-prop interval: Observed successes $n\hat{p} \ge 10$ and failures .