AP Statistics Inference Cheat Sheet: Every Procedure

Every AP Statistics inference problem comes down to three decisions: is the data categorical (proportions) or quantitative (means), how many groups are you comparing, and do you need a confidence interval or a significance test? Answer those and the procedure, formula, and conditions follow. This cheat sheet covers the z-procedures for proportions and t-procedures for means in the revised AP Statistics course, with the conditions and conclusion language that earn full credit.

One note before you start: AP Statistics was revised for 2026–27. Chi-square goodness-of-fit tests and inference for the slope of a regression line were removed from the course, so older cheat sheets that include them are out of date. Check your course materials and the AP Central course page if you are unsure what your class covers.

How do you choose the right inference procedure?

Work through these questions in order.

1. What type of variable? A yes/no or category outcome means proportions. A numerical measurement means means.
2. How many samples or groups? One group, two independent groups, or one group measured twice (paired)?
3. What is the question asking? "Estimate" or "how large" calls for a confidence interval. "Is there convincing evidence" calls for a significance test.

Paired or two-sample?

This is the most common mistake on the exam. Data are paired when each value in one group is naturally linked to a specific value in the other: the same person before and after, twins, or the left and right hands of the same subject. If the two groups are separate people with no one-to-one link, use a two-sample procedure.

Formulas for every procedure

Every confidence interval follows the same shape: statistic ± (critical value) × (standard error). Every test statistic follows (statistic − parameter under H₀) / (standard error).

Note the subtle difference for one proportion: the interval uses p̂ in the standard error, but the test uses the hypothesized value p₀, because the test assumes H₀ is true.

Common z* critical values

For t-procedures, the critical value t* depends on degrees of freedom. Use your calculator or the table in the reference information.

Conditions for inference

Conditions are where students lose the most points. State each condition and support it with a specific fact from the problem.

For two-sample procedures, check each condition for both groups, and the samples or treatment groups must be independent of each other. For paired data, check the normality condition on the differences, not on the original two sets of values.

What full-credit condition checks look like

• Weak: "Random: yes."
• Strong: "Random: the problem states the 400 students were randomly selected from the school."
• Weak: "Large counts is met."
• Strong: "Large counts: np₀ = 400(0.5) = 200 ≥ 10 and n(1 − p₀) = 200 ≥ 10."

Worked example: a one-proportion z-test

A principal claims that half of all students at a large school support a later start time. In a random sample of 400 students, 176 support it. Is there convincing evidence that less than half of all students support it, at α = 0.05?

State. H₀: p = 0.50, Hₐ: p < 0.50, where p is the true proportion of all students at the school who support a later start time. Use α = 0.05.

Plan. One-sample z-test for a population proportion.

• Random: random sample of students.
• 10%: 400 is less than 10% of all students at a large school.
• Large counts: 400(0.50) = 200 ≥ 10 and 400(0.50) = 200 ≥ 10.

Do. p̂ = 176/400 = 0.44. z = (0.44 − 0.50) / √(0.50 × 0.50 / 400) = −0.06 / 0.025 = −2.4. p-value ≈ 0.0082.

Conclude. Because the p-value of 0.0082 is less than α = 0.05, we reject H₀. There is convincing evidence that less than half of all students at this school support a later start time.

How to write conclusions that earn credit

A complete test conclusion has three parts: the comparison (p-value vs. α), the decision (reject or fail to reject H₀), and the answer in context, phrased in terms of the alternative hypothesis.

• If p-value < α: "We reject H₀. There is convincing evidence that [Hₐ in context]."
• If p-value ≥ α: "We fail to reject H₀. There is not convincing evidence that [Hₐ in context]."

Never write "we accept H₀" or "this proves H₀ is true." Failing to reject means the data are not strong enough to rule out H₀.

Type I errors, Type II errors, and power

The probability of a Type I error equals α. Power increases when you increase the sample size, increase α, or when the true parameter is farther from the null value. Exam questions often ask for a consequence of an error in context, so practice describing what would actually happen.

Quick checklist before you submit

1. Did you define the parameter in context, with "true" and "all"?
2. Did you name the procedure exactly?
3. Did you support every condition with evidence from the problem?
4. Did you compare the p-value to α, or tie your conclusion to the interval?
5. Is your conclusion in context and about the population, not the sample?

For the full exam strategy, read how to get a 5 on AP Statistics, and estimate your standing with the AP Statistics score calculator.

How Ivystar can help

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