Inference for Means

Name: AP Statistics Unit 7: Inference for Means
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10–18% of the exam4 lessons · 56 min15 terms

What this unit covers

The topics below follow the published Statistics course framework for Unit 7. This unit is worth 10–18% of the exam, so budget your time against that rather than against how long the unit takes to teach.

t-distributionsCIs for meanst-testsPaired data

Lessons in this unit

The t-Distribution13 min · 3 objectivesExplain why the t-distribution is used instead of z for means · Describe how the t-distribution changes with degrees of freedom · Identify the degrees of freedom for a one-sample t procedure
Confidence Intervals for a Mean14 min · 3 objectivesConstruct a one-sample t-interval for a population mean · Verify the conditions for t procedures, including the Normal/large-sample condition · Interpret the interval in context
Significance Tests for a Mean15 min · 3 objectivesState hypotheses and compute a one-sample t test statistic · Find a p-value using the t-distribution and make a decision · Connect the confidence interval and two-sided test
Paired Data & Two-Sample Means14 min · 3 objectivesRecognize paired data and analyze the differences with a one-sample t procedure · Distinguish a paired design from two independent samples · Set up inference for a difference of two independent means

Formulas in Unit 7

One-sample t statistic and df

t = (x-bar − μ₀) / (s / sqrt(n)) · df = n − 1

Same structure as the z statistic, but s replaces σ and the reference distribution is t with n − 1 degrees of freedom. Larger n → t looks more like z.

One-sample t-interval for a mean

x-bar ± t\* · ( s / sqrt(n) ) with df = n − 1

t\* depends on the confidence level and df. For the same confidence level, t\* is larger than z\*, making t-intervals wider — the price of not knowing σ.

One-sample t test for a mean

t = (x-bar − μ₀) / ( s / sqrt(n) ) with df = n − 1

Compare the p-value from the t-distribution to α. Reject H₀ if p-value < α; otherwise fail to reject. For a two-sided Hₐ, double the one-tail area.

Paired and two-sample t procedures

paired: t = (x-bar_diff − 0) / (s_diff / sqrt(n)), df = n − 1 · two-sample: t = (x-bar₁ − x-bar₂) / sqrt( s₁²/n₁ + s₂²/n₂ )

Paired reduces to a one-sample t on the differences. The two-sample standard error adds the two variance contributions (independence). Use technology for the two-sample df.

Every term in Unit 7

All 15 terms we publish for Inference for Means, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Why t rather than z: The population standard deviation is unknown and estimated by s, which adds variability. The t distribution has heavier tails to account for it.
Degrees of freedom for one sample: df = n − 1. As df grows, the t distribution approaches the standard normal.
Conditions for t procedures: Random sample, 10% condition, and Normal/Large Sample — the population is normal, n ≥ 30, or a graph of the data shows no strong skew or outliers.
One-sample t-interval: x̄ ± t*(s/√n), interpreted as capturing the true population mean in context.
One-sample t-test statistic: t = (x̄ − μ₀)/(s/√n), compared to a t distribution with n − 1 degrees of freedom.
Matched pairs t-test: Compute the difference for each pair and run a one-sample t-test on those differences. Using a two-sample test here is a standard and costly error.
Two-sample t-interval: (x̄₁ − x̄₂) ± t*√(s₁²/n₁ + s₂²/n₂), with technology supplying the degrees of freedom.
Deciding one-sample or two-sample: Two independent groups means two-sample; two measurements on the same or paired subjects means matched pairs.
Robustness of t procedures: They perform well even when the population is not exactly normal, provided the sample is large or the data show no strong skew or outliers.
Choosing a significance level: Use a smaller α when a false positive is costly, and a larger one when missing a real effect is worse.
Reading a t-table: Find the row for your degrees of freedom, then the column for your confidence level or tail probability. Between-row values are rounded conservatively downward.
Checking normality from a sample: Graph the data. A dotplot or boxplot showing no strong skew and no outliers satisfies the condition for a moderate sample.
Why the 10% condition exists: Sampling without replacement makes observations slightly dependent; keeping the sample under 10% of the population makes the effect negligible.
Interpreting a t-interval for a difference: If the interval contains zero, there is no convincing evidence the two means differ. Say this in context rather than stating it abstractly.
Paired data recognition: Two measurements on the same subject, or naturally matched subjects. If you can subtract within a pair meaningfully, the design is paired.

What examiners penalize here

For small samples, you must address the shape condition explicitly: state that a dotplot/boxplot/stemplot of the data shows no strong skew or outliers, so it is plausible the population is approximately Normal. Skipping this on free response loses the conditions point.
Spot paired data by asking: is there a *natural one-to-one link* between the two measurements (same subject twice, twins, matched pairs)? If yes, take differences and run a **one-sample t**. Only use a two-sample t when the two groups are genuinely independent.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

38 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Statistics study guideAll units, weights and score data in one place All 9 Statistics unitsThe full course

Questions about this unit

How much of the AP Statistics exam is Unit 7?

Unit 7, Inference for Means, is worth 10–18% of the Statistics multiple-choice section according to the published course framework. Across all 9 units that makes it a substantial share — heavier than an even split would give it.

What topics are covered in Statistics Unit 7?

Inference for Means covers t-distributions, CIs for means, t-tests and Paired data. We publish 15 terms with definitions for this unit, all of them on this page.

How should I study Statistics Unit 7?

Read the 4 lessons below first — about 55 minutes — then drill the 15 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 9 units of AP Statistics

Unit names, topics and exam weights follow the published College Board course framework for AP Statistics. AP® is a trademark registered by the College Board, which does not endorse this site.