What it covers: Confidence intervals and significance tests for a single mean, paired (matched-pairs) data, and the difference between two means, all using the t-distribution.
Exam weight: About 10–20% of the AP Statistics exam.
The big question: Given a sample mean, what can we conclude about the population mean when the population standard deviation is unknown?
Statistical practices: Formulate Questions (P1), Collect Data (P2), Analyze Data (P3), Interpret Results (P4).
Key topics at a glance
Sampling Distribution of x̄
Mean = μ, standard deviation = σ/√n, approximately normal when the population is normal or n is large (CLT). This is the foundation for all mean inference.
Why t, Not z
When σ is unknown you use s, so you use the t-distribution — heavier tails, with n − 1 degrees of freedom for one sample.
Conditions
Random sample or randomized experiment; 10% (n < 10% of the population); Normal/Large Sample (population normal, n ≥ 30, or graph shows no strong skew/outliers).
One-Sample t-Interval
x̄ ± t*·(s/√n). The standard error is s/√n; t* uses n − 1 degrees of freedom.
One-Sample t-Test
H₀: μ = μ₀ vs. Hₐ. Statistic t = (x̄ − μ₀)/(s/√n), df = n − 1; find the p-value, then decide vs. α.
Paired (Matched Pairs)
When data are paired, work with the differences: run a one-sample t-procedure on d̄ with s_d and df = n − 1. Hypotheses use μ_d.
Two-Sample t
Interval: (x̄₁ − x̄₂) ± t*·√(s₁²/n₁ + s₂²/n₂). Test statistic uses the same standard error. If a difference interval contains 0, no significant difference.
Interpret & Robustness
Interval: 'C% confident the interval captures the true mean.' t-procedures are robust to mild non-normality, especially for large samples.
The key terms you must know
Sampling distribution of x̄ — mean μ, SD σ/√n, approximately normal under the conditions.
t-distribution — used when σ is unknown; heavier tails, set by degrees of freedom.
Degrees of freedom — n − 1 for a one-sample or paired t-procedure.
Standard error of the mean — s/√n, the estimated SD of x̄.
Random / 10% / Normal-Large Sample — the three conditions for mean inference.
One-sample t-interval — x̄ ± t*·(s/√n).
One-sample t-test — t = (x̄ − μ₀)/(s/√n).
Paired (matched-pairs) data — matched observations analyzed through their differences.
Mean difference μ_d — the parameter in a paired t-procedure.
Two-sample t-procedures — inference on μ₁ − μ₂ from two independent samples.
Robustness — t-procedures stay fairly accurate under mild non-normality.
Margin of error — t*·(standard error); shrinks as n grows.
Key themes to remember
Means use t because σ is unknown. The t-distribution accounts for estimating the spread from the sample.
Everything flows from the sampling distribution of x̄. Its center μ and spread σ/√n drive every interval and test.
Paired is a one-sample procedure on differences. Recognize matched data and reduce it to a single list of differences.
Check conditions and interpret in context. Random, 10%, Normal/Large Sample — then talk about the real variable and population.
Intervals and tests agree. A mean outside a C% interval is rejected by the matching two-sided test at α = 1 − C.
Common exam traps
Use t, not z, when σ is unknown. Almost every real mean problem uses the t-distribution.
Standard error is s/√n, not s. Don't forget to divide by √n.
Paired vs. two-sample is a design question. Matched subjects ⇒ paired (one-sample t on differences); independent groups ⇒ two-sample t.
Never 'accept' H₀. Fail to reject it.
A confidence interval is about the mean, not individuals. It does not say where most data fall.
Check the Normal/Large Sample condition — small samples need a roughly symmetric graph with no outliers.
If a two-mean interval contains 0, there is no convincing evidence the means differ.