Unit 7: Inference for Quantitative Data: Means
AP Statistics: 51 practice questions with detailed explanations.
Unit Study Guide
Executive Summary
When σ is unknown, t procedures handle means. The t-distribution's heavier tails account for the extra uncertainty of estimating σ with s.
The t-distribution
Symmetric, centered at 0, with heavier tails than z. Shape depends on degrees of freedom (n - 1 for one sample). As df grows, t approaches z. t* is larger than z* for the same confidence — wider intervals for small samples.
One-sample procedures
Standard error: s / √(n). Interval: x-bar ± t*·(s/√(n)). Test statistic: t = (x-bar - mu0) / (s/√(n)) with df = n - 1. Conditions: random sample, independence (10% rule), and a roughly normal population or n ≥ 30 (robustness).
Two samples
Compare mu1 - mu2 with two-sample t; the standard error combines both groups' variances. Matched pairs use ONE sample of differences — pairing removes subject-to-subject variation and boosts power. df for paired tests is nₚairs - 1.
Choosing procedures
One mean, unknown σ → one-sample t. Two independent means → two-sample t. Before/after on the same subjects → paired t. Intervals that exclude 0 suggest a significant difference; containing 0 suggests none at that level.
Exam traps
df is n - 1 (one sample) or n - 2 (slopes), not n. t* > z*, shrinking toward z* as n grows. Paired designs are NOT two independent samples.