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Unit 4 · Inference for Means Flashcards Cheat Sheet Essentials Visual Review MC Practice FRQ Practice

AP Statistics Unit 4 Visual Review

A topic-by-topic visual walkthrough of Unit 4: Inference for Quantitative Data — t-based confidence intervals and significance tests for means, paired data, and two-sample means.

← Back to Unit 4 hub
TOPIC 4.1 Sampling Distribution of x̄ // center & spread of the sample mean mean of x̄ = μ SD of x̄ = σ / √n Normal if the population is Normal OR n ≥ 30 (CLT) Why we use t, not z In practice σ is UNKNOWN, so we estimate it with the sample SD s. That extra uncertainty means we use the t-distribution instead of the Normal. The t-distribution Bell-shaped but with HEAVIER tails than the Normal. df = n − 1 As df → ∞, the t-curve approaches the Normal. x̄ centered at μ, SD = σ/√n; use the t-distribution (df = n−1) since σ is unknown. The Review Hub · AP Statistics Unit 4 TOPIC 4.2 Confidence Interval for a Mean x̄ ± t*·( s / √n ) Worked example (95%) x̄ = 50, s = 8, n = 25, df = 24: t* ≈ 2.064 (df=24, 95%) ME = 2.064·(8/5) = 3.30 CI = 50 ± 3.30 = (46.7, 53.3) t* is bigger than z* for the same confidence. Conditions (for a one-sample t) • RANDOM sample • 10%: n ≤ 10% of the population • NORMAL/Large: population Normal or n ≥ 30, or graph shows no strong skew. PAIRED data: run a one-sample t on the differences. One-sample t-interval: x̄ ± t*·(s/√n), df = n − 1. The Review Hub · AP Statistics Unit 4 TOPIC 4.3 Interpreting a Mean Interval Interpret the interval "We are 95% confident the true MEAN [quantity] of [population] is between 46.7 and 53.3 [units]." Always name μ, the population, and the units. Same confidence-level meaning as for proportions. Test a claimed mean If a claimed value μ₀ is OUTSIDE the interval, it's not a plausible value. A 95% CI ↔ a two-sided test at α = 0.05. Wider interval = more confidence but less precision. To halve the margin of error, quadruple n Because ME ∝ 1/√n, cutting ME in half requires 4× the sample size — precision gets expensive. State confidence in the true mean μ in context; a value outside the CI is implausible. The Review Hub · AP Statistics Unit 4 TOPIC 4.4 Setting Up a Test for a Mean // hypotheses about the population mean μ H₀: μ = μ₀ Hₐ: μ < μ₀, μ > μ₀, or μ ≠ μ₀ Same framework as proportions Define μ in context, choose Hₐ's direction, and set α before looking at data. Test uses the t-distribution (df = n − 1) because σ is unknown. Paired data → differences For before/after or matched pairs, compute each pair's DIFFERENCE and test μ_diff. H₀: μ_diff = 0 A paired t-test is a one-sample t on the differences. State H₀: μ = μ₀ and Hₐ; paired data → test the mean difference μ_diff. The Review Hub · AP Statistics Unit 4 TOPIC 4.5 Carrying Out a Test for a Mean t = ( x̄ − μ₀ ) / ( s / √n ), df = n − 1 Worked example H₀: μ = 100, x̄ = 104, s = 12, n = 36: t = (104−100)/(12/6) = 4/2 = 2.0, df = 35 Find the p-value = tail area on the t-curve (double it for a ≠ alternative). The 4-step conclusion 1. Hypotheses & α. 2. One-sample t-test; check conditions. 3. Compute t & the p-value. 4. Compare to α; conclude IN CONTEXT. The SE uses the sample SD s (not σ). One-sample t-test: t = (x̄−μ₀)/(s/√n), then find the p-value (df = n−1). The Review Hub · AP Statistics Unit 4 TOPIC 4.6 Difference of Two Means // sampling distribution of x̄₁ − x̄₂ mean = μ₁ − μ₂ SE = √( s₁²/n₁ + s₂²/n₂ ) Two INDEPENDENT samples Variances add under the root (independent groups) — use each sample's own s. This is the UNPOOLED (two-sample) t — the AP default. Conditions & df Two independent random samples; 10% condition; Normal/Large for each. df from technology (or use the smaller of n₁−1, n₂−1 as a conservative estimate). x̄₁−x̄₂ centered at μ₁−μ₂; SE = √(s₁²/n₁ + s₂²/n₂). The Review Hub · AP Statistics Unit 4 TOPIC 4.7 CI for a Difference of Means (x̄₁ − x̄₂) ± t*·√( s₁²/n₁ + s₂²/n₂ ) Worked example (95%) x̄₁=52, s₁=6, n₁=40; x̄₂=48, s₂=7, n₂=45. SE = √(6²/40 + 7²/45) ≈ 1.41 diff = 4; CI = 4 ± t*·1.41 ≈ (1.2, 6.8) with t* ≈ 1.99 Interval above 0 → group 1's mean is higher. Two-sample t-interval The estimate is x̄₁ − x̄₂; the margin uses both samples' variability. t* depends on the confidence level and df (technology gives the df). Define which group is #1 before interpreting. Two-sample t-interval: (x̄₁−x̄₂) ± t*·√(s₁²/n₁ + s₂²/n₂). The Review Hub · AP Statistics Unit 4 TOPIC 4.8 Interpreting a Two-Mean Interval Does the interval contain 0? A CI for μ₁ − μ₂ tests whether the two population means differ. 0 inside → no significant difference · 0 outside → significant difference Read the direction Interval entirely POSITIVE → μ₁ > μ₂. Interval entirely NEGATIVE → μ₁ < μ₂. Contains 0 → could go either way. The sign of the difference tells you which is larger. Interpret in context "We are 95% confident the true difference in mean [quantity] (group1 − group2) is between __ and __ [units]." Tie the conclusion back to the real groups. If a two-mean interval contains 0, the means aren't significantly different. The Review Hub · AP Statistics Unit 4 TOPIC 4.9 Setting Up a Two-Mean Test H₀: μ₁ = μ₂ (i.e. μ₁ − μ₂ = 0) Hₐ: μ₁ ≠ μ₂, μ₁ > μ₂, or μ₁ < μ₂ Two-sample t-test Compares the means of two independent groups. Unlike proportions, means are NOT pooled on the AP exam — use each s separately. Define both populations & which is group 1. Independent vs. paired TWO-SAMPLE: separate, unrelated groups. PAIRED: each subject measured twice, or matched — use a ONE-sample t on differences. Confusing the two is a common exam mistake. Ask: are the two data sets linked one-to-one? H₀: μ₁=μ₂; use a two-sample t (means are NOT pooled). The Review Hub · AP Statistics Unit 4 TOPIC 4.10 Carrying Out a Two-Mean Test t = ( x̄₁ − x̄₂ ) / √( s₁²/n₁ + s₂²/n₂ ) Worked example x̄₁=52, s₁=6, n₁=40; x̄₂=48, s₂=7, n₂=45: t = (52−48)/√(6²/40+7²/45) = 4/1.41 ≈ 2.84 Find the p-value on the t-curve (double it for ≠); use technology's df. Decision & conclusion p-value ≤ α → reject H₀ "There is convincing evidence the two means differ (in context)." Otherwise, "not enough evidence." Interval & test agree at matching confidence/α. Two-sample t-test: t = (x̄₁−x̄₂)/√(s₁²/n₁ + s₂²/n₂), then the p-value. The Review Hub · AP Statistics Unit 4
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How to use the visual review

Spend 30 seconds per slide before clicking next. Look at the diagram, then ask yourself: "Could I state this condition, choose t over z, or interpret this interval from memory?"

Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.

This is great for review the night before the exam — fast, visual, and covers every idea you need to recognize in Unit 4.