AP Statistics | Premium Practice Exam

Unit 1 · Exploring One-Variable Data

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Describe distributions with shape, center, spread, outliers (SOCS).

Mean: x̄ = Σxᵢ / n
Median: middle value (sorted)
IQR = Q3 − Q1
Outlier rule: < Q1 − 1.5·IQR or > Q3 + 1.5·IQR
s² = Σ(xᵢ − x̄)² / (n−1) [sample variance]

Skewed right → mean > median; skewed left → mean < median
Median & IQR are resistant; mean & SD are not
Boxplot: min, Q1, Q2 (median), Q3, max

ExampleData: 2, 4, 4, 6, 8. x̄ = 4.8, Median = 4, IQR = 4. Outlier fence: Q1=3, Q3=7 → lower <−3, upper >13. No outliers.

Unit 2 · Exploring Two-Variable Data

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Scatterplot: direction, form, strength. LSRL minimizes Σ(residual)².

ŷ = a + bx
b = r·(Sy/Sx) slope
a = ȳ − b·x̄ y-intercept
r = correlation (−1 ≤ r ≤ 1)
r² = coefficient of determination (% variation explained)

Residual = observed − predicted = y − ŷ
LSRL always passes through (x̄, ȳ)
r measures linear association only
Residual plot: random scatter → linear model appropriate

Unit 3 · Collecting Data

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Experiments vs. observational studies. Only well-designed experiments can establish causation.

SRS: every individual has equal probability of selection
Stratified: divide into strata, SRS within each
Cluster: divide into clusters, randomly select entire clusters
Systematic: select every kᵗʰ unit after random start
Experiment principles: randomization, control, replication, blinding
Confounding variable: associated with both explanatory & response
Block design: reduces variability by grouping similar units

Unit 4 · Probability

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P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A | B) = P(A ∩ B) / P(B)
Independent: P(A ∩ B) = P(A)·P(B)
Complement: P(Aᶜ) = 1 − P(A)
Bayes: P(A|B) = P(B|A)·P(A) / P(B)

Mutually exclusive: P(A ∩ B) = 0 ≠ independent
Law of large numbers: sample proportions → true probability

Unit 5 · Random Variables & Distributions

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Discrete RV: μ = Σ[x·P(x)], σ² = Σ[(x−μ)²·P(x)]
Linear transform: μ(a+bX) = a+b·μX, σ(a+bX) = |b|·σX
Sum: μ(X+Y) = μX+μY, σ²(X+Y) = σ²X+σ²Y (if independent)
Binomial: P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ
μ = np, σ = √(np(1−p))
Geometric: P(X=k) = (1−p)ᵏ⁻¹·p, μ = 1/p

Binomial conditions: BINS (Binary, Independent, fixed N, Same p)
Normal approx to Binomial: np≥10 and n(1−p)≥10

Unit 6 · Sampling Distributions

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Sampling dist of x̄: μ(x̄) = μ, σ(x̄) = σ/√n
Sampling dist of p̂: μ(p̂) = p, σ(p̂) = √[p(1−p)/n]
CLT: for large n, x̄ ~ N(μ, σ/√n)
Conditions for p̂ normality: np≥10, n(1−p)≥10, n≤10%·N

Unit 7 · Inference for Means

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One-sample t: t = (x̄ − μ₀) / (s/√n), df = n−1
CI for μ: x̄ ± t*·(s/√n)
Paired t: treat differences as one sample
Two-sample t: t = (x̄₁−x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Conditions: SRS/random, Normal (n≥30 or pop normal), Independence (n ≤ 10% N)
Wider CI: lower confidence level, larger n → narrower
Margin of error = t* · SE

Unit 8 · Inference for Proportions

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One-prop z: z = (p̂ − p₀) / √[p₀(1−p₀)/n]
CI for p: p̂ ± z*·√[p̂(1−p̂)/n]
Two-prop z: z = (p̂₁−p̂₂) / √[p̂c(1−p̂c)(1/n₁+1/n₂)]
p̂c = (X₁+X₂)/(n₁+n₂) [pooled]

CI uses p̂; hypothesis test uses p₀ (null)
Use pooled p̂c for two-prop significance test only

Unit 9 · Chi-Square Tests

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χ² = Σ (O − E)² / E
Goodness of fit: df = k − 1
Homogeneity / Independence: df = (r−1)(c−1)
Expected cell: E = (row total × col total) / grand total

All expected counts ≥ 5 (condition)
GOF: one sample, compare to claimed distribution
Homogeneity: multiple populations, same categorical variable
Independence: one sample, two categorical variables

Unit 10 · Inference for Regression

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H₀: β = 0 (no linear relationship)
t = b / SEb, df = n−2
CI for β: b ± t*·SEb
s = √[Σresiduals² / (n−2)] [residual std dev]

Conditions: L-I-N-E-R (Linear, Independent, Normal errors, Equal variance, Random)
s measures typical residual size
Reject H₀ → statistically significant linear relationship