AP Statistics Premium Practice Exam

Unit 1

Exploring One-Variable Data

Mean: x̄ = (Σxᵢ) / n
Sample SD: s = √[Σ(xᵢ − x̄)² / (n−1)]
IQR = Q3 − Q1
Outlier fences: Q1 − 1.5·IQR and Q3 + 1.5·IQR
z-score: z = (x − μ) / σ
Percentile (Normal): use z-table

Mean is sensitive to outliers; median is resistant.
Skewed right → mean > median; Skewed left → mean < median.
Adding constant c to all values: mean/median shift by c; SD unchanged.
Multiplying by constant c: mean, median, SD all multiply by c.
Normal distribution: 68-95-99.7 Rule (1σ / 2σ / 3σ).

Example

A dataset has Q1 = 20, Q3 = 40. Is 65 an outlier?

IQR = 20; upper fence = 40 + 1.5(20) = 70. Since 65 < 70, NOT an outlier.

Unit 2

Exploring Two-Variable Data

Correlation: r (no units, −1 ≤ r ≤ 1)
LSRL: ŷ = a + bx
Slope: b = r · (sᵧ / sₓ)
y-intercept: a = ȳ − b·x̄
Coefficient of determination: r²
Residual = Actual − Predicted = y − ŷ

r measures only linear association; not causation.
r² = proportion of variation in y explained by x.
Residual plot: random scatter = good linear model.
Influential point: removing it greatly changes LSRL.
Lurking variable can create misleading association.

Example

r = 0.8, sₓ = 2, sᵧ = 5, x̄ = 10, ȳ = 30. Find LSRL.

b = 0.8 × (5/2) = 2; a = 30 − 2(10) = 10. So ŷ = 10 + 2x.

Unit 3

Collecting Data

SRS: Every individual equally likely to be chosen.
Stratified: Population divided into strata; SRS from each.
Cluster: Divide into clusters; randomly select entire clusters.
Systematic: Every kth individual after a random start.
Observational: No treatment assigned; cannot establish causation.
Experiment: Researcher assigns treatments; can establish causation.
Control/Placebo/Blind/Double-blind: Reduce confounding.
Blocking: Group experimental units by a known variable before randomizing.
Replication: Repeat experiment enough times to reduce chance variation.

Unit 4

Probability

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)
Binomial: P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ
Binomial mean: μ = np; SD: σ = √(np(1−p))
Geometric: P(X=k) = (1−p)ᵏ⁻¹ · p; mean = 1/p
E(X) = Σ[xᵢ · P(xᵢ)]
Var(X+Y) = Var(X) + Var(Y) [if independent]

Example

A fair coin is tossed 4 times. P(exactly 3 heads)?

P = C(4,3)·(0.5)³·(0.5)¹ = 4·0.125·0.5 = 0.25

Unit 5

Sampling Distributions

Sampling dist of p̂: mean = p, SD = √(p(1−p)/n)
Sampling dist of x̄: mean = μ, SD = σ/√n (= SE)
Central Limit Theorem: n ≥ 30 → x̄ approx Normal
10% condition: n ≤ 0.10N (for independence)
Large counts: np ≥ 10 and n(1−p) ≥ 10 (for proportions)

SE of x̄ decreases as n increases (by factor of √n).
CLT applies regardless of the shape of the population distribution when n is large.
Sampling variability ≠ bias; bias = consistent under/overestimation.

Unit 6

Inference for Proportions

CI for p: p̂ ± z* · √(p̂(1−p̂)/n)
z* = 1.645 (90%), 1.960 (95%), 2.576 (99%)
z-test: z = (p̂ − p₀) / √(p₀(1−p₀)/n)
Two-prop z-test: z = (p̂₁−p̂₂) / √[p̂ₒ(1−p̂ₒ)(1/n₁+1/n₂)]
p̂ₒ (pooled) = (x₁+x₂)/(n₁+n₂)

Example

n = 100, p̂ = 0.62. Find 95% CI.

SE = √(0.62·0.38/100) = 0.0485; CI: 0.62 ± 1.96(0.0485) = (0.525, 0.715)

Unit 7

Inference for Means

CI for μ: x̄ ± t* · (s/√n), df = n−1
t-test: t = (x̄ − μ₀) / (s/√n)
Paired t: use d̄ = mean of differences, sᵈ = SD of differences
Two-sample t: t = (x̄₁−x̄₂) / √(s₁²/n₁ + s₂²/n₂)
Conditions: Random, 10% condition, Normal/Large sample

Use t-procedures when σ is unknown (always in practice).
Larger df → t-distribution closer to Normal.
Paired design reduces variability; more powerful than two-sample.

Unit 8

Chi-Square Tests

χ² = Σ[(O − E)² / E]
Goodness-of-Fit: df = categories − 1
Independence/Homogeneity: df = (r−1)(c−1)
Expected cell: E = (row total × col total) / grand total
Conditions: all expected counts ≥ 5

Goodness-of-Fit: one sample, compare to claimed distribution.
Independence: one sample, two categorical variables.
Homogeneity: multiple samples, one categorical variable.
χ² is always right-tailed.

Unit 9

Inference for Regression Slopes

t-test for slope: t = b / SEb, df = n−2
CI for β: b ± t* · SEb
H₀: β = 0 (no linear relationship)
SEb appears in computer output (standard error of slope)

Conditions (LINE): Linear, Independent, Normal residuals, Equal variance.
Check residual plot for linearity and equal variance.
Normal probability plot of residuals checks normality.
p-value < α → reject H₀ → significant linear relationship.

AP Statistics

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