Essential Practice — 5 Exam Sets

Strata = non-overlapping subgroups. An SRS is drawn from each stratum.

Upper fence = Q3 + 1.5 × IQR. Any value above the fence is an outlier.

Compute z = (80−75)/5 = 1.0. Then look up P(Z < 1.0) in the standard normal table.

σ_x̄ = σ / √n. This is called the standard error.

If A and B are independent: P(A ∩ B) = P(A) · P(B)

LLN: the sample mean converges to the population mean with larger n.

Residual = observed − predicted = y − ŷ

R² = proportion of variation in y explained by the linear relationship with x.

Use t when σ is unknown (estimated by s) and the distribution is approximately normal or n is large.

Two groups, one proportion each → two-proportion z-test.

ME = z* · √(p̂(1−p̂)/n). For 95%, z* ≈ 1.96.

Power = 1 − β (where β = P(Type II error)). It is the probability of correctly rejecting a false H₀.

P(B|A) = P(A ∩ B) / P(A)

Same subjects measured twice → take differences → use a one-sample t-test on differences (matched pairs).

GOF tests whether a single categorical variable's observed distribution matches a claimed distribution.

Remember SOCS: Shape, Outliers, Center, Spread. AP graders expect all three key features.

This is a statistical phenomenon, not a learning effect. It occurs whenever r < 1.

Adding a constant shifts the distribution but doesn't change spread. SD measures spread.

Blocks group similar experimental units together before random assignment to treatments.

Continuous: can take any value in an interval (measured). Discrete: countable values.

For Binomial: μ = np

CLT applies to any population shape when n is large enough (commonly n ≥ 30).

t = (x̄ − μ₀) / (s / √n)

A placebo controls for the psychological effect of receiving a treatment (placebo effect).

Mutually exclusive means P(A ∩ B) = 0. Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

P(X≥2) = 1 − P(X≤1) = 1 − [P(X=0) + P(X=1)]

A residual plot (residuals vs. fitted values) should show no pattern if the linear model is appropriate.

Higher confidence → larger z* or t* → wider interval. More confidence requires a wider net.

The tail pulls the mean toward it. Right-skewed = long right tail → mean > median.

Double-blind: neither participants nor evaluators know who received which treatment.

For 90% CI, α=0.10, and you need z* such that P(−z* < Z < z*) = 0.90. Look up the 95th percentile of the standard normal.

Larger n → smaller SE → larger test statistic → smaller p-value → higher power.

Systematic sampling: choose a random start, then select every k-th member.

P(Z < z) = 0.84. Using the 68-95-99.7 rule: 84% is halfway between 68% and 95% bands → about z=1.0.

Residual = observed − predicted. Positive → actual y is above the line.

Used for boxplots. Think: Min, Q1, Q2(Median), Q3, Max.

This ensures the sampling distribution of p̂ is approximately normal.

Adding a constant b doesn't change spread. Multiplying by a scales the SD by |a|, so Var scales by a².

df = n − 1 for a one-sample t-test.

Independence: knowing A occurred gives no information about B.

CI = x̄ ± t* · (s/√n). Compute ME = 2.064 × (15/5) = 2.064 × 3.

R² = r². The coefficient of determination.

p̂_c = (x₁ + x₂) / (n₁ + n₂)

P-value assumes H₀ is true. It's NOT the probability H₀ is true.

Voluntary response: people choose whether to respond, often only those with strong opinions do.

Influential points are usually high-leverage points (extreme x values). Removing them noticeably changes the regression line.

Multiplication Rule: P(A ∩ B) = P(A) · P(B|A)

r = ±√(R²). The sign of r matches the sign of the slope.

Two-sample t: comparing means of two independent groups. Not the same subjects (that's matched pairs).

Multiplying by c: mean × c, SD × c, Var × c².

The large counts condition: np ≥ 10 and n(1−p) ≥ 10.

Two-sided p-value = 2 × P(Z > |z|).

Expected = (row total × column total) / grand total

Bayes: update a prior probability given new evidence. P(A|B) from P(B|A).

The 97.5th percentile of Z is z≈1.96. Use x = μ + z·σ.

Width ∝ 1/√n. If n×4, then √n×2, so width is halved.

σ = √(np(1−p)) for binomial.

Only randomized controlled experiments allow causal conclusions. Observational studies show association only.

Side-by-side boxplots show center, spread, and outliers for multiple groups simultaneously.

The sampling distribution of x̄ is centered at the population mean μ. This makes x̄ an unbiased estimator.

Least-squares = minimize sum of squared residuals (SSE).

40 and 60 are exactly 1 SD below and above the mean (z = −1 and z = 1). Use the Empirical Rule.

P(B|A) = P(A ∩ B) / P(A). Here P(Science ∩ Math) = 15/n, P(Math) = 40/n.

Smaller α = more stringent standard. Harder to reject H₀ → fewer false positives (lower Type I error) but more false negatives (higher Type II error) → lower power.

Cluster sampling: randomly select groups (clusters), then include all members of selected clusters.

p̂ = 110/200 = 0.55. ME = 1.96 × √(0.55 × 0.45/200) ≈ 1.96 × 0.0352 ≈ 0.069.

E[aX + b] = a·E[X] + b

The direction of skew is the direction of the tail, not the bulk of the data.

Compute d = x₁ − x₂ for each pair, then test H₀: μ_d = 0.

Expected = (row total × column total) / grand total = (50 × 60) / 200

SE(p̂) = √(p(1−p)/n). Under H₀, use p₀; for CIs, use p̂.

Type II error (β): miss = fail to detect a real effect. Type I error (α): false alarm.

Extrapolation = prediction beyond the range of the data. This can produce unreliable predictions.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 13/52 + 12/52 − 3/52

df = (rows − 1)(columns − 1) = (3−1)(4−1)

The control group provides a baseline, allowing us to isolate the effect of the treatment.

A larger SD = more spread out (more variable) distribution.

E[X − Y] = E[X] − E[Y], always (whether independent or not).

CLT: regardless of population shape, the sampling distribution of x̄ becomes more normal as n increases.

Greater confidence requires a wider interval to be more certain of capturing μ.