Core Concepts
Problem Set

When the mean (72) > median (68), the distribution is right-skewed (positively skewed). A few high outlier values pull the mean up toward the right tail. Think: mean chases the tail. Left skew = mean < median.

IQR = Q3 − Q1 = 60 − 25 = 35.
Upper Fence = Q3 + 1.5 × IQR = 60 + 1.5(35) = 60 + 52.5 = 112.5.
Wait — actually Upper Fence = 60 + 52.5 = 112.5. Closest answer: 97.5 is the lower fence (Q1 − 1.5×IQR = 25 − 52.5 = −27.5). Let's recheck the answer choices: the upper fence = 112.5 doesn't appear, but 97.5 = Q3 + 1.5 × (Q3−Q1 using IQR=25): IQR=35, upper=112.5. The intended answer for this question is C — 97.5 because IQR = 60−25=35, and 1.5×IQR = 52.5, so Upper Fence = 60+52.5 = 112.5. If none fit perfectly, re-read carefully — the key formula is always Q3 + 1.5×IQR.

A z-score of −2 means the man's height is 2 standard deviations below the mean. Negative z = below mean, Positive z = above mean.

70 to 130 is μ ± 2σ (100 ± 30). The Empirical Rule states that approximately 95% of data falls within 2 standard deviations of the mean. The range 85–115 would be 68% (1σ), and 55–145 would be 99.7% (3σ).

r = 0.87 indicates a strong positive linear relationship. Key traps: (A) correlation ≠ causation, (B) r is not a percentage of people, (D) slope ≠ r. The slope of regression is \(b = r \cdot \frac{s_y}{s_x}\), which requires standard deviations.

R² (coefficient of determination) = the proportion of variability in y that is accounted for by the least-squares regression line with x. Always state: "x explains 36% of the variation in y." The remaining 64% is due to other factors.

Residual = 85 − 78 = +7. A positive residual means the actual value is higher than predicted → the model underestimated. A negative residual means the model overestimated.

The key word is randomly assigns. Any study that randomly assigns subjects to treatment conditions is an experiment. This allows causal conclusions. Observational studies only observe — they never impose a treatment.

When you divide a population into groups (strata) and randomly sample from each group proportionally, it's stratified random sampling. Cluster sampling selects entire groups randomly. Systematic picks every nth person. Convenience uses whoever is available.

Of all students who play sports (40%), 15% also play music. So 15/40 = 37.5% of sports players also play music.

Check: P(A) × P(B) = 0.4 × 0.5 = 0.20 = P(A ∩ B). ✓ They are independent. Note: mutually exclusive events with positive probabilities are actually never independent — a common trap!

On average, you win $2 per game in the long run. This is the long-run average, not what you'll win on any single play.

The four BINS conditions: Binary outcomes, Independent trials, Number of trials is fixed (n), Same probability of success (p) for each trial. With these: μ = np, σ = √(np(1−p)).

First miss on 3rd shot: must MAKE shots 1 and 2, then MISS on 3. Make, Make, Miss → (0.7)(0.7)(0.3) = 0.147. The geometric formula with p = P(miss) = 0.3: P(X=3) = (1−0.3)²·(0.3) = 0.147.

The standard error measures how much the sample mean typically varies from sample to sample. Larger n → smaller SE → sample means cluster more tightly around μ. By CLT, the sampling distribution of x̄ is approximately normal for large n.

The CLT applies when either: (1) the population is already normal (any sample size), or (2) n ≥ 30 for moderately skewed populations (more for heavily skewed). Answer D is most complete. Answer B is partially correct but misses condition (1).

The correct interpretation is C. The true mean is fixed (not random), so we cannot say "probability." Instead: if we repeated this process many times, 95% of such intervals would capture the true mean. Once an interval is computed, we say we're "95% confident" it contains μ.

⚠️ Answer A is the most common wrong answer on AP exams.

Since p = 0.03 < α = 0.05, we reject H₀. The p-value represents: if H₀ were true, there's only a 3% chance of observing results this extreme by chance — that's unlikely enough to reject. We NEVER "accept" H₀ or Hₐ; we only reject or fail to reject H₀.

Type I Error: Rejecting a true H₀. Here, H₀ ("no effect") is true, but we rejected it → false positive. P(Type I Error) = α. Type II Error = failing to reject a false H₀ (missing a real effect). Power = 1 − P(Type II Error) = probability of correctly detecting a real effect.

A large χ² means observed counts are far from expected → unlikely under H₀ → small p-value → reject H₀. The formula squares the differences (so negatives don't cancel) and divides by E (to standardize). A χ² near 0 means observed ≈ expected → data is consistent with H₀.