Topic A: Probability
Lessons 1–4 · Conditional probability, independence, and experiments.
01–04 — Lessons 1–4
Conditional & Compound Probability
- P(A|B): probability of A given B has occurred = P(A∩B)/P(B).
- Independent events: P(A|B) = P(A). Knowing B gives no info about A.
- Multiplication rule: P(A∩B) = P(A)·P(B|A).
- Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B).
- Complement: P(not A) = 1 − P(A).
P(rain) = 0.3, P(umbrella | rain) = 0.9 → P(rain ∩ umbrella) = 0.3 × 0.9 = 0.27
Practice — Lessons 1–4
Q1 · Multiple Choice
If P(A) = 0.4 and P(B) = 0.5 and A, B are independent, what is P(A∩B)?
Q2 · Multiple Choice
P(A|B) = P(A) tells us that A and B are:
Topic B: Normal Distribution
Lessons 5–9 · Properties of normal distributions and z-scores.
05–09 — Lessons 5–9
The Normal Distribution & Z-Scores
Many real-world distributions are approximately bell-shaped (normal).
- Normal distribution: symmetric, bell-shaped, mean = median = mode.
- 68-95-99.7 rule: 68% within 1σ, 95% within 2σ, 99.7% within 3σ.
- Z-score: z = (x − μ)/σ — how many std. deviations from the mean.
- A z-score of 0 is exactly at the mean; ±1 is one std dev away.
- Use z-tables or calculator to find percentages.
SAT scores: μ = 1000, σ = 200. Score 1200: z = (1200−1000)/200 = 1.0 → top ~16%.
Practice — Lessons 5–9
Q3 · Multiple Choice
A normal distribution has μ = 70 and σ = 10. What percentage of data falls between 60 and 80?
Q4 · Short Answer
Score = 85, μ = 75, σ = 5. What is the z-score?
Topic C: Statistical Inference
Lessons 10–13 · Sampling distributions, margin of error, and drawing conclusions.
10–13 — Lessons 10–13
Sampling & Statistical Inference
Statistical inference uses sample data to draw conclusions about a population.
- Sample mean (x̄): estimate of the population mean (μ).
- Sampling variability: different samples give different estimates.
- Margin of error: accounts for the variability — typically ±2·(σ/√n).
- Confidence interval: range where the true parameter likely falls.
- Larger sample size → smaller margin of error → more precise estimate.
Sample of 100, x̄ = 52%, σ = 5%: margin of error ≈ 2(5/10) = 1% → interval (51%, 53%).
Practice — Lessons 10–13
Q5 · Multiple Choice
A larger sample size results in:
Q6 · Multiple Choice
A confidence interval of (42%, 58%) for a proportion means: