Stochastics: binomial and normal distributions, hypothesis testing
What this domain tests
Stochastics in the Abitur covers Bernoulli chains and the binomial distribution, expected value and standard deviation, the normal approximation, and one- or two-sided hypothesis tests on a proportion p. Contexts are quality control, surveys, medical tests and games; the arithmetic is done with the allowed calculator, so marks are lost on setting up the model, not on multiplying.
Rules and concepts
- Binomial model B(n, p): P(X = k) = C(n, k)·p^k·(1 − p)^(n−k); valid only for fixed n, constant p and independent trials.
- P(X ≥ k) = 1 − P(X ≤ k − 1); "mindestens", "höchstens", "mehr als" and "genau" each map to a different interval.
- E(X) = n·p, Var(X) = n·p·(1 − p), σ = √Var.
- Normal approximation when σ > 3 (Laplace condition), with continuity correction: P(X ≥ k) ≈ 1 − Φ((k − 0.5 − μ)/σ).
- Hypothesis test: H₀: p ≤ p₀ (or p ≥ p₀) against H₁; rejection region from P(X ≥ k | p = p₀) ≤ α. Two-sided tests split α/2 on each side.
- Error type I: rejecting a true H₀, probability at most α, chosen by you. Error type II: not rejecting a false H₀; it depends on the actual p and needs a specific alternative to compute.
Worked example
A supplier claims at most 10% of parts are defective. In a sample of n = 200, 28 are defective. Test at α = 5%. H₀: p ≤ 0.10, H₁: p > 0.10. Under p = 0.10: μ = 20, σ = √(200·0.1·0.9) = √18 ≈ 4.24, and σ > 3, so the normal approximation is allowed. With continuity correction, z = (27.5 − 20)/4.24 ≈ 1.77, giving a p-value of about 0.04 < 0.05. Reject H₀: the data give significant evidence at the 5% level that the defect rate exceeds 10%.
Traps
- Reading α as "the probability that H₀ is true"; it is only the tolerated error of rejecting a true H₀.
- Forgetting the tail direction; "at least 28" and "more than 28" differ by one unit, which flips borderline decisions.
- Using the normal approximation when σ ≤ 3, or omitting the 0.5 continuity correction.
- Treating drawing without replacement as binomial; p must stay constant.
- Rounding the critical value the wrong way - the rejection region must keep the actual error ≤ α.