Type I vs Type II error (and why sample size matters)

Every hypothesis test can go wrong in two ways:

	H₀ is actually true	H₀ is actually false
You reject H₀	❌ Type I error (false alarm). Probability = α	✅ Correct. Probability = power = 1 − β
You don't reject H₀	✅ Correct	❌ Type II error (missed signal). Probability = β

Type I (α) means concluding there is a difference when there isn't. In quality terms this is the producer's risk: rejecting a good lot, or adjusting a process that was fine.
Type II (β) means missing a difference that is really there. In quality terms this is the consumer's risk: accepting a bad lot, or missing a real process shift.

You choose α (often 0.05). β depends on how big the real difference is, how noisy the process is (σ), and your sample size. A bigger sample lowers β without raising α.

Worked example

A filling process should average 50 g with σ = 2 g. You take n = 16 bottles and run a two-sided z test at α = 0.05. Suppose the true mean has actually drifted to 51 g. How likely are you to catch it?

Standard error = σ / √n = 2 / √16 = 0.5 Fail-to-reject zone = 50 ± 1.96 × 0.5 = 49.02 to 50.98 If the true mean is 51: β = P(49.02 < x̄ < 50.98) = Φ((50.98 − 51)/0.5) − Φ((49.02 − 51)/0.5) = Φ(−0.04) − Φ(−3.96) ≈ 0.484 − 0.000 = 0.484 Power = 1 − β ≈ 0.516

So with 16 bottles you would miss a 1 g drift almost half the time. To get 90% power (β = 0.10) for a 1 g shift:

n = ((z₁₋α/₂ + z₁₋β) × σ / δ)² = ((1.96 + 1.2816) × 2 / 1)² = 42.03 → round UP to 43

Common exam traps

Mixing up the labels. Type I is the false alarm (α, producer's risk). Type II is the miss (β, consumer's risk).
Thinking "fail to reject" means "accept". Not rejecting H₀ doesn't prove it is true. The test may simply have had too little power.
Thinking a lower α is always better. At a fixed sample size, lowering α raises β. Only a larger n (or less noise) lowers both.
Rounding the sample size down. Always round up: 42.03 becomes 43.
Using 1.645 for a two-sided test. At α = 0.05, two-sided uses 1.96 and one-sided uses 1.645.

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