False alarm vs missed effect: A hospital smoke alarm has two ways to fail. It can go off when there's no fire (a false alarm), or stay silent when there is a fire (a miss). Hypothesis tests have exactly the same two errors.
Type I error — reject a TRUE H₀ ('false alarm'). You conclude there's an effect when really there isn't.
Type II error — fail to reject a FALSE H₀ ('miss'). You conclude there's no effect when really there is.
The four possible outcomes form a 2×2 grid (shown below).
| H₀ is true | H₀ is false | |
|---|---|---|
| Reject H₀ | Type I error (α) | ✓ correct |
| Don't reject H₀ | ✓ correct | Type II error (β) |
Their probabilities have names: P(Type I error) = α, the significance level itself. Setting α = 5% means you accept a 5% chance of a false alarm.
P(Type II error) = β. This is NOT just '1 − α' — you can only compute β once you assume a specific alternative value (e.g. 'what if the true mean is really 106?').
IB-style question — identify the error
A new fertiliser is tested. H₀: it makes no difference to yield; H₁: it increases yield. In reality the fertiliser does nothing, but the trial happens to give an unusually good sample and the farmer concludes it works.
What type of error is this?
Step by step
- H₀ ('no difference') is actually TRUE here.
- The farmer rejected H₀ (concluded it works).
- Rejecting a true H₀ is a Type I error.
Final answer
A Type I error — rejecting a true H₀ (a false alarm: the fertiliser is useless but the test says it works).
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The critical region is where you reject H₀: Instead of a p-value, a test can use a critical (rejection) region: a set of outcomes so extreme that you reject H₀ if your result lands there.
P(Type I) = α = P(landing in the critical region WHEN H₀ is true).
To find a Type II error probability you assume H₀ is false at a given alternative value, then:
P(Type II) = β = P(landing OUTSIDE the critical region GIVEN the alternative is true).
The key is which distribution you use: α uses the H₀ distribution; β uses the alternative distribution.
IB-style question — binomial Type I and Type II
A coin is spun 20 times to test H₀: p = 0.5 against H₁: p > 0.5, where p = P(heads). The rule is: reject H₀ if 15 or more heads appear. Let X ~ B(20, p).
(a) Find the probability of a Type I error.
(b) Given the coin is actually biased with p = 0.8, find the probability of a Type II error.
Step by step
- (a) Type I = reject a true H₀, i.e. X ≥ 15 when p = 0.5. Use the GDC binomial cdf.
- (b) Type II = fail to reject H₀ (X ≤ 14) when the alternative p = 0.8 is true. Switch the distribution to B(20, 0.8).
Final answer
(a) P(Type I) ≈ 0.0207. (b) P(Type II) ≈ 0.196. Note α was computed under p = 0.5, but β under p = 0.8.
IB-style question — normal critical region and Type II
Bags should weigh μ = 100 g, with known population sd σ = 15 g. A sample of n = 25 bags is taken and the test is H₀: μ = 100 against H₁: μ > 100 at the 5% level, so H₀ is rejected if the sample mean x̄ is large.
(a) Find the critical value of x̄.
(b) If the true mean is actually 106 g, find the probability of a Type II error.
Step by step
- (a) Standard error = σ/√n = 15/5 = 3. The upper 5% critical z is 1.645, so x̄ must exceed 100 + 1.645×3.
- (b) Type II = NOT rejecting (x̄ < 104.9) when the true mean is 106. Use the alternative distribution: x̄ ~ N(106, 3²).
- Read the probability from the GDC normal cdf.
Final answer
(a) Critical value x̄ ≈ 104.9 g. (b) P(Type II) ≈ 0.361 — a 36% chance of missing the real over-weight.