Mathfolis

Type I, Type II Errors, and Power

Unit 6 · Inference for Categorical Data: Proportions

What AP Stats asks here

A hypothesis test can fail in two ways: reject a true H0H_0 (Type I, false positive, rate α\alpha) or fail to reject a false H0H_0 (Type II, false negative, rate β\beta). Power is 1−β1 - \beta — the probability of correctly detecting an alternative. AP problems require describing each error in context and listing the levers that raise power.

Error types

Type I — reject true H0H_0
P(Type I)=αP(\text{Type I}) = \alpha
Type II — fail to reject false H0H_0
P(Type II)=βP(\text{Type II}) = \beta
Power
Power=1−β=P(reject H0∣p=pa)\text{Power} = 1 - \beta = P(\text{reject } H_0 \mid p = p_a)

Levers that raise power

Increase n
shrinks SE, sharper detection\text{shrinks SE, sharper detection}
Increase α\alpha
larger rejection region (costs Type I)\text{larger rejection region (costs Type I)}
Larger true effect
easier to detect\text{easier to detect}
AP Tip: Which error matters more depends on consequences. For a fire alarm Type II is fatal; for criminal trials Type I is the worse error. Pick α with the cost asymmetry in mind.
Caution: Power is a function of the true parameter, not a single number. A power claim is only meaningful with a stated alternative.
Type 1

Describe Type I / II in context

Type I = false positive. Type II = false negative. Always tied to the specific H0H_0 and HaH_a in the problem.

Example 1
A medical test uses H0H_0: patient does not have the disease. Which choice correctly describes the Type II error? (A) The test concludes the patient has the disease but the patient does not — a false positive. (B) The test concludes the patient does not have the disease but the patient does — a false negative. (C) The test concludes nothing. (D) The test concludes the patient has the disease and the patient does — a true positive.

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Type 2

α\alpha–β\beta trade-off

α\alpha and β\beta are on a seesaw at fixed n. Only increasing n breaks the trade-off.

Example 2
A test currently uses α=0.05\alpha = 0.05 with β=0.20\beta = 0.20 at a specific alternative (power = 0.80). Researchers tighten to α=0.01\alpha = 0.01 with no change in n. What happens to β\beta and power? (A) β\beta falls; power rises. (B) Both stay the same — only the decision threshold changed. (C) β\beta rises; power falls. (D) Power becomes 0.

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Type 3

Ways to increase power

Larger n, larger α\alpha, larger true effect, and a one-sided alternative (when justified) all raise power.

Example 3
Which change increases the power of a one-proportion test? (A) Decrease n. (B) Decrease α\alpha. (C) Increase n. (D) Decrease the true effect size.

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