Solutions — Type I and Type II Errors — Grade 12 Specialist Maths

State the definition of a Type I error and give the probability of it occurring in terms of α. Fluency

A Type I error occurs when the null hypothesis H₀ is rejected even though it is actually true.

It is also called a “false positive” — the test incorrectly concludes that there is a significant effect when there is not.

P(Type I error) = α, the significance level of the test.

For example, at α = 0.05, there is a 5% chance of making a Type I error.

State the definition of a Type II error and give its probability in terms of β. Fluency

A Type II error occurs when the null hypothesis H₀ is not rejected even though H₀ is actually false.

It is also called a “false negative” or “missed detection” — the test fails to identify a real effect that exists.

P(Type II error) = β.

The value of β depends on: the true value of the parameter, the sample size, and the significance level.

A test is conducted at α = 0.05. What is the probability of making a Type I error? What is the probability of correctly not rejecting a true H₀? Fluency

P(Type I error) = α = 0.05

P(correctly not rejecting a true H₀) = 1 − α = 1 − 0.05 = 0.95

This means that when H₀ is true, 95% of the time we correctly retain it, and 5% of the time we incorrectly reject it.

Complete the decision table: fill in “Type I error”, “Type II error”, “Correct decision” for the four possible outcomes of a hypothesis test. Fluency

	H₀ is true	H₀ is false
Reject H₀	Type I error (probability = α)	Correct decision (power = 1 − β)
Fail to reject H₀	Correct decision (probability = 1 − α)	Type II error (probability = β)

A medical test screens for a disease. Describe a Type I and Type II error in context. Which is more serious? Understanding

H₀: patient does not have the disease H₁: patient has the disease

Type I error: Rejecting H₀ when H₀ is true — concluding the patient has the disease when they do not. This is a false positive. The patient would undergo unnecessary further testing or treatment, causing anxiety and expense.

Type II error: Failing to reject H₀ when H₀ is false — concluding the patient does not have the disease when they actually do. This is a false negative. The patient would not receive treatment they need, potentially allowing the disease to progress.

Type II error is generally more serious in this context, because failing to diagnose and treat a disease can have life-threatening consequences. For serious diseases, researchers often accept a higher α (more false positives) in order to minimise β (false negatives) and maximise power.

A hypothesis test at α = 0.05 has power 0.82 when H₀ is false. Find β and interpret it. Understanding

Power = 1 − β

0.82 = 1 − β

β = 1 − 0.82 = 0.18

Interpretation: When H₀ is actually false, there is an 18% probability that the test fails to detect this (Type II error). Equivalently, the test correctly identifies the false H₀ 82% of the time.

A researcher reduces α from 0.05 to 0.01 with fixed sample size. How does this affect Type I error, Type II error, and power? Understanding

(a) Type I error probability: Decreases from 0.05 to 0.01. Making it harder to reject H₀ means fewer false positives.

(b) Type II error probability (β): Increases. By making it harder to reject H₀, the test now has a higher chance of failing to detect a real effect (more false negatives).

(c) Power (= 1 − β): Decreases. With the same data and a stricter threshold, the test is less likely to correctly identify a false null hypothesis.

Key principle: For a fixed sample size, reducing α and reducing β simultaneously is impossible — the only way to do both is to increase the sample size n.

In 200 tests on correctly-filling machines at α = 0.05, how many Type I errors are expected? Understanding

Since the machines are correctly filling (H₀ is true), the only error possible is a Type I error.

P(Type I error) = α = 0.05

Expected number of Type I errors = 200 × 0.05 = 10

So about 10 out of 200 correctly-functioning machines would be incorrectly flagged as faulty by the test.

Quality control test: n = 25, X ~ N(100, 36) under H₀. Reject when &x̄ < 97. Find α, then β when μ = 96, then power. Problem Solving

Under H₀: μ = 100, σ = 6, n = 25.

SE = σ/√n = 6/√25 = 6/5 = 1.2

(a) α = P(&x̄ < 97 | μ = 100)

Z = (97 − 100) / 1.2 = −3/1.2 = −2.5

α = P(Z < −2.5) ≈ 0.0062

(b) β = P(&x̄ ≥ 97 | μ = 96)

Under H₁ (μ = 96): Z = (97 − 96) / 1.2 = 1/1.2 ≈ 0.833

β = P(&x̄ ≥ 97 | μ = 96) = P(Z ≥ 0.833) = 1 − Φ(0.833) ≈ 1 − 0.7977 ≈ 0.2023

(c) Power = 1 − β ≈ 1 − 0.2023 = 0.7977

The test has about 79.8% power to detect the machine is underfilling when μ = 96 g.

Clinical trial: α = 0.01, power = 0.70. Answer four parts about errors and improving the test. Problem Solving

(a) P(concluding treatment works when it doesn’t) = P(Type I error) = α = 0.01

(b) P(failing to detect real effect) = P(Type II error) = β = 1 − power = 1 − 0.70 = 0.30

(c) Doubling the sample size: increases the power (closer to 1) and decreases β. A larger sample gives more precise estimates of the population parameter, making it easier to detect a real effect. The critical region shifts to better detect differences from H₀.

(d) In a medical context, a Type I error means approving an ineffective treatment. This wastes resources, may expose patients to unnecessary side effects, and delays access to effective treatments. Using α = 0.01 instead of 0.05 makes this 5 times less likely to occur, protecting patient safety. The stricter threshold is appropriate when the consequences of a false positive are severe.

Type I and Type II Errors — Full Worked Solutions