Type I and Type II Errors (AI HL)
Every hypothesis test can go wrong in one of two ways: rejecting a null hypothesis that was actually true, or failing to reject one that was actually false. These are called Type I and Type II errors, and the IB expects you to define both precisely, in context, and to explain how changing the sample size or significance level trades one off against the other. It's part of the broader Hypothesis Testing topic.
24 questions on this sub-topic.
The two errors
Covered under IB syllabus reference AHL4.18: Type I and Type II errors, including calculations of their probabilities.
Type I error
Reject \(H_0\) when \(H_0\) is true
Its probability equals the significance level \(\alpha\) - a false alarm.
Type II error
Fail to reject \(H_0\) when \(H_0\) is false
Its probability is denoted \(\beta\) - a missed effect. Power is \(1-\beta\), the probability of correctly rejecting a false \(H_0\).
Need the full syllabus wording and formula-booklet reference table? See Hypothesis Testing.
Worked examples
A researcher tests \(H_0: \mu = 20\) against \(H_1: \mu \neq 20\) at the 5% significance level.
(a) Define a Type I error in context.
(b) State the probability of making a Type I error.
(c) Define a Type II error in context.
(d) Explain how increasing the sample size affects the probability of a Type II error.
Worked solution
(a) Rejecting \(H_0\) (concluding \(\mu \neq 20\)) when in fact \(\mu = 20\). A1
(b) 0.05 (5%) A1
(c) Failing to reject \(H_0\) (concluding \(\mu = 20\)) when in fact \(\mu \neq 20\). A1
(d) Increasing sample size decreases the probability of a Type II error (greater power to detect a difference). R1
The power of a hypothesis test is \(1 - \beta\), where \(\beta\) is the probability of a Type II error. Explain what high power means and how it can be achieved.
Worked solution
High power means the test is likely to correctly reject a false \(H_0\). A1
Power is increased by using a larger sample size, M1
or by increasing the significance level \(\alpha\). A1
Common mistakes
- Defining the errors in the abstract instead of in context. "Rejecting a true null hypothesis" alone often loses the mark - the answer should say what that means for the actual scenario, e.g. "concluding the mean has changed when it hasn't."
- Confusing which error the significance level controls. \(\alpha\) is the probability of a Type I error by definition, not an estimate of it - it's set by the test designer before the data is collected.
- Assuming lowering \(\alpha\) always improves the test. A smaller significance level reduces the Type I error rate but increases the Type II error rate (lower power) - the two trade off against each other unless the sample size also increases.
Ready to practise properly?
24 Type I and Type II error questions, marked instantly like the real exam.
Quick answers
What is a Type I error?
A Type I error is rejecting \(H_0\) when \(H_0\) is actually true. Its probability equals the significance level, denoted \(\alpha\).
What is a Type II error?
A Type II error is failing to reject \(H_0\) when \(H_0\) is actually false. Its probability is denoted \(\beta\). See the parent topic's GDC guidance for running the underlying tests.