What's the Difference Between a Type I Error and a Type II Error?

A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected.

This is HL-only content from AHL 4.18: T and Z tests, Type I and Type II errors.

How the Two Errors Arise

A hypothesis test begins with a null hypothesis, $H_0$, and an alternative hypothesis, $H_1$. The test leads to one of two decisions: reject $H_0$, or find insufficient evidence to reject $H_0$.

The probability of a Type I error is the significance level, $\alpha$. For example, using a $5\%$ significance level means accepting a $0.05$ probability of rejecting $H_0$ when it is actually true.

The probability of correctly rejecting a false null hypothesis is the power of the test:

$$\text{Power}=1-\beta$$

Contextual Example

Suppose $H_0$ states that a new medicine has no effect.

• A Type I error means concluding that the medicine works when it actually does not.
• A Type II error means failing to detect that the medicine works when it actually does.

For a fixed sample size, reducing $\alpha$ generally makes rejection more difficult and may increase $\beta$. Increasing the sample size can improve the test's power without necessarily increasing $\alpha$.

A common misconception is to write “accept $H_0$.” A hypothesis test does not prove $H_0$; the correct wording is “there is insufficient evidence to reject $H_0$.”

Exam Technique

In an IB Math AI HL response, define the error using the specific context, not only “rejecting a true hypothesis.” Identify what $H_0$ claims, state the incorrect decision, and describe its real-world consequence. These ideas can appear on any GDC-active HL paper, particularly in extended hypothesis-testing questions.