Type I and Type II Errors

The Two Types of Errors

| | $H_0$ True | $H_0$ False | |---|---|---| | Reject $H_0$ | Type I Error ❌ | Correct decision ✅ | | Fail to reject $H_0$ | Correct decision ✅ | Type II Error ❌ |

Type I Error (False Positive)

Rejecting $H_0$ when it is actually true.

$P(\text{Type I Error}) = \alpha$

Example: Concluding a drug works when it actually doesn't.

Type II Error (False Negative)

Failing to reject $H_0$ when it is actually false.

$P(\text{Type II Error}) = \beta$

Example: Concluding a drug doesn't work when it actually does.

Consequences in Context

Always describe errors in context on the AP exam:

• Type I: "We would conclude that [Ha in context] when in reality [H0 in context]."
• Type II: "We would fail to find evidence that [Ha in context] when in reality [Ha is true]."

The Relationship Between $\alpha$ and $\beta$

• Decreasing $\alpha$ → increases $\beta$ (fewer Type I errors, more Type II errors)
• Increasing $\alpha$ → decreases $\beta$ (more Type I errors, fewer Type II errors)
• There's always a trade-off!

Power

Power = probability of correctly rejecting $H_0$ when it is false

$\text{Power} = 1 - \beta = P(\text{Reject } H_0 | H_0 \text{ is false})$

Higher power = better test (more likely to detect a real effect)

Factors That Affect Power

| Factor | Effect on Power | |--------|----------------| | Increase $\alpha$ | ↑ Power (but more Type I errors) | | Increase $n$ | ↑ Power (more data = better detection) | | Increase true effect size | ↑ Power (larger difference easier to detect) | | Decrease $\sigma$ | ↑ Power (less noise = clearer signal) |

Choosing $\alpha$

Consider the consequences:

• If Type I error is very costly → use smaller $\alpha$ (e.g., 0.01)
• If Type II error is very costly → use larger $\alpha$ (e.g., 0.10)
• Medical testing: Missing a disease (Type II) is often worse → larger $\alpha$
• Criminal justice: Convicting an innocent person (Type I) is worse → smaller $\alpha$

AP Tip: You will be asked to describe Type I and Type II errors in context. Don't just say "rejecting a true null hypothesis" — explain what that means for the specific problem.