Type I and Type II Errors

Two types of errors in hypothesis testing

Type I Error (False Positive): Rejecting $H_0$ when it's actually true. Probability = $\alpha$ (significance level).

"Convicting an innocent person."

Type II Error (False Negative): Failing to reject $H_0$ when it's actually false. Probability = $\beta$.

"Letting a guilty person go free."

The tradeoff

• Decreasing $\alpha$ (harder to reject) → increases $\beta$ (more likely to miss a real effect).
• The only way to decrease both: increase sample size.

Power

Power = $1 - \beta$ = probability of correctly rejecting a false $H_0$. We want power $\geq 0.80$ (80%).

Visual representation

Two bell curves overlap: one is the null distribution ($H_0$ true), the other is the alternative ($H_a$ true). The threshold (critical value) divides the space:

• $\alpha$ = area of null curve beyond the threshold (right tail)
• $\beta$ = area of alternative curve before the threshold (left of threshold)

Try it in the visualization

Two overlapping distributions are drawn. Drag the threshold to see $\alpha$ and $\beta$ change in opposite directions. Increase the effect size to separate the curves further (reduces both errors).