Type I & II Errors (Power) – AP Statistics

Learning Objectives

Type I Error ($\alpha$): False Positive. Convicting an innocent person. We rejected $H_0$ but $H_0$ was actually true. Controlled directly by your Alpha level!

Type II Error ($\beta$): False Negative. Letting a guilty person go free. The alternative $H_a$ was completely true, but our sample failed to clear the $\alpha$ hurdle.

Power ($1-\beta$): True Positive! The probability we successfully reject $H_0$ when $H_a$ is in fact true. As effect size (distance) grows, Power surges.

Tags

Type IType IIPowerAlpha

       ERROR RELATIONSHIP

       $\text{Power} + \text{Type II } (\beta) = 1$

Null ($H_0$) Assumed

True ($H_a$) Reality

Decision Threshold

Alpha Level ($\alpha$) = 0.05

True Reality Geometry

Effect Size (Shift $H_a$) = 10

Sample Size ($n$) = 10

DECISION PROBABILITIES

Type I Error ($\alpha$): 5.0%

ASSUMING $H_a$ IS TRUE TODAY:

POWER ($1-\beta$): 45.0%

Type II Error ($\beta$): 55.0%

The See-Saw Effect: If you slide $\alpha$ down to strictly protect against False Positives (convicting innocents), the vertical line shifts heavily right. This brutally crushes your Power and explodes your Type II Error (letting the guilty walk free). The only way to win BOTH is to increase $n$ and shrink the curves!