We assume $H_0$ is true (cyan curve) and find a critical value. If the truth is actually $H_A$ (purple curve), where does our critical value place us?
Error Matrix
Type I Error ($\alpha$): Reject $H_0$ when $H_0$ is True. (False Positive).
Type II Error ($\beta$): Fail to Reject $H_0$ when $H_A$ is True. (False Negative).
Power ($1 - \beta$): Correctly Rejecting $H_0$ when $H_A$ is True.
The Golden Rule
Decreasing $\alpha$ directly increases $\beta$. Only by increasing Sample Size ($n$) can you decrease BOTH $\alpha$ and $\beta$ simultaneously (by narrowing the curves).
Tags
hypothesis testtype I errortype II errorpoweralpha
Distributions & Probabilities
$H_0$: Null Sampling Dist.
Mean $\mu_0$:
0.00
Standard Error ($SE$):
Critical Val ($x^*$):
Type I Error ($\alpha$):
$H_A$: Alternative Sampling Dist.
True Mean $\mu_a$:
Type II Error ($\beta$):
Power ($1-\beta$):
Significance Level $\alpha$
Null Mean $\mu_0$
Alternative Mean $\mu_a$
Population Std Dev ($\sigma$)
Sample Size ($n$)
This model assumes a One-Tailed Upper (Right) Hypothesis Test: $H_A: \mu > \mu_0$.