Errors in Hypothesis Tests¶

Type I & Type II Errors¶

• Type I = False positive
  • Innocent but judged guilty
• Type II = False negative
  • Guilty but judged innocent

Probability of Type I Error¶

• If significance level $\alpha$ is known, $P(Type\ I\ error) = \alpha$
• If unknown, $P(Type\ I\ error)=$ probability of being in the critical region given $H_0$ is true

Probability of Type II Error¶

• $P(Type\ II\ error) = P($not in the critical region, given the actual population parameter is true$)$

Reduce Probability of Type II Error¶

• $\uparrow n$
• $\uparrow \alpha$
• $\downarrow \sigma$ of the hypothesis test
• Actual population parameter is farther from the null population parameter

Relationship Between Possibilities of Type I & II Errors¶

• $\uparrow \alpha \implies \uparrow P(Type\ I), \downarrow P(Type\ II)$
• $\downarrow \alpha \implies \downarrow P(Type\ I), \uparrow P(Type\ II)$
• $\uparrow n \implies \downarrow P(Type\ I), \downarrow P(Type\ II)$

Power of A Test¶

The probability that it will correctly reject a false null hypothesis

$Power = 1 - P(Type\ II\ error)$