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

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