Minimizing Chance of Type I and Type II Errors

O.J. Simpson trial: the situation O.J. is assumed innocent. Evidence collected

O.J. Simpson trial: jury decisions In criminal trial: The evidence does not warrant rejecting the assumption of innocence. Behave as if O.J. is innocent. In civil trial: The evidence does warrant rejecting the assumption of innocence. Behave as if O.J. is guilty. Was an error made in either trial?

Errors in Trials If O.J. is innocent, then an error was made in the civil trial. If O.J. is guilty, then an error was made in the criminal trial.

Errors in Hypothesis Testing

Definitions: Types of Errors Type I error: The null hypothesis is rejected when it is true. Type II error: The null hypothesis is not rejected when it is false. There is always a chance of making one of these errors. We’ll want to minimize the chance of doing so!

Example: Grade inflation? H0: μ = 2.7 HA: μ > 2.7 n = 36 s = 0.6 and Data Random sample of students Decision Rule Set significance level α = 0.05. If p-value < 0.05, reject null hypothesis.

If X-bar is 2.865 … Reject null since p-value is (just barely!) smaller then 0.05.

If X-bar is 2.95 … Reject null since p-value is smaller then 0.05.

If X-bar is 3.00 … Reject null since p-value is smaller then 0.05.

Alternative Decision Rule “Reject if p-value  0.05” is equivalent to “reject if the sample average, X-bar, is larger than 2.865” X-bar > 2.865 is called “rejection region.”

Type I Error

Minimize chance of Type I error... … by making significance level  small. Common values are  = 0.01, 0.05, or 0.10. “How small” depends on seriousness of Type I error. Decision is up to researcher.

P(Type I Error) in trials Criminal trials: “Beyond a reasonable doubt”. Jurors must unanimously vote guilty. Significance level  set at 0.001, say. Very small chance of Type I error. Civil trials: “Preponderance of evidence.” 9 out of 12 jurors must vote guilty. Significance level  set at 0.10, say. Larger chance of a Type I error.

Example: Serious Type I Error New Drug A is supposed to reduce blood pressure by more than 15 mm Hg. H0: μ = 15 versus HA: μ > 15 Drug A can have serious side effects, so don’t want patients on it unless μ > 15. Implication of Type I error: Expose patients to serious side effects without other benefit. Set  = P(Type I error) to be small  0.01

Example: Not so serious Type I Error Grade inflation? H0: μ = 2.7 vs. HA: μ > 2.7 Type I error: claim average GPA is more than 2.7 when it really isn’t. Implication: Instructors grade harder. Students get unhappy. Set  = P(Type I error) at, say, 0.10.

Type II Error Type II Error is made when we fail to reject the null when the alternative is true. When Type I error is made smaller, Type II error is made larger. When Type II error is made smaller, Type I error is made larger. Inverse relationship