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Minimizing Chance of Type I and Type II Errors

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

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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?

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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.

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Errors in Hypothesis Testing

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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!

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Example: Grade inflation? H 0 : μ = 2.7 H A : μ > 2.7 Random sample of students n = 36 s = 0.6 and Data Decision Rule Set significance level α = If p-value < 0.05, reject null hypothesis.

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

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

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

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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 > is called “rejection region.”

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Type I Error

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

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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.

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Example: Serious Type I Error New Drug A is supposed to reduce blood pressure by more than 15 mm Hg. H 0 : μ = 15 versus H A : 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

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Example: Not so serious Type I Error Grade inflation? H 0 : μ = 2.7 vs. H A : 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.

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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

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