1. The Language of Statistical Decision Making Lecture 2 Section 1.3 Mon, Sep 5, 2005

2. Statistical Significance Statistically significant – The difference between the claim of the null hypothesis and the data is large enough that it cannot reasonably be attributed to chance. Statistically significant – The difference between the claim of the null hypothesis and the data is large enough that it cannot reasonably be attributed to chance.

3. Possible Errors We might reject H 0 when it is true. We might reject H 0 when it is true. This is a Type I error. This is a Type I error. We might accept H 0 when it is false. We might accept H 0 when it is false. This is a Type II error. This is a Type II error. See Making Intelligent Errors, by Walter Williams. See Making Intelligent Errors, by Walter Williams.Making Intelligent ErrorsMaking Intelligent Errors

4. Decisions and Errors Correct Type I Error Correct Type II Error State of Nature H 0 trueH 0 false Accept H 0 Reject H 0 Decision

5. Example Consider the hypotheses: Consider the hypotheses: H 0 : A category 5 hurricane will not hit New Orleans in the next 25 years. H 0 : A category 5 hurricane will not hit New Orleans in the next 25 years. H 1 : A category 5 hurricane will hit New Orleans in the next 25 years. H 1 : A category 5 hurricane will hit New Orleans in the next 25 years. What are the negative consequences of each type of error? What are the negative consequences of each type of error? Which type of error is more serious? Which type of error is more serious? Which should get the benefit of the doubt? Which should get the benefit of the doubt?

6. Example We should probably reverse the roles: We should probably reverse the roles: H 0 : A category 5 hurricane will hit New Orleans in the next 25 years. H 0 : A category 5 hurricane will hit New Orleans in the next 25 years. H 1 : A category 5 hurricane will not hit New Orleans in the next 25 years. H 1 : A category 5 hurricane will not hit New Orleans in the next 25 years.

7. Example Consider the hypotheses: Consider the hypotheses: H 0 : An asteroid will not hit New Orleans in the next 25 years. H 0 : An asteroid will not hit New Orleans in the next 25 years. H 1 : An asteroid will hit New Orleans in the next 25 years. H 1 : An asteroid will hit New Orleans in the next 25 years. What are the negative consequences of each type of error? What are the negative consequences of each type of error? Which type of error is more serious? Which type of error is more serious? Which should get the benefit of the doubt? Which should get the benefit of the doubt?

8. Example These are fine the way they are: These are fine the way they are: H 0 : An asteroid will not hit New Orleans in the next 25 years. H 0 : An asteroid will not hit New Orleans in the next 25 years. H 1 : An asteroid will hit New Orleans in the next 25 years. H 1 : An asteroid will hit New Orleans in the next 25 years.

9. Significance Level Significance Level – The likelihood of rejecting H 0 when it is true, i.e., the likelihood of committing a Type I error. Significance Level – The likelihood of rejecting H 0 when it is true, i.e., the likelihood of committing a Type I error.  – The likelihood of a Type I error.  – The likelihood of a Type I error.  – The likelihood of a Type II error.  – The likelihood of a Type II error. That is,  is the significance level. That is,  is the significance level. The quantity 1 –  is called the power of the test. The quantity 1 –  is called the power of the test.

10. Significance Level The value of  is determined by our criteria for rejecting the null hypothesis (which we haven’t talked about yet). The value of  is determined by our criteria for rejecting the null hypothesis (which we haven’t talked about yet). If we demand overwhelming evidence against H 0 before rejecting it, then  will be small. If we demand overwhelming evidence against H 0 before rejecting it, then  will be small. If we demand little evidence against it, then  will be large. If we demand little evidence against it, then  will be large.

11. Significance Level Generally speaking, as  increases,  decreases. Generally speaking, as  increases,  decreases. That is, if we demand overwhelming evidence against H 0 (i.e., for H 1 ) before rejecting it, then  will be large. That is, if we demand overwhelming evidence against H 0 (i.e., for H 1 ) before rejecting it, then  will be large. If we demand little evidence against H 0, then  will be small. If we demand little evidence against H 0, then  will be small.

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