2. Goals… Define what is meant by a Type I error. Define what is meant by a Type II error. Define what is meant by the power of a test. Identify the relationship between the power of a test and a Type II error. List four ways to increase the power of a test.

3. Errors in Statistical Inferene

4. Type I Error We make a “Type I Error” when we incorrectly reject H0. We make a “Type II Error” when we incorrectly fail to reject Ho.

5. TRUTH ABOUT H0 H0 is trueHo is False (Ha is true) Decision about H0 Reject H0. TYPE I ERROR Correct Decision Fail to reject H0 Correct Decision TYPE II ERROR

6. TRUTH ABOUT H0 H0 is true The water contains copper levels greater than 1.3 mg/L (UNSAFE) Ho is False (Ha is true) The water contains copper levels less than 1.3 mg/L (SAFE) Decision about H0 Reject H0 State the water is safe. TYPE I ERROR Correct Decision Fail to reject H0 Report that the water is unsafe. Correct Decision TYPE II ERROR

7. TRUTH ABOUT H0 H0 is true The water contains copper levels greater than 1.3 mg/L (UNSAFE) Ho is False (Ha is true) The water contains copper levels less than 1.3 mg/L (SAFE) Decision about H0 Reject H0 State the water is safe. TYPE I ERROR Correct Decision Inform the EPA that the water is safe to drink when it is, in fact, safe. Fail to reject H0 Report that the water is unsafe. Correct Decision Inform the EPA that the water is NOT safe to drink when it is, in fact, not safe. TYPE II ERROR

8. TRUTH ABOUT H0 H0 is true The water contains copper levels greater than 1.3 mg/L (UNSAFE) Ho is False (Ha is true) The water contains copper levels less than 1.3 mg/L (SAFE) Decision about H0 Reject H0 State the water is safe. TYPE I ERROR Inform the EPA that the water is safe to drink when it is actually UNSAFE! Correct Decision Inform the EPA that the water is safe to drink when it is, in fact, safe. Fail to reject H0 Report that the water is unsafe. Correct Decision Inform the EPA that the water is NOT safe to drink when it is, in fact, not safe. TYPE II ERROR Inform the EPA that the water is not safe to drink even though it really WAS safe.

9. TYPE I ERROR Inform the EPA that the water is safe to drink when it is actually UNSAFE! TYPE II ERROR Inform the EPA that the water is not safe to drink even though it really WAS safe. Consequence? People drink unsafe water and could get sick  People are unable to use this valuable water source and the government imposes water restrictions 

10. Which error do you believe is more serious? Why? If you had to choose and alpha level of α = 0.1, 0.05, or 0.01 which would you choose? Why? TYPE I ERROR Inform the EPA that the water is safe to drink when it is actually UNSAFE! TYPE II ERROR Inform the EPA that the water is not safe to drink even though it really WAS safe. People drink unsafe water and could get sick  People are unable to use this valuable water source and the government imposes water restrictions 

11. Mel Mel N. Colly is interested in whether or not his new treatment for depressed patients is decreasing his patients’ rating of depression. Suppose all of his depressed patients have a mean depression score of 8 with a standard deviation of 4. Mel chooses a random sample of 30 depressed patients treated with his innovative approach and determines that the mean depression score for these individuals is 7.5. Does the treatment decrease depression? H 0 : μ = 8 H a : μ < 8

12. Mel H 0 : μ = 8 H a : μ < 8 Describe and give the consequences of a Type I error and a Type II error. TYPE I ERROR Description: Mel concludes that the mean depression score post-treatment is less than 8 even through it this is not the case. Consequence: Mel uses the treatment on depressed patients even though it has no effect on them (or it could increase depression!) rather than researching a better treatment that could really help. Patients waste time and money on a useless (or harmful) treatment.

13. Mel H 0 : μ = 8 H a : μ < 8 Describe and give the consequences of a Type I error and a Type II error. TYPE II ERROR Description: Mel concludes that the mean depression score post-treatment is still 8 (or higher), but in reality post-treatment scores are less than 8. Consequence: Mel does not use the effective treatment on his patients and these individual miss out on the opportunity to decrease their level of depression and improve their lives.

14. Mel Which error do you believe is more serious? Why? If you had to choose and alpha level of α = 0.1, 0.05, or 0.01 which would you choose? Why? TYPE I ERROR Description: Mel concludes that the mean depression score post-treatment is less than 8 even through it this is not the case. Consequence: Mel uses the treatment on depressed patients even though it has no effect on them rather than researching a better treatment that could really help. Patients waste time and money on a useless (or harmful) treatment. TYPE II ERROR Description: Mel concludes that the mean depression score post-treatment is still 8 (or higher), but in reality post- treatment scores are less than 8. Consequence: Mel does not use the effective treatment on his patients and these individual miss out on the opportunity to decrease their level of depression and improve their lives.

15. Error Probabilities H0 is trueHa is true Reject H0 α (0.05) TYPE I ERROR Correct Decision Fail to reject H0 Correct Decision TYPE II ERROR We decide to use α = 0.05 What does that mean? The probability of a Type I error = α

16. Error Probabilities H0 is trueHa is true Reject H0 α TYPE I ERROR Correct Decision Fail to reject H0 Correct Decision β TYPE II ERROR The probability of a Type II error is β

17. Error Probabilities H0 is true Ha is true Reject H0 α TYPE I ERROR 1-β Correct Decision Fail to reject H0 Correct Decision (1-α) β TYPE II ERROR Which box do we WANT to fall in?

18. Error Probabilities H0 is true Ha is true Reject H0 α TYPE I ERROR 1-β Correct Decision Fail to reject H0 Correct Decision (1-α) β TYPE II ERROR 1-β is called the POWER of the test

19. Error Probabilities μ = 8 (H0 is true) μ < 8 (Ha is true) Reject H0 α =0.05 TYPE I ERROR Power = 1-β = 0.166 Correct Decision Fail to reject H0 Correct Decision (1-α) β = 0.834 TYPE II ERROR Mel wants to use an alpha level of 0.05. Find the probability of a Type I error The probability of a Type II error is 0.834. What is the Power of the test?

21. Power: You do NOT need to calculate β or Power by hand (except in the way we just did in the chart). You just need to understand the concept. Power is the probability that you will CORRECTLY reject the null hypothesis. A perfect study would have VERY high power There are a few ways to increase power… http://bcs.whfreeman.com/tps3e/

22. HOMEWORK!!! Classwork: Error and Power Worksheet Homework: 11.59, 11.62, 11.64, 11.69 (full test)