1. Errors in Hypothesis Tests

2. When you perform a hypothesis test you make a decision: When you make one of these decisions, there is a possibility that you could be wrong! That you made an error! reject H 0 or fail to reject H 0

3. There are two decisions that we make; reject or fail to reject. Each could possibly be a wrong decision; therefore, there are two types of errors.

4. Type I error When you reject the null hypothesis that is really true Denoted by  –Is the level of significance of the test  

5. Fail to rejectReject Type II error When you fail to reject the null hypothesis when it is false Denoted by    Type I – reject a true H 0 Type II – fail to reject a false H 0 The x-bar is really part of the H a curve, but we mistake it as being part of the H 0 curve The x-bar is really part of the H 0 curve, but we mistake it as being part of the Ha curve

6. H 0 True H 0 False Reject Fail to reject Type I error Correct Type II error Suppose H 0 is true & we fail to reject it, what type of decision was made? Suppose H 0 is false & we reject it, what type of decision was made? Suppose H 0 is true & we reject it, what type of decision was made? Suppose H 0 is false & we fail to reject it, what type of decision was made?  

7. How do we word statements of type I & type II errors? “We decide this decision when in reality this is true.” You replace the red, underlined words with words from context!

8. Consequences – are NOT the definitions of type I & II errors. They are what happens as a result of making that incorrect decision.

9. Consider a murder trial: What are the hypotheses? Type I error – Consequence: Type II error – Consequence: Decide the defendant is guilty when really innocent An innocent person goes to prison Decide defendant is not guilty when really guilty A guilty person goes free H 0 : defendant is innocent H a : defendant is guilty Which of these errors does our society believe to be worse? Type I - that is why there must be evidence beyond a reasonable doubt! We don’t want to send innocent people to jail! What are H 0 : H a : What is a type I error? What is a type II error? What is a consequence of a type I error? What is a consequence of a type II error?

10. Facts: Every time you make a decision, you have potentially made an error.  &  are inversely related 00  aa  Fail to reject H 0 Reject H 0 As  decreases,  increases As  increases,  decreases

11. Facts continued: The seriousness of the error types is determined by the specific situations. –Depending upon the situation type I or type II may be the more serious. DO NOTWe often DO NOT know if an error is made in real life. –Except for cases like Firestone tires Drugs like: Phen-phen & Vioxx Someone made an error with these products

12. Lay’s Chip Company decides to accept a truckload of potatoes based upon results from a sample of potatoes from the truckload. What are the hypotheses? Type I error? Type II error? From the supplier’s viewpoint, which is more serious? From the chip company’s viewpoint, which is more serious? Decide the potatoes are bad when they really are good Decide the potatoes are good when they really are bad H 0 : potatoes good H a : potatoes bad A type I error A type II error Sometimes, the seriousness depends upon the person’s point-of-view

13. Most people would agree that the type II error would be more serious because it would endanger the river’s ecosystem. Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150 degrees F, there will be no negative effects on the river’s ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge above 150 degrees F, fifty water samples will be taken at randomly selected times, and the temperature of each sample recorded. What are the hypotheses? What are the Type I and II errors? Which is more serious? H 0 :  = 150 H a :  >150 Type I : Decide the temperature is above 150° when it’s really below. Type II:Decide the temperature isn’t above 150° when it’s really above.

14. A doctor is considering a new medication to help fight infections. However, the medication has the possibility of being highly toxic to the patient. You will test the medication to determine toxicity. What are the hypotheses? What are the Type I & II errors? Which is more serious? H 0 : medicine is not toxic H a : medicine is toxic Type I: decide medicine is toxic when it really isn’t Type II : decide medicine isn’t toxic when it really is Most would consider a type II error more serious since people could be harmed.

15. A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces). Identify the following decisions: We decide the mean weight of hamburgers is less than 4 ounces when in fact they really are less. Correct decision!

16. A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces). Identify the following decisions: We decide the mean weight of hamburgers is not less than 4 ounces when in fact they really are less. Type II error

17. How does one decide what a level to use? largest   After assessing the consequences of type I and type II errors, identify the largest  that is tolerable for the problem. Use that  level for your level of significance.

18. Power of a test

19. power The power of a test (against a specific alternative value) Is the probability that the test will reject the null hypothesis when the alternative is true In practice, we carry out the test in hope of showing that the null hypothesis is false, so high power is important

20. H 0 True H 0 False Reject Fail to reject Type I Correct Type II Power   Suppose H 0 is true – what if we decide to fail to reject it? Suppose H 0 is false – what if we decide to reject it? Suppose H 0 is true – what if we decide to reject it? Suppose H 0 is false – what if we decide to fail to reject it? We correctly reject a false H 0 !

21. A researcher selects a random sample of size 49 from a population with standard deviation  = 35 in order to test at the 1% significance level the hypothesis: H 0 :  = 680 H a :  > 680 What is the probability of committing a Type I error?  =.01

22. H 0 :  = 680 H a :  > 680 For what values of the sample mean would you reject the null hypothesis? 691.63

23. H 0 :  = 680 H a :  > 680 If H 0 is rejected, suppose that  a is 700. What is the probability of committing a Type II error? What is the power of the test? =.0471 Power = 1 -.0471 =.9529

24. H 0 :  = 680 H a :  > 680 If H 0 is rejected, suppose that  a is 695. What is the probability of committing a Type II error? What is the power of the test? Normalcdf(-10^99,691.63,695,5) =.2502 Power = 1 -.2502 =.7498

25. Reject H 0 Fail to Reject H 0 Power = 1 -  00  aa 

26. What happens to , , & power when the sample size is increased? Reject H 0 Fail to Reject H 0 00  aa 

27. Facts: The researcher is free to determine the value of . The experimenter cannot control , since it is dependent on the alternate value. The ideal situation is to have  as small as possible and power close to 1. (Power >.8)  powerAs  increases, power increases. (But also the chance of a type I error has increased!) sample sizeBest way to increase power, without increasing , is to increase the sample size

28. A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken from a well. Identify the decision: a) You decide that the water is not safe to drink when in fact it is safe. Type I Error

29. A water quality control board reports that water is unsafe for drinking if the mean nitrate concentration exceeds 30 ppm. Water specimens are taken from a well. Identify the decision: b) You decide that the water is not safe to drink when in fact it is not safe. Correct – Power!!

30. Bottles of a popular cola are suppose to contain 300 ml of cola. A consumer group believes the company is under-filling the bottles. (Assume  = 50 with n = 30) Find the power of this test against the alternative  = 296 ml.