Statistics_TestingHypothesis presentation
- 1. S TAT I S T I C A L M E T H O D S I N E X P E R I M E N TA L D E S I G N C O U R S E
- 2. C O N T E N T S O F T H E W E E K • Testing Hypotheses
- 3. I N T R O D U C T O R Y E X A M P L E • A manufacturer of emergency equipment asserts that a respirator that it makes delivers pure air for 75 minutes on average. A government regulatory agency is charged with testing such claims, in this case to verify that the average time is not less than 75 minutes. To do so it would select a random sample of respirators, compute the mean time that they deliver pure air, and compare that mean to the asserted time 75 minutes.
- 4. T E S T I N G H Y P O T H E S E S In the sampling that we have studied so far, the goal has been to estimate a population parameter. However, the sampling done by the government agency has a somewhat different objective, not so much to estimate the population mean μ as to test an assertion—or a hypothesis—about it, namely, whether it is as large as 75 or not. The agency is not necessarily interested in the actual value of μ, just whether it is as claimed. Their sampling is done to perform a test of hypotheses, the subject of this chapter.
- 5. T E S T I N G H Y P O T H E S E S 8.1: The Elements of Hypothesis Testing 8.2: Large Sample Tests for a Population Mean 8.3: The Observed Significance of a Test 8.4: Small Sample Tests for a Population Mean 8.5: Large Sample Tests for a Population Proportion
- 6. L E A R N I N G O B J E C T I V E S To understand the logical framework of tests of hypotheses. To learn basic terminology connected with hypothesis testing To learn fundamental facts about hypothesis testing
- 7. 8 . 1 E L E M E N T S O F H Y P O T H E S I S T E S T I N G Types of hypotheses A hypothesis about the value of a population parameter is an assertion about its value. As in the introductory example we will be concerned with testing the truth of two competing hypotheses, only one of which can be true.
- 8. T Y P E S O F H Y P O T H E S E S
- 9. H Y P O T H E S I S T E S T I N G
- 10. H Y P O T H E S I S T E S T I N G
- 14. Thus in order to make the null and alternative hypotheses easy for the student to distinguish, in every example and problem in this text we will always present one of the two competing claims about the value of a parameter with an equality. The claim expressed with equality is the null hypothesis. This is the same as always stating the null hypothesis in the least favorable light. So in the introductory example about the respirators, we stated the manufacturer’s claim as “the average is 75 minutes” instead of the perhaps more natural “the average is at least 75 minutes,” essentially reducing the presentation of the null hypothesis to its worst case. The first step in hypothesis testing is to identify the null and alternative hypotheses.
- 15. T H E L O G I C O F H Y P O T H E S I S T E S T I N G
- 16. T H E L O G I C O F H Y P O T H E S I S T E S T I N G
- 17. T H E L O G I C O F H Y P O T H E S I S T E S T I N G
- 18. T H E R E J E C T I O N R E G I O N
- 19. T H E R E J E C T I O N R E G I O N
- 20. T H E R E J E C T I O N R E G I O N
- 21. E X A M P L E
- 22. E X A M P L E
- 23. T H E D E C I S I O N P R O C E D U R E
- 24. T W O T Y P E S O F E R R O R S
- 25. T W O T Y P E S O F E R R O R S
- 26. S TA N D A R D I Z I N G T H E T E S T S TAT I S T I C
- 27. W H E N T H E T E S T S T A T I S T I C H A S S T U D E N T ’ S t - D I S T R I B U T I O N W H E N T H E T E S T S T A T I S T I C H A S T H E S T A N D A R D N O R M A L D I S T R I B U T I O N
- 28. S Y S T E M A T I C H Y P O T H E S I S T E S T I N G P R O C E D U R E : C R I T I C A L V A L U E A P P R O A C H
- 29. 8 . 2 L A R G E S A M P L E T E S T S F O R A P O P U L A T I O N M E A N Z =
- 30. S T A N D A R D I Z E D T E S T S T A T I S T I C S F O R L A R G E S A M P L E H Y P O T H E S I S T E S T S C O N C E R N I N G A S I N G L E P O P U L A T I O N M E A N
- 31. S T A N D A R D I Z E D T E S T S T A T I S T I C S F O R L A R G E S A M P L E H Y P O T H E S I S T E S T S C O N C E R N I N G A S I N G L E P O P U L A T I O N M E A N
- 32. E X A M P L E
- 33. Figure : Rejection Region and Test Statistic for "Example 8.2
- 34. K E Y TA K E A W AY
- 35. 8 . 3 T H E O B S E R V E D S I G N I F I C A N C E O F A T E S T
- 37. C O M P U T A T I O N A L D E F I N I T I O N O F T H E O B S E R V E D S I G N I F I C A N C E O F A T E S T O F H Y P O T H E S E S
- 38. T H E P - V A L U E A P P R O A C H T O H Y P O T H E S I S T E S T I N G P R O C E D U R E :
- 39. E X A M P L E
- 40. Test Statistic for "Example 8.3.2"
- 41. F I G U R E 7 . 1 . 5
- 42. K E Y TA K E A W AY
- 43. 8 . 4 S M A L L S A M P L E T E S T S F O R A P O P U L A T I O N M E A N
- 44. S T A N D A R D I Z E D T E S T S T A T I S T I C S F O R S M A L L S A M P L E H Y P O T H E S I S T E S T S C O N C E R N I N G A S I N G L E P O P U L A T I O N
- 45. S T A N D A R D I Z E D T E S T S T A T I S T I C S F O R S M A L L S A M P L E H Y P O T H E S I S T E S T S
- 46. E X A M P L E
- 47. S O L U T I O N
- 48. S O L U T I O N
- 50. K E Y TA K E A W AY