1. Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Hypotheses Test (I) Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Examples of hypothesis tests Based on historical records, does female students really perform better in statistics class than male students? Data from a simulated AR(2) process.

3. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Do you see an increasing/decreasing trend?

4. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University The trends are apparent.

5. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

6. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

7. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University An AR(2) process Stationary process

8. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University An AR(2) process Stationary process

9. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University What is hypothesis test

10. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Null and alternative hypotheses Two hypotheses and are defined as: (Null hypothesis) (Alternative hypothesis) A procedure for deciding whether to accept (or more precisely, fail to reject) the hypothesis or to accept the hypothesis (or reject ) is called a test procedure or simply a test.

11. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Simple and Composite Hypotheses

12. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Guidelines for Hypothesis Testing 1. When testing a hypothesis concerning the value of some parameter, the statement of equality will always be included in H 0. In this way H 0 pinpoints a specific numerical value that could be the actual value of. This value is called the null value and is denoted by 0. 2. Whatever is to be detected or supported is the alternative hypothesis. 3. It is hoped that the evidence leads us to reject H 0 and thereby to accept H 1.

13. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University The Critical Region and Test Statistics Consider the following hypotheses test: Suppose that we are given a random sample of size n,, from a distribution with parameter. Let S denote the sample space of the n- dimensional random vector.

14. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University In order to carry out the test we can partition the sample space S into two disjoint subsets S o and S 1. Subset S o contains the values of X for which we will accept, and subset S 1 contains the values of X for which we will reject. The subset for which will be rejected is called the critical region of the test.

15. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Why is it fixed?

16. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Critical region – fixed due to specification of H o, distribution of the test statistic, and the level of significance.

17. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University The Power Function Either the critical region or the fixed interval for the test statistic of a test is independent of the test parameter. However, the value of is unknown, therefore the probability that a test will reject is a function of, and is denoted by, i.e.,

18. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University for where C is the critical region of. The function is called the power function of the test.

19. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

20. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

21. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

22. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

23. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Types of error

24. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

25. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

26. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

27. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

28. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

29. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Making a test have a specific significance level

30. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

31. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

32. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Example

33. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

34. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

35. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

36. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

37. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

38. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Power function of the test C=6

39. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Power functions C=6 C=7 C=8

40. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Now, let s set the size of the random sample n = 20 and conduct the same test. Let.

41. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

42. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

43. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

44. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

45. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

46. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

47. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Example Suppose that is a random sample of size n and we wish to test the hypotheses:

48. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

49. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

50. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

51. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

52. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

53. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

54. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

55. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

56. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

57. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

58. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

59. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Noncentral t distribution in R

60. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

61. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University

62. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Non-centrality parameter ncp=0 ncp=1,-1 ncp=2,-2 ncp=3,-3