1. STATISTICS HYPOTHESES TEST (I) Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Examples of hypothesis tests Based on historical records, do female students really perform better in statistics class than male students? 9/3/2015 2 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

3. What is hypothesis test 9/3/2015 3 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

4. 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 ”. 9/3/2015 4 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

5. Simple and Composite Hypotheses 9/3/2015 5 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6. 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. 9/3/2015 6 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

7. 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. 9/3/2015 7 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

8. Why is it fixed? 9/3/2015 8 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

9. Critical region – fixed due to specification of H o, distribution of the test statistic, and the level of significance. 9/3/2015 9 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

10. The Power Function 9/3/2015 10 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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17. Types of error 9/3/2015 17 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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21. Making a test have a specific significance level 9/3/2015 21 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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24. Example 9/3/2015 24 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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26. 9/3/2015 26 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. Why? The probability density function of the test statistic is known.

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30. Power function of the test C=6 9/3/2015 30 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

31. Power functions C=6 C=7 C=8 9/3/2015 31 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

32. Now, let ’ s set the size of the random sample n = 20 and conduct the same test. Let. 9/3/2015 32 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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39. Example Suppose that is a random sample of size n and we wish to test the hypotheses: 9/3/2015 39 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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51. Noncentral t distribution in R 9/3/2015 51 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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54. Non-centrality parameter ncp=0 ncp=1,-1 ncp=2,-2 ncp=3,-3 9/3/2015 54 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

55. 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. 9/3/2015 55 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

56. A confidence interval is just the flip side of a hypothesis test. If the hypothesis test fails to reject H 0, then the parameter from H 0 is definitely within the confidence interval. 9/3/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 56

57. Review of the confidence interval & acceptance interval 9/3/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 57

58. Using the acceptance interval for hypothesis test 9/3/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 58 One-sided acceptance interval One-sided confidence interval Critical region The highest probability of committing a type-one error. H o is true H 1 is true Test statistic There exists a dual relationship between a hypothesis test and its corresponding confidence interval estimation.