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?
5/15/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

4. Null and alternative hypotheses
Two hypotheses Ho and H1 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”.
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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

5. Simple and Composite Hypotheses
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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
5/15/2018
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 So and S1. Subset So contains the values of X for which we will accept , and subset S1 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.
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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

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

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

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

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

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

30. Power function of the test
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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 5/15/2018
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.
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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

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Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

55. Guidelines for Hypothesis Testing
When testing a hypothesis concerning the value of some parameter , the statement of equality will always be included in H0. In this way H0 pinpoints a specific numerical value that could be the actual value of . This value is called the null value and is denoted by 0.
Whatever is to be detected or supported is the alternative hypothesis.
It is hoped that the evidence leads us to reject H0 and thereby to accept H1.
5/15/2018
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 H0, then the parameter from H0 is definitely within the confidence interval.
5/15/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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

58. Using the acceptance interval for hypothesis test
One-sided acceptance interval One-sided confidence interval
Using the acceptance interval for hypothesis test
H1 is true
Ho is true
There exists a dual relationship between a hypothesis test and its corresponding confidence interval estimation.
Test statistic
Critical region
The highest probability of committing a type-one error.
5/15/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.