1. Dept of Bioenvironmental Systems Engineering National Taiwan University Lab for Remote Sensing Hydrology and Spatial Modeling STATISTICS Hypotheses Test (II) One-Sample Tests on the Mean and Variance 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 Tests on Mean

3. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University (1) 2 is known

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

5. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University (2) 2 is unknown

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 (3) 2 is known

8. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University (4) 2 is unknown

9. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Tests on Variance

10. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University (1) is known

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

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

13. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University (2) is unknown

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

15. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University (3) is known

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

17. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University (4) is unknown

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

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 T-test in R

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

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 Paired t-test The paired t test provides an hypothesis test of the difference between population means for a pair of random samples whose differences are approximately normally distributed.

26. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University If the population from which paired differences to be analyzed by a paired t test were sampled violate one or more of the paired t test assumptions, the results of the analysis may be incorrect or misleading. For example, if the assumption of independence for the paired differences is violated, then the paired t test is simply not appropriate.

27. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University Note that the two values that make up each paired difference need not be independent, and in fact are expected to be correlated, such as before and after measurements. If you treat paired data as coming from two independent samples, such as doing an inappropriate two- sample unpaired t test instead of a paired t test, then you may sacrifice power.

28. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University If the assumption of normality is violated, or outliers are present, then the paired t test may not be the most powerful test available, and this could mean the difference between detecting a true difference or not. A nonparametric test or employing a transformation may result in a more powerful test. For example, if the distribution of the paired differences is not symmetric, a transformation may produce symmetry.

29. Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering National Taiwan University The test statistic is calculated as: Two-sided test One-sided test

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