1. STATISTICS INTERVAL ESTIMATION Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Estimating the mean Point estimation – – Increasing the sample size reduces the standard deviation of our estimate. – If the sample size is given (we are given a random sample), how do we see (interpret) our sample estimate? How do we (or in what sense) judge whether our estimate is close to the population mean? [uncertainties] Interval estimation 10/31/2015 2 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

3. Distribution of sample means Given a particular random sample, a=? n1n1 n2n2 10/31/2015 3 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

4. Confidence Interval  is called the confidence coefficient. 10/31/2015 4 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

5. One-sided CI 10/31/2015 5 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

6. Example 10/31/2015 6 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

7. constitutes a random interval and is a confidence interval for. 10/31/2015 7 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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10. Remarks 10/31/2015 10 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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15. Let ’ s recall the procedures of determining before drawing any random sample: 10/31/2015 15 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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17. Methods of Finding Confidence Intervals - The Pivotal Quantity Method 10/31/2015 17 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

18. Now, if for each possible sample value 10/31/2015 18 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

19. Remarks 10/31/2015 19 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

20. skipped 10/31/2015 20 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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22. Definition of Location parameter 10/31/2015 22 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

23. Example of location parameter 10/31/2015 23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

24. Definition of scale parameter 10/31/2015 24 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

25. Example of scale parameter 10/31/2015 25 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

26. Example 10/31/2015 26 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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30. In the above example, the confidence coefficient is determined after the random sample is obtained. It is not pre-determined (out of our control). What if we want to find the confidence interval of Θ with a pre-determined confidence coefficient? 10/31/2015 30 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

31. Review of sampling distributions 10/31/2015 31 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

32. Normal distributions 10/31/2015 32 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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36. Chi-square distribution 10/31/2015 36 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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38. Standard normal and chi-square distributions 10/31/2015 38 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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40. Student ’ s t-distribution Student’s t distribution with k degrees of freedom 10/31/2015 40 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

41. Back to discussion on confidence intervals 10/31/2015 41 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

42. Confidence interval for the mean 10/31/2015 42 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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49. Confidence interval for the variance Given a random sample from a normal distribution with mean and variance. We want to determine the confidence interval of. 10/31/2015 49 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

50. (i) is unknown is a pivotal quantity and has a Chi-square distribution with d.o.f. n-1. 10/31/2015 50 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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53. (ii) if is known is a pivotal quantity and has a Chi-square distribution with d.o.f. n. 10/31/2015 53 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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55. Confidence Interval for Difference in Means 10/31/2015 55 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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58. has a t-distribution with d.o.f. (m+n-2) 10/31/2015 58 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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60. confidence interval of is 10/31/2015 60 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

61. Some pivotal functions for samples of size n 10/31/2015 61 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

62. Confidence interval for a population proportion, p Let X be a random variable with binomial density Binom(n, p). A random number of X, say x, is given, and we want to find a 95% confidence interval for p. As n approaches infinity, X can be approximated by a normal distribution with mean np and variance npq, i.e., 10/31/2015 62 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

63. Two-sided confidence interval of the population proportion p is given by 10/31/2015 63 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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65. For large n, 10/31/2015 65 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

66. Determining required sample size for a test with desired margin of errors In estimation of the population proportion, if we like the margin of error to be no greater than , how large should the sample size be? The margin of error is. However, we do not know in advance. 10/31/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 66

67. Confidence interval for difference in means 10/31/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 67

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69. 10/31/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 69 Confidence interval for difference in means when the population variances are different and unknown.