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

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Exponential distribution (negative exponential distribution) Mean rate of occurrence in a Poisson process. 12/6/2011 2 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Exponential distribution, =0.5 Sample size=10, 50 random samples 12/6/2011 3 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Exponential distribution, =0.5 Sample size=50, 50 random samples 12/6/2011 4 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Variation of sample means 6 random samples (sample size n= 50) of a normal density with unit variance and an unknown mean [N(, 1)]. 6 random samples (sample size n= 500) of a normal density with unit variance and the same unknown mean [N(, 1)]. Examine the variation of sample means. 12/6/2011 5 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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n=50 n=500 12/6/2011 6 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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? Interval estimation 12/6/2011 7 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Distribution of sample means a=? n1n1 n2n2 12/6/2011 8 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Confidence Interval is called the confidence coefficient. 12/6/2011 9 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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One-sided CI 12/6/2011 10 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Example 12/6/2011 11 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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constitutes a random interval and is a confidence interval for. 12/6/2011 12 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Remarks 12/6/2011 15 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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12/6/2011 19 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Let s recall the procedures of determining before drawing any random sample: 12/6/2011 20 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Methods of Finding Confidence Intervals - The Pivotal Quantity Method 12/6/2011 22 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Now, if for each possible sample value 12/6/2011 23 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Remarks 12/6/2011 24 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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skipped 12/6/2011 25 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Definition of Location parameter 12/6/2011 27 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Example of location parameter 12/6/2011 28 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Definition of scale parameter 12/6/2011 29 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Example of scale parameter 12/6/2011 30 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Example 12/6/2011 31 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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? 12/6/2011 35 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Review of sampling distributions 12/6/2011 36 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Normal distributions 12/6/2011 37 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Chi-square distribution 12/6/2011 41 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Standard normal and chi- square distributions 12/6/2011 44 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Student s t-distribution Students t distribution with k degrees of freedom 12/6/2011 46 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Back to discussion on confidence intervals 12/6/2011 47 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Confidence interval for the mean 12/6/2011 48 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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. 12/6/2011 55 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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(i) is unknown is a pivotal quantity and has a Chi-square distribution with d.o.f. n-1. 12/6/2011 56 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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(ii) if is known is a pivotal quantity and has a Chi-square distribution with d.o.f. n. 12/6/2011 59 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Confidence Interval for Difference in Means 12/6/2011 61 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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has a t-distribution with d.o.f. (m+n-2) 12/6/2011 64 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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confidence interval of is 12/6/2011 66 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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Some pivotal functions for samples of size n 12/6/2011 67 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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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., 12/6/2011 68 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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However, we cannot find since p is unknown. 12/6/2011 69 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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For large n, 12/6/2011 70 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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