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STATISTICS INTERVAL ESTIMATION
Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University
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Estimating the mean Point estimation – Interval 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 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Distribution of sample means
Given a particular random sample, a=? n2 12/2/2018 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. 12/2/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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One-sided CI 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Example 12/2/2018 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 .
12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Remarks 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 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:
12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 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
12/2/2018 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
12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Remarks 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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skipped 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Definition of Location parameter
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Example of location parameter
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Definition of scale parameter
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Example of scale parameter
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Example 12/2/2018 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? 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Review of sampling distributions
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Normal distributions 12/2/2018
Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Chi-square distribution
12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 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
12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Student’s t-distribution
Student’s t distribution with k degrees of freedom 12/2/2018 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
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Confidence interval for the mean
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 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 . 12/2/2018 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.
12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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is a pivotal quantity and has a Chi-square distribution with d.o.f. n.
(ii) if is known is a pivotal quantity and has a Chi-square distribution with d.o.f. n. 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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12/2/2018 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
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has a t-distribution with d.o.f. (m+n-2)
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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confidence interval of is
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Some pivotal functions for samples of size n
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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., 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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Two-sided confidence interval of the population proportion p is given by
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12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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For large n, 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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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. 12/2/2018 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
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12/2/2018 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 when the population variances are different and unknown.
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Acceptance Intervals A normal distribution with expected value 60 and standard deviation 40. Find an interval (a, b) which satisfies Confidence interval of a parameter Parameter estimation Acceptance interval of a statistic Hypothesis test 12/2/2018 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.
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