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Unformatted text preview: STAT503 Spring 2009 Lecture Notes: Chapter 6 1 Chapter 6: Confidence Intervals February 7, 2009 6.1 Statistical Estimation Our first venture into statistical inference. Our goals are to 1. give an estimate of a feature of the population; and, 2. assess the precision of the estimate . Consider taking a sample of size n from a large population of seeds of the princess bean Phaseotus vulgaris and recording the seed weights. is the population mean for all seeds in this population is the population SD for all seeds in the population. We usually do not know these so we estimate them : The sample mean y estimates . The sample SD s estimates . If the seed weights follow a normal distribution then we know that Y Normal( ,/ n ). Our goal is to construct an interval around y such that we are, say, 95% confident it contains the true . This is called a 95% confidence interval (C.I.). 6.2 Standard Error of the Mean If Y Normal( , ) then Y Normal( ,/ n ). A reasonable estimate for Y = / n is s/ n . This estimate is called the standard error of the mean and is denoted SE y or just SE: SE y = s n . Example: Suppose we took a sample of size 20 from the population of princess beans above and found y = 450mg and s = 100mg. The standard error of y would be SE y = 100 20 = 22 . 36068 . Chapter6.tex; Last Modified: February 7, 2009 (W. Sharabati) STAT503 Spring 2009 Lecture Notes: Chapter 6 2 Comments: The standard error can be interpreted as typical error we make when using the sample mean as an estimate of the population mean. Roughly speaking, the difference between and y is rarely more than a few standard errors . Indeed, we expect y to be within 1 standard error of quite often. Do not confuse the sample standard deviation SD, which is s , with the standard error SE: SD describes the dispersion in the sample data. As n gets large, s . SE describes the dispersion in y . As n gets large, SE . 6.3 Confidence Interval for The Idea of Confidence Interval Imagine an invisible man walking a visible dog on an invisible leash . The leash is spring loaded so that the dog is within 1 SE from the man about twothirds of the time, and within 2 SEs from the man about 95% of the time. Therefore, the interval: dog 2SE would contain the man most of the time. We can observe y (the dog) and, together with the SE, we construct an interval that we hope contains what we cannot see , the population mean (the man). The Math In general, we want to construct a (1 )100% confidence interval for ....
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