Chapter 7.1-7.2

Chapter 7.1-7.2 - Statistics 511: Statistical Methods Dr....

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Unformatted text preview: Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 12: Confidence Intervals Devore: Section 7.1-7.2 March, 2011 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Motivation Why do we need a confidence interval? Because with each new sample we have a new parameter estimate (e.g. new sample mean).... Which one do we choose? We do not know the true mean and do not know how close each one is to . Thus, we want to have some degree of precision reported together with an estimate Suppose our X = 10 . We want to say something like...With probability 95% the true mean is between 9 and 11 March, 2011 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Basic properties of Confidence Intervals Consider normal population distribution with known We want to estimate unknown The problem is purely illustrative; in practice, mean is usually known before the variance (standard deviation) We know that X is normally distributed with mean and standard deviation / n . March, 2011 Page 3 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Because the area under the normal curve between- 1 . 96 and 1 . 96 is . 95 , we have P (- 1 . 96 Z 1 . 96) = P- 1 . 96 X- / n 1 . 96 = 0 . 95 Simple algebra tells us that P X- 1 . 96 n < < X + 1 . 96 n = 0 . 95 March, 2011 Page 4 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 The meaning of the confidence interval The event in parentheses above is a random interval with the left endpoint X- 1 . 96 n and right endpoint X + 1 . 96 n . It is centered at sample mean X . For a given sample X 1 = x 1 ,...,X n = x n , we compute the observed sample mean x and substitute it in the definition of our random interval instead of X . The resulting fixed interval is called 95% confidence interval (CI). The usual way to express it is either to say that x- 1 . 96 n , x + 1 . 96 n is a 95% CI for March, 2011 Page 5 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Alternatively, we say that x- 1 . 96 n x + 1 . 96 n with 95% A more concise expression is x 1 . 96 n March, 2011 Page 6 Statistics 511: Statistical Methods...
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Chapter 7.1-7.2 - Statistics 511: Statistical Methods Dr....

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