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Unformatted text preview: Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 12: Confidence Intervals Devore: Section 7.17.2 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 μ and do not know how close each one is to μ . • 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 σ • We want to estimate unknown μ • 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 μ and standard deviation σ/ √ 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 μ σ/ √ n ≤ 1 . 96 = 0 . 95 • Simple algebra tells us that P ¯ X 1 . 96 σ √ n < μ < ¯ X + 1 . 96 σ √ 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 σ √ n and right endpoint ¯ X + 1 . 96 σ √ 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 σ √ n , ¯ x + 1 . 96 σ √ n is a 95% CI for μ March, 2011 Page 5 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 • Alternatively, we say that ¯ x 1 . 96 σ √ n ≤ μ ≤ ¯ x + 1 . 96 σ √ n with 95% • A more concise expression is ¯ x ± 1 . 96 σ √ n March, 2011 Page 6 Statistics 511: Statistical Methods...
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 Fall '08
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