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# Lecture10standard - Statistics 511 Statistical Methods Dr...

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 12: Confidence Intervals Devore: Section 7.1-7.2 March, 2011 Page 1

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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

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Because the area under the normal curve between - 1 . 96 and 1 . 96 is 0 . 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

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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 Dr. Levine Purdue University Spring 2011

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Lecture10standard - Statistics 511 Statistical Methods Dr...

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