confsum - Statistics Confidence intervals summary sheet...

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Unformatted text preview: Statistics: Confidence intervals summary sheet Introduction • Suppose that we take a SRS of size n from a population with mean μ and variance σ 2 . We know that the probability distribution of the sample mean ¯ X is at least approximately ¯ X ∼ N ( μ,σ 2 /n ) for sufficiently large n . Thus we can make statements of probability about where ¯ X is likely to be. How does this help us decide where μ is likely to be? • The standardised version of the sample mean is ¯ X- μ σ/ √ n ∼ N (0 , 1). Thus we can determine from tables that P ( ¯ X- μ σ √ n ≤ 1 . 96) = 0 . 975 , P (- 1 . 96 < ¯ X- μ σ √ n ≤ 1 . 96) = 0 . 95 . • This probability statement can be rearranged into P ( ¯ X- 1 . 96 σ √ n < μ ≤ ¯ X + 1 . 96 σ √ n ) = 0 . 95 , so that μ lies in the interval ( ¯ X- 1 . 96 σ √ n , ¯ X +1 . 96 σ √ n ) with probability 95%. This probability statement is about μ rather than ¯ X . However it is useless as a tool for inference because ¯ X here is a random variable – remember: probability calculations concern random variables, and we cannot replace ¯ X by one of its observed values here and still call it a probability statement. • Suppose that we do replace ¯ X by ¯ x , its observed value calculated from data. What is the status of the interval (¯ x- 1 . 96 σ √ n , ¯ x + 1 . 96 σ √ n )? It is called a confidence interval for μ . This confidence interval is at 95% because the theoretical probability statement from which the confidence interval evolved had probability 0.95....
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confsum - Statistics Confidence intervals summary sheet...

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