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Unformatted text preview: Public Health 6450 – Fall 2011 Andrew Mugglin and Lynn Eberly Division of Biostatistics School of Public Health University of Minnesota [email protected] Part 08 Review Revisiting Important Concepts When σ Is Unknown Where are we going? Previously: • Using the Central Limit Theorem (Chapter 5.2) • Interval estimates for μ when σ is known (Chapter 6.1) Current topics: • Revisiting some important concepts • Interval estimates for μ when σ is unknown (Chapter 7.1) (Interval estimates for the Binomial will be covered in Ch. 8.) Mugglin and Eberly PubH 6450 Fall 2011 Part 08 2 / 29 Review Revisiting Important Concepts When σ Is Unknown Point estimates vs. interval estimates • For a population parameter of interest (e.g., μ ), a point estimate (e.g., ¯ X ) is a statistic calculated from an observed random sample. • The best way to estimate a parameter can often be chosen based on the estimate’s properties, such as unbiasedness , consistency (Law of Large Numbers), etc. This is why we use ¯ X to estimate μ , ˆ p to estimate p , and s 2 to estimate σ 2 . • A point estimate is a random variable, because it is computed from a random sample. Therefore we need to describe its distribution (e.g., its variability) to convey the uncertainty in the point estimate. • This variability is often quantified via the confidence interval . Mugglin and Eberly PubH 6450 Fall 2011 Part 08 3 / 29 Review Revisiting Important Concepts When σ Is Unknown CI for population mean Definition A 100 C % confidence interval for the mean μ of a normal population with known variance σ 2 , based on a random sample of size n , is ¯ X ± z * σ √ n , where z * is the value of the standard normal variable such that Pr( z * ≤ Z ≤ z * ) = C . This formula can also be used when n is large enough that the CLT applies. For a 90% CI, use z * = 1 . 645. For a 95% CI, use z * = 1 . 96. For a 99% CI, use z * = 2 . 576. Mugglin and Eberly PubH 6450 Fall 2011 Part 08 4 / 29 Review Revisiting Important Concepts When σ Is Unknown Things to remember • Correct interpretation: if we take many samples from the same population and use this method to construct a 95% CI based on each sample, then on average , 95% of the CIs will include the (true) population parameter. • CIs demonstrate the uncertainty in our point estimate ¯ X . • The confidence level itself (e.g., 95%) may or may not be correct, depending on whether any assumptions were violated. Violation of CLT assumptions (e.g., too small n for nonnormal data) or a biased sample or correlated (nonindependent) observations could lead to CIs with incorrect coverage for your parameter of interest. • CAUTION: Many research papers will report summary statistics as estimate ± st.dev. , which is not the same as a CI computed from estimate ± margin of error ....
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 Fall '10
 AndyMugglin
 Normal Distribution, Standard Deviation, Variance, Eberly

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