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part9 - Public Health 6450 – Fall 2011 Andrew Mugglin and...

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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 09 Review Revisiting Important Concepts Hypothesis Testing Where are we going? Previously: • Interval estimates for μ when σ is known (use z * and σ ) • Interval estimates for μ when σ is unknown (use t * and s ) Current topics: • Revisiting important concepts • Introduction to hypothesis testing: Ch.6.2 • Hypothesis tests for μ when σ is known: Ch.6.2 (use z and σ ) • Hypothesis tests for μ when σ is unknown: Ch.7.1 (use t and s ) (Interval estimates and hypothesis testing for the Binomial will be covered in Ch. 8.) Mugglin and Eberly PubH 6450 Fall 2011 Part 09 2 / 53 Review Revisiting Important Concepts Hypothesis Testing Standardizing to a normal distribution • We compute probabilities that X (or ¯ X ) takes a range of values by first standardizing to a standard normal Z . • The probability that X < a is Pr ( X < a ) = Pr X- μ σ < a- μ σ = Pr Z < a- μ σ . • When we have an i . i . d . sample of X s, then Pr ( ¯ X < a ) = Pr ¯ X- μ σ/ √ n < a- μ σ/ √ n = Pr Z < a- μ σ/ √ n . Mugglin and Eberly PubH 6450 Fall 2011 Part 09 3 / 53 Review Revisiting Important Concepts Hypothesis Testing Computing CIs for the mean • When the population variance σ 2 is known, the 100 C % confidence interval for μ is given by: ¯ X- z * σ √ n , ¯ X + z * σ √ n where Pr(- z * ≤ Z ≤ z * ) = C . • When the population variance σ 2 is unknown, the 100 C % confidence interval for μ is given by: ¯ X- t * s √ n , ¯ X + t * s √ n where Pr(- t * ≤ t n- 1 ≤ t * ) = C . • For 95% CI, z * = 1 . 96 and t * depends on n- 1. Mugglin and Eberly PubH 6450 Fall 2011 Part 09 4 / 53 Review Revisiting Important Concepts Hypothesis Testing Back to arrow (3) Mugglin and Eberly PubH 6450 Fall 2011 Part 09 5 / 53 Review Revisiting Important Concepts Hypothesis Testing True State of Nature Parameters: μ , σ 2 , β , ρ , θ, p, … Observations Sample Statistics: Mugglin and Eberly PubH 6450 Fall 2011 Part 09 6 / 53 Review Revisiting Important Concepts Hypothesis Testing Statistical inference • Statistical inference is the process of drawing some conclusion about a population parameter using the information contained in a sample. • We can use a CI to make a statement about the population parameter. • Example: In Part 7, we calculated the 90% confidence interval for mean birthweight among female alcohol-exposed infants as (2.35,2.65). We can say that “We are 90% confident that the mean birthweight for female alcohol-exposed infants lies between 2.35 kg and 2.65 kg.” • A CI is one tool for making statistical inference....
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part9 - Public Health 6450 – Fall 2011 Andrew Mugglin and...

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