# part7 - Public Health 6450 Fall 2011 Andrew Mugglin and...

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Public Health 6450 – Fall 2011 Andrew Mugglin and Lynn Eberly Division of Biostatistics School of Public Health University of Minnesota [email protected] Part 07

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Review Revisiting important concepts Point and Interval Estimates Where are we going? Previously: Binomial distribution: deﬁnition (Chapter 5.1) Sampling distribution of the Binomial mean (Chapter 5.1) Sampling distribution of any mean (Chapter 5.2) Current topics: Revisiting some important concepts Moving towards statistical inference: point and interval estimates (Chapter 6.1) Mugglin and Eberly PubH 6450 Fall 2011 Part 07 2 / 31
Review Revisiting important concepts Point and Interval Estimates Binomial mean and variance Recall: for S 1 , S 2 ,..., S n iid Bernoulli ( p ), X = n X i =1 S i ∼ B ( n , p ) with E [ X ] = μ x = np and Var [ X ] = σ 2 x = np (1 - p ). Remember that “ iid ” means “independent and identically distributed.” Mugglin and Eberly PubH 6450 Fall 2011 Part 07 3 / 31

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Review Revisiting important concepts Point and Interval Estimates Central Limit Theorem for the Binomial For large n , X approx N ± np , p np (1 - p ) ² ˆ p approx N p , r p (1 - p ) n ! When designing a study (in particular, ﬁguring out how big n should be), it is useful to know that the variance of a Binomial is largest for p 0 . 5 and smallest for p close to 0 or 1. If your study goal is to estimate p , will values of p close to 0.5 require a bigger n or a smaller n , compared to values of p near 0 or 1? Mugglin and Eberly PubH 6450 Fall 2011 Part 07 4 / 31
Review Revisiting important concepts Point and Interval Estimates CLT for the Binomial What is happening to the distribution of X as n gets bigger? [Source: www.marn.cc.ca.us/ npsomas] Mugglin and Eberly PubH 6450 Fall 2011 Part 07 5 / 31

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Review Revisiting important concepts Point and Interval Estimates Survey Respondants: Using the CLT for the Binomial You do a statewide telephone survey on public school levies of 1500 adults and ﬁnd only 170 black respondents. You are concerned that black people may be underrepresented in your sample. To check this, you decide to compare the sample with the 2000 US Census: 23,772,494 out of 209,128,094 adults called themselves “Black or African-American.” (Pretend the Census has complete coverage.) What is the probability of getting a sample of size 1500 with 170 or fewer blacks? Using the CLT, we know that X N ± , ² . Answers: μ x = 170 . 5115, σ x = 12 . 2934 (Why?) Mugglin and Eberly PubH 6450 Fall 2011 Part 07 6 / 31
Revisiting important concepts Point and Interval Estimates Survey Respondants (cont’d) We want to ﬁnd: Pr( X 170) = Pr ± X - μ x σ x 170 - μ x σ x ² = = Pr ( Z ≤ - 0 . 04) = 0 . 484 from SAS/R/Excel/Table Is it unusual to get only 170 black respondents in a sample of 1500 adults? Mugglin and Eberly

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## This note was uploaded on 11/21/2011 for the course PUBH 6450 taught by Professor Andymugglin during the Fall '10 term at Minnesota.

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part7 - Public Health 6450 Fall 2011 Andrew Mugglin and...

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