Chapter 7 Estimates and Sample Sizes Example: Public opinion polls provide a common source of data that are analyzed as proportions.

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1 Chater 7 Estimates ad Samle Sizes Examle: Public oiio olls rovide a commo source of data that are aalyzed as roortios. 1

2 The true oulatio roortio of adults atiowide who believe that Obama s eergy olicies will brig dow gas rices, desigated as, is estimated by 70 ˆ How reliable is the estimator ˆ? Ca we characterize the reliability (or ucertaity)? The true oulatio roortio of adults atiowide who feel that gas rices have caused a fiacial hardshi i their household i July 08 versus Jue 08, desigated as Jul ad Ju, resectively, are estimated by ˆ ad ˆ Jul Are the true values of Jul ad Ju the same or are they differet? Ju 3 4

3 We eed to kow the samlig distributio of ˆ---if we were to draw samles of,339 eole over ad over, ad each time calculate a ew value of ˆ, what would the distributio look like? Solutio: view ˆ as the mea umber of successes er trial over the trials: 1. Assig a success a value of 1 ad a failure a value of 0. Defie X to be the sum of all samle observatios 3. X is the mea umber of successes i trials From sectio 6-6, if X is a biomial RV, as X will be aroximately ormally distributed with ad q X What ca we say about? X will also be aroximately ormally distributed for large sice dividig by does ot chage the shae of the distributio. X 1 X 1 1 E E X Var X Var X Rule of thumb for assumig aroximate ormality is: 5 ad q 5; or 5 ad qˆ

4 Proerties of the Samlig Distributio of ˆ 1. Mea of samlig distributio equals mea of samled oulatio ˆ. Stadard deviatio of samlig distributio equals 1 ˆ 3. Stadard error of samlig distributio equals E ˆ SE ˆ ˆ 1 ˆ 1.Estimatig a oulatio roortio Requires SRS reality will more likely be that we have some sort of radom samle ad most imortatly the samle is reresetative of the oulatio of iterest Coditios for a biomial distributio are satisfied Coditios for ormal aroximatio are satisfied Defie oulatio roortio x qˆ 1 samle roortio of x successes i a samle of size.poit Estimate A sigle value (statistic) used to estimate a oulatio arameter. 7 8

5 3.Iterval Estimate (cofidece iterval) A rage of values used to estimate the true value of a oulatio arameter. Abbreviated as CI Associated with a cofidece level Cofidece level defied as 1, where 1 is the roortio of times the CI does cotai the oulatio arameter assumig that the estimatio rocess is reeated a large umber of times Also called degree of cofidece or cofidece coefficiet What is alha? comlemet of the CL 4.Critical values The umber o the borderlie searatig samle statistics that are likely to occur from those that are ulikely to occur a cutoff value The umber z --ositive z value that is the vertical boudary searatig a area I the right tail of the stadard ormal distributio. Also have Examle stadard ormal curve z. Commo choices: 0.90 with with with 0.01 Ofte exressed as a ercetage 9 10

6 5.Margi of Error Deoted E with robability 1, the maximum likely differece betwee the observed samle roortio ˆ ad the true value of the oulatio roortio q ˆ ˆ E z SE ˆ z Use E to costruct a CI: E E E E, E 6.Geeral iterval estimate A geeral format for all cofidece itervals is: oit estimator E oit estimate (critical value SE(oit estimate)) Examle 1 7.Iterretig a CI Must be careful. We are 95% cofidet that the true oulatio roortio falls i Not: there is a 95% robability that the true oulatio roortio falls betwee Not: there is a 95% chace that the true oulatio roortio falls betwee After you have used samle data to costruct the iterval, the iterval either cotais the truth or it does ot. Cofidece comes from the method/rocedure. Let s look at a simulatio 1. Selected 100,000 ideedet trials from a biomial distributio with = 1 ad = 0.51 this became the oulatio. Radomly samled = 1000 observatios from the oulatio 3. Calculated ˆ ad SE( ˆ) 4. Created 95% CI for 5. Reeated stes more times ad lotted results 11 1

7 Drew = 1000 from a oulatio with = 0.51 lower.cl hat uer.cl [1,] [,] [3,] [4,] [5,] [6,] [7,] [8,] [9,] [10,] [11,] [1,] [13,] [14,] [15,] [16,] [17,] [18,] [19,] [0,] Simulatios =

8 8.Determiig samle size We ca use the exressio for margi of error to hel determie how large of a samle (miimum samle size) we eed to obtai a estimate of with a articular level of cofidece. z E z E z E q ˆ ˆ q ˆ ˆ q ˆ ˆ If we are i the rocess of desigig a study, where do we get a estimate of ˆ From a ilot study From the literature From a exert Whe i doubt, use ˆ 0.5--the variace for a biomial radom variable is maximum whe 0.5 Practice 1 The geetics ad IVF Istitute coducted a cliical trial of the YSORT method desiged to icrease the robability of coceivig a boy. Durig the study, 51 babies were bor to arets usig the YSORT method, ad 39 of them were boys. Use the samle data to costruct a 99% CI estimate of the ercetage of boys bor to arets usig the YSORT method. Iterret the iterval. Based o the result, does the YSORT method aear to be effective? Why or why ot? Examle 15 16

9 Practice The music idustry must adjust to the growig ractice of cosumers dowloadig sogs istead of buyig CDs. It therefore becomes imortat to estimate the roortio of sogs that are curretly dowloaded. How may radomly selected sog urchases must be surveyed to determie the ercetage that was obtaied by dowloadig? Assume that we wat to be 95% cofidet that the samle ercetage is withi oe ercetage oit of the true oulatio ercetage of sogs that are dowloaded. Relative Frequecy