L important application of normal distribution is to

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l Important application of normal distribution is to approximate the binomial for large n l See example from Keller, Fig. 9.8, p. 310, where n =20, p =0.5 l Do you think approximation would be better or worse if p =0.2?
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19 Normal approximation to the binomial… l Denote binomial rv by X B l Know that E ( X B)= np =10 & Var( X B)= np(1-p)= 5 l Natural to choose approximating normal rv as X N~ N (10,5) l How good is the approximation? l Consider P (10≤ X B ≤12)=.1762+.1602+.1201=.4565 while P (10≤ X N ≤12)= P (0≤Z≤0.89)=.3133 l What went wrong with the approximation? l Would you approximate P ( X B=10) by P ( X N=10)?
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Normal approximation to the binomial ... l Need a continuity correction (Keller p. 311) l Would approximate P ( X B=10) by P (9.5≤ X N ≤10.5) l In general approximate l P ( X B ≤ x) by P ( X N ≤x+0.5) l P ( X B ≥ x) by P ( X N ≥ x-0.5) l Now reconsider approximation l P (10≤ X B ≤12)=.4565 l Use P (9.5≤ X N ≤12.5) instead of P (10≤ X N ≤12) l Does this improve the approximation? 20
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Normal approximation to the binomial… 21 0.4565 0.4557 Clearly 4557 . 3686 . 0871 . ) 12 . 1 0 ( ) 0 22 . 0 ( ) 12 . 1 22 . 0 ( 5 10 5 . 12 5 10 5 . 9 ) 5 . 12 5 . 9 ( = + = + - = - = - - - = Z P Z P Z P X P X P N σ μ
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22 Airline meals l On today’s flight all 160 passengers offered a lunch choice of beef or chicken l Past data indicates 60% choose beef over chicken l Passenger choices appear to be independent l On this flight what is the probability that more than 110 passengers will choose beef?
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23 Estimation l Inferential statistics l Extracting information about population parameters on basis of sample statistics l “Past data indicates 60% choose beef over chicken” l What does a sample mean tell us about the population mean? l In practical situations parameters are unknown because they are difficult or impossible to determine l Using sample statistics may be only practical alternative
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24 Estimation… l Estimators l Consider a generic parameter θ that characterizes the pdf, f(x) , of a rv X l Suppose X 1, X 2, …, X n is a sample of size n from f(x) l A statistic is any function of sample data l An estimator is a statistic whose purpose is to estimate a parameter or some function thereof l A point estimator is simply a formula (rule) for combining sample information to produce a single number to estimate θ
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Estimation ... l Estimators are random variables (because they are a function of rv’s X 1, X 2, …, X n ) l Examples of point estimators l Sample mean is a point estimator for the population mean l Sample variance is an estimator of the population variance l Why does it make no sense to expect an estimator to always produce an estimate equal to the parameter of interest?
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