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S231 Lecture 4 Cyntha

# S231 Lecture 4 Cyntha - Harris/Decima Poll Example(a public...

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Harris/Decima Poll Example (a public opinion poll) Semi-annual poll on Post-Secondary Education conducted by Harris/Decima company under contract from CAUT in Nov. 2010 (telephone poll of 2000 °representative± adults). Twenty-six percent of respondents agreed and 48 % disagreed with the following statement: °University and college teachers earn too much±. Harris/Decima declared the result to be accurate within ° 2 . 2 % , 19 times out of 20 (light green: Nov 2009, blue: Nov. 2010.) Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 14 / 45

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Maximum Likelihood Estimation 3 What does " accurate within ° 2 . 2 % , 19 times out of 20" mean and how was it obtained? Suppose Y represents the number of individuals who, in a randomly selected group of n persons, agreed with the statement. Then Y is Binomial random variable with p.f. f ( y ; θ ) = ( n y ) θ y ( 1 ± θ ) n ± y for y = 0 , 1 , ..., n where θ represents the fraction of the entire population (what population?) that agree. Observed data: y = 520, n = 2000 Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 17 / 45
Maximum Likelihood Estimation 3 What does " accurate within ° 2 . 2 % , 19 times out of 20" mean and how was it obtained? Suppose Y represents the number of individuals who, in a randomly selected group of n persons, agreed with the statement. Then Y is Binomial random variable with p.f. f ( y ; θ ) = ( n y ) θ y ( 1 ± θ ) n ± y for y = 0 , 1 , ..., n where θ represents the fraction of the entire population (what population?) that agree. Observed data: y = 520, n = 2000 Likelihood function: L ( θ ) = ( 2000 520 ) θ 520 ( 1 ± θ ) 2000 ± 520 for 0 < θ < 1 Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 17 / 45

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Maximum Likelihood Estimation 3 What does " accurate within ° 2 . 2 % , 19 times out of 20" mean and how was it obtained? Suppose Y represents the number of individuals who, in a randomly selected group of n persons, agreed with the statement. Then Y is Binomial random variable with p.f. f ( y ; θ ) = ( n y ) θ y ( 1 ± θ ) n ± y for y = 0 , 1 , ..., n where θ represents the fraction of the entire population (what population?) that agree. Observed data: y = 520, n = 2000 Likelihood function: L ( θ ) = ( 2000 520 ) θ 520 ( 1 ± θ ) 2000 ± 520 for 0 < θ < 1 max θ L ( θ ) = L ( ˆ θ ) where ˆ θ = y / n (show this!) Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 17 / 45
Maximum Likelihood Estimation 3 What does " accurate within ° 2 . 2 % , 19 times out of 20" mean and how was it obtained? Suppose Y represents the number of individuals who, in a randomly selected group of n persons, agreed with the statement. Then Y is Binomial random variable with p.f. f ( y ; θ ) = ( n y ) θ y ( 1 ± θ ) n ± y for y = 0 , 1 , ..., n where θ represents the fraction of the entire population (what population?) that agree. Observed data: y = 520, n = 2000 Likelihood function: L ( θ ) = ( 2000 520 ) θ 520 ( 1 ± θ ) 2000 ± 520 for 0 < θ < 1 max θ L ( θ ) = L ( ˆ θ ) where ˆ θ = y / n (show this!) Maximum likelihood estimate of θ is ˆ θ = 520 / 2000 or 26 % Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 17 / 45

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Likelihood Function for Harris Decima Poll From the graph, the interval suggested by the pollsters, 26 % ° 2 . 2 % or [ 23 . 8 , 28 . 2 ]
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