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22 estimating the population proportion l recall

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Unformatted text preview: 22 Estimating the population proportion l Recall binomial rv X l Number of successes in n trials l Know that E( X )= np & Var( X )= np(1-p) l Now consider the rv X / n l This is simply the sample proportion of successes l It is an unbiased estimator of the population proportion l Why? l It has a variance of p(1-p)/n l Why? 23 Estimating the population proportion… l Also recall that for large n, X is approximately normal l Thus the sample proportion is also approximately normal for large n using the CLT l Approximate sampling distribution follows immediately l As do confidence intervals & hypothesis tests ) 1 , ( ~ / ) 1 ( ˆ and ) 1 ( , ˆ then ˆ Let N n p p p P Z n p p p ~N P n X P-- = - = Predicting election results l How do political commentators predict election results so soon after polls close & before all votes are counted? l They use data from exit polls and early returns as samples l Conduct inference on true (as yet unknown) proportion of voters supporting the incumbent A l In a 2 candidate race an exit poll of 400 produces 212 people who say they voted for A l What do you conclude about A’s re-election? 24 25 Predicting election results... l Set the problem up as an hypothesis test l A will win if s/he gains more than 50% of the vote & will lose if <50% of votes are cast in his/her favour l So p= 0.5 is the limit, above which s/he will win; this sets our null hypothesis l Is there sufficient evidence to refute this null? ! ! call to close too error, of margin within 1151 . ) 2 . 1 ( 2 . 1 400 / 25 . 5 . 53 . and 400 5 . 5 . , 5 . ˆ then .53 400 212 ˆ 5 . : 5 . : 1 ⇒ = =- =- = × = = = Z P value p Z ~N P P p H p H 26 Predicting election results… l In hypothesis testing example assumed H to be true l Allows calculation of the variance of the sampling distribution l What happens when constructing a confidence interval for p? l Use sample proportion to estimate p l Test statistic will remain approximately normal than.5 less are that of values includes CI 98% our .47,.59) ( or 400 / ) 47 . 53 (. 33 . 2 53 . is for CI 98% the & 33 . 2 , .02 For / ˆ 1 ˆ ˆ 01 . 2 p p z n ) P- ( P z P α/ ⇒ × × ± = = ± α 27 Progress report #6 l Have the statistical tools that underpin a large range of statistical procedures l Know about some key distributions, eg Binomial, Uniform, Normal & t l Introduced basics of point & interval estimation l Introduced basics of constructing & conducting hypothesis tests l All of this has been done in a univariate context l Most interesting problems involve multivariate situations where relationships between variables is important l Takes us back to correlation & regression l Previously introduced as descriptive devices l Now consider inference in a regression context...
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22 Estimating the population proportion l Recall binomial...

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