The denition of conditional probability and the law

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Unformatted text preview: t n voters at random, where n is much smaller than the total population of voters. For example, a given poll might survey n = 200 voters to estimate the fraction of voters, within a population of several thousand voters, that favor a certain proposition. The resulting estimate of p would be p = X , where X denotes the number of the n sampled n voters who favor the proposition. We shall discuss how a confidence interval with a given level of confidence could be determined for this situation. Assuming the population is much larger than n, it is reasonable to model X as a binomial random variable with parameters n and p. The mean of X is np so Chebychev’s inequality yields that for any constant a > 0 : P { X − np| ≥ aσ } ≤ 1 . a2 X aσ −p ≥ n n 1 . a2 Another way to put it, is: P Still another way to put this, using σ = P p∈ p−a ≤ np(1 − p), is: p(1 − p) ,p + a n p(1 − p) n ≥1− 1 . a2 (2.11) 2.10. THE LAW OF TOTAL PROBABILITY, AND BAYES FORMULA 47 In (2.11), p is the fixed proportion to be estimated. It is treated as...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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