The denition of conditional probability and the law

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online