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Unformatted text preview: MTH4106 Introduction to Statistics Notes 5 Spring 2012 Estimating a Proportion Suppose that there is a population of N items, of which M have Type A and N- M have Type B. Put p = M N = proportion having Type A . We take a random sample of size n . Let X be the number of Type A in the sample. If the sampling is done with replacement then X ∼ Bin ( n , p ) . If the sampling is done without replacement then X ∼ Hg ( n , M , N ) . Usually we assume that n is much smaller than M and N- M , so that X is approximately Bin ( n , p ) . Let Y = X n = sample proportion of Type A . Then E ( Y ) = E X n = 1 n E ( X ) = 1 n × np = p and Var ( Y ) = Var X n = 1 n 2 Var ( X ) = 1 n 2 × npq , where q = 1- p , = pq n . We do not know the true value of p , so we estimate it using Y . We write Y = estimator = a random variable y = the value of Y from our sample ˆ p = the estimated value of p , so that ˆ p = y ....
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This note was uploaded on 03/12/2012 for the course MTH 4106 taught by Professor R.a.bailey during the Spring '12 term at Queen Mary, University of London.
- Spring '12