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Chapter 7 Point Estimation Nutan S. Mishra Department of Mathematics and Statistics University of South Alabama Estimation of population mean When population mean µ is unknown it is estimated by the sample mean X . Also we have seen in the last chapter that X is an unbiased and consistent estimator of the population mean That is E( X ) =µ and var( X ) 0 as n∞ Standard deviation of X = σ/√n Estimated standard deviation of X = s /√n Also known as estimate of standard error of sample mean The difference between X and µ is called error in estimation. Point Estimation Sample mean x assigns a single value to the unknown value of µ . Thus it is called a point estimator. Similarly the sample standard deviation s is a point estimator for the unknown value of population standard deviation σ. The value of x is bound to be different from the population mean µ The difference between the two is called sampling error Or error in estimation Error of estimate E = | X - µ| is the error of estimate. To examine this error use the fact that for large n x ~ approximate N(0,1) n 1-α α/2 α/2 -4 -3 -2 -1 -Zα/2 P(-Zα/2 < x n < Zα/2) = 1-α 0 1 2 Zα/2 3 4 Maximum error of estimate x P(-Zα/2 < < Zα/2) = 1-α n Is equivalent to| x | ≤ Zα/2 n Thus the maximum error of the estimate is E = Zα/2 .σ/√n Thus for given value of n, σ and α we can compute the maximum error in estimation. Where α is the probability of error E or more. 1-α is probability that error will be smaller than E Determining sample size If the maximum allowable error is specified with its probability of occurrence E = Zα/2 .σ/√n Then the sample size can be computed by plugging in the other quantities in the above equation. This method requires prior knowledge of σ and an assumption that n is large. Determining sample size If σ is unknown, and we assume that population is approximately normal then x ~ t-distribution with (n-1)d.f. t s n E = tα/2 .s/√n Exercise 7.2(a) X~ B(n, p) The population parameter p is unknown and x/n is an estimator of p That is pˆ x / n To show that x/n is an unbiased estimator of p That is to show that E ( p ˆ) p