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Chapter 7 Point Estimation

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 .

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Chapter 7 Point Estimation

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  1. Chapter 7Point Estimation Nutan S. Mishra Department of Mathematics and Statistics University of South Alabama

  2. Estimation of population mean When population mean µ is unknown it is estimated by the sample mean . Also we have seen in the last chapter that is an unbiased and consistent estimator of the population mean That is E( ) =µ and var( ) 0 as n∞ Standard deviation of = σ/√n Estimated standard deviation of = s /√n Also known as estimate of standard error of sample mean The difference between and µ is called error in estimation.

  3. Point Estimation Sample mean 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 is bound to be different from the population mean µ The difference between the two is called sampling error Or error in estimation

  4. 1-α α/2 α/2 -Zα/2 Zα/2 Error of estimate E = | - µ| is the error of estimate. To examine this error use the fact that for large n ~ approximate N(0,1) P(-Zα/2 < < Zα/2) = 1-α

  5. Maximum error of estimate P(-Zα/2 < < Zα/2) = 1-α Is equivalent to ≤ Zα/2 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

  6. 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.

  7. Determining sample size If σ is unknown, and we assume that population is approximately normal then ~ t-distribution with (n-1)d.f. E = tα/2 .s/√n

  8. Exercise 7.2(a) X~ B(n, p) The population parameter p is unknown and x/n is an estimator of p That is To show that x/n is an unbiased estimator of p That is to show that

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