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# lecture 32 - Limits of Point estimation In point estimation...

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Limits of Point estimation In point estimation, statistic is used to estimate population parameter. Statistic is a function on random sample. From a particular sample, value of that statistic is used to estimate the parameter. This is not the actual value of the parameter. The estimate was obtained at random and could be any number. Most of the times, the point estimator is a cont. r.v., then even probability of getting actual population parameter is 0.

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Interval estimation In stead of considering a statistic as a point estimator, we may use random intervals to trap the parameter. In this case, the end points of the interval are r.v.s but we can talk about the probability that it traps the parameter value. Definition : A 100(1- α )% confidence interval (0< α <1) for a parameter θ is a random interval [L 1 , L 2 ] such that ). 1 ( ] [ 2 1 α θ - = L L P
Notes : Here θ is not a random variable. Its value is fixed, though unknown. The interval is random. The meaning of the probability is that for several choices of this interval, proportion of those which will contain θ is (1- α ). From a particular sample we can read off a particular confidence interval. Here we can say with confidence (1- α )100% that this interval contains θ . Since here nothing remains random, we talk of confidence rather than probability.

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Confidence interval of mean Assume the population X is normal . We find the (1- α )100% confidence interval for population mean μ when population variance σ 2 is known . We know : sample mean f8e5 X of a random sample of size n from X has normal distribution with mean μ and Variance σ 2 /n . Thus [ ] ). 1 ( / / ). 1 ( / | | 2 / 2 / 2 / α σ μ σ α σ μ α α α - = + < < - - = < - n z X n z X P z n X P
Exercise 7.4.3 When an experiment is conducted, the following observations on the normal random variable X, the survival time (in months)under the new treatment, result : 8.0 13.6 13.2 13.6 12.5 14.2 14.9 14.5 13.4 8.6 11.5 16.0 14.2 19.0 17.9 17.0 X has a s.d. of 3 months. f8e5 x=13.88, z 0.025 =1.96, so 95% C.I. is [12.41, 15.35].

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Impracticality of assumptions In practice, we face 2 problems in application of this C.I. formula and need some remedies. The population is not normal Population variance is unknown.
Central Limit Theorem Let X 1 ,X 2 , ...... ,X n be a random sample of size n from any distribution with mean μ and variance σ 2 . Then for large n , is approximately normal with mean μ and variance σ 2 /n.

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