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Question: please give detailed steps thank you...

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Please give detailed steps. Thank you.

5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Lets denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average) where n Σ Xi (the sample mean) a. Show which of μι, μ2, μ3 and is an unbiased estimators of μ, if any. For the biased estimators, calculate thier bias. (Recall that for θ-an esti- mator of parameter θ. Bias(8) E(0)-θ)b. Show which of μ1, μ2, μ3 and μ4 is a consistent estimator of of μ,if any. C. Suppose that the population variance is known: σ2-5 Which of these estimators would you prefer if you had a sample of size n 5? Does your answer depend on the true value of the population mean μ? What if you had a sample of size n-10,000? Hint: With a small sample size we prefer an unbiased estimator. Among the biased ones, we compare the mean squared error of the estimators, where MSE(θ)-var(θ) + [bias(0)12 In a large sample we prefer a consistent estimator.

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