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2. [20] All observations come from a population with mean #1,. and variance 0'3. A sample of size n is taken, and the sample mean is computed. (Recall that the Expected Value 2 and Variance of the sample mean are #3,, and of, respectively.) Suppose an \"extra\" observation yn+1 is \"discovered\". Rather than recalculate the sample mean using all n + 1 observations the researcher proposes the following: I yn+l = Y y + n (a) What is the expected value of pig)? Is a; an unbiased estimator of pay? Why or why not? [5] 05(n+1) . [5] \"2 (b) Show that the variance of a; is (c) Compare the variance of pi, with the variance of 17 (when only the rst 11. observa- tions are used to compute 17'). Which is smaller? [5] ((1) Show that My is a consistent estimator of pay. [5]