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a. If ? ̅1 is the mean of a random sample of size n from a...

a. If ? ̅1 is the mean of a random sample of size n from a normal population with mean ? and variance ?1 2 and ? ̅2 is the mean of a random sample of size n from a normal population with mean ? and variance ?2 2, and the two samples are independent, show that ?? ̅1 + (1 − ?)? ̅2 where 0 ≤ ? ≤ 1 is an unbiased estimator of ?.

b. Find the value of ? that makes the variance of the estimator a minimum.

c. Find the efficiency of the estimator in part (a), with ? = 1/2  relative to this estimator with ? equal to your answer from part (b).

d. Show that the estimator in (a) is consistent.

e. Is the estimator in part (a) a sufficient estimator of ?? (Show all work.)

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