This is a more general version of Problem C.1. Let Y1, Y2, …, Yn be n pairwise uncorrelated random variables with common mean m and common variance 2. Let Y denote the sample average. (i) Define the class of linear estimators of m by Wa a1Y1 a2Y2 … anYn,
where the ai are constants. What restriction on the ai is needed for Wa to be an unbiased estimator of m?
(ii) Find Var(Wa).
(iii) For any numbers a1, a2, …, an, the following inequality holds: (a1 a2 … an)2/n a1 a2 … an. Use this, along with parts (i) and (ii), to show that
2 2 2
Var(Wa) Var(Y) whenever Wa is unbiased, so that Y is the best linear unbiased estimator. [Hint: What does the inequality become when the ai satisfy the restriction from part (i)]

The gilbert instrument corporation is considering replacing the wood steamer it currently uses to shape guitar sides. the steamer has 6 years of remaining life. if kept, the steamer will have depreciaiton expenses of $650 for five years and $325 for the sixthyear. its current book value is $3,575, and it can be sold on an internet auction site for$4,150 at this time. if the old steamer is not replaced, it can be sold for $800 at the endof its useful life. gilbert is considering purchasing the side steamer 3000, a higher-end steamer, whichcosts $12,000 and has an estimated useful life of 6 years with an estimated salvage value of$1,500. this steamer falls into the macrs 5-year class, so the applicable depreciationrates are 20.00%, 32.00%, 19.20%, 11.52%, 11.52%, and 5.76%. the new steamer is fasterand allows for an output expansion, so sales would rise by $2,000 per year; the newmachine's much greater efficiency would reduce operating expenses by $1,900 per year. to support the greater sales, the new machine would require that inventories increase by$2,900, but accounts payable would simultaneously increase by $700. gilbert's marginalfederal-plus-state tax rate is 40%, and its wacc is 15%.a. should it replace the old steamer? b. npv of replace = $2,083.51