Suppose there is a very tight political race and we want to gauge support for candidate A. Let Mn = 21-1X; denote the proportion of polled people supporting candidate A and let p denote the true proportion. Further suppose we want to poll enough people to ensure that we are within +1% of p 95% of the time. Stated mathematically, we would like to find n such that: P(Mn - p > 0.01) < 0.05 (1) when E(Xi) = p. (a) Show that the variance of Bernoulli(p) is maximized when p=0.5.
(b) Using the worst-case variance from part a) and a Normal approximation, find the n such that (1) is satisfied.
(c) Again using the worst-case variance, use Chebychev's inequality to find a bound for n.

The treasurer for pittsburgh iron works wishes to use financial futures to hedge her interest rate exposure. she will sell five treasury futures contracts at $139,000 per contract. it is july and the contracts must be closed out in december of this year. long-term interest rates are currently 7.30 percent. if they increase to 9.50 percent, assume the value of the contracts will go down by 20 percent. also if interest rates do increase by 2.2 percent, assume the firm will have additional interest expense on its business loans and other commitments of $149,000. this expense, of course, will be separate from the futures contracts. a. what will be the profit or loss on the futures contract if interest rates increase to 9.50 percent by december when the contract is closed out

Afirm’s production function is represented by q(m,r) = 4m 3/4r1/3, where q denotes output, m raw materials, and r robots. the firm is currently using 6 units of raw materials and 12 robots. according to the mrts, in order to maintain its output level the firm would need to give up 2 robots if it adds 9 units of raw materials. (a) true (b) false

Google started as one of many internet search engines, amazon started as an online book seller, and ebay began as a site where people could sell used personal items in auctions. these firms have grown to be so large and dominant that they are facing antitrust scrutiny from competition regulators in the us and elsewhere. did these online giants grow by fairly beating competition, or did they use unfair advantages? are there any clouds on the horizon for these firms -- could they face diseconomies of scale or diseconomies of scope as they continue to grow? if so, what factors may limit their continued growth?