Suppose there is a very tight political race and we want to gauge support for candidate A. Let Mn = 1/n Σ Xi 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 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.

Required:
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.

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The management of a private investment club has a fund of $250,000 earmarked for investment in stocks. to arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk (x), medium risk (y), and low risk (z). management estimates that high risk stocks will have a rate of return of 15%/year; medium risk stocks, 10%/year; and low risk stocks, 6%/year. the amount of money invested in low risk stocks is to be twice the sum of the amount invested in stocks of the other two categories. if the investment goal is to have a rate of return of 9% on the total investment, determine how much the club should invest in each type of stock. (assume that all the money available for investment is invested.)