2. (10 points) (The Weak Law of Large Numbers) In order to estimate f, the true

Question

2. (10 points) (The Weak Law of Large Numbers) In order to estimate f, the true fraction of smokers in a large population, Alvin selects n people at random. His estimator Mn is obtained by dividing Sn, the number of smokers in his sample, by N, i.e., Mn Sn/n. Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that where and ? are some prespecified tolerances. Determine how the value of n recommended by the Chebyshev inequality changes in the following cases. (a) (5 points) The value of e is reduced to half its original value (b) (5 points) The value of ? is reduced to half its original value

Explanation / Answer

a. Using the chebyshev’s inequality,

P (|Mn – f| ? ?) ? Var (Mn)/2

Assuming that the samples are drawn independently from the population, we have

Var (Mn)/2 = Var (Sn)/n^2

Var (Mn)/2 = ?^2/n^2

Var (Mn)/2 = ?

It is clear that if ? is reduced to half its original value, the number of samples should be quadrupled.

b. From part a, we see that if the probability of ? is reduced to ?/2, then the number of samples should be doubled.

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