In order to estimate f, the true fraction of smokers in a large population, Alvi

Question

In order to estimate f, the true fraction of smokers in a large population, Alvin selects n people at random. His estimator M_n is obtained by dividing S_n, the number of smokers in his sample, by n, i.e. M_n = S_n/n. Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that P[|M_n - f| greaterthanorequalto elementof] lessthanorequalto delta, where elementof and delta are some prespecified tolerances. Determine how the value of n recommended by the Chebyshev inequality changes in the following cases. (a) The value of elementof is reduced to half its original value. (b) The probability delta is reduced to half its original value.

Explanation / Answer

Solution:

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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