The voters in California are voting for a new governor. Of the voters in Califor

Question

The voters in California are voting for a new governor. Of the voters in California, a proportion p will vote for Frank, and a proportion 1 - p will vote for Tony. In an election poll, a number of voters are asked for whom they will vote. Let X_i be the indicator random variable for the event "the ith person interviewed will vote for Frank." That is, X_i = 1 if the ith person interviewed will vote for Frank, and X_i = 0 if the ith person interviewed will not vote for Frank. A model for the election poll is that the people to be interviewed are selected in such a way that the indicator random variables X_1, X_2, .. are independent and have a Ber(p) distribution. (a) Suppose we use X bar _n to predict p. According to Chebyshev's inequality, how large should n be (how many people should be interviewed) such that the probability that X bar _n is within 0.2 of the "true" p is at least 0.97? (b) Answer the same question, but now X bar _n should be within 0.1 of p. (c) Answer the question from part (a), but now the probability should be at least 0.95. (d) Now compute the number n so that X bar _n will be within 0.02 of p with probability 0.95. (This is often the standard level of accuracy for polling.)

Explanation / Answer

(a) Chebyshev's inequality

Pr ( l X - l >= k) = 1/k2

Here in this case standard deviation = sqrt [p(1-p)/ N] so

as maximum value of p(1-p) is 1/4 as p(1-p) <= 1/4

so < = sqrt [1/4N]

<= 1/2 sqrt (1/N)

so, as we are calculating the largest N then = 0.5 sqrt (1/N)

now by Chebyshev's inequality

Pr ( l X - l >= k) = 1 - 0.9 = 0.1 = 1/k2

so k2 = 1/0.1 = 10

k = 3.1622

so 2k = 0.2

k = 0.1

= 0.1/ 3.1622 = 0.5 sqrt (1/N)

so N = 250

(b) now answer when it is within 0.1 of p

2k = 0.1

so here n = 4 * 250 = 1000

(c) Here probability has been changed to 0.95

Pr ( l X - l >= k) = 1 - 0.95 = 0.05 = 1/k2

so k2 = 1/0.05 = 20

k = 4.4721

so 2k = 0.2

k = 0.1

= 0.1/ 4.4721= 0.5 sqrt (1/N)

so N = 500

(d) so here

2k = 0.02

k = 0.01

= 0.01/ 4.4721= 0.5 sqrt (1/N)

so N = 50000

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