A typical opinion poll surveys about 1000 adults. Suppose that the sampling fram

Question

A typical opinion poll surveys about 1000 adults. Suppose that the sampling frame contains 100 million adults including yourself, and that an SRS of 1000 adults is chosen from the frame. What is the probability that you are selected to be in the sample? Now suppose that 2000 such samples are selected, each sample selected independently of the others. What is the probability that you will not be in any of the samples? How many samples must be selected for you to have a 0.5 probability of being in at least one sample?

Explanation / Answer

Here the sample size is 1000 adults and population size is 100 million.

a) Required probability is,

p = 1000 / 100 million = 1/100,000 = 0.00001

b) The probability that you will not be in any of the sample is, (1 - p)^n

Here n = 2000

Here p = 0.00001

Probability = (1 - 0.00001)^2000 = 0.9802

c) Here we have given the probability and we have to calculate sample size.

i.e. symbolically it can be written as,

(1-p)^a = 0.5

Here p = 0.00001

take ln on bothsides,

ln[ (1-p)^a ] = ln(0.5)

a*ln(1-p) = -0.6931

a*ln(1-0.00001) = -0.6931

a*ln(0.99999) = -0.6931

a = -0.6931 / ln(0.99999)

a = -0.6931/-1.00001E-05

a = 69314.37 which is approximately equal to 69314.37

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