Question

You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable estimate for the population proportion. Your would like to be 99% confident that you estimate is within 2% of the true population proportion. How large of a sample size is required?

Answer 1

Solution

For a population proportion p, let a sample size of n be drawn and the resulting sample proportion be p^

The Confidence Interval for the population proportion(p) is given as

p^ ± z_α/2 * √(p^(1-p^)/n)

Given margin of error should be within 2%, therefore:

z_α/2 * √(p^(1-p^)/n) <= 0.02

->We now need to estimate n from the above inequality

->For a 99% Confidence Level, z_α/2 = 2.58

->For the smallest sample (n), we need to maximize p^(1-p^) for which p^ = 0.5

Therefore,

2.58 * √(0.5(1-0.5)/n) <= 0.02

Solving for n

n >= 2.58² *0.5 *0.5 / 0.02²

n >= 4160.25

The smallest value of n is 4161 which is the required sample size