Question

Problem 1: A pollster would like to estimate the fraction p of people in a population who intend to vote for a particular candidate. How large must a random sample be in order to be at least 95% certain that the fraction of positive answers in the sample is within 0.02 of the true p

Answer 1

Solution,

Given that,

$\hat p$ = 0.5

1 - $\hat p$ = 1 - 0.5 = 0.5

margin of error = E = 0.02

At 95% confidence level

$\alpha$ = 1 - 95%  

$\alpha$ = 1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z_0.025 = 1.96

Sample size = n = (Z_{$\alpha$ / 2} / E )² * $\hat p$ * (1 - $\hat p$ )

= (1.96 / 0.02 )² * 0.5 * 0.5

= 2401

Sample size = n = 2401