Question

A political campaign manager wishes to survey a number of voters to estimate the proportion of those who are in favor of his candidate. If a previous survey shows that 55% of registered voters plans to vote for his candidate, what is the minimum sample size required to make his survey accurate with a 95% confidence level and a margin of error of 2.5%?

Answer 1

The formula for error:

$E=z\cdot \sqrt{\cfrac{\hat{p}\cdot (1-\hat{p})}{n}}.$

Here

$E\ -$ the error, $E=0.025$ ;

$z\ -$ z-score, for 95% confidence level $z=1.96$ ;

$\hat{p}\ -$ the sample proportion, $\hat{p}=0.55;$

$n\ -$ the sought sample size.

So,

$n=\cfrac{z^2\cdot\hat{p}\cdot (1-\hat{p})}{E^2}=\\ =\cfrac{1.96^2\cdot0.55\cdot (1-0.55)}{0.025^2}=1521.3.$

The minimum sample size is 1522.