Question

We are interested in conducting a study to determine the percentage of voters of a state would vote for the incumbent governor.

What is the minimum sample size needed to estimate the population proportion with a margin of error of .05 or less at 95% confidence?

Answer 1

The minimum sample size is N=1537.

Explanation:

We want to know the sample size to estimate the proportion of voters with a margin of error of 0.05, with a 95% of confidence.

The margin of error can be defined as:

`e=UL-LL=(p+z*\sqrt{(p(1-p))/(N)}) -(p-z*\sqrt{(p(1-p))/(N)})=2z\sqrt{(p(1-p))/(N)}`

We can calculate N from this

`e=2z\sqrt{(p(1-p))/(N)}\\\\(p(1-p))/(N)=((e)/(2z))^2\\\\N= ((2z)/(e))^2*p(1-p)`

The sample size will be calculated for p=0.5, which is the proportion that requires the larger sample size (maximum variance).

The z-value for a 95% CI is z=1.96.

The minimum sample size is then

`N= ((2z)/(e))^2*p(1-p)=((2*1.96)/(0.05))^2*0.5(1-0.5)=6146.56*0.5*0.5=1536.64`

The minimum sample size is N=1537.