Sample size choice with binary outcome

Question

Our civic association is planning to implement a survey asking our neighborhood one true false question. I would like to make sure that we obtain an adequate sample size.

Let's assume N = 2500 people, and we want to be 90% confident of our results with margin of error no more than 5%.

We would like to know the minimum sample size. Is there a simple formula to apply? If so can you please share it?

If so, is there an easy way to stratify the sample by some categorical variable such as race. For this example let's assume 25% of the neighborhood is part of a minority group. What is the minimum number of minority households that must be included?

Answer 1

Including a finite correction factor, the margin of error $m$ for binary data is given by $$m = z_{\alpha/2} \sqrt{{p(1-p)} \over {n}} \sqrt{{{N-n} \over {N-1}}},$$ where $N$ is the population size, $n$ is the sample size, $p$ is the success probability (say, the chance of a voter giving a true answer), and $z_{\alpha/2}$ is the appropriate quantile from the standard normal distribution for the desired confidence.

Solving for $n$ we find $$n ={ {z^2_{\alpha/2} \ p (1-p)N} \over {z^2_{\alpha/2} \ p(1-p)+(N-1)m^2} } $$ Now $p$ is unknown, but the worst case is $p={{1} \over {2}},$ so using that we have $$n ={ {z^2_{\alpha/2} \ N} \over {z^2_{\alpha/2} \ +4(N-1)m^2} } $$

Using your numbers ($z_{\alpha/2}=1.6449$), I get a sample size of $n=244.2,$ so use $n=245.$

Ignoring the finite correction factor would have led to a sample size of $n=272.$

For your second question, I'm not sure what you are asking. The minimum number of minority households to sample may be more of a political issue than statistical.