I have a population that is in size equals to 65,536. The population follows a uniform distribution and is composed of discrete, ordered integer numbers. From this population I wish to identify the appropriate sample n so that it has a specific error of e.g., 5% and a confidence level of 95%.
I have some questions on how I can correctly compute the sample size. The Margin of Error (MOE) formula is:
$\epsilon = z_{\gamma} \times \sqrt{\sigma^2/n}$.
Given that I know the desired error, I can solve for $n$ and identify the sample size. But here I have some questions
(1) $σ^2$: Since the population follows a discrete uniform distribution, can I compute $σ^2$ as: $(N^2-1)/12 = 357,913,941.25 \text{ for N} = 65,536$ ? [1]
(2) $z_γ$: Regarding this value, can I use the the normal distribution table and compute it for $γ = 95$, as shown in [2]? Or since my population follows discrete uniform distribution I have to do some kind of conversion first? In this case, how can I do it?
(3) Given that (1) and (2) are computed and I can finally solve the error equation for n as: $e = z_γ \times \sqrt{σ^2/n} \leftrightarrow \\ n = z_{γ}^{2} \times σ^2/e^2 $
It is of course possible that the sample size will not be an integer number. In this case I should compute the ceiling of the result, correct? I.e., $\lceil{z_{γ}^{2} \times σ^2/e^2}\rceil$.
Thank you in advance