Sample Size Determination When Population is Unknown I was looking at the following formula for determining the sample size:

Here are what the following terms in the equation stand for:

This equation looks straightforward - if the population size, critical Z value, sample proportion and margin of error is know - apparently the sample size can be calculated.
My Question: But what if we don't know the true population size? What happens if we only to have a sample, and that's it?
For example: Suppose there is a new disease that recently appears and researchers only have a sample of 100 people with this disease. Suppose the researchers don't have enough money to study all of these 100 people, and they want to find out the the smallest number of people they need to study within these 100 people. The researchers want to find out the average level of a specific protein people with this disease have - but it costs money to measure the protein level for each person. At the same time, the researchers don't know how many people in the world have this disease (i.e. population is unknown).
Is there any formula that can be used to determine how many of these 100 people need to be measured to determine the average protein level associated with this disease, such that the results are statistically significant - when the true population is unknown? Would this formula require some distributional assumption for the protein levels, i.e. does the protein level have to be normally distributed?
Thanks

You could just drop the finite population correction and use $Z^{2}p(1-p)/e^2$ which is correct* as $N\to \infty$; for finite populations it will be slightly too large but unless $n$ is a substantial fraction of $N$ (several percent say), the effect of ignoring it is not worth worrying about.
*(assuming you're doing the test that formula was designed for, which is a proportions test. Note that this formula doesn't seem to match the scenario you subsequently describe -- which was not about proportions)
