Select a sample size form the infinite population I have a list of every store chain in the US with at least 100 locations in the US.
a) What number of those do I have to pick in order to have a representative sample of the number of locations across the country or in a specific location?
b) What number do we pick to have reasonable confidence that we have a representative sample of the number of locations in the US, if we do not know how many chains exist overall?
I am trying to identify the sample size based on the confidence level and that is what I have done so far:
I selected a random number for the data sample size: let's say n= 100, 50, or 25. I derived confidence level based on that sample size and the confidence levels I have been getting were almost the same, So is there other way to approach this?
Any information would be helpful!
 A: There are two issues here, (1) Getting a confidence interval of the desired length and (2) getting a representative (unbiased) sample of the population.
For (1) it is often enough to use a formula to find the $n$ that will result in a confidence of the length you want.
For (2) the task is more difficult. If your list is comprehensive, then a simple random sample from it may be all you need to estimate the proportion of such chains that have at least 200 stores across the country. That is,
$P(X \ge 200\,|\,X \ge 100),$ where $X$ is the number of stores a chain has.
However, a random sample from your list will probably not give an unbiased estimate of the proportion of large $(X \ge 100)$ chains that have
a store in a specific location, such as Sikeston, MO. A small chain $(X < 100)$ serving mostly the south and midwest of the US may be much more likely to
serve a particular small town in MO than will a huge chain with most of its locations concentrated in the contiguous 11 western US states. But a few very large chains have locations almost everywhere (e.g, McDonald's, Walmart, certain drugstores, and truck rental companies, etc.)
