# 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?

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.)