Sample size from population?

Question

This is probably very rudimentary maths, but given a strict population size ($N = 20$ for example), is the sample size any number $<N$? For use in calculation of confidence intervals using a population size, all of the formulas use $n$ and not $N$, meaning I need a sample size rather than population size.

$$N = 20;\qquad n = 1, 2, 3, \dots , 19$$ $$N = 10;\qquad n = 1, 2, 3, \dots , 9$$

Is this right? This isn't really a maths question, but moreso a semantics question; is a sample size just taken from a population? A sample size of $n = 5$ can come from a population size of $N = 10$?

Basically: for determining confidence intervals of a population, is the sample size used in calculations an arbitrary number as long as it fits within the amount of the population?

edit; according to the textbook (and online sources), a sample should be an approximation of a population, so $n = N - 1$? Math is my weakest area, so I apologize if this is silly.

Answer 1

I'm guessing what you're looking for is a confidence interval for the population size $N$.

Your question says "using a population size" rather than "for a population size", so you're not making it altogether clear.

The formulas for confidence intervals found in elementary textbooks are usually based samples in which the observations are mutually independent. If it's done that way, then the sample size can exceed the population size. The sequence $2,1,5,3,3,5,1$ of size $7$ could be taken, and in this case, not all of the members of the population are included in the sample ($4$ is not).

One can do better by sampling without replacement, so that no member of the population is used more than once in the sample. In that case, the sample size cannot exceed the population size. But the formulas for confidence intervals are messier and harder to derive in that case.

Answer 2

If you know the size $N$ of the population, you can take an arbitrary sample $n < N$. In your example, if $N = 20$, it is valid to take $n = N - 1 = 19$. Note that this is an arbitrary sample, so $n$ can have any combination of groups of 19 elements out of a total of 20 elements.

For example, if each element is identified with a number (for simplicity) you can take a first sample ${1, ..., 19}$, or take another sample ${2, ..., 20}$, or another sample ${1 , 2, ..., 17, 18, 20}$ (item $19$ is missing in this sample). That's what I mean by combinations of groups of 19 elements out of 20 (the number of combinations in a group of k elements out of n is what is known as the "binomial coefficient").

In summary, a sample of $n$ elements is a subset of a population of $N$ elements (in practice, $N$ is not usually known with certainty because it is usually very large, which is why we work with samples of size $n < N$, where $n$ is chosen from a suitably large but familiar size)