# Philosophy of sampling for experiments: finite versus infinite

What is the difference between a finite population and an infinite one - when you are designing an experiment (sample/power and interpretation of the results)?

Say a company has a database of 20,000 customers. Given that a response to some stimulus is relatively small, and a meaningful minimum detectable difference is also small, if you run a power analysis for a 2 sample proportion, you may find that you need 2 groups of 15,000 for the experiment.

Do you quit and say you cant experiment on this population? Or do you (somehow) instead treat the population as finite and run a power analysis that way? What are the implications?

What I would like to know, and wanted to add this detail in case the last part of the question wasn't completely clear - is what the difference is between

1. The inference with assuming an infinite population, say the inference from a logistic regression model with glm() in R.

2. The inference with assuming a finite population, say inference from a logistic regression model with svyglm in R?

Will #1 allow inference about the wider population / data generating process and #2 only allow inference about that particular population (which is assumed fixed)?

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You can do the power analysis assuming a finite population. Because the variance of the estimate goes to $0$ as the sample size gets close to the population size, this makes a big difference. The variance for a binomial proportion based on the infinite population assumption would be $$p(1-p)/n$$ where $n$ is the sample size. But if the population size is $N$ it will be $$[p(1-p)/n][1-n/N].$$ The finite population correction factor, $(1-n/N)$, will make it go to $0$ as $n$ approaches $N$ rather than be $p(1-p)/N$ that you would get for a single proportion assumong an infinite population. For your two sample problem the formula is a little more complicated but the idea is the same.

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I'm very interested in the two sample problem if you have a reference! –  B_Miner Sep 11 '12 at 22:24
Specifically, a test for two independent proportions in a finite sample (where that N is known). –  B_Miner Sep 11 '12 at 22:39
Don't have a reference but if the population size is N=N$_1$+N$_2$ where N$_1$ and N$_2$ are the respective population sizes for the two groups and the true proportions are P$_1$ and P$_2$ respectively then the difference of the 2 estimates in the case of independent samples has variance P$_1$(1-P$_1$)/n$_1$+P$_2$(1-P$_2$)/n$_2$ where n$_1$ and n$_2$ are the respective sample sizes from the two subpopulations without the finite population correction. with the finite population correction you would have [P$_1$(1-P$_1$)/n$_1$][1-n$_1$/N$_1$]+[P$_2$(1-P$_2$)/n$_2$]][1-n$_2$/N$_2$]. –  Michael Chernick Sep 11 '12 at 22:43
Thanks Michael, didn't know adjusting the typical formula for two independent proportions would be so easy! Why dont statistical packages have that option? –  B_Miner Sep 11 '12 at 23:02
I think you can find this in the survey sampling texts and the literature. Statistical packages designed for survey sampling may also have it. –  Michael Chernick Sep 11 '12 at 23:04