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When calculating confidence intervals for population parameters, the population size is never a factor, rather sample size and the estimated parameter are used.

It seems to me very counter-intuitive that to assert with a certain confidence that a certain view has certain probability, one requires the same sample size regardless of whether the population is 2K persons or 2M persons.

Why is the confidence of estimated parameters of normal distributions independent of the population size?

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2 Answers 2

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This is an approximation. It's inherent in this entire approach that the population is regarded as effectively infinite. Thus, samples from it are not regarded as samples from a finite number of individuals, which would imply a discrete distribution, but as samples from a continuous distribution that idealizes the actual population. For instance, whereas the actual distribution of heights of humans in the world is a discrete distribution that probably looks somewhat similar to a Gaussian distribution when viewed from afar, the approach replaces this by a continuous distribution, in this case perhaps a Gaussian distribution. Instead of regarding this as an approximation, one could also take the view that the population itself is just a big sample from a more abstract distribution, something like "how the heights of people with this particular genetic mix would be distributed if infinitely many of them were born", or "how the political views of people exposed to these particular circumstances and these particular ads would be distributed if there were infinitely many of them".

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I might be wrong, but I think sample size is a function of population, but the formula is typically given in the limit as $pop\rightarrow\infty$.

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