Statistics question

[jedusor]

Can anyone explain to me why the t-statistic approaches the z-score as n approaches infinity, rather than as n approaches N? Is it just assumed that an unknown population size is infinite? I asked the teacher, and she didn't know. It's not important to actual calculations, of course, because there's obviously no way for n to be greater than N; I'm just curious.

Comments

[persona.livejournal.com]

(U) I think you've hit the nail on the head: for discrete populations, the estimated variance equals the actual population variance when n equals N. I think n going to infinity may become important in streaming data settings, where the population variance may be well-defined even if N isn't. Whether infinity is well-defined is open to debate.

[hahathor.livejournal.com]

Two disclaimers: 1) It's been ages since I addressed this problem 2) I'm drunk

But I've been writing reasoned sober shit about statistics all fucking day for my company blog - so I shall ramble....

Generally the hypothetical population you're interested in is not the actual population that exists. So, for example, if you're interested in, say, the mean height of American men, and there are N men alive in the US right now, you're actually not interested in the mean of those N, because those N are actually a sample of sorts of all possible American men that could exist. And if you took a sample of all N men, you'd actually have N-1 df, because (since there's a finite number of men), there is no freedom for the height of that last man - his height is determined by the observed height of the first N-1 men, and the actual mean height of the population. Of course, as soon as new men are born, or die, or move into or out of the US, your "population" changes, which is further evidence that N isn't actually the population of interest.

I'll be curious to learn in the morning if this made ANY sense at all.

[jedusor.livejournal.com]

Okay, I can see that. I hadn't thought of it in practical terms, I guess. :P

Is there a term for the hypothetical population?