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Can I use a T-Statistic to construct a Confidence Interval for a sample collected from a large population and without knowledge about the population's standard deviation? I know the "model" answer is that a sample, even if the sample size $n$ is small, would have to use a Normal Z statistics if it was collected from a large population. But I have used a T-Statistic in a problem like this one because earlier, I felt that it would be more appropriate to use a T-Statistic since I'm having the standard mean error estimated by the sample standard deviation. And now, I am worried that I've done wrong.

So, can I use a T-Statistic to construct a Confidence Interval for a sample collected from a large population and without knowledge about the population's standard deviation? In my case, $n=8$. If no, why would it be wrong to use a T-Statistic in this case? If yes, also, what are the reasons for being correct to use?

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If you're asking whether you can construct a confidence interval for the mean (or some other statistic) using a small sample from a large population then the answer is yes. In fact, the basic theory says that such a population is infinite in size. That is: you could repeatedly redraw mutually independent samples of size 8 from this population again and again. Estimating the standard deviation from the sample is common practice. Using the T-distribution's percentiles is the correct way to construct the confidence interval for your mean.

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  • $\begingroup$ Is there such a theory that says that if a sample is from a large population, I would have to use a Z-Statistic(even if the standard deviation was estimated from the sample) when constructing the Confidence Interval for the mean? A friend told me that there is such a theory and being not confident in my own knowledge, I thought I have done wrongly. $\endgroup$ – Carven Apr 24 '12 at 15:32
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    $\begingroup$ No, the size of the population, once it gets "large", doesn't matter, only the size of the sample. As the sample size gets larger, the distribution of the T-statistic approaches the distribution of the Z-statistic (the t distribution approaches the Normal distribution), so at some point people just use the Z-statistic. $\endgroup$ – jbowman Apr 24 '12 at 15:34

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