Can I use T-Statistic for constructing a Confidence Interval for sample from large population?

Question

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?

Answer 1

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.