When doing a simple random sample to estimate population mean for some statistic, how do I know whether sampling happens with or without replacement?
It feels wrong to use replacement, because 1) my AP stats teacher never does that and 2) I might use someone's data twice in the average.
But on the other hand, the proof that the statistic is an unbiased estimator of the mean is $$E(X)=E(X_1)+\cdots +E(X_n)=\mu+\cdots+\mu$$ which implies $$E\left(\frac{X}{n}\right)=\frac{n\mu}{n}=\mu$$ But doesn't this assume that the $n$ statistics $X_i$ are independent of each other? And isn't that only true if we replace after each sample?