Difference between true standard deviation and population standard deviation

Question

What exactly is the difference between the two? Is it true that if you have been given the true standard deviation, you have to use the Z-score and if you have been given sample standard deviation, you have to use the t-score?

Thanks

Answer 1

If you have $n\ge2$ independent samples $X_1,\,\cdots,\,X_n$ of a mean-$\mu$, SD-$\sigma$ distribution, $\bar{X}:=\frac1n\sum_iX_i$ is an unbiased estimator of $\mu$, and $\frac{1}{n-1}\sum_i(X_i-\bar{X})^2$ is an unbiased estimator of $\sigma^2$ (whose square root we denote $s$), and so is $\frac1n\sum_i(X_i-\mu)^2$ (but you can't use it without knowing $\mu$ exactly). If the sampled distribution is normal, $Z:=\frac{\bar{X}-\mu}{\sigma}$ is too, but $T:=\frac{\bar{X}-\mu}{s}$ is $t$-distributed.