I understand that when drawing from a normal distribution with a known population mean and standard deviation subtracting this mean and dividing by this standard deviation will give z-scores that follow a standard normal distribution.
When you don't know the population standard deviation and take the sample standard deviation instead you will get t-scores that follow a t-distribution.
My question: In all of the explanations I have read so far nobody loses a word about the mean. In the first case, it is assumed that the population mean is known, and in the second case? Doesn't it make a difference whether it is known or only estimated based on the drawn samples? If not why not?