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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?

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The true mean unknown in both cases. That is the reason of testing hypotheses on it and/or crunching confidence intervals.

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  • $\begingroup$ So even in the first case it is unknown? Do you have a source for that? $\endgroup$
    – vonjd
    Feb 6 at 9:43
  • $\begingroup$ ... having thought about it, I think it has to be because otherwise the mean of the resulting distribution wouldn't be 0! $\endgroup$
    – vonjd
    Feb 6 at 10:00
  • $\begingroup$ The hypothesized mean is known. The true mean not. $\endgroup$
    – Michael M
    Feb 6 at 10:10

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