hypothesis testing: why you know population SD without knowing the population mean?

Question

When making inference about mean, i.e. when constructing a confidence interval for approximating population mean, in exercises there always provides a population SD, a.k.a. sigma, for us to solve the problem. But I highly doubt why can this be the case. As long as we know the sigma, that implies we already know the mean (according the the formula of how sigma is calculated), then what's the purpose of estimating an interval for the population mean if we already know its value?

Or I'm doubting if I have some misunderstanding about the origin of sigma in the first place: does it suggests it's possible to get sigma without knowing population mean? If so, how?

p.s. I know we can use sample SD to approximate sigma if sample size is large, but I'm not talking about this case.

Answer 1

The case with known population SD is simpler than the case with unknown population SD, and for this reason it is often taught first. However you are right that this is not usually realistic. In a standard course the case with unknown SD should be treated soon afterwards, and students should be told that known SD is not (normally) realistic and the unknown SD case will be treated later.

On top of that, asymptotically ($n\to\infty$) just plugging in an estimator of the unknown SD is often equivalent to the known SD case.

Another remark is that knowing the true population $\sigma$ does not necessarily imply you know the mean. Think of a measurement instrument of which the general precision is known, however it is used to measure something you don't know yet. Then you may know the SD but not the mean. (In reality the precision of the instrument is not normally precisely known, but may have been estimated very precisely from earlier data from other measurements, and therefore be treated as known in the new problem.)