I'm currently trying to solve an exercise in statistics.
I have got many hundred observations for some measured fluctations.
In a normal Q-Q plot, I have plotted all the observations and a line has been drawed. This is all done by software.
As I can see, the plotted observations do not really differ from a straight line, maybe only a little bit at the beginning of the line and at the end of line. Therefore, as I understand, I can assume that the data (the observations) follow a normal distribution.
However, in the exercise, I'm told to make use of the central limit theorem. But as I understand, this theorem should only be used if I can't assume a normal distribution (if the plotted observations differed from a straight line).
Did I misunderstand something?
In simple words, the Central Limit Theorem says that the sum of a sufficiently large number of weakly dependent quantities has a distribution close to normal, regardless of their initial distribution.
So
you don't need any assumptions on prior distributions, and it is really the main key point of CLT.
UPDATE (Thanks to comment by Karl): You don't need any special assumptions on prior distribution while it has well-defined finite mean $\mu$ and variance $\sigma^2$. There exist examples of such distributions which don't satisfy this requirement.