the fundamental issue here is ... is it reasonably to expect ... that when
you are making some inference about a population mean ... that you will
KNOW the variance in the population?
i suspect that the answer is no ... in all but the most convoluted cases
... or, to say it another way ... in 99.99% (or more) of the cases where we
talk about making an inference about the mean in a population ... we have
no more info about the variance than we do the mean ... ie, X bar is the
best we can do as an estimate of mu ... and, S^2 is the best we can do as
an estimate of sigma squared ...
this is why i personally don't like to start with the case where you assume
that you know sigma ... as a "simplification" ... since it is totally
unrealistic
start with the realistic case ... even if it takes a bit more "doing" to
explain it
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