I know that, but I see a few problems.

In part two do you really want a CI for

or

?

THEN, I think your instructor messed up with the relationship between the pop variances.

It works either way, but the algebra is nicer if we switch the 2 and the 3.

You start with the pt estimator that I already mentioned.

is unbiased for

and it's variance is

It's normal hence

Next use the relationship between the sigma's

BUT I keep asking, are they known?

If they are this is the test stat and you use a st normal table putting half of alpha in each table.

MORE likely the sigma's are unknown and you use the relationship and derive a t stat.

Here you need to pool the sample variances and obtain the chi-square distribution which replaces the

s