Originally Posted by
matheagle I know that, but I see a few problems.
In part two do you really want a CI for $\displaystyle \mu_1-\mu_2$
or $\displaystyle 3\mu_1-2\mu_2$?
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.
$\displaystyle 3\bar X-2\bar Y$ is unbiased for $\displaystyle 3\mu_1-2\mu_2$
and it's variance is $\displaystyle 9\sigma^2_1+4\sigma^2_2 $
It's normal hence
$\displaystyle {(3\bar X-2\bar Y)-(3\mu_1-2\mu_2)\over \sqrt{9\sigma^2_1+4\sigma^2_2}}\sim N(0,1)$
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 $\displaystyle \sigma^2$s