Hi math stack exchange,
I came across the following question and found it quite interesting and am struggling to solve it. I haven't seen anything like it because it is both two sample and has sigma known. If anyone could give me some insight as to how to begin looking at the problem would be appreciated
Let $X_1, \ldots , X_n$ and $Y_1, \ldots , Y_n$ be independent random samples from the $N(\mu_{X}, 2)$ and $N(\mu_{Y} , 2)$ distributions, respectively. Calculate the minimal value of $n$ so that one can be 95% confident the interval $[\bar{X} − \bar{Y}− 1, \bar{X} − \bar{Y}+ 1]$ contains the true value of $\mu_X − \mu_Y$.
My idea is to set up the usual confidence interval for one side for a sigma unknown as the following form:
$1= \sqrt{(\sigma^2/n) +(\sigma^2/n)}\cdot 1.96$
$1= \sigma\cdot \sqrt{1/n +1/n}\cdot 1.96$
$1=2\cdot(2/n)\cdot 1.96$ ($2$ here at the start subbed in for sigma
And then solve for $n$?
Does this approach work?
Thanks