 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2008/o/o5/i/solution

Independence of the two samples makes their sample means independent normal random variables. A difference of independent normal random variables is normal, with the means subtracted and the variances added. Thus

\overset{ˉ}{X} - \overset{ˉ}{Y} \sim N (μ_{X} - μ_{Y}, σ^{2} (1/ m + 1/ n)) .

 New to topics? Read the docs here!