Given that the random variables [latex]X[/latex] and [latex]Y[/latex] have distributions [latex]N(60, 25)[/latex] and [latex]N(54, 64)[/latex] respectively, and that [latex]\bar{X}[/latex] and [latex]\bar{Y}[/latex] denotes the means of [latex]7[/latex] independent observations of [latex]X[/latex] and [latex]11[/latex] independent observations of [latex]Y[/latex] respectively.
State the sampling distributions of
(i)
[latex]\bar{X}-\bar{Y}[/latex]
(ii)
[latex]2\bar{X}+5\bar{Y}[/latex]
(iii)
[latex]\frac{{{X}{1}}+{{X}{2}}+{{X}{3}}-{{Y}{1}}-{{Y}{2}}-…-{{Y}{5}}}{8}[/latex]