Question: Let \(X=\left( {{X}_{1}},{{X}_{2}},...,{{X}_{{{n}_{1}}}} \right)\), \(Y=\left( {{Y}_{1}},{{Y}_{2}},...,{{Y}_{{{n}_{2}}}} \right)\) be independent random samples from the normal distributions \(N\left( {{\mu }_{X}},\sigma _{X}^{2} \right)\), \(N\left( {{\mu }_{Y}},\sigma _{Y}^{2} \right)\).

a) Normalize \(\bar{X}-\bar{Y}\)

b) Compute the realization of the normalized random variable \(\bar{X}-\bar{Y}\). You know that \(\bar{X}\) = 140 was computed from a realization with the range 60, \(\bar{Y}\) = 137 was computed from a realization with the range 50, \({{X}_{i}}\tilde{\ }N\left( 142,20 \right)\) and \({{X}_{i}}\tilde{\ }N\left( 135,18 \right)\).