JPJC Sampling Tutorial Q4

\(X\) and \(Y\) are random variables with \(Y\) being defined as \(Y=aX+b\), where \(a\) and \(b\) are positive constants, and \(\mathrm{E}\left( X \right)=0\) and \(\mathrm{Var}\left( X \right)=\frac{4}{5}\). It is given that \(\mathrm{E}\left( Y \right)=50\) and \(\mathrm{Var}\left( Y \right)=80\). Find \(a\) and \(b\). A random sample consists of \(160\) independent observations of \(Y\). Find the probability that the sample mean lies between \(49.0\) and \(50.5\).[6]