2019 CJC P2 Q10

The Mathematics examination score of a randomly chosen student in Group \(A\) is \(X\), where \(X\) follows a normal distribution with mean \(55\) and standard deviation \(\sigma\). The Mathematics examination score of a randomly chosen student in Group \(B\) is \(Y\), where \(Y\) follows a normal distribution with mean \(45\) and standard deviation \(10\).

1. It is known that \(2\mathrm{P}\left( X>45 \right)=5\mathrm{P}\left( X>65 \right)\). Show that \(\sigma =17.7\), correct to \(3\) significance figures.[3]
2. Find the probability that total score of \(3\) randomly selected students from Group \(A\) differ from \(4\) times the score of a randomly selected student from Group \(B\) by at most \(10\).[4]
3. Find the probability that the mean score of \(3\) randomly selected students from Group \(A\) and \(4\) randomly selected students from Group \(B\) is at least \(50\).[3]

The Mathematics examination scores of students in Group \(C\) are found to have a mean of \(40\) and standard deviation of \(25\). Explain why the examination scores of students in Group \(C\) are unlikely to be normally distributed.[1]