2022 EJC J2 MYE P2 Q10

In this question you should state the parameters of any distributions you use.

A teacher, while analysing the performance of a large cohort for a recent quiz, finds that the score of each student from the entire cohort follows the distribution \(\mathrm{N}\left( 50,{{\sigma }^{2}} \right)\). He also finds that \(95\%\) of the students scored between \(36\) and \(64\).

1. Find the value of \({{\sigma }^{2}}\).[2]

You may assume \(\sigma =7\) for the remaining parts of the question.

2. Three students are selected at random from the cohort. Find the probability that
   1. the sum of the scores of the first two students is less than \(1.5\) times the score of the third student,[3]
   2. the lowest score of the three students is more than \(53\).[2]
3. Find the least integer \(n\) such that there is at least a \(95\%\) chance that the average score of \(n\) randomly chosen students is at least \(48\).[3]
4. \(15\) students are selected at random. Find the probability that less than half of them scored at most \(55\) for the quiz.[3]