2205 TJC P2 Q8

An examination consists of two parts: a written paper and a lab-based practical. Marks obtained by a randomly chosen candidate follow normal distribution with means and standard deviations as shown in the following table.

It is given that the \(95\)th percentile score for the written paper is \(85\) marks.

1. Show that the value of \(\sigma\) is \(13.983\), correct to \(5\) significant figures.[1]
2. Find the probability that the difference between the marks obtained in the written paper and in the lab-based practical of a randomly chosen candidate is at least \(9\).[3]

The overall mark obtained for the examination is the total of \(60\)% of the mark obtained from written paper and \(40\)% of the mark obtained from lab-based practical.

3. Find the probability that the overall mark of a randomly chosen candidate is less than \(60\).[3]
4. Find the smallest value of \(n\) such that the probability that the mean overall mark of \(n\) randomly chosen candidates being at least \(58\) is at least \(0.95\).[3]
5. State an assumption needed for the calculations in part (b) to (d) to be valid and explain why the assumption may not be valid in practice.[2]