Bob measured the heights of 63 students. After analysis, he conjectured that the height, \(H\) , of the students could be modelled by a normal distribution with mean 166.5 cm and standard deviation 5 cm.

(a)     Based on this assumption, estimate the number of these students whose height is at least 170 cm.

Later Bob noticed that the tape he had used to measure the heights was faulty as it started at the 5 cm mark and not at the zero mark.

(b)     What are the correct values of the mean and variance of the distribution of the heights of these students?

(a)     \(H{\text{ \~ N}}\)(\(166.5\), \({5^2}\) )

\({\text{P}}(H \geqslant 170) = 0.242...\)     (M1)(A1)

\(0.242... \times 63 = 15.2\)     A1

so, approximately \(15\) students

(b)     correct mean: \(161.5\) (cm)     A1

variance remains the same, i.e. 25 (cm²)     A2

[6 marks]

A surprising number of students lacked the basic knowledge of the normal distribution and were unable to answer the first part of this question. Those students who showed a knowledge of the topic tended to answer the question well. In part (b) many students either had a misunderstanding of the difference between variance and standard deviation, or did not read the question properly.