The heights of students in a single year group in a large school can be modelled by a normal distribution.

It is given that 40% of the students are shorter than 1.62 m and 25% are taller than 1.79 m.

Find the mean and standard deviation of the heights of the students.

let the heights of the students be $X$

${\text{P}}(X < 1.62) = 0.4,{\text{ P}}(X > 1.79) = 0.25$     M1

Note:     Award M1 for either of the probabilities above.

${\text{P}}\left( {Z < \frac{{1.62 - \mu }}{\sigma }} \right) = 0.4,{\text{ P}}\left( {Z < \frac{{1.79 - \mu }}{\sigma }} \right) = 0.75$     M1

Note:     Award M1 for either of the expressions above.

$\frac{{1.62 - \mu }}{\sigma } =  - 0.2533 \ldots ,{\text{ }}\frac{{1.79 - \mu }}{\sigma } = 0.6744 \ldots$     M1A1

Note:     A1 for both values correct.

$\mu  = 1.67{\text{ (m)}},{\text{ }}\sigma  = 0.183{\text{ (m)}}$     A1A1

Note:     Accept answers that round to 1.7 (m) and 0.18 (m).

Note: Accept answers in centimetres.

[6 marks]

A large number of good solutions in this question, although candidates failing on the question failed at different stages. A number did not standardise the distribution correctly, and there were others who were unable to correctly solve the simultaneous equations. There were a notable number of otherwise good candidates who were unable to attempt the question, even though it is of a very standard type.