This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$P(Y < 42.52) = 0.6940$ (M1)

either $P\left(Z < \frac{{30.31 - \alpha }}{\beta }\right) = 0.1180$ or $P\left(Z < \frac{{42.52 - \alpha }}{\beta }\right) = 0.6940$ (M1)

$\frac{{30.31 - \alpha }}{\beta } = \underbrace {{\Phi ^{ - 1}}(0.1180)}_{ - 1.1850 \ldots }$ (A1)

$\frac{{42.52 - \alpha }}{\beta } = \underbrace {{\Phi ^{ - 1}}(0.6940)}_{0.5072 \ldots }$ (A1)

attempting to solve simultaneously (M1)

$\alpha = 38.9$ and $\beta = 7.22$ (A1)

[6 marks]

$P(\alpha - 1.2\beta < Y < \alpha + 1.2\beta)$ (or equivalent eg. $2P(\alpha < Y < \alpha + 1.2\beta)$) (M1) $= 0.770$ A1

**Note: **Award (M1)A1 for $P(-1.2 < Z < 1.2) = 0.770$. [2 marks]