Find $\mu$ and $\sigma$.

Find ${\text{P}}\left( {\left| {X - \mu } \right| < 1.2\sigma } \right)$.

${\text{P}}(X < 42.52) = 0.6940$    (M1)

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

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

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

attempting to solve simultaneously     (M1)

$\mu  = 38.9$ and $\sigma  = 7.22$     A1

[6 marks]

${\text{P}}(\mu  - 1.2\sigma  < X < \mu  + 1.2\sigma )$ (or equivalent eg. $2{\text{P}}(\mu  < X < \mu  + 1.2\sigma )$)     (M1)

$= 0.770$    A1

Note: Award (M1)A1 for ${\text{P}}( - 1.2 < Z < 1.2) = 0.770$.

[2 marks]