| Date | November 2016 | Marks available | 2 | Reference code | 16N.2.hl.TZ0.8 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 8 | Adapted from | N/A |
Question
A random variable \(X\) is normally distributed with mean \(\mu \) and standard deviation \(\sigma \), such that \({\text{P}}(X < 30.31) = 0.1180\) and \({\text{P}}(X > 42.52) = 0.3060\).
Find \(\mu \) and \(\sigma \).
Find \({\text{P}}\left( {\left| {X - \mu } \right| < 1.2\sigma } \right)\).
Markscheme
\({\text{P}}(X < 42.52) = 0.6940\) (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]
