Standard +0.8 This question requires recognizing the symmetry property of the normal distribution to find μ = 60, then using the given probability to find σ ≈ 2.8125, and finally calculating P(G > 65) using standardization. While the symmetry insight is key, the multi-step calculation and need to work backwards from probabilities to parameters makes this moderately harder than a standard normal distribution question.
Standardise 65 with their \(\mu\) and \(\sigma\), allow \(\sqrt{\text{or}}\) cc errors
\(= 0.0458\)
A1
Final answer, a.r.t. 0.046, c.w.o.
[6]
$\mu = 60$ | B1 | $\mu = 60$ stated or implied, can be written down
$\frac{63.8 - \mu}{\sigma} = \Phi^{-1}(0.9) = 1.282$ | M1 | Standardise 63.8 or 56.2 with $\sigma$, allow $\sqrt{\text{or}}$ cc errors, equate to $\Phi^{-1}$ 1.282 (or 1.281 or 1.28) seen
$\sigma = 2.96(4)$ | A1 | $\sigma$, in range [2.96, 2.97], can be implied by what follows, not $\sigma^2$
$1 - \Phi\left(\frac{65-60}{2.964}\right) = 1 - \Phi(1.687)$ | M1 | Standardise 65 with their $\mu$ and $\sigma$, allow $\sqrt{\text{or}}$ cc errors
$= 0.0458$ | A1 | Final answer, a.r.t. 0.046, c.w.o.
| [6] |
The random variable $G$ has a normal distribution. It is known that
$$\text{P}(G < 56.2) = \text{P}(G > 63.8) = 0.1.$$
Find P($G > 65$). [6]
\hfill \mbox{\textit{OCR S2 2012 Q3 [6]}}