2011 H2 Mathematics Paper 2 Question 5

The Normal Distribution

Answers

μ=53.7,  σ=8.32{\mu = 53.7, \; \sigma = 8.32}

Full solutions

X∼N(μ,σ2)Z=X−μσ∼N(0,1)\begin{gather*} X \sim \textrm{N}(\mu, \sigma^2) \\ Z = \frac{X-\mu}{\sigma} \sim \textrm{N}(0, 1) \end{gather*}
P(X<40.0)=0.05P(Z<40−μσ)=0.05\begin{align*} \textrm{P}(X < 40.0) &= 0.05 \\ \textrm{P}\left(Z < \frac{40-\mu}{\sigma}\right) &= 0.05 \\ \end{align*}
40−μσ=−1.6449\frac{40-\mu}{\sigma} = -1.6449
μ−1.6449σ=40\begin{equation} \mu -1.6449\sigma = 40 \end{equation}
P(X<70.0)=0.975P(Z<70−μσ)=0.975\begin{align*} \textrm{P}(X < 70.0) &= 0.975 \\ \textrm{P}\left(Z < \frac{70-\mu}{\sigma}\right) &= 0.975 \\ \end{align*}
70−μσ=1.9600\frac{70-\mu}{\sigma} = 1.9600
μ+1.9600σ=70\begin{equation} \mu + 1.9600\sigma = 70 \end{equation}
Solving (1){(1)} and (2){(2)} simultaneously,
μ=53.7 (3 sf)  ■σ=8.32 (3 sf)  ■\begin{align*} \mu &= 53.7 \textrm{ (3 sf)} \; \blacksquare \\ \sigma &= 8.32 \textrm{ (3 sf)} \; \blacksquare \\ \end{align*}