Problem Statement
A random variable $X$ is normally distributed with mean $\mu$ and standard deviation $\sigma$.
It is known that $P(X < 45) = 0.12$ and $P(X > 72) = 0.05$.
**(a)** [6 marks] Find the value of $\mu$ and the value of $\sigma$, giving your answers to three significant figures.
Verified Solution & Marking Scheme
Part (a) [M1]
Use InvNorm to find standardized $z$-values: for $P(Z < z_1) = 0.12 \implies z_1 = -1.174986$
Part (a) [M1]
For $P(Z > z_2) = 0.05 \implies P(Z < z_2) = 0.95 \implies z_2 = 1.644853$
Part (a) [A1]
Set up simultaneous equations: $\frac{45 - \mu}{\sigma} = -1.174986 \implies 45 - \mu = -1.175\sigma$
Part (a) [A1]
$\frac{72 - \mu}{\sigma} = 1.644853 \implies 72 - \mu = 1.645\sigma$
Part (a) [M1]
Subtract equations: $(72 - 45) = (1.644853 - (-1.174986))\sigma \implies 27 = 2.8198\sigma \implies \sigma = 9.575$
Part (a) [A1]
Substitute back: $\mu = 72 - 1.644853(9.575) = 56.25$. Therefore, $\mu = 56.3$ and $\sigma = 9.58$ (to 3 s.f.)