Problem Statement
The mass $X$ of adult male hedgehogs in an English sanctuary is normally distributed with unknown mean $\mu$ and standard deviation $\sigma$.
It is given that $P(X < 650) = 0.12$ and $P(X > 920) = 0.05$.
**(a)** [3 marks] Write down two standardized equations involving $\mu$ and $\sigma$.
**(b)** [4 marks] Find the value of $\mu$ and the value of $\sigma$.
Verified Solution & Marking Scheme
Part (a) [M1]
Use GDC invNorm: $z_1 = \text{invNorm}(0.12) = -1.174986$
Part (a) [A1]
Equation 1: $\frac{650 - \mu}{\sigma} = -1.175 \implies 650 - \mu = -1.175\sigma$
Part (a) [M1]
For $P(X > 920) = 0.05 \implies P(X < 920) = 0.95 \implies z_2 = \text{invNorm}(0.95) = 1.64485$
Part (a) [A1]
Equation 2: $\frac{920 - \mu}{\sigma} = 1.645 \implies 920 - \mu = 1.645\sigma$
Part (b) [M1]
Subtract Eq 1 from Eq 2: $(920 - 650) = (1.64485 - (-1.17499))\sigma \implies 270 = 2.81984\sigma$
Part (b) [A1]
$\sigma = \frac{270}{2.81984} = 95.7\text{ grams}$
Part (b) [M1, A1]
$\mu = 920 - 1.64485(95.75) = 762.5\text{ grams}$ (accept $\mu = 763, \sigma = 95.7$)