Problem Statement
The weight of avocados produced on an orchard is normally distributed with mean $\mu$ and standard deviation $\sigma$. It is known that $12\%$ of avocados weigh less than $140\text{ g}$, and $5\%$ of avocados weigh more than $195\text{ g}$.
**(a)** [3 marks] Write down two simultaneous equations in terms of $\mu$ and $\sigma$ using standard normal $z$-values.
**(b)** [3 marks] Hence, find the values of $\mu$ and $\sigma$ correct to 3 significant figures.
Verified Solution & Marking Scheme
Part (a) [M1]
Find $z_1$ such that $\Phi(z_1) = 0.12 \implies z_1 = \text{invNorm}(0.12) \approx -1.17499$
Part (a) [M1]
Find $z_2$ such that $P(Z > z_2) = 0.05 \implies \Phi(z_2) = 0.95 \implies z_2 = \text{invNorm}(0.95) \approx 1.64485$
Part (a) [A1]
Equations: $\frac{140 - \mu}{\sigma} = -1.175 \implies 140 - \mu = -1.175\sigma$ and $\frac{195 - \mu}{\sigma} = 1.645 \implies 195 - \mu = 1.645\sigma$
Part (b) [M1]
Subtract first equation from second: $(195 - 140) = (1.64485 - (-1.17499))\sigma \implies 55 = 2.81984\sigma$
Part (b) [A1]
$\sigma = \frac{55}{2.81984} \approx 19.5\text{ g}$ (3 s.f.)
Part (b) [A1]
$\mu = 140 + 1.17499(19.505) \approx 163\text{ g}$ (3 s.f.)