Problem Statement
The masses of avocados harvested from an orchard follow a normal distribution $X \sim N(\mu, \sigma^2)$. It is known that $10\%$ of avocados weigh less than $130\text{ g}$ and $5\%$ weigh more than $210\text{ g}$.
**(a)** [3 marks] Write down two simultaneous equations in terms of $\mu$ and $\sigma$ using standard normal variables.
**(b)** [4 marks] Hence, calculate the value of the mean $\mu$ and standard deviation $\sigma$ correct to 3 significant figures.
Verified Solution & Marking Scheme
Part (a) [M1]
Using inverse normal: $P(Z < z_1) = 0.10 \implies z_1 = -1.2816$
Part (a) [A1]
$\frac{130 - \mu}{\sigma} = -1.2816 \implies \mu - 1.2816\sigma = 130$
Part (a) [M1]
$P(Z > z_2) = 0.05 \implies P(Z < z_2) = 0.95 \implies z_2 = 1.6449$
Part (a) [A1]
$\frac{210 - \mu}{\sigma} = 1.6449 \implies \mu + 1.6449\sigma = 210$
Part (b) [M1]
Subtracting equations: $(1.6449 + 1.2816)\sigma = 210 - 130 = 80$
Part (b) [A1]
$2.9265\sigma = 80 \implies \sigma = 27.336 \approx 27.3\text{ g}$
Part (b) [A1]
$\mu = 130 + 1.2816(27.336) = 165.03 \approx 165\text{ g}$