May/June 2026 Paper 52

The lengths of the leaves of a certain type of tree are normally distributed with mean $\mu \text{cm}$ and standard deviation $\sigma \text{cm}$. It is known that 20% of the leaves have a length less than 3.60 cm and 70% of the leaves have a length greater than 5.40 cm. Find the value of $\mu$ and the value of $\sigma$.

The lengths of 600 randomly chosen leaves of a different type of tree are normally distributed with mean 4.63 cm and standard deviation 0.55 cm. How many of these 600 leaves would you expect to have a length within 0.6 cm of the mean length?