2015 AJC P2 Q7

A nursery supplies plants to Evergreen Pte. Ltd. The heights, in centimetres, of plants supplied may be taken to follow the distribution\(\mathrm{N}\left( \mu ,{{\sigma }^{2}} \right)\).

Over the years, Evergreen rejects \(6\%\) of the plants supplied as too short and \(4\%\) as too tall. It is known that Evergreen only accepts plants of heights between \(25\) cm and \(91\) cm.

Show that \(\sigma =20\) correct to the nearest integer. Hence find the median height of the plants accepted by Evergreen.[5]

Evergreen inspects the plants supplied in batches of \(10\). Find the probability that a total of at most \(4\) plants are rejected in two randomly chosen batches.[2]

Three children, Anna, Bella and Cinderella, each chooses a plant at random from the nursery. Find the probability that the average height of the three plants is at least twice the average height of the plants chosen by Anna and Bella.[3]