3. A study was made of the heights of boys of different ages in Lancashire.
The study concluded that the heights of 13 year-old boys are normally distributed with a mean of 156 cm and a variance of \(73 \mathrm {~cm} ^ { 2 }\).
Find the probability that a 13 year-old boy chosen at random will be
- more than 165 cm tall,
- between 156 and 165 cm tall.
The study also concluded that the heights of 14 year-old boys are normally distributed with a mean of 160 cm and a variance of \(79 \mathrm {~cm} ^ { 2 }\).
One 13 year-old and one 14 year-old boy are chosen at random.
- Find the probability that both boys are more than 165 cm tall.
- State, with a reason, whether the probability that the combined height of the two boys is more than 330 cm is more or less than your answer to part (c).
(2 marks)