Every Saturday, a particular community holds a ‘Puzzle’ event to raise money for a new Leisure Centre. Competitors attempt to solve a puzzle as quickly as possible.

Last Saturday, 600 competitors took part. The times taken to complete the puzzle were normally distributed with mean 32.4 minutes and standard deviation 2.5 minutes.

(a) How many competitors would you expect to have times within 1.2 minutes of the mean time? [4]

In this Saturday’s event, 60% of the competitors had times less than 36.0 minutes.

(b) 9 competitors who took part in this Saturday’s event are selected at random. Find the probability that at least 2 and fewer than 8 of these competitors had times less than 36.0 minutes. [3]

(c) 80 competitors who took part in this Saturday’s event are selected at random. Use a suitable approximation to find the probability that more than 50 of these competitors had times less than 36.0 minutes. [5]