BMX racing is a sprint sport where races range from around 30 to 45 seconds depending on the length of the track. Given races are over in such a short time, the start - commonly referred to as gate starts due to everyone lining up on a gate, are critical to success in the sport. A coach with a particular interest in data analysis coaching an elite athlete records data at a local race track for the time it takes their athlete to reach the bottom of the starting hill. The coach has found that the hill times for their athlete follow a normal distribution with mean 2.3 seconds and standard deviation 0.02 seconds.

(a) [2 marks] What hill time distinguishes the slowest 10% of hill times?

(b) [2 marks] What is the probability that a randomly selected hill time is less than 2.27 seconds?

(c) [4 marks] Let X be the number of hill times that are less than 2.27 seconds from 10 separate starts. Assuming the athlete is well rested between starts, what distribution might you consider X to follow. Justify your response by indicating any assumptions you have made.

(d) [3 marks] The coach is unable to make it to training one night and asks for the athlete to complete 10 gate starts. The coach asks the athlete to review the times and let him know how many out of 10 were faster than 2.27 seconds. The athlete advises the coach that on 4 of the 10 starts they were faster than 2.27 seconds. Calculate the probability of the athlete beating a hill time of 2.27 seconds on 4 or more of the 10 gate starts.

(e) [2 marks] With reference to the probability in part (d), do you believe the athlete?