2013 CJC P2 Q12

The waiting times, in minutes, for orders of Chili Crab and Pepper Crab at a popular hawker stall are modelled as having independent normal distributions with means and standard deviations as shown in the following table:

1. If the probability that one randomly chosen order of Pepper Crab takes a longer waiting time than one randomly chosen order of Chili Crab is \(0.290\), show that \(\mu =12.0\), correct to \(3\) significant figures.[3]

Customers who drive park their cars at a nearby car park at \(\$0.05\) per minute. The walking time between the car park and the hawker stall is negligible. The time taken by a randomly chosen customer to finish his meal is normally distributed with mean \(20\) minutes and standard deviation \(4\) minutes.

2. Find the probability that a randomly chosen customer who parked at the car park, bought one order of Pepper Crab, finished his meal and immediately drove off, spent more than \(\$2\) on car park charges.[2]
3. A random sample of \(n\) customers who each make one order of Pepper Crab is taken. Given that the probability that their mean waiting time exceeds \(10.5\) minutes is at most \(0.0385\), find the smallest possible value of \(n\). Explain briefly if any assumption is needed for the above calculation.[4]