2013 TJC P2 Q9

A particular brand of orange juice is sold in bottles of two sizes. The amount in each bottle, in millilitres, is normally distributed as shown in the table below.

1. It is given that \(80%\) of small bottles contains between \(300\) ml and \(312\) ml of orange juice. Find the value of \(\sigma \).[2]
2. Find the probability that the average amount of orange juice in three randomly chosen small bottles is more than half the amount of orange juice in a randomly chosen medium bottle. [3]
3. Clara bought fifteen medium bottles of orange juice for a party. Find the least number of paper cups each of maximum capacity \(120\) ml that she must buy such that the probability that there are sufficient paper cups to contain the total amount of orange juice is at least \(95\%\). Assume that the paper cups are not to be reused and there is no wastage of orange juice. [4]