2012 AJC MYCT P2 Q4

The owner of a fruit stall, Mr Lee, bought a large batch of durians from a durian farm.

The weights (in kg) of a sample of six durians from this batch are

\(1.0\),        \(1.1\),       \(1.3\),       \(1.7\),       \(a\),       \(b\).

It is known that the unbiased estimates for the population mean and variance based on this sample are \(1.25\)kg and \(0.095\)kg\(^{2}\) respectively.

1. Find the values of \(a\) and \(b\).[4]
2. Assuming that the weight of durians from Robert’s farm follows a normal distribution with mean \(1.60\)kg and standard deviation \(0.3\)kg, carry out an appropriate test at the \(5%\) level to find out whether there is significant evidence that this batch of durians is not taken from Robert’s farm.[4]

The mean and standard deviation of the weight of papayas from Robert’s farm are \(1.2\)kg and \(0.45\)kg respectively. Mr Lee bought a batch of papayas from Robert’s farm. Find the least number of papayas in the batch such that the probability of the average weight exceeding \(1.3\)kg is less than \(0.05\).[3]