2017 HCI P2 Q11 [Modified]

Yummy Berries Farm produces blueberries and raspberries packed in boxes.

1. Yummy Berries Farm claims that the mean mass, \(x\) grams, of each box of blueberries is no less than \(125\) grams. After receiving complaints from customers, the Consumers Association of Singapore (CASE) took a random sample of \(50\) boxes of blueberries from Yummy Berries Farm and the mass of each box was recorded. The data obtained are summarised in the table.
   
   1. Find the unbiased estimates of the population mean and variance.
   2. Test, at the \(10\%\) level of significance, whether Yummy Berries Farm has overstated its claim.
      State, giving a reason, whether any assumptions about the masses of boxes of blueberries are needed in order for the test to be valid.
2. The masses of boxes of raspberries, each of \(y\) grams, are assumed to have a mean of \(170\) grams with standard deviation \(15\) grams. CASE took a large random sample of \(n\) boxes of raspberries and the mean mass of boxes of raspberries from the sample is found to be \(165\) grams. A test is to be carried out at the \(5\%\) level of significance to determine if the mean mass of the boxes of raspberries is not \(170\) grams.
   Find the minimum number of boxes of raspberries to be taken for which the result of the test would be to reject the null hypothesis.