The volumes of juice in bottles of Apricola are normally distributed. In a random sample of 8 bottles, the volumes of juice, in millilitres, were found to be as follows.

332,  334,  330,  328,  331,  332,  329,  333

a) Find unbiased estimates of the population mean and variance.

A random sample of 50 bottles of Apricola gave unbiased estimates of 331 millilitres and $4.20{\text{ }}millilitre{s^2}$ for the population mean and variance respectively.

b) Use this sample of size 50 to calculate a 98% confidence interval for the population mean.

c) The manufacturer claims that the mean volume of juice in all bottles is 333 millilitres. State, with a reason, whether your answer to part (a) supports this claim.