Question 11 OR alternative.

A farmer grows two different types of cherries, Type \(A\) and Type \(B\). He assumes that the masses of each type are normally distributed. He chooses a random sample of 8 cherries of Type \(A\). He finds that the sample mean mass is 15.1 g and that a \(95\%\) confidence interval for the population mean mass, \(\mu\) g, is \(13.5\leqslant\mu\leqslant16.7\).

(i) Find an unbiased estimate for the population variance of the masses of cherries of Type \(A\).

The farmer now chooses a random sample of 6 cherries of Type \(B\) and records their masses as follows.

(ii) Test at the \(5\%\) significance level whether the mean mass of cherries of Type \(B\) is less than the mean mass of cherries of Type \(A\). You should assume that the population variances for the two types of cherry are equal.