N2015 P2 Q8

A market stall sells pineapples which have masses that are normally distributed. The stall owner claims that the mean mass of the pineapples is at least \(0.9\text{ kg}\). Nur buys a random selection of \(8\) pineapples from the stall. The \(8\) pineapples have masses, in \(\text{kg}\), as follows.

\[0.80\quad1.00\quad0.82\quad0.85\quad0.93\quad0.96\quad0.81\quad0.89\]

Find unbiased estimates of the population mean and variance of the mass of pineapples. Test at the \(10\%\) level of significance whether there is any evidence to doubt the stall owner's claim.[7]