6. A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet.
The data on the height, \(V\) cm, of the 80 boys who had always eaten a vegetarian diet is summarised by
$$\Sigma V = 10367 , \quad \Sigma V ^ { 2 } = 1350314 .$$
- Calculate unbiased estimates of the mean and variance of \(V\).
The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and \(96.24 \mathrm {~cm} ^ { 2 }\) respectively.
- Stating your hypotheses clearly, test at the \(1 \%\) level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet.
(8 marks)