In a particular country, large numbers of ducks live on lakes \(A\) and \(B\). The mass, in kg, of a duck on lake \(A\) is denoted by \(x\) and the mass, in kg, of a duck on lake \(B\) is denoted by \(y\). A random sample of 8 ducks is taken from lake \(A\) and a random sample of 10 ducks is taken from lake \(B\). Their masses are summarised as follows. $$\Sigma x = 10.56 \quad \Sigma x ^ { 2 } = 14.1775 \quad \Sigma y = 12.39 \quad \Sigma y ^ { 2 } = 15.894$$ A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\).

1. Test, at the \(10 \%\) significance level, whether the scientist's claim is justified. You should assume that both distributions are normal and that their variances are equal.
   A second random sample of 8 ducks is taken from lake \(A\) and their masses are summarised as $$\Sigma x = 10.24 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.294$$ where \(\bar { x }\) is the sample mean. The scientist now claims that the population mean mass of ducks on lake \(A\) is greater than \(p \mathrm {~kg}\). A test of this claim is carried out at the \(10 \%\) significance level, using only this second sample from lake \(A\). This test supports the scientist's claim.
2. Find the greatest possible value of \(p\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.