Question 11 OR alternative.

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.

\[\sum x=10.56\quad \sum x^2=14.1775\quad \sum y=12.39\quad \sum y^2=15.894\]

A scientist claims that ducks on lake \(A\) are heavier on average than ducks on lake \(B\).

(i) 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

\[\sum x=10.24\quad\text{and}\quad\sum(x-\bar{x})^2=0.294,\]

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.

(ii) Find the greatest possible value of \(p\).