2 The manager of a large country estate is preparing to plant an area of woodland. He orders a large number of saplings (young trees) from a nursery. He selects a random sample of 12 of the saplings and measures their heights, which are as follows (in metres).
$$\begin{array} { l l l l l l l l l l l l }
0.63 & 0.62 & 0.58 & 0.56 & 0.59 & 0.62 & 0.64 & 0.58 & 0.55 & 0.61 & 0.56 & 0.52
\end{array}$$
- The manager requires that the mean height of saplings at planting is at least 0.6 metres. Carry out the usual \(t\) test to examine this, using a \(5 \%\) significance level. State your hypotheses and conclusion carefully. What assumption is needed for the test to be valid?
- Find a \(95 \%\) confidence interval for the true mean height of saplings. Explain carefully what is meant by a \(95 \%\) confidence interval.
- Suppose the assumption needed in part (i) cannot be justified. Identify an alternative test that the manager could carry out in order to check that the saplings meet his requirements, and state the null hypothesis for this test.