1. (a) Explain briefly why it is often useful to take a sample from a population.
   (b) Suggest a suitable sampling frame for a local council to use to survey attitudes towards a proposed new shopping centre.
2. A certain type of lettuce seed has a \(12 \%\) failure rate for germination. In a new sample of 25 genetically modified seeds, only 1 did not germinate.
   Clearly stating your hypotheses, test, at the \(1 \%\) significance level, whether the GM seeds are better.
3. A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5 .
   (a) Find \(\mathrm { P } ( X = 0 )\).
   (b) In 100 measurements, the value 0 occurs three times. Find the highest significance level at which you should reject the original hypothesis in favour of \(\lambda < 5\).
4. The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function

$$\begin{array} { l l } \mathrm { f } ( t ) = k ( 10 - t ) & 0 \leq t \leq 10
\mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$ (a) Sketch the graph of \(\mathrm { f } ( t )\) and find the value of \(k\).
(b) Find the mean value of \(T\).
(c) Find the 95th percentile of \(T\).
(d) State whether you consider this function to be a sensible model for \(T\) and suggest how it could be modified to provide a better model.