Rolls of cloth delivered to a factory contain defects at an average rate of 2 per metre. A quality assurance manager selects a random sample of 15 metres of cloth from each delivery to test whether or not there is evidence that $\lambda > 0.3$. The criterion that the manager uses for rejecting the hypothesis that $\lambda = 0.3$ is that there are 9 or more defects in the sample.

\begin{enumerate}[label=(\alph*)]
\item Find the size of the test. [2]
\end{enumerate}

Table 1 gives some values, to 2 decimal places, of the power function of this test.

\includegraphics{figure_5}

\begin{enumerate}[label=(\alph*)]
\item[(b)] Find the value of $r$. [2]
\end{enumerate}

The manager would like to design a test, of whether or not $\lambda > 0.3$, that uses a smaller length of cloth. He chooses a length of 10 m and requires the probability of a type I error to be less than 10\%.

\begin{enumerate}[label=(\alph*)]
\item[(c)] Find the criterion to reject the hypothesis that $\lambda = 0.3$ which makes the test as powerful as possible. [2]

\item Hence state the size of this second test. [1]
\end{enumerate}

Table 2 gives some values, to 2 decimal places, of the power function for the test in part (c).

\includegraphics{figure_5_table2}

\begin{enumerate}[label=(\alph*)]
\item[(e)] Find the value of $s$. [2]

\item Using the same axes, on graph paper draw the graphs of the power functions of these two tests. [4]

\item[(g)] State the value of $\lambda$ where the graphs cross.
\begin{enumerate}[label=(\roman*)]
\item Explain the significance of $\lambda$ where the graphs cross. [2]
\end{enumerate}
\end{enumerate}

There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm. Deliveries of cloth with $\lambda = 0.3$ are unusable.

\begin{enumerate}[label=(\alph*)]
\item[(h)] Suggest, giving your reasons, which test manager should adopt. [2]
\end{enumerate}

\hfill \mbox{\textit{Edexcel S4  Q5 [17]}}