5. Rolls of cloth delivered to a factory contain defects at an average rate of $\lambda$ 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.

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

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$\lambda$ & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\
\hline
Power & 0.15 & 0.34 & $r$ & 0.72 & 0.85 & 0.92 & 0.96 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
\item Find the value of $r$.

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 $10 \%$.
\item Find the criterion to reject the hypothesis that $\lambda = 0.3$ which makes the test as powerful as possible.
\item Hence state the size of this second test.

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

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$\lambda$ & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\
\hline
Power & 0.21 & 0.38 & 0.55 & 0.70 & $s$ & 0.88 & 0.93 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{center}
\end{table}
\item Find the value of $s$.
\item Using the same axes, on graph paper draw the graphs of the power functions of these two tests.
\item \begin{enumerate}[label=(\roman*)]
\item State the value of $\lambda$ where the graphs cross.
\item Explain the significance of $\lambda$ being greater than this value.

The cost of wrongly rejecting a delivery of cloth with $\lambda = 0.3$ is low. Deliveries of cloth with $\lambda > 0.7$ are unusual.
\end{enumerate}\item Suggest, giving your reasons, which the test manager should adopt.\\
(2)
\end{enumerate}

\hfill \mbox{\textit{Edexcel S4 2006 Q5 [17]}}