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.

1. 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]

   \(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
   Power	0.15	0.34	\(r\)	0.72	0.85	0.92	0.96

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. 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 \(10 \%\).
3. Find the criterion to reject the hypothesis that \(\lambda = 0.3\) which makes the test as powerful as possible.
4. 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]

   \(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
   Power	0.21	0.38	0.55	0.70	\(s\)	0.88	0.93

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
5. Find the value of \(s\).
6. Using the same axes, on graph paper draw the graphs of the power functions of these two tests.
7. 1. State the value of \(\lambda\) where the graphs cross.
   2. 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.
8. Suggest, giving your reasons, which the test manager should adopt.
   (2)