4. A proportion \(p\) of letters sent by a company are incorrectly addressed and if \(p\) is thought to be greater than 0.05 then action is taken.
Using \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\), a manager from the company takes a random sample of 40 letters and rejects \(\mathrm { H } _ { 0 }\) if the number of incorrectly addressed letters is more than 3 .

1. Find the size of this test.
2. Find the probability of a Type II error in the case where \(p\) is in fact 0.10 Table 1 below gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]

   \(p\)	0.075	0.100	0.125	0.150	0.175	0.200	0.225
   Power	0.35	\(s\)	0.75	0.87	0.94	0.97	0.99

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
3. Write down the value of \(s\). A visiting consultant uses an alternative system to test the same hypotheses. A sample of 15 letters is taken. If these are all correctly addressed then \(\mathrm { H } _ { 0 }\) is accepted. If 2 or more are found to have been incorrectly addressed then \(\mathrm { H } _ { 0 }\) is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and \(\mathrm { H } _ { 0 }\) is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise \(\mathrm { H } _ { 0 }\) is accepted.
4. Find the size of the test used by the consultant. \section*{Question 4 continues on page 8} For your convenience Table 1 is repeated here \begin{table}[h]

   \(p\)	0.075	0.100	0.125	0.150	0.175	0.200	0.225
   Power	0.35	\(s\)	0.75	0.87	0.94	0.97	0.99

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table} Figure 1 shows the graph of the power function of the test used by the consultant. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8dfc721d-4782-4482-9976-11189370f3b7-07_1712_1673_660_130} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}
5. On Figure 1 draw the graph of the power function of the manager's test.
   (2)
6. State, giving your reasons, which test you would recommend.
   (2)