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 .
\begin{enumerate}[label=(\alph*)]
\item Find the size of this test.
\item 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]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$p$ & 0.075 & 0.100 & 0.125 & 0.150 & 0.175 & 0.200 & 0.225 \\
\hline
Power & 0.35 & $s$ & 0.75 & 0.87 & 0.94 & 0.97 & 0.99 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
\item 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.
\item 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]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$p$ & 0.075 & 0.100 & 0.125 & 0.150 & 0.175 & 0.200 & 0.225 \\
\hline
Power & 0.35 & $s$ & 0.75 & 0.87 & 0.94 & 0.97 & 0.99 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}

Figure 1 shows the graph of the power function of the test used by the consultant.

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

\hfill \mbox{\textit{Edexcel S4 2011 Q4 [12]}}