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 H$_0$: $p = 0.05$ and H$_1$: $p > 0.05$, a manager from the company takes a random sample of 40 letters and rejects H$_0$ if the number of incorrectly addressed letters is more than 3.

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

\item Find the probability of a Type II error in the case where $p$ is in fact 0.10
[2]
\end{enumerate}

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

\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}

Table 1

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{2}
\item Write down the value of $s$.
[1]
\end{enumerate}

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 H$_0$ is accepted. If 2 or more are found to have been incorrectly addressed then H$_0$ is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and H$_0$ is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise H$_0$ is accepted.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{3}
\item Find the size of the test used by the consultant.
[3]
\end{enumerate}

Question 4 continues on page 8

\includegraphics{figure_1}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{4}
\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  Q4 [12]}}