A manager in a sweet factory believes that the machines are working incorrectly and the proportion $p$ of underweight bags of sweets is more than 5\%. He decides to test this by randomly selecting a sample of 5 bags and recording the number $X$ that are underweight. The manager sets up the hypotheses H$_0$: $p = 0.05$ and H$_1$: $p > 0.05$ and rejects the null hypothesis if $x > 1$.

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

\item Show that the power function of the test is
$$1 - (1-p)^4(1+4p)$$
[3]
\end{enumerate}

The manager goes on holiday and his deputy checks the production by randomly selecting a sample of 10 bags of sweets. He rejects the hypothesis that $p = 0.05$ if more than 2 underweight bags are found in the sample.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{2}
\item Find the probability of a Type I error using the deputy's test.
[2]
\end{enumerate}

Question 3 continues on page 12

The table below gives some values, to 2 decimal places, of the power function for the deputy's test.

\begin{tabular}{|c|c|c|c|c|}
\hline
$p$ & 0.10 & 0.15 & 0.20 & 0.25 \\
\hline
Power & 0.07 & $s$ & 0.32 & 0.47 \\
\hline
\end{tabular}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{3}
\item Find the value of $s$.
[1]
\end{enumerate}

The graph of the power function for the manager's test is shown in Figure 1.

\includegraphics{figure_1}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{4}
\item On the same axes, draw the graph of the power function for the deputy's test.
[1]

\item (i) State the value of $p$ where these graphs intersect.

(ii) Compare the effectiveness of the two tests if $p$ is greater than this value.
[2]
\end{enumerate}

The deputy suggests that they should use his sampling method rather than the manager's.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{6}
\item Give a reason why the manager might not agree to this change.
[1]
\end{enumerate}

\hfill \mbox{\textit{Edexcel S4  Q3 [12]}}