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 \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\) and rejects the null hypothesis if \(x > 1\).
Find the size of the test.
Show that the power function of the test is
$$1 - ( 1 - p ) ^ { 4 } ( 1 + 4 p )$$
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.
Find the probability of a Type I error using the deputy's test.
\section*{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.
\(p\)
0.10
0.15
0.20
0.25
Power
0.07
\(s\)
0.32
0.47
Find the value of \(s\).
The graph of the power function for the manager's test is shown in Figure 1.
\begin{figure}[h]
On the same axes, draw the graph of the power function for the deputy's test.
State the value of \(p\) where these graphs intersect.
Compare the effectiveness of the two tests if \(p\) is greater than this value.
The deputy suggests that they should use his sampling method rather than the manager's.
Give a reason why the manager might not agree to this change.