3. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2 . The firm believes that the appointment of a new salesman will increase the number of houses sold. The firm tests its belief by recording the number of houses sold, \(x\), in the week following the appointment. The firm sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 2\) and \(\mathrm { H } _ { 1 } : \lambda > 2\), where \(\lambda\) is the mean number of houses sold per week, and rejects the null hypothesis if \(x \geqslant 3\)

1. Find the size of the test.
2. Show that the power function for this test is $$1 - \frac { 1 } { 2 } e ^ { - \lambda } \left( 2 + 2 \lambda + \lambda ^ { 2 } \right)$$ The table below gives the values of the power function to 2 decimal places. \begin{table}[h]

   \(\lambda\)	2.5	3.0	3.5	4.0	5.0	7.0
   Power	0.46	\(r\)	0.68	\(s\)	0.88	0.97

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
3. Calculate the values of \(r\) and \(s\).
4. Draw a graph of the power function on the graph paper provided on page 6
5. Find the range of values of \(\lambda\) for which the power of this test is greater than 0.6 For your convenience Table 1 is repeated here.

   \(\lambda\)	2.5	3.0	3.5	4.0	5.0	7.0
   Power	0.46	\(r\)	0.68	\(s\)	0.88	0.97

   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Table 1} \includegraphics[alt={},max width=\textwidth]{4f096806-33da-453f-a4c1-12be20d1a96d-06_2125_1603_614_166}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-07_72_47_2615_1886}