Sequential or two-stage test design

5. Water is tested at various stages during a purification process by an environmental scientist. A certain organism occurs randomly in the water at a rate of \(\lambda\) every 10 ml . The scientist selects a random sample of 20 ml of water to check whether there is evidence that \(\lambda\) is greater than 1 . The criterion the scientist uses for rejecting the hypothesis that \(\lambda = 1\) is that there are 4 or more organisms in the sample of 20 ml .

1. Find the size of the test.
2. When \(\lambda = 2.5\) find P (Type II error). A statistician suggests using an alternative test. The statistician's test involves taking a random sample of 10 ml and rejecting the hypothesis that \(\lambda = 1\) if 2 or more organisms are present but accepting the hypothesis if no organisms are in the sample. If only 1 organism is found then a second random sample of 10 ml is taken and the hypothesis is rejected if 2 or more organisms are present, otherwise the hypothesis is accepted.
3. Show that the power of the statistician's test is given by $$1 - \mathrm { e } ^ { - \lambda } - \lambda ( 1 + \lambda ) \mathrm { e } ^ { - 2 \lambda }$$ Table 1 below gives some values, to 2 decimal places, of the power function of the statistician’s test. \begin{table}[h] \end{table} Table 1 Figure 1 shows a graph of the power function for the scientist's test.
4. On the same axes draw the graph of the power function for the statistician’s test. Given that it takes 20 minutes to collect and test a 20 ml sample and 15 minutes to collect and test a 10 ml sample
5. show that the expected time of the statistician's test is slower than the scientist's test for \(\lambda \mathrm { e } ^ { - \lambda } > \frac { 1 } { 3 }\)
6. By considering the times when \(\lambda = 1\) and \(\lambda = 2\) together with the power curves in part (e) suggest, giving a reason, which test you would use.
   (2) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{399f7507-4878-45ad-b77e-02ebd807ed75-10_1185_1157_1452_392} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{399f7507-4878-45ad-b77e-02ebd807ed75-11_81_47_2622_1886}

2. The cloth produced by a certain manufacturer has defects that occur randomly at a constant rate of \(\lambda\) per square metre. If \(\lambda\) is thought to be greater than 1.5 then action has to be taken. Using \(\mathrm { H } _ { 0 } : \lambda = 1.5\) and \(\mathrm { H } _ { 1 } : \lambda > 1.5\) a quality control officer takes a \(4 \mathrm {~m} ^ { 2 }\) sample of cloth and rejects \(\mathrm { H } _ { 0 }\) if there are 11 or more defects. If there are 8 or fewer defects she accepts \(\mathrm { H } _ { 0 }\). If there are 9 or 10 defects a second sample of \(4 \mathrm {~m} ^ { 2 }\) is taken and \(\mathrm { H } _ { 0 }\) is rejected if there are 11 or more defects in this second sample, otherwise it is accepted.

1. Find the size of this test.
2. Find the power of this test when \(\lambda = 2\)