Find power function or power value

4. The number of accidents that occur at a crossroads has a mean of 3 per month. In order to improve the flow of traffic the priority given to traffic is changed. Colin believes that since the change in priority the number of accidents has increased. He tests his belief by recording the number of accidents \(x\) in the month following the change. Colin sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 3\) and \(\mathrm { H } _ { 1 } : \lambda > 3\), where \(\lambda\) is the mean number of accidents per month, and rejects the null hypothesis if \(x > 4\).

1. Find the size of the test. The table gives the values of the power function of the test to two decimal places.

   \(\lambda\)	4	5	6	7
   Power	\(r\)	0.56	\(s\)	0.83

2. Calculate the value of \(r\) and the value of \(s\).
3. Comment on the suitability of the test when \(\lambda = 4\).

5. Rolls of cloth delivered to a factory contain defects at an average rate of \(\lambda\) per metre. A quality assurance manager selects a random sample of 15 metres of cloth from each delivery to test whether or not there is evidence that \(\lambda > 0.3\). The criterion that the manager uses for rejecting the hypothesis that \(\lambda = 0.3\) is that there are 9 or more defects in the sample.

1. Find the size of the test. Table 1 gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]

   \(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
   Power	0.15	0.34	\(r\)	0.72	0.85	0.92	0.96

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find the value of \(r\). The manager would like to design a test, of whether or not \(\lambda > 0.3\), that uses a smaller length of cloth. He chooses a length of 10 m and requires the probability of a type I error to be less than \(10 \%\).
3. Find the criterion to reject the hypothesis that \(\lambda = 0.3\) which makes the test as powerful as possible.
4. Hence state the size of this second test. Table 2 gives some values, to 2 decimal places, of the power function for the test in part (c). \begin{table}[h]

   \(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
   Power	0.21	0.38	0.55	0.70	\(s\)	0.88	0.93

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
5. Find the value of \(s\).
6. Using the same axes, on graph paper draw the graphs of the power functions of these two tests.
7. 1. State the value of \(\lambda\) where the graphs cross.
   2. Explain the significance of \(\lambda\) being greater than this value. The cost of wrongly rejecting a delivery of cloth with \(\lambda = 0.3\) is low. Deliveries of cloth with \(\lambda > 0.7\) are unusual.
8. Suggest, giving your reasons, which the test manager should adopt.
   (2)

5. The number of tornadoes per year to hit a particular town follows a Poisson distribution with mean \(\lambda\). A weatherman claims that due to climate changes the mean number of tornadoes per year has decreased. He records the number of tornadoes \(x\) to hit the town last year. To test the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 7\) and \(\mathrm { H } _ { 1 } : \lambda < 7\), a critical region of \(x \leq 3\) is used.

1. Find, in terms \(\lambda\) the power function of this test.
2. Find the size of this test.
3. Find the probability of a Type II error when \(\lambda = 4\).

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}