Comment on test validity or assumptions

4. The number of emergency plumbing calls received per day by a local council was recorded over a period of 80 days. The results are summarised in the table below.

1. Show that the mean number of emergency plumbing calls received per day is 3.5 A council officer suggests that a Poisson distribution can be used to model the number of emergency plumbing calls received per day. He uses the mean from the sample above and calculates the expected frequencies shown in the table below.

   \(\boldsymbol { x }\)	0	1	2	3	4	5	6	7	8 or more
   Expected frequency	2.42	8.46	14.80	\(r\)	15.10	10.57	6.17	3.08	\(s\)

2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
3. Test, at the \(5 \%\) level of significance, whether or not the Poisson distribution is a suitable model for the number of emergency plumbing calls received per day. State your hypotheses clearly.

3. The number of accidents on a particular stretch of motorway was recorded each day for 200 consecutive days. The results are summarised in the following table.

1. Show that the mean number of accidents per day for these data is 1.6 A motorway supervisor believes that the number of accidents per day on this stretch of motorway can be modelled by a Poisson distribution. She uses the mean found in part (a) to calculate the expected frequencies for this model. Her results are given in the following table.

   Number of accidents	0	1	2	3	4	5 or more
   Frequency	40.38	64.61	\(r\)	27.57	11.03	\(s\)

2. Find the value of \(r\) and the value of \(s\), giving your answers to 2 decimal places.
3. Stating your hypotheses clearly, use a \(10 \%\) level of significance to test the motorway supervisor's belief. Show your working clearly.
   

The number of accidents, \(x\), that occur each day on a motorway are recorded over a period of 40 days. The results are shown in the following table.

Number of accidents	0	1	2	3	4	5	6	\(\geqslant 7\)
Observed frequency	3	5	8	10	5	7	2	0

\begin{enumerate}[label=(\roman*)] \item Show that the mean number of accidents each day is 2.95 and calculate the variance for this sample. Explain why these values suggest that a Poisson distribution might fit the data. [3] \item A Poisson distribution with mean 2.95, as found from the data, is used to calculate the expected frequencies, correct to 2 decimal places. The results are shown in the following table.

Number of accidents	0	1	2	3	4	5	6	\(\geqslant 7\)
Observed frequency	3	5	8	10	5	7	2	0
Expected frequency	2.09	6.18	9.11	8.96	6.61	3.90	1.92	1.23

Show how the expected frequency of 6.61 for \(x = 4\) is obtained. [2] \item Test at the 5% significance level the goodness of fit of this Poisson distribution to the data. [7] \end{enumerate]

The number of times per day a computer fails and has to be restarted is recorded for 200 days. The results are summarised in the table. Test whether or not a Poisson model is suitable to represent the number of restarts per day. Use a 5\% level of significance and state your hypothesis clearly. (Total 12 marks)

The number of goals scored by a football team is recorded for 100 games. The results are summarised in Table 1 below.

Number of goals	Frequency
0	40
1	33
2	14
3	8
4	5

Table 1

1. Calculate the mean number of goals scored per game. [2]

The manager claimed that the number of goals scored per match follows a Poisson distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2.

Number of goals	Expected Frequency
0	34.994
1	\(r\)
2	\(s\)
3	6.752
\(\geqslant 4\)	2.221

Table 2

1. Find the value of \(r\) and the value of \(s\) giving your answers to 3 decimal places. [3]
2. Stating your hypotheses clearly, use a 5\% level of significance to test the manager's claim. [7]

The number of hurricanes per year in a particular region was recorded over 80 years. The results are summarised in Table 1 below.

No of hurricanes, \(h\)	0	1	2	3	4	5	6	7
Frequency	0	2	5	17	20	12	12	12

Table 1

1. Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution. [2]
2. Show that the mean number of hurricanes per year from Table 1 is 4.4875 [2]
3. Use the answer in part (b) to calculate the expected frequencies \(r\) and \(s\) given in Table 2 below to 2 decimal places. [3]

\(h\)	0	1	2	3	4	5	6	7 or more
Expected frequency	0.90	4.04	\(r\)	13.55	\(s\)	13.65	10.21	13.39

Table 2

1. Test, at the 5\% level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly. [6]

Breakdowns on a certain stretch of motorway were recorded each day for 80 consecutive days. The results are summarised in the table below.

Number of breakdowns	0	1	2	\(>2\)
Frequency	38	32	10	0

It is suggested that the number of breakdowns per day can be modelled by a Poisson distribution. Using a 5% level of significance, test whether or not the Poisson distribution is a suitable model for these data. State your hypotheses clearly. [13]

A physicist believes that the number of particles emitted by a radioactive source with a long half-life can be modelled by a Poisson distribution. She records the number of particles emitted in 80 successive 5-minute periods and her results are shown in the table below.

1. Comment on the suitability of a Poisson distribution for this situation. [3 marks]
2. Show that an unbiased estimate of the mean number of particles emitted in a 5-minute period is 1.2 and find an unbiased estimate of the variance. [5 marks]
3. Explain how your answers to part (b) support the fitting of a Poisson distribution. [1 mark]
4. Stating your hypotheses clearly and using a 5\% level of significance, test whether or not these data can be modelled by a Poisson distribution. [11 marks]

The manager of a hockey team studies last season's results and puts forward the theory that the number of goals scored per match by her team can be modelled by a Poisson distribution with mean 2.0. The number of goals scored during the season are summarised below.

Goals scored	0	1	2	3	4 or more
Frequency	6	11	15	10	8

1. State suitable hypotheses to carry out a goodness of fit test. [1]
2. Carry out a \(\chi^2\) goodness of fit test on this data set, using a 5% level of significance and draw a conclusion in context. [9]