Chi-squared goodness of fit: Poisson

4 A scientist is investigating the numbers of a particular type of butterfly in a certain region. He claims that the numbers of these butterflies found per square metre can be modelled by a Poisson distribution with mean 2.5. He takes a random sample of 120 areas, each of one square metre, and counts the number of these butterflies in each of these areas. The following table shows the observed frequencies together with some of the expected frequencies using the scientist's Poisson distribution.

1. Find the values of \(p\) and \(q\), correct to 2 decimal places.
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test the scientist's claim.

2 The number of breakdowns on a particular section of road is recorded each day over a period of 90 days. It is suggested that the number of breakdowns follows a Poisson distribution with mean 3.5. The data is summarised in the table, together with some of the expected frequencies resulting from the suggested Poisson distribution.

1. Complete the table.
2. Carry out a goodness of fit test, at the 10\% significance level, to determine whether or not \(\operatorname { Po } ( 3.5 )\) is a good fit to the data.

3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).

1. Complete the table.
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable. \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-07_2726_35_97_20}

3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).

1. Complete the table.
2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable. \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-07_2726_35_97_20}

4 An experiment by Lord Rutherford at Cambridge in 1909 involved measuring the numbers of \(\alpha\)-particles emitted during radioactive decay. The following table shows emissions during 2608 intervals of 7.5 seconds. It is given that the mean number of particles emitted per interval, calculated from the data, is 3.87 , correct to 3 significant figures.

1. Find the contribution to the \(\chi ^ { 2 }\) value of the frequency of 273 corresponding to \(x = 6\) in a goodness of fit test for a Poisson distribution.
2. Given that no cells need to be combined, state why the number of degrees of freedom is 10 .
3. Given also that the calculated value of \(\chi ^ { 2 }\) is 13.0 , correct to 3 significant figures, carry out the test at the 10\% significance level.

4

1. A researcher is investigating the feeding habits of bees. She sets up a feeding station some distance from a beehive and, over a long period of time, records the numbers of bees arriving each minute. For a random sample of 100 one-minute intervals she obtains the following results.

   Number of bees	0	1	2	3	4	5	6	7	\(\geqslant 8\)
   Number of intervals	6	16	19	18	17	14	6	4	0

   1. Show that the sample mean is 3.1 and find the sample variance. Do these values support the possibility of a Poisson model for the number of bees arriving each minute? Explain your answer.
   2. Use the mean in part (i) to carry out a test of the goodness of fit of a Poisson model to the data.
2. The researcher notes the length of time, in minutes, that each bee spends at the feeding station. The times spent are assumed to be Normally distributed. For a random sample of 10 bees, the mean is found to be 1.465 minutes and the standard deviation is 0.3288 minutes. Find a \(95 \%\) confidence interval for the overall mean time.

3

1. A medical researcher is looking into the delay, in years, between first and second myocardial infarctions (heart attacks). The following table shows the results for a random sample of 225 patients.

   Delay (years)	\(0 -\)	\(1 -\)	\(2 -\)	\(3 -\)	\(4 - 10\)
   Number of patients	160	40	13	9	3

   The mean of this sample is used to construct a model which gives the following expected frequencies.

   Delay (years)	\(0 -\)	\(1 -\)	\(2 -\)	\(3 -\)	\(4 - 10\)
   Number of patients	142.23	52.32	19.25	7.08	4.12

   Carry out a test, using a \(2.5 \%\) level of significance, of the goodness of fit of the model to the data.
2. A further piece of research compares the incidence of myocardial infarction in men aged 55 to 70 with that in women aged 55 to 70 . Incidence is measured by the number of infarctions per 10000 of the population. For a random sample of 8 health authorities across the UK, the following results for the year 2010 were obtained.

   Health authority	A	B	C	D	E	F	G	H
   Incidence in men	47	56	15	51	45	54	50	32
   Incidence in women	36	30	30	47	54	55	27	27

   A Wilcoxon paired sample test, using the hypotheses \(\mathrm { H } _ { 0 } : m = 0\) and \(\mathrm { H } _ { 1 } : m \neq 0\) where \(m\) is the population median difference, is to be carried out to investigate whether there is any difference between men and women on the whole.
   1. Explain why a paired test is being used in this context.
   2. Carry out the test using a \(10 \%\) level of significance.

2 Scientists researching into the chemical composition of dust in space collect specimens using a specially designed spacecraft. The craft collects the particles of dust in trays that are made up of a large array of cells containing aerogel. The aerogel traps the particles that penetrate into the cells.

1. For a random sample of 100 cells, the number of particles of dust in each cell was counted, giving the following results.

   Number of particles	0	1	2	3	4	5	6	7	8	9	\(10 +\)
   Frequency	4	7	10	20	17	15	10	9	5	3	0

   It is thought that the number of particles collected in each cell can be modelled using the distribution Poisson(4.2) since 4.2 is the sample mean for these data. Some of the calculations for a \(\chi ^ { 2 }\) test are shown below. The cells for 8,9 and \(10 +\) particles have been combined.

   Number of particles

   Observed frequency

   Expected frequency

   Contribution to \(X ^ { 2 }\)

   5	6	7	\(8 +\)
   15	10	9	8
   16.33	11.44	6.86	6.39
   0.1083	0.1813	0.6676	0.4056

   Complete the calculations and carry out the test using a \(10 \%\) significance level to see whether the number of particles per cell may be modelled in this way.
2. The diameters of the dust particles are believed to be distributed symmetrically about a median of 15 micrometres \(( \mu \mathrm { m } )\). For a random sample of 20 particles, the sum of the signed ranks of the diameters of the particles smaller than \(15 \mu \mathrm {~m} \left( W _ { - } \right)\)is found to be 53 . Test at the \(5 \%\) level of significance whether the median diameter appears to be more than \(15 \mu \mathrm {~m}\).

4 The numbers of call-outs per day received by a fire station for a random sample of 255 weekdays were recorded as follows. The mean number of call-outs per day for these data is 0.6 . A Poisson model, using this sample mean of 0.6 , is fitted to the data, and gives the following expected frequencies (correct to 3 decimal places).

1. Using a \(5 \%\) significance level, carry out a test to examine the goodness of fit of the model to the data. The time \(T\), measured in days, that elapses between successive call-outs can be modelled using the exponential distribution for which \(\mathrm { f } ( t )\), the probability density function, is $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 , \\ \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 , \end{cases}$$ where \(\lambda\) is a positive constant.
2. For the distribution above, it can be shown that \(\mathrm { E } ( T ) = \frac { 1 } { \lambda }\). Given that the mean time between successive call-outs is \(\frac { 5 } { 3 }\) days, write down the value of \(\lambda\).
3. Find \(\mathrm { F } ( t )\), the cumulative distribution function.
4. Find the probability that the time between successive call-outs is more than 1 day.
5. Find the median time that elapses between successive call-outs.

A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table. It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion. The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.

1. Show that \(p = 46.304\), correct to 3 decimal places, and find \(q\).
2. Carry out a goodness of fit test at the \(10 \%\) significance level.

10 Roberto owns a small hotel and offers accommodation to guests. Over a period of 100 nights, the numbers of rooms, \(x\), that are occupied each night at Roberto's hotel and the corresponding frequencies are shown in the following table.

1. Show that the mean number of rooms that are occupied each night is 3.25 .
   The following table shows most of the corresponding expected frequencies, correct to 2 decimal places, using a Poisson distribution with mean 3.25.

   Number of rooms occupied \(( x )\)	0	1	2	3	4	5	6	\(\geqslant 7\)
   Observed frequency	4	9	18	26	20	16	7	0
   Expected frequency	3.88	12.60	20.48	22.18	18.02	11.72

2. Show how the expected value of 22.18 , for \(x = 3\), is obtained and find the expected values for \(x = 6\) and for \(x \geqslant 7\).
3. Use a goodness-of-fit test at the \(5 \%\) significance level to determine whether the Poisson distribution is a suitable model for the number of rooms occupied each night at Roberto's hotel.

8 The numbers of a particular type of laptop computer sold by a store on each of 100 consecutive Saturdays are summarised in the following table. Fit a Poisson distribution to the data and carry out a goodness of fit test at the \(2.5 \%\) significance level.

8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table. Fit a Poisson distribution to the data, and test its goodness of fit at the \(5 \%\) significance level.

8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table. Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.

10 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.

1. 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.
   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

2. Show how the expected frequency of 6.61 for \(x = 4\) is obtained.
3. Test at the \(5 \%\) significance level the goodness of fit of this Poisson distribution to the data.

The number of puncture repairs carried out each week by a small repair shop is recorded over a period of 40 weeks. The results are shown in the following table.

1. Calculate the mean and variance for the number of repairs in a week and comment on the possible suitability of a Poisson distribution to model the data.
   Records over a longer period of time indicate that the mean number of repairs in a week is 1.6 . The following table shows some of the expected frequencies, correct to 3 decimal places, for a period of 40 weeks using a Poisson distribution with mean 1.6.

   Number of repairs in a week	0	1	2	3	4	5	\(\geqslant 6\)
   Expected frequency	8.076	12.921	10.337	5.513	2.205	\(a\)	\(b\)

2. Show that \(a = 0.706\) and find the value of the constant \(b\).
3. Carry out a goodness of fit test of a Poisson distribution with mean 1.6, using a \(10 \%\) significance level.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table. Fit a Poisson distribution to the data, and test its goodness of fit at the 5\% significance level.

8 A team of researchers have reason to believe that the number of calls received in randomly chosen 10-minute intervals to a call centre can be well modelled by a Poisson distribution. To test this belief the researchers record the number of telephone calls received in 60 randomly chosen 10-minute intervals. The results, together with relevant calculations, are shown in the following table.

1. Calculate the mean of the observed number of calls received.
2. Calculate the variance of the observed number of calls received.
3. Comment on what your answers to parts (a) and (b) suggest about the proposed model.
4. Explain why it is necessary to combine some cells in the table.
5. Show how the values 15.506 and 0.793 in the table were obtained.
6. Carry out the test, at the \(5 \%\) significance level. In the light of the result of the test, the team consider that a different model is appropriate. They propose the following improved model: $$P ( R = r ) = \begin{cases} \frac { 1 } { 60 } ( a + ( 2 - r ) b ) & r = 0,1,2,3,4 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are integers.
7. Use at least three of the observed frequencies to suggest appropriate values for \(a\) and \(b\). You should consider more than one possible pair of values, and explain which pair of values you consider best. (Do not carry out a goodness-of-fit test.)

5 Some bird-watchers study the song of chaffinches in a particular wood. They investigate whether the number, \(N\), of separate bursts of song in a 5 minute period can be modelled by a Poisson distribution. They assume that a burst of song can be considered as a single event, and that bursts of song occur randomly. \section*{(a) State two further assumptions needed for \(N\) to be well modelled by a Poisson distribution.} The bird-watchers record the value of \(N\) in each of 60 periods of 5 minutes. The mean and variance of the results are 3.55 and 5.6475 respectively.
(b) Explain what this suggests about the validity of a Poisson distribution as a model in this context. The complete results are shown in the table. The bird-watchers carry out a \(\chi ^ { 2 }\) goodness of fit test at the \(5 \%\) significance level.
(c) State suitable hypotheses for the test.
(d) Determine the contribution to the test statistic for \(n = 3\).
(e) The total value of the test statistic, obtained by combining the cells for \(n \leqslant 1\) and also for \(n \geqslant 6\), is 9.202 , correct to 4 significant figures. Complete the goodness of fit test.
(f) It is known that chaffinches are more likely to sing in the presence of other chaffinches. Explain whether this fact affects the validity of a Poisson model for \(N\).

1. The number of emails per hour received by a helpdesk were recorded. The results for a random sample of 80 one-hour periods are shown in the table.

1. Show that the mean number of emails per hour in the sample is 3 The manager believes that the number of emails per hour received could be modelled by a Poisson distribution. The following table shows some of the expected frequencies.

   Number of emails per hour	Expected Frequencies
   0	\(r\)
   1	11.949
   2	17.923
   3	17.923
   4	13.443
   5	\(s\)
   \(\geqslant 6\)	\(t\)

2. Find the values of \(r , s\) and \(t\), giving your answers to 3 decimal places.
3. Using a 10\% significance level, test whether or not a Poisson model is reasonable. You should clearly state your hypotheses, test statistic and the critical value used.

1. The number of jobs sent to a printer per hour in a small office is recorded for 120 hours. The results are summarised in the following table.

1. Show that the mean number of jobs sent to the printer per hour for these data is 1.75 The office manager believes that the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. The office manager uses the mean given in part (a) to calculate the expected frequencies for this model. Some of the results are given in the following table.

   Number of jobs	0	1	2	3	4	5 or more
   Expected frequency	20.85	36.49	31.93	\(r\)	\(s\)	3.95

2. Show that the value of \(s\) is 8.15 to 2 decimal places.
3. Find the value of \(r\) to 2 decimal places. The value of \(\sum \frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) for the first four frequencies in the table is 1.43
4. Test, at the \(5 \%\) level of significance, whether or not the number of jobs sent to the printer per hour can be modelled using a Poisson distribution. Show your working clearly, stating your hypotheses, test statistic and critical value.

6. An area of grass was sampled by placing a \(1 \mathrm {~m} \times 1 \mathrm {~m}\) square randomly in 100 places. The numbers of daisies in each of the squares were counted. It was decided that the resulting data could be modelled by a Poisson distribution with mean 2. The expected frequencies were calculated using the model. The following table shows the observed and expected frequencies.

1. Find values for \(r , s\) and \(t\).
2. Using a \(5 \%\) significance level, test whether or not this Poisson model is suitable. State your hypotheses clearly. An alternative test might have been to estimate the population mean by using the data given.
3. Explain how this would have affected the test.
   (2)

5. 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)

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

\begin{table}[h] \end{table}

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

   \(h\)	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

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
4. Test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly.

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