Frequency distribution and Poisson fit

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.

1. State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below.

   \(x\)	0	1	2	3	4	5	\(> 5\)
   Frequency, \(f\)	18	39	20	12	8	3	0

2. Show that the sample mean is 1.62 and calculate the sample variance.
3. Do your calculations in part (ii) support the suggestion that a Poisson distribution is a suitable model for the distribution of \(X\) ? Explain your answer. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 1.62 .
4. Find \(\mathrm { P } ( X = 2 )\). Comment on your answer in relation to the data in the table.
5. Find the probability that at least ten cars arrive in a period of 50 seconds during weekday lunchtimes.
6. Use a suitable approximating distribution to find the probability that no more than 550 cars arrive in a randomly chosen period of one hour during weekday lunchtimes.

9 JUNE 2005
Morning
1 hour 30 minutes
Additional materials:
Answer booklet
Graph paper
MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes

• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• You are permitted to use a graphical calculator in this paper.

• The number of marks is given in brackets [ ] at the end of each question or part question.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
• Final answers should be given to a degree of accuracy appropriate to the context.
• The total number of marks for this paper is 72.

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.

1. State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below. Carry out a test at the \(5 \%\) level of significance to examine whether there is any association between type of customer and type of drink. State carefully your null and alternative hypotheses.

2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.

1. Show that the sample mean is 1.84 and calculate the sample variance.
2. Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
3. Find the probability of
   (A) no sultanas in a biscuit,
   (B) at least two sultanas in a biscuit.
4. Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
5. Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
6. Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.

9 The number of visitors arriving at an art exhibition is recorded for each 10 -minute period of time during the ten hours that it is open on a particular day. The results are as follows.

1. Calculate the mean and variance for this sample and explain whether your answers support a suggestion that a Poisson distribution might be a suitable model for the number of visitors in a 10-minute period.
2. Use an appropriate Poisson distribution to find the two expected frequencies missing from the following table.

   Number of visitors in a 10-minute period	0	1	2	3	4	5	6	7	8	\(\geqslant 9\)
   Expected number of 10 -minute periods	1.10		8.79		11.72	9.38	6.25	3.57	1.79	1.28

3. Test, at the \(10 \%\) significance level, the goodness of fit of this Poisson distribution to the data.

1. The manager of a supermarket is investigating the number of complaints per day received from customers.

A random sample of 180 days is taken and the results are shown in the table below.

1. Calculate the mean and the variance of these data.
2. Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of complaints per day. The manager uses a Poisson distribution with mean 3 to model the number of complaints per day.
3. For a randomly selected day find, using the manager's model, the probability that there are
   1. at least 3 complaints,
   2. more than 4 complaints but less than 8 complaints. A week consists of 7 consecutive days.
4. Using the manager's model and a suitable approximation, show that the probability that there are less than 19 complaints in a randomly selected week is 0.29 to 2 decimal places.
   Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.) A period of 13 weeks is selected at random.
5. Find the probability that in this period there are exactly 5 weeks that have less than 19 complaints.
   Show your working clearly.

1. A student is investigating the numbers of cherries in a Rays fruit cake. A random sample of Rays fruit cakes is taken and the results are shown in the table below.

1. Calculate the mean and the variance of these data.
2. Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of cherries in a Rays fruit cake. The number of cherries in a Rays fruit cake follows a Poisson distribution with mean 1.5 A Rays fruit cake is to be selected at random. Find the probability that it contains
3. 1. exactly 2 cherries,
   2. at least 1 cherry. Rays fruit cakes are sold in packets of 5
4. Show that the probability that there are more than 10 cherries, in total, in a randomly selected packet of Rays fruit cakes, is 0.1378 correct to 4 decimal places. Twelve packets of Rays fruit cakes are selected at random.
5. Find the probability that exactly 3 packets contain more than 10 cherries. \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

3 A glassware factory produces a large number of ornaments each week. Just before they leave the factory, all the ornaments are checked and some may be found to be defective. The Quality Assurance Manager of the factory wishes to model the number of defective ornaments that are found each week using a Poisson distribution. The numbers of defective ornaments found each week in a period of 40 weeks are shown in Table 3.1. \begin{table}[h] \end{table} You are given that summary statistics for the data are \(\sum f = 40 , \sum \mathrm { rf } = 84\) and \(\sum \mathrm { r } ^ { 2 } \mathrm { f } = 256\).

1. By using the summary statistics to determine estimates for the mean and variance of the number of defective ornaments produced by the factory each week, explain how the data support the suggestion that the number of defective ornaments produced each week can be modelled using a Poisson distribution. The Quality Assurance Manager is asked by the head office to carry out a chi-squared hypothesis test for goodness of fit based on a \(\operatorname { Po } ( 2 )\) distribution.
2. Table 3.2, which is incomplete, gives observed frequency, probability, expected frequency and chi-squared contribution. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 3.2}

   No. of defective ornaments in a week, \(r\)	Observed frequency	Probability	Expected frequency	Chi-squared contribution
   0	2	0.13534	5.4134	2.15232
   1	14
   2	13	0.27067		0.43620
   3	5		7.2179
   \(\geqslant 4\)	6	0.14288		0.01421

   \end{table}
   1. Complete the copy of the table in the Printed Answer Booklet.
   2. Carry out the test at the \(10 \%\) significance level.
3. On one occasion a fork-lift truck in the factory drops a crate containing eight ornaments and all of them are subsequently found to be defective. Explain why the Poisson model cannot model defects occurring in this manner.

In a packet of 40 biscuits, the number of currants in each biscuit is as follows

1. Find the mean and variance of the random variable \(X\) representing the number of currants per biscuit. [4 marks]
2. State an appropriate model for the distribution of \(X\), giving two reasons for your answer. [2 marks]

Another machine produces biscuits with a mean of 1.9 currants per biscuit.

1. Determine which machine is more likely to produce a biscuit with at least two currants. [5 marks]