5.02m Poisson: mean = variance = lambda

Chai packs china mugs into cardboard boxes. Chai's manager suspects that breakages occur at random times and that the number of breakages may follow a Poisson distribution. He takes a small sample of observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a standard deviation of 1.5.

1. Explain how this information tends to support the manager's suspicion. [2]

The manager now takes a larger sample and claims that the numbers of breakages in a one-hour period follow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods are summarised in the following table.

Number of breakages	0	1	2	3	4	5	6	7 or more
Frequency	21	33	46	31	23	16	10	0

The mean number of breakages calculated from this sample is 2.5.

1. Use the data from this larger sample to carry out a goodness of fit test, at the 10% significance level, to test the claim. [8]

A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:

1. Show that the mean number of errors per page in this sample of pages is 2. [2]
2. Find the variance of the number of errors per page in this sample. [2]
3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution. [1]

Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2.

1. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]

A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.

1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]

Calculate the probability that a randomly selected sample square contains

1. no sheep, [1]
2. more than 2 sheep. [4]

A sheepdog has been sent into the field to round up the sheep.

1. Explain why the model may no longer be applicable. [1]

In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.

1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]

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.

Number of accidents	0	1	2	3	4	5
Frequency	47	57	46	35	9	6

1. Show that the mean number of accidents per day for these data is 1.6 [1]

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	\(s\)

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

The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).

1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
3. At a busy time of the day, \(\lambda = 30\).
   1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
   2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]

An electrical retailer gives customers extended guarantees on washing machines. Under this guarantee all repairs in the first 3 years are free. The retailer records the numbers of free repairs made to 80 machines.

1. Show that the sample mean is 0.4375. [1]
2. The sample standard deviation \(s\) is 0.6907. Explain why this supports a suggestion that a Poisson distribution may be a suitable model for the distribution of the number of free repairs required by a randomly chosen washing machine. [2]

The random variable \(X\) denotes the number of free repairs required by a randomly chosen washing machine. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 0.4375.

1. Find P\((X = 1)\). Comment on your answer in relation to the data in the table. [4]
2. The manager decides to monitor 8 washing machines sold on one day. Find the probability that there are at least 12 free repairs in total on these 8 machines. You may assume that the 8 machines form an independent random sample. [3]
3. A launderette with 8 washing machines has needed 12 free repairs. Why does your answer to part (iv) suggest that the Poisson model with mean 0.4375 is unlikely to be a suitable model for free repairs on the machines in the launderette? Give a reason why the model may not be appropriate for the launderette. [3]

The retailer also sells tumble driers with the same guarantee. The number of free repairs on a tumble drier in three years can be modelled by a Poisson distribution with mean 0.15. A customer buys a tumble drier and a washing machine.

1. Assuming that free repairs are required independently, find the probability that
   1. the two appliances need a total of 3 free repairs between them,
   2. each appliance needs exactly one free repair.
   [5]

An electrician records the number of repairs of different types of appliances that he makes each day. His records show that over 40 working days he repaired a total of 180 CD players.

1. Explain why a Poisson distribution may be suitable for modelling the number of CD players he repairs each day and find the parameter for this distribution. [4 marks]
2. Find the probability that on one particular day he repairs
   1. no CD players,
   2. more than 6 CD players. [3 marks]
3. Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]

A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.

1. Suggest a suitable distribution for modelling the number of sales made per hour on a weekday morning and state the value of any parameters needed. [1 mark]
2. Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning. [1 mark]
3. Find the probability that on a weekday morning the shop sells
   1. more than 4 pairs in a one-hour period,
   2. no pairs in a half-hour period,
   3. more than 4 pairs during each hour from 9 am until noon. [6 marks]

The area manager visits the shop on a weekday morning, the day after an advert appears in a local paper. In a one-hour period the shop sells 7 pairs of shoes, leading the manager to believe that the advert has increased the shop's sales.

1. Stating your hypotheses clearly, test at the 5\% level of significance whether or not there is evidence of an increase in sales following the appearance of the advert. [4 marks]

An ornithologist believes that on average 4.2 different species of bird will visit a bird table in a rural garden when 50 g of breadcrumbs are spread on it.

1. Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria. [1 mark]
2. Explain why the parameter used with this model may need to be changed if
   1. 50 g of nuts are used instead of breadcrumbs,
   2. 100g of breadcrumbs are used.
   [2 marks]

A bird table in a rural garden has 50 g of breadcrumbs spread on it. Find the probability that

1. exactly 6 different species visit the table, [2 marks]
2. more than 2 different species visit the table. [4 marks]

It is believed that the number of sets of traffic lights that fail per hour in a particular large city follows a Poisson distribution with a mean of 3. Find the probability that

1. there will be no failures in a one-hour period, [1 mark]
2. there will be more than 4 failures in a 30-minute period. [3 marks]

Using a suitable approximation, find the probability that in a 24-hour period there will be

1. less than 60 failures, [5 marks]
2. exactly 72 failures. [4 marks]

The manager of a supermarket receives an average of 6 complaints per day from customers. Find the probability that on one day she receives

1. 3 complaints, [3 marks]
2. 10 or more complaints. [2 marks]

The supermarket is open on six days each week.

1. Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [4 marks]

A rugby player scores an average of 0.4 tries per match in which he plays.

1. Find the probability that he scores 2 or more tries in a match. [5 marks]

The team's coach moves the player to a different position in the team believing he will then score more frequently. In the next five matches he scores 6 tries.

1. Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of an increase in the mean number of tries the player scores per match as a result of playing in a different position. [5 marks]

Faults occur in a roll of material at a rate of \(\lambda\) per m\(^2\). To estimate \(\lambda\), three pieces of material of sizes 3 m\(^2\), 7 m\(^2\) and 10 m\(^2\) are selected and the number of faults \(X_1\), \(X_2\) and \(X_3\) respectively are recorded. The estimator \(\hat{\lambda}\), where $$\hat{\lambda} = k(X_1 + X_2 + X_3)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X_1\), \(X_2\) and \(X_3\) and find the value of \(k\). [4]
2. Find Var(\(\hat{\lambda}\)). [3]

A random sample of \(n\) pieces of this material, each of size 4 m\(^2\), was taken. The number of faults on each piece, \(Y\), was recorded.

1. Show that \(\frac{1}{4}\bar{Y}\) is an unbiased estimator of \(\lambda\). [2]
2. Find Var(\(\frac{1}{4}\bar{Y}\)). [3]
3. Find the minimum value of \(n\) for which \(\frac{1}{4}\bar{Y}\) becomes a better estimator of \(\lambda\) than \(\hat{\lambda}\). [2]

Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.

1. State these two assumptions. [2]
2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]

Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\mathrm{Po}(0.8)\).

1. 1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
   2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\mathrm{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]

The random variable \(X\) has the binomial distribution B(12, 0·3). The independent random variable \(Y\) has the Poisson distribution Po(4). Find

1. \(E(XY)\), [2]
2. Var\((XY)\). [6]

On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution Po(120).

1. Find the probability that at least 28 orders are received in a randomly chosen 10-minute period. [2]
2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received. [3]
3. State a necessary assumption for the validity of your calculation in part (b). [1]

Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.

1. State these two assumptions. [2]
2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]

Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\text{Po}(0.8)\).

1. 1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
   2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\text{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]