5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!

Andreia's secretary makes random errors in his work at an average rate of 1.7 errors every 100 words.

1. Find the probability that the secretary makes fewer than 2 errors in the next 100 -word piece of work. Andreia asks the secretary to produce a 250 -word article for a magazine.
2. Find the probability that there are exactly 5 errors in this article. Andreia offers the secretary a choice of one of two bonus schemes, based on a random sample of 40 pieces of work each consisting of 100 words. In scheme \(\mathbf { A }\) the secretary will receive the bonus if more than 10 of the 40 pieces of work contain no errors. In scheme \(\mathbf { B }\) the bonus is awarded if the total number of errors in all 40 pieces of work is fewer than 56
3. Showing your calculations clearly, explain which bonus scheme you would advise the secretary to choose. Following the bonus scheme, Andreia randomly selects a single 500 -word piece of work from the secretary to test if there is any evidence that the secretary's rate of errors has decreased.
4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, find the critical region for this test.

A plumbing company receives call-outs during the working day at an average rate of 2.4 per hour.

1. Find the probability that the company receives exactly 7 call-outs in a randomly selected 3 -hour period of a working day. The company has enough staff to respond to 28 call-outs in an 8 -hour working day.
2. Show that the probability that the company receives more than 28 call-outs in a randomly selected 8 -hour working day is 0.022 to 3 decimal places. In a random sample of 100 working days each of 8 hours,
3. 1. find the expected number of days that the company receives more than 28 call-outs,
   2. find the standard deviation of the number of days that the company receives more than 28 call-outs,
   3. use a Poisson approximation to estimate the probability that the company receives more than 28 call-outs on at least 6 of these days.

1. During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8

The number of cyclists passing the same point in a 10 -minute interval travelling westbound can be modelled by a Poisson distribution with mean 3

1. Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption. Given that exactly 12 cyclists pass the point in a 10 -minute interval,
2. find the probability that at least 11 are travelling eastbound. After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14
3. Test, at the \(5 \%\) level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.
   State your hypotheses clearly.

Rowan and Alex are both check-in assistants for the same airline. The number of passengers, \(R\), checked in by Rowan during a 30-minute period can be modelled by a Poisson distribution with mean 28

1. Calculate \(\mathrm { P } ( R \geqslant 23 )\) The number of passengers, \(A\), checked in by Alex during a 30-minute period can be modelled by a Poisson distribution with mean 16, where \(R\) and \(A\) are independent. A randomly selected 30-minute period is chosen.
2. Calculate the probability that exactly 42 passengers in total are checked in by Rowan and Alex. The company manager is investigating the rate at which passengers are checked in. He randomly selects 150 non-overlapping 60-minute periods and records the total number of passengers checked in by Rowan and Alex, in each of these 60-minute periods.
3. Using a Poisson approximation, find the probability that for at least 25 of these 60-minute periods Rowan and Alex check in a total of fewer than 80 passengers. On a particular day, Alex complains to the manager that the check-in system is working slower than normal. To see if the complaint is valid the manager takes a random 90-minute period and finds that the total number of people Rowan checks in is 67
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the system is working slower than normal. You should state your hypotheses and conclusion clearly and show your working.

Xena catches fish at random, at a constant rate of 0.6 per hour.

1. Find the probability that Xena catches exactly 4 fish in a 5 -hour period. The probability of Xena catching no fish in a period of \(t\) hours is less than 0.16
2. Find the minimum value of \(t\), giving your answer to one decimal place. Independently of Xena, Zion catches fish at random with a mean rate of 0.8 per hour.
   Xena and Zion try using new bait to catch fish. The number of fish caught in total by Xena and Zion after using the new bait, in a randomly selected 4-hour period, is 12
3. Use a suitable test to determine, at the \(5 \%\) level of significance, whether or not there is evidence that the rate at which fish are caught has increased after using the new bait. State your hypotheses clearly and the \(p\)-value used in your test.

1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.

The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907

1. Find the value of \(A\) The tablecloths are sold in packets of 20
   A randomly selected packet is taken.
2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
   The random variable \(X\) represents the number of these tablecloths that have no faults.
3. Find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
   The piece of cloth has area 30 square metres and is found to have 6 faults.
5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
6. Write down the \(p\)-value for the test used in part (e).

1. Table 1 below shows the number of car breakdowns in the Snoreap district in each of 60 months.

\begin{table}[h] \end{table} Anja believes that the number of car breakdowns per month in Snoreap can be modelled by a Poisson distribution. Table 2 below shows the results of some of her calculations. \begin{table}[h] \end{table}

1. State suitable hypotheses for a test to investigate Anja's belief.
2. Explain why Anja has changed the label of the final column to \(\geqslant 5\)
3. Showing your working clearly, complete Table 2
4. Find the value of \(\frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) when the number of car breakdowns is
   1. 1
   2. 3
5. Explain why Anja used 3 degrees of freedom for her test. The test statistic for Anja's test is 6.54 to 2 decimal places.
6. Stating the critical value and using a \(5 \%\) level of significance, complete Anja's test.

1. A manager keeps a record of accidents in a canteen.

Accidents occur randomly with an average of 2.7 per month. The manager decides to model the number of accidents with a Poisson distribution.

1. Give a reason why a Poisson distribution could be a suitable model in this situation.
2. Assuming that a Poisson model is suitable, find the probability of
   1. at least 3 accidents in the next month,
   2. no more than 10 accidents in a 3-month period,
   3. at least 2 months with no accidents in an 8-month period. One day, two members of staff bump into each other in the canteen and each report the accident to the manager. The canteen manager is unsure whether to record this as one or two accidents. Given that the manager still wants to model the number of accidents per month with a Poisson distribution,
3. state
   The manager introduces some new procedures to try and reduce the average number of accidents per month. During the following 12 months the total number of accidents is 22 The manager claims that the accident rate has been reduced.
4. Use a \(5 \%\) level of significance to carry out a suitable test to assess the manager's claim.
   You should state your hypotheses clearly and the \(p\)-value used in your test.

1. Two car hire companies hire cars independently of each other.

Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.

1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
   In a random sample of 600 new cars produced at the factory,
4. 1. find the mean of the number of cars with a defect,
   2. find the variance of the number of cars with a defect.
5. 1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
   2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}

The discrete random variable \(X\) follows a Poisson distribution with mean 1.4

1. Write down the value of
   1. \(\mathrm { P } ( \mathrm { X } = 1 )\)
   2. \(\mathrm { P } ( \mathrm { X } \leqslant 4 )\) The manager of a bank recorded the number of mortgages approved each week over a 40 week period.

      Number of mortgages approved	0	1	2	3	4	5	6
      Frequency	10	16	7	4	2	0	1

   (b) Show that the mean number of mortgages approved over the 40 week period is 1.4 The bank manager believes that the Poisson distribution may be a good model for the number of mortgages approved each week. She uses a Poisson distribution with a mean of 1.4 to calculate expected frequencies as follows.

   Number of mortgages approved	0	1	2	3	4	5 or more
   Expected frequency	9.86	r	9.67	4.51	1.58	s

2. Find the value of r and the value of s giving your answers to 2 decimal places. The bank manager will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution.
3. Calculate the test statistic and state the conclusion for this test. State clearly the degrees of freedom and the hypotheses used in the test. \section*{Q uestion 4 continued} \section*{Q uestion 4 continued}

Indre works on reception in an office and deals with all the telephone calls that arrive. Calls arrive randomly and, in a 4-hour morning shift, there are on average 80 calls.

1. Using a suitable model, find the probability of more than 4 calls arriving in a particular 20 -minute period one morning. Indre is allowed 20 minutes of break time during each 4-hour morning shift, which she can take in 5 -minute periods. When she takes a break, a machine records details of any call in the office that Indre has missed. One morning Indre took her break time in 4 periods of 5 minutes each.
2. Find the probability that in exactly 3 of these periods there were no calls. On another occasion Indre took 1 break of 5 minutes and 1 break of 15 minutes.
3. Find the probability that Indre missed exactly 1 call in each of these 2 breaks.

1. The number of customers entering Jeff's supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.
Using this information,

1. 1. find the probability that exactly 26 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning,
   2. find the probability that at least 21 customers enter Jeff's supermarket during a randomly selected 1-hour period one morning. A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff's supermarket over a randomly selected 40-minute period is found to be 10
2. Test, at the 5\% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff's supermarket. State your hypotheses clearly. A further randomly selected 20 -minute period is observed and the hypothesis test is repeated. Given that the true rate of customers entering Jeff's supermarket is now 1 every 5 minutes,
3. calculate the probability of a Type II error.

On a weekday, a garage receives telephone calls randomly, at a mean rate of 1.25 per 10 minutes.

1. Show that the probability that on a weekday at least 2 calls are received by the garage in a 30 -minute period is 0.888 to 3 decimal places.
2. Calculate the probability that at least 2 calls are received by the garage in fewer than 4 out of 6 randomly selected, non-overlapping 30-minute periods on a weekday. The manager of the garage randomly selects 150 non-overlapping 30-minute periods on weekdays.
   She records the number of calls received in each of these 30-minute periods.
3. Using a Poisson approximation show that the probability of the manager finding at least 3 of these 30 -minute periods when exactly 8 calls are received by the garage is 0.664 to 3 significant figures.
4. Explain why the Poisson approximation may be reasonable in this case. The manager of the garage decides to test whether the number of calls received on a Saturday is different from the number of calls received on a weekday. She selects a Saturday at random and records the number of telephone calls received by the garage in the first 4 hours.
5. Write down the hypotheses for this test. The manager found that there had been 40 telephone calls received by the garage in the first 4 hours.
6. Carry out the test using a \(5 \%\) level of significance.

During the summer, mountain rescue team \(A\) receives calls for help randomly with a rate of 0.4 per day.

1. Find the probability that during the summer, mountain rescue team \(A\) receives at least 19 calls for help in 28 randomly selected days. The leader of mountain rescue team \(A\) randomly selects 250 summer days from the last few years.
   She records the number of calls for help received on each of these days.
2. Using a Poisson approximation, estimate the probability of the leader finding at least 20 of these days when more than 1 call for help was received by mountain rescue team \(A\). Mountain rescue team \(A\) believes that the number of calls for help per day is lower in the winter than in the summer. The number of calls for help received in 42 randomly selected winter days is 8
3. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the number of calls for help per day is lower in the winter than in the summer. State your hypotheses clearly. During the summer, mountain rescue team \(B\) receives calls for help randomly with a rate of 0.2 per day, independently of calls to mountain rescue team \(A\). The random variable \(C\) is the total number of calls for help received by mountain rescue teams \(A\) and \(B\) during a period of \(n\) days in the summer.
   On a Monday in the summer, mountain rescue teams \(A\) and \(B\) each receive a call for help. Given that over the next \(n\) days \(\mathrm { P } ( C = 0 ) < 0.001\)
4. calculate the minimum value of \(n\)
5. Write down an assumption that needs to be made for the model to be appropriate.

Telephone calls arrive at a call centre randomly, at an average rate of 1.7 per minute. After the call centre was closed for a week, in a random sample of 10 minutes there were 25 calls to the call centre.

1. Carry out a suitable test to determine whether or not there is evidence that the rate of calls arriving at the call centre has changed.
   Use a \(5 \%\) level of significance and state your hypotheses clearly. Only 1.2\% of the calls to the call centre last longer than 8 minutes.
   One day Tiang has 70 calls.
2. Find the probability that out of these 70 calls Tiang has more than 2 calls lasting longer than 8 minutes. The call centre records show that \(95 \%\) of days have at least one call lasting longer than 30 minutes.
   On Wednesday 900 calls arrived at the call centre and none of them lasted longer than 30 minutes.
3. Use a Poisson approximation to estimate the proportion of calls arriving at the call centre that last longer than 30 minutes.

1. The number of errors made by a secretary is modelled by a Poisson distribution with a mean of 2.4 per 100 words.

A 100-word piece of work completed by the secretary is selected at random.

1. Find the probability that
   1. there are exactly 3 errors,
   2. there are fewer than 2 errors. After a long holiday, a randomly selected piece of work containing 250 words completed by the secretary is examined to see if the rate of errors has changed.
2. Stating your hypotheses clearly, and using a \(5 \%\) level of significance, find the critical region for a suitable test.
3. Find P (Type I error) for the test in part (b)

1. Bacteria are randomly distributed in a river at a rate of 5 per litre of water. A new factory opens and a scientist claims it is polluting the river with bacteria. He takes a sample of 0.5 litres of water from the river near the factory and finds that it contains 7 bacteria. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether there is evidence that the level of pollution has increased.

\section*{Q uestion 1 continued}

3 Over a long period Jenny counts the number of trolleys used at her local supermarket between 10 am and 10.20 am each day. She finds that the mean number of trolleys used between these times on a weekday is 40.00. You should assume that the use of trolleys occurs randomly, independently of one another, and at a constant average rate.

1. Calculate the probability that, on a randomly chosen weekday, the number of trolleys used between these times is between 32 and 50 inclusive.
2. Write down an expression for the probability that, on a randomly chosen weekday, exactly 5 trolleys are used during a time period of \(t\) minutes between 10 am and 10.20 am. Jenny carries out this process for seven consecutive days. She finds that the mean number of trolleys used between 10 am and 10.20 am is 35.14 and the variance is 91.55 .
3. Explain why this suggests that the distribution of the number of trolleys used between these times on these seven consecutive days is not well modelled by a Poisson distribution.
4. Give a reason why it might not be appropriate to apply the Poisson model to the total number of trolleys used between these times on seven consecutive days.

1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).

1. Find
   1. \(\mathrm { P } ( X + Y ) > 40\),
   2. \(\operatorname { Var } ( 2 X - Y )\).
   3. State a necessary assumption for your calculations in part (i) to be valid.

2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).

1. It is given that \(W\) has a Poisson distribution.
   1. Write down the standard deviation of \(W\).
   2. Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
   3. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).

6 The discrete random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\).
Use an algebraic method to find the range of values of \(\lambda\) for which the single most likely value of \(R\) is 7. [7]

4 Leyla investigates the number of shoppers who visit a shop between 10.30 am and 11 am on Saturday mornings. She makes the following assumptions.

• Shoppers visit the shop independently of one another.
• The average rate at which shoppers visit the shop between these times is constant.
  1. State an appropriate distribution with which Leyla could model the number of shoppers who visit the shop between these times.

Leyla uses this distribution, with mean 14, as her model.

Calculate the probability that, between 10.35 am and 10.50 am on a randomly chosen Saturday, at least 10 shoppers visit the shop. Leyla chooses 25 Saturdays at random.

Find the expected number of Saturdays, out of 25, on which there are no visitors to the shop between 10.35 am and 10.50 am .

In fact on 5 of these Saturdays there were no visitors to the shop between 10.35 am and 10.50 am . Use this fact to comment briefly on the validity of the model that Leyla has used.

3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, \(X\), for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, \(Y\), for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.

1. Calculate:
   1. \(\mathrm { P } ( X \leqslant 3 )\);
   2. \(\quad \mathrm { P } ( Y = 5 )\).
2. The total daily demand for the device at Joe's two garages is denoted by \(T\).
   1. Write down the distribution of \(T\), stating any assumption that you make.
   2. Determine \(\mathrm { P } ( 6 < T < 12 )\).
   3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
   4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least \(99 \%\) of days.

2 The number of telephone calls per day, \(X\), received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, \(Y\), received by Candice may be modelled by a Poisson distribution with mean 6.0.

1. For any particular day, find:
   1. \(\mathrm { P } ( X = 3 )\);
   2. \(\quad \mathrm { P } ( Y \geqslant 5 )\).
2. 1. Write down the distribution of \(T\), the total number of telephone calls and e-mails per day received by Candice.
   2. Determine \(\mathrm { P } ( 7 \leqslant T \leqslant 10 )\).
   3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
      (2 marks)