5.02l Poisson conditions: for modelling

5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .

1. Show that the variance of the number of faulty screws in a packet is 1.104 .
2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.

2 The number of orders arriving at a shop during an 8-hour working day is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 25.2 )\).

1. State two assumptions that are required for the Poisson model to be valid in this context.
2. 1. Find the probability that the number of orders that arrive in a randomly chosen 3-hour period is between 3 and 5 inclusive.
   2. Find the probability that, in two randomly chosen 1 -hour periods, exactly 1 order will arrive in one of the 1 -hour periods, and at least 2 orders will arrive in the other 1 -hour period. [4]
3. The shop can only deal with a maximum of 120 orders during any 36-hour period. Use a suitable approximating distribution to find the probability that, in a randomly chosen 36-hour period, there will be too many orders for the shop to deal with.

6 The number of sports injuries per month at a certain college has a Poisson distribution. In the past the mean has been 1.1 injuries per month. The principal recently introduced new safety guidelines and she decides to test, at the \(2 \%\) significance level, whether the mean number of sports injuries has been reduced. She notes the number of sports injuries during a 6-month period.

1. Find the critical region for the test and state the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. During the 6 -month period there are a total of 2 sports injuries. Carry out the test.
4. Assuming that the mean remains 1.1 , calculate the probability that there will be fewer than 30 sports injuries during a 36-month period.

3 A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.

1. Assume that the owner is correct.
   1. Find the probability that there will be at least 4 hits during a 10-minute period.
   2. Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period.
      A friend agrees that the website receives, on average, 0.3 hits per minute. However, she notices that the number of hits during the day-time ( 9.00 am to 9.00 pm ) is usually about twice the number of hits during the night-time ( 9.00 pm to 9.00 am ).
2. 1. Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution.
   2. Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time.

7 A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 4.2 )\).

1. State two assumptions required for the Poisson model to be valid.
2. Find the probability that
   1. at least 1 patient will arrive in a 15-minute period,
   2. fewer than 4 patients will arrive in a 10-minute period.
   3. The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough vaccine for 370 patients. Use a suitable approximation to find the probability that this will not be enough vaccine for that week.

1 The number of new enquiries per day at an office has a Poisson distribution. In the past the mean has been 3 . Following a change of staff, the manager wishes to test, at the \(5 \%\) significance level, whether the mean has increased.

1. State the null and alternative hypotheses for this test. The manager notes the number, \(N\), of new enquiries during a certain 6 -day period. She finds that \(N = 25\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \geqslant 25 ) = 0.0683\).
2. What conclusion should she draw?

5 The probability that a new car of a certain type has faulty brakes is 0.008 . A random sample of 520 new cars of this type is chosen, and the number, \(X\), having faulty brakes is noted.

1. Describe fully the distribution of \(X\) and describe also a suitable approximating distribution. Justify this approximating distribution.
2. Use your approximating distribution to find
   1. \(\mathrm { P } ( X > 3 )\),
   2. the smallest value of \(n\) such that \(\mathrm { P } ( X = n ) > \mathrm { P } ( X = n + 1 )\).

1 It is known that \(1.2 \%\) of rods made by a certain machine are bent. The random variable \(X\) denotes the number of bent rods in a random sample of 400 rods.

1. State the distribution of \(X\).
2. State, with a reason, a suitable approximate distribution for \(X\).
3. Use your approximate distribution to find the probability that the sample will include more than 2 bent rods.

7 A Lost Property office is open 7 days a week. It may be assumed that items are handed in to the office randomly, singly and independently.

1. State another condition for the number of items handed in to have a Poisson distribution. It is now given that the number of items handed in per week has the distribution \(\operatorname { Po } ( 4.0 )\).
2. Find the probability that exactly 2 items are handed in on a particular day.
3. Find the probability that at least 4 items are handed in during a 10-day period.
4. Find the probability that, during a certain week, 5 items are handed in altogether, but no items are handed in on the first day of the week.

6 Accidents on a particular road occur at a constant average rate of 1 every 4.8 weeks.

1. State, in context, one condition for the number of accidents in a given period to be modelled by a Poisson distribution.
   Assume now that a Poisson distribution is a suitable model.
2. Find the probability that exactly 4 accidents will occur during a randomly chosen 12-week period.
3. Find the probability that more than 3 accidents will occur during a randomly chosen 10 -week period.
4. Use a suitable approximating distribution to find the probability that fewer than 30 accidents will occur during a randomly chosen 2 -year period ( \(104 \frac { 2 } { 7 }\) weeks).

6 On average a motorway police force records one car that has run out of petrol every two days.

1. (a) Using a Poisson distribution, calculate the probability that, in one randomly chosen day, the police force records exactly two cars that have run out of petrol.
   (b) Using a Poisson distribution and a suitable approximation to the binomial distribution, calculate the probability that, in one year of 365 days, there are fewer than 205 days on which the police force records no cars that have run out of petrol.
2. State an assumption needed for the Poisson distribution to be appropriate in part (i), and explain why this assumption is unlikely to be valid.

1 A roller-coaster ride has a safety system to detect faults on the track.

1. State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day. Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.
2. Find the probability that on a randomly chosen day there are
   (A) no faults,
   (B) at least 2 faults.
3. Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults. There is also a separate safety system to detect faults on the roller-coaster train itself. Faults are detected by this system at an average rate of 0.05 per day, independently of the faults detected on the track. You may assume that a Poisson distribution is also suitable for modelling the number of faults detected on the train.
4. State the distribution of the total number of faults detected by the two systems in a period of 10 days. Find the probability that a total of 5 faults is detected in a period of 10 days.
   [0pt]
5. The roller-coaster is operational for 200 days each year. Use a suitable approximating distribution to find the probability that a total of at least 50 faults is detected in 200 days. [5]

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.

3 The number of calls received at an office per 5 minutes is modelled by a Poisson distribution with mean 3.2.

1. Find the probability of
   (A) exactly one call in a 5 -minute period,
   (B) at least 6 calls in a 5 -minute period.
2. Find the probability of
   (A) exactly one call in a 1 -minute period,
   (B) exactly one call in each of five successive 1-minute periods.
3. Use a suitable approximating distribution to find the probability of at most 45 calls in a period of 1 hour. Two assumptions required for a Poisson distribution to be a suitable model are that calls arrive

8 In a large city the number of traffic lights that fail in one day of 24 hours is denoted by \(Y\). It may be assumed that failures occur randomly.

1. Explain what the statement "failures occur randomly" means.
2. State, in context, two different conditions that must be satisfied if \(Y\) is to be modelled by a Poisson distribution, and for each condition explain whether you think it is likely to be met in this context.
3. For this part you may assume that \(Y\) is well modelled by the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( Y = 7 ) = \mathrm { P } ( Y = 8 )\). Use an algebraic method to calculate the value of \(\lambda\) and hence calculate the corresponding value of \(\mathrm { P } ( Y = 7 )\).

9 The managers of a car breakdown recovery service are discussing whether the number of breakdowns per day can be modelled by a Poisson distribution. They agree that breakdowns occur randomly. Manager \(A\) says, "it must be assumed that breakdowns occur at a constant rate throughout the day".

1. Give an improved version of Manager \(A\) 's statement, and explain why the improvement is necessary.
2. Explain whether you think your improved statement is likely to hold in this context. Assume now that the number \(B\) of breakdowns per day can be modelled by the distribution \(\operatorname { Po } ( \lambda )\).
3. Given that \(\lambda = 9.0\) and \(\mathrm { P } \left( B > B _ { 0 } \right) < 0.1\), use tables to find the smallest possible value of \(B _ { 0 }\), and state the corresponding value of \(\mathrm { P } \left( B > B _ { 0 } \right)\).
4. Given that \(\mathrm { P } ( B = 2 ) = 0.0072\), show that \(\lambda\) satisfies an equation of the form \(\lambda = 0.12 \mathrm { e } ^ { k \lambda }\), for a value of \(k\) to be stated. Evaluate the expression \(0.12 \mathrm { e } ^ { k \lambda }\) for \(\lambda = 8.5\) and \(\lambda = 8.6\), giving your answers correct to 4 decimal places. What can be deduced about a possible value of \(\lambda\) ?

2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.

1. State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute. You may assume that these conditions are satisfied.
2. Find the probability that, during one minute, Jess sees
   (A) exactly one meteor,
   (B) at least 4 meteors.
3. Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
4. Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
5. Jess watches the shower for \(t\) minutes. She wishes to be at least \(99 \%\) certain that she will see one or more meteors. Find the smallest possible integer value of \(t\).

2 A radioactive source is decaying at a mean rate of 3.4 counts per 5 seconds.

1. State conditions for a Poisson distribution to be a suitable model for the rate of decay of the source. You may assume that a Poisson distribution with a mean rate of 3.4 counts per 5 seconds is a suitable model.
2. State the variance of this Poisson distribution.
3. Find the probability of
   (A) exactly 3 counts in a 5 -second period,
   (B) at least 3 counts in a 5 -second period.
4. Find the probability of exactly 40 counts in a period of 60 seconds.
5. Use a suitable approximating distribution to find the probability of at least 40 counts in a period of 60 seconds.
6. The background radiation rate also, independently, follows a Poisson distribution and produces a mean count of 1.4 per 5 seconds. Find the probability that the radiation source together with the background radiation give a total count of at least 8 in a 5 -second period.

2 At a drive-through fast food takeaway, cars arrive independently, randomly and at a uniform average rate. The numbers of cars arriving per minute may be modelled by a Poisson distribution with mean 0.62.

1. Briefly explain the meaning of each of the three terms 'independently', 'randomly' and 'at a uniform average rate', in the context of cars arriving at a fast food takeaway.
2. Find the probability of at most 1 car arriving in a period of 1 minute.
3. Find the probability of more than 5 cars arriving in a period of 10 minutes.
4. State the exact distribution of the number of cars arriving in a period of 1 hour.
5. Use a suitable approximating distribution to find the probability that at least 40 cars arrive in a period of 1 hour.

2 Manufacturing defects occur in a particular type of aluminium sheeting randomly, independently and at a constant average rate of 1.7 defects per square metre.

1. Explain the meaning of the term 'independently' and name the distribution that models this situation.
2. Find the probability that there are exactly 2 defects in a sheet of area 1 square metre.
3. Find the probability that there are exactly 12 defects in a sheet of area 7 square metres. In another type of aluminium sheet, defects occur randomly, independently and at a constant average rate of 0.8 defects per square metre.
4. A large box is made from 2 square metres of the first type of sheet and 2 square metres of the second type of sheet, chosen independently. Show that the probability that there are at least 8 defects altogether in the box is 0.1334 . A random sample of 100 of these boxes is selected.
5. State the exact distribution of the number of boxes which have at least 8 defects.
6. Use a suitable approximating distribution to find the probability that there are at least 20 boxes in the sample which have at least 8 defects.

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.

7 The lifetime, in hours, of a 'Trulite' light bulb is a random variable \(T\). The probability density function f of \(T\) is given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 \\ \lambda \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \end{cases}$$ where \(\lambda\) is a positive constant. Given that the mean lifetime of Trulite bulbs is 2000 hours, find the probability that a randomly chosen Trulite bulb has a lifetime of at least 1000 hours. A particular light fitting has 6 randomly chosen Trulite bulbs. Find the probability that no more than one of these bulbs has a lifetime less than 1000 hours. By using new technology, the proportion of Trulite bulbs with very short lifetimes is to be reduced. Find the least value of the new mean lifetime that will ensure that the probability that a randomly chosen Trulite bulb has a lifetime of no more than 4 hours is less than 0.001 .

5 The manager of an emergency response hotline believes that calls are made to the hotline independently and at constant average rate throughout the day. From a small random sample of the population, the manager finds that the mean number of calls made in a 1-hour period is 14.4. Let \(R\) denote the number of calls made in a randomly chosen 1-hour period.

1. Using evidence from the small sample, state a suitable distribution with which to model \(R\). You should give the value(s) of any parameter(s).
2. In this part of the question, use the distribution and value(s) of the parameter(s) from your answer to part (a).
   1. Find \(\mathrm { P } ( R > 20 )\).
   2. Given that \(\mathrm { P } ( \mathrm { R } = \mathrm { r } ) > \mathrm { P } ( \mathrm { R } = \mathrm { r } + 1 )\), show algebraically that \(r > 13.4\).
   3. Hence write down the mode of the distribution. The manager also finds, from records over many years, that the modal value of \(R\) is 10 .
3. Use this result to comment on the validity of the distribution used in part (b).
4. Assume now that the type of distribution used in part (b) is valid. Find the range(s) of values of the parameter(s) of this distribution that would correspond to the modal value of \(R\) being 10.

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 \(N\).