5.02i Poisson distribution: random events model

1. The number of telephone calls per hour received by a business is a random variable with distribution \(\operatorname { Po } ( \lambda )\).

Charlotte records the number of calls, \(C\), received in 4 hours. A test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 1.5\) is carried out. \(\mathrm { H } _ { 0 }\) is rejected if \(C > 10\)

1. Write down the alternative hypothesis.
2. Find the significance level of the test. Given that \(\mathrm { P } ( C > 10 ) < 0.1\)
3. find the largest possible value of \(\lambda\) that can be found by using the tables.
   

5. A school photocopier breaks down randomly at a rate of 15 times per year.

1. Find the probability that there will be exactly 3 breakdowns in the next month.
2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
4. Find the probability that the head teacher will buy a new photocopier.

The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8

1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera. A car has been caught speeding by this camera.
2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined \(\pounds 60\)
4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)

4. Accidents occur randomly at a crossroads at a rate of 0.5 per month. A researcher records the number of accidents, \(X\), which occur at the crossroads in a year.

1. Find \(\mathrm { P } ( 5 \leqslant X < 7 )\) A new system is introduced at the crossroads. In the first 18 months, 4 accidents occur at the crossroads.
2. Test, at the \(5 \%\) level of significance, whether or not there is reason to believe that the new system has led to a reduction in the mean number of accidents per month. State your hypotheses clearly.

3.

1. State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. The number of reported first aid incidents per week at an airport terminal has a Poisson distribution with mean 3.5
2. Find the modal number of reported first aid incidents in a randomly selected week. Justify your answer. The random variable \(X\) represents the number of reported first aid incidents at this airport terminal in the next 2 weeks.
3. Find \(\mathrm { P } ( X > 5 )\)
4. Given that there were exactly 6 reported first aid incidents in a 2 week period, find the probability that exactly 4 were reported in the first week.
5. Using a suitable approximation, find the probability that in the next 40 weeks there will be at least 120 reported first aid incidents.
   

1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)

The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).

1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
2. 1. Write down suitable hypotheses for this test.
   2. Describe a suitable test statistic that the manager should use.
   3. Explain what is meant by the critical region for this test.
3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
4. Find the actual significance level for this test.

4. A sweet shop produces different coloured sweets and sells them in bags. The proportion of green sweets produced is \(p\) Each bag is filled with a random sample of \(n\) sweets. The mean number of green sweets in a bag is 4.2 and the variance is 3.57

1. Find the value of \(n\) and the value of \(p\) The proportion of red sweets produced by the shop is 0.35
2. Find the probability that, in a random sample of 25 sweets, the number of red sweets exceeds the expected number of red sweets. The shop claims that \(10 \%\) of its customers buy more than two bags of sweets. A random sample of 40 customers is taken and 1 customer buys more than two bags of sweets.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy more than two bags of sweets is less than the shop's claim. State your hypotheses clearly.
   

5. A delivery company loses packages randomly at a mean rate of 10 per month. The probability that the delivery company loses more than 12 packages in a randomly selected month is \(p\)

1. Find the value of \(p\) The probability that the delivery company loses more than \(k\) packages in a randomly selected month is at least \(2 p\)
2. Find the largest possible value of \(k\) In a randomly selected month,
3. find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
4. Find the probability that at least 10 packages were lost in each of these two months.
5. Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.

1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
3. Find the value of \(C\)
4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7

1. Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
2. Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of \(n \mathrm { ml }\) of the pond water is 0.9394 correct to 4 significant figures.
3. 1. Show that \(n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0\)
   2. Hence find the value of \(n\) After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
4. Using a suitable test, at the \(1 \%\) level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528

1. Find the value of \(x\)
2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".

3 A photocopier in a school is known to break down at random at a mean rate of 8 times per week.

1. Give a reason why a Poisson distribution could be used to model the number of breakdowns. The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.
2. Write down suitable null and alternative hypotheses that the headteacher should use. The refurbished photocopier was monitored for the first week after it was installed.
3. Using a \(5 \%\) level of significance, find the critical region to test whether the rate of breakdowns has now changed.
4. Find the actual significance level of a test based on the critical region from part (c). During the first week after it was installed there were 4 breakdowns.
5. Comment on this finding in the light of the critical region found in part (c).

A shop sells shoes at a mean rate of 4 pairs of shoes per hour on a weekday.

1. Suggest a suitable distribution for modelling the number of sales of pairs of shoes made per hour on a weekday.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Find the probability that on a weekday the shop sells
   1. more than 4 pairs of shoes in a one-hour period,
   2. more than 4 pairs of shoes in each of 3 consecutive one-hour periods. The area manager visits the shop on a weekday, the day after an advert for the shop appears in a local paper. In a one-hour period during the manager's visit, the shop sells 7 pairs of shoes. This leads the manager to believe that the advert has increased the shop's sales of pairs of shoes.
4. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of an increase in sales of pairs of shoes following the appearance of the advert.

1. A company produces steel cable.

Defects in the steel cable produced by this company occur at random, at a constant rate of 1 defect per 16 metres. On one day the company produces a piece of steel cable 80 metres long.

1. Find the probability that there are at most 5 defects in this piece of steel cable. The company produces a piece of steel cable 80 metres long on each of the next 4 days.
2. Find the probability that fewer than 2 of these 4 pieces of steel cable contain at most 5 defects. The following week the company produces a piece of steel cable \(x\) metres long.
   Using a normal approximation, the probability that this piece of steel cable has fewer than 26 defects is 0.5398
3. Find the value of \(x\)

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. The length of pregnancy for a randomly selected pregnant sheep is \(D\) days where

$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$ Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,

1. find the value of \(\sigma\) Qiang selects 25 pregnant sheep at random from a large flock.
2. Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days. Charlie takes 200 random samples of 25 pregnant sheep.
3. Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.

7. Flaws occur at random in a particular type of material at a mean rate of 2 per 50 m .

1. Find the probability that in a randomly chosen 50 m length of this material there will be exactly 5 flaws. This material is sold in rolls of length 200 m . Susie buys 4 rolls of this material.
2. Find the probability that only one of these rolls will have fewer than 7 flaws. A piece of this material of length \(x \mathrm {~m}\) is produced. Using a normal approximation, the probability that this piece of material contains fewer than 26 flaws is 0.5398
3. Find the value of \(x\).

2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8

1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.

1. During a typical day, a school website receives visits randomly at a rate of 9 per hour.

The probability that the school website receives fewer than \(v\) visits in a randomly selected one hour period is less than 0.75

1. Find the largest possible value of \(v\)
2. Find the probability that in a randomly selected one hour period, the school website receives at least 4 but at most 11 visits.
3. Find the probability that in a randomly selected 10 minute period, the school website receives more than 1 visit.
4. Using a suitable approximation, find the probability that in a randomly selected 8 hour period the school website receives more than 80 visits.
   

2. The random variable \(X \sim \mathrm {~B} ( 10 , p )\)

1. 1. Write down an expression for \(\mathrm { P } ( X = 3 )\) in terms of \(p\)
   2. Find the value of \(p\) such that \(\mathrm { P } ( X = 3 )\) is 16 times the value of \(\mathrm { P } ( X = 7 )\) The random variable \(Y \sim \operatorname { Po } ( \lambda )\)
2. Find the value of \(\lambda\) such that \(\mathrm { P } ( Y = 3 )\) is 5 times the value of \(\mathrm { P } ( Y = 5 )\) The random variable \(W \sim \mathrm {~B} ( n , 0.4 )\)
3. Find the value of \(n\) and the value of \(\alpha\) such that \(W\) can be approximated by the normal distribution, \(\mathrm { N } ( 32 , \alpha )\)

At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.

1. Find the probability that there are no signal failures on a randomly selected day.
2. Find the probability that there is at least 1 signal failure on each of the next 3 days.
3. Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures. Repair works are carried out on the line. After these repair works, the number, \(f\), of signal failures in a 32-day period is recorded. A test is carried out, at the \(5 \%\) level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.
4. State the hypotheses for this test.
5. Find the largest value of \(f\) for which the null hypothesis should be rejected.

4. In a large population, past records show that 1 in 200 adults has a particular allergy. In a random sample of 700 adults selected from the population, estimate

1. 1. the mean number of adults with the allergy,
   2. the standard deviation of the number of adults with the allergy. Give your answer to 3 decimal places. A doctor claims that the past records are out of date and the proportion of adults with the allergy is higher than the records indicate. A random sample of 500 adults is taken from the population and 5 are found to have the allergy. A test of the doctor's claim is to be carried out at the 5\% level of significance.
2. 1. State the hypotheses for this test.
   2. Using a suitable approximation, carry out the test.
      (6) It is also claimed that \(30 \%\) of those with the allergy take medication for it daily. To test this claim, a random sample of \(n\) people with the allergy is taken. The random variable \(Y\) represents the number of people in the sample who take medication for the allergy daily. A two-tailed test, at the \(1 \%\) level of significance, is carried out to see if the proportion differs from 30\% The critical region for the test is \(Y = 0\) or \(Y \geqslant w\)
3. Find the smallest possible value of \(n\) and the corresponding value of \(w\)

2. John weaves cloth by hand. Emma believes that faults are randomly distributed in John's cloth at a rate of more than 4 per 50 metres of cloth. To check her belief, Emma takes a random sample of 100 metres of the cloth and finds that it contains 14 faults.

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, Emma's belief. Armani also weaves cloth by hand. He knows that faults are randomly distributed in his cloth at a rate of 4 per 50 metres of cloth. Emma decides to buy a large amount of Armani's cloth to sell in pieces of length \(l\) metres. She chooses \(l\) so that the probability of no faults in a piece is exactly 0.9
2. Show that \(l = 1.3\) to 2 significant figures. Emma sells 5000 of these pieces of cloth of length 1.3 metres. She makes a profit of \(\pounds 2.50\) on each piece of cloth that does not contain any faults but a loss of \(\pounds 0.50\) on any piece that contains at least one fault.
3. Find Emma's expected profit.

5. Cars stop at a service station randomly at a rate of 3 every 5 minutes.

1. Calculate the probability that in a randomly selected 10 minute period,
   1. exactly 7 cars will stop at the service station,
   2. more than 7 cars will stop at the service station. Using a normal approximation, the probability that more than 40 cars will stop at the service station during a randomly selected \(n\) minute period is 0.2266 correct to 4 significant figures.
2. Find the value of \(n\).