In a study, samples of soil were collected during the summer. Soil samples of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) were collected for analysis. The study found that there were, on average, 11 earthworms per sample.
a) Explain briefly the conditions under which a Poisson distribution could be used to model the number of earthworms per sample.
b) In July, pupils at a primary school are asked to dig a smaller hole, \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 10 \mathrm {~cm}\), and to count the number of earthworms they find. Calculate the probability that the pupils find exactly 5 earthworms.
c) In the autumn, the average number of earthworms per sample is greater than in the summer. The probability that, in the autumn, there are fewer than 13 earthworms in a soil sample of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) is close to \(36 \%\). Find the mean number of earthworms, to the nearest whole number, per \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) soil sample in the autumn.
Jessica is studying the relationship between hip girth, \(h \mathrm {~cm}\), and thigh girth, \(t \mathrm {~cm}\), for American adults who are physically active. She takes a random sample of 11 people from a very large dataset which she has downloaded into a spreadsheet software package. The results are shown below.
| \(h ( \mathrm {~cm} )\) | \(98 \cdot 6\) | \(112 \cdot 1\) | \(97 \cdot 9\) | \(110 \cdot 2\) | \(89 \cdot 2\) | \(111 \cdot 7\) | \(87 \cdot 0\) | \(94 \cdot 7\) | \(100 \cdot 4\) | \(104 \cdot 0\) | \(88 \cdot 4\) |
| \(t ( \mathrm {~cm} )\) | \(48 \cdot 3\) | \(87 \cdot 2\) | \(55 \cdot 2\) | \(68 \cdot 0\) | \(48 \cdot 5\) | \(63 \cdot 2\) | \(49 \cdot 5\) | \(55 \cdot 7\) | \(59 \cdot 1\) | \(64 \cdot 0\) | \(52 \cdot 4\) |
a) Jessica notes that, for the thigh girth data, the lower quartile is 49.5 and the upper quartile is \(64 \cdot 0\).
i) Show that 87.2 should be classified as an outlier for \(t\).
ii) Give a reason why Jessica might exclude the outlier.
iii) Give a reason why Jessica might include the outlier.
Jessica decides to exclude the outlier and produces the following scatter diagram.
\section*{Thigh girth versus Hip girth}
\includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-04_647_1250_1439_404}
b) Interpret, in context, the correlation in the data shown in the diagram.
The equation of the regression line of \(t\) on \(h\) for this sample is
$$t = 0.69 h - 11.26$$
c) Interpret the gradient of the regression line in this context.
d) Use your knowledge of large data sets and spreadsheet software packages to suggest a way in which Jessica could improve her investigation.
A company, Run4Lyfe, sponsors an athletic event. The organisers of the event claim that \(70 \%\) of the participants know the name of the sponsoring company. Run4Lyfe is concerned that the proportion, \(p\), of participants knowing the name of the sponsoring company is less than \(70 \%\). They decide to survey 60 randomly selected participants to carry out a significance test.
a) State suitable hypotheses for carrying out the test.
b) i) Explain what is meant by the critical region for this test.
ii) Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, \(5 \%\).
iii) Given that 40 participants out of the 60 in the sample know the name of the company, complete the significance test.
c) State, with a reason, how you would advise Run4Lyfe with regards to sponsoring the event next year.
The fertility rate for a country is the average number of children that are born to a woman over her lifetime. The graphs and table below show some data on the fertility rates for 197 countries in the years 1914 and 2014.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Fertility rates in 1914}
\includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_671_1483_593_283}
\end{figure}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Fertility rates in 2014}
\includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_616_1219_1434_287}
\end{figure}
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Decreases in fertility rates from 1914 to 2014}
\includegraphics[alt={},max width=\textwidth]{77c62e6d-58e4-42d3-9982-5a8325e8e826-06_476_613_2270_388}
\end{figure}
| Minimum value | - 0.71 | | Lower quartile | 2.08 | | Median | 3.19 | | Upper quartile | 3.94 | | Maximum value | 6.49 |
a) Comment on the shapes of the distributions of fertility rates for 1914 and 2014.
b) Interpret the minimum value, \(- 0 \cdot 71\), in the boxplot.
You are also given the following information:
| Country | | | | France | | 1.98 | | Ethiopia | | 4.4 |
ii) Find the best possible estimate for the decrease in the fertility rate from 1914 to 2014 for Ethiopia.
iii) Give one possible reason why the answers to i) and ii) are so different.
iv) Explain why these estimates may not be very accurate.
\section*{Section B: Mechanics}
The diagram below shows a vehicle of mass 1300 kg towing a trailer of mass 500 kg by means of a light horizontal tow bar. The vehicle is moving forward along a straight horizontal road such that a constant resistance of magnitude 650 N acts on the vehicle and a constant resistance of magnitude 320 N acts on the trailer. The vehicle's engine produces a constant driving force of \(F \mathrm {~N}\).
\includegraphics[max width=\textwidth, alt={}]{77c62e6d-58e4-42d3-9982-5a8325e8e826-08_158_851_781_609}
Given that the acceleration of the vehicle and trailer is \(0.85 \mathrm {~ms} ^ { - 2 }\), show that \(F = 2500\) and determine the tension in the tow bar.
CAIE
Further Paper 4
2021
June
Q5
10 marks
Standard +0.3
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.
- 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.
- 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]
Edexcel
S2
Q6
15 marks
Standard +0.3
A doctor expects to see, on average, 1 patient per week with a particular disease.
- Suggest a suitable model for the distribution of the number of times per week that the doctor sees a patient with the disease. Give a reason for your answer. [3]
- Using your model, find the probability that the doctor sees more than 3 patients with the disease in a 4 week period. [4]
The doctor decides to send information to his patients to try to reduce the number of patients he sees with the disease. In the first 6 weeks after the information is sent out, the doctor sees 2 patients with the disease.
- Test, at the 5\% level of significance, whether or not there is reason to believe that sending the information has reduced the number of times the doctor sees patients with the disease. State your hypotheses clearly. [6]
Medical research into the nature of the disease discovers that it can be passed from one patient to another.
- Explain whether or not this research supports your choice of model. Give a reason for your answer. [2]
Edexcel
S2
2011
January
Q6
16 marks
Standard +0.3
Cars arrive at a motorway toll booth at an average rate of 150 per hour.
- Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute.
[2]
- State clearly any assumptions you have made by suggesting this model.
[2]
Using your model,
- find the probability that in any given minute
- no cars arrive,
- more than 3 cars arrive.
[3] - In any given 4 minute period, find \(m\) such that P(\(X > m\)) = 0.0487
[3]
- Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period.
[6]
Edexcel
S2
Specimen
Q4
11 marks
Standard +0.3
A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:
| No. of errors | 0 | 1 | 2 | 3 | 4 | 5 | | No. of pages | 37 | 65 | 60 | 49 | 27 | 12 |
- Show that the mean number of errors per page in this sample of pages is 2. [2]
- Find the variance of the number of errors per page in this sample. [2]
- 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.
- Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a 5\% level of significance. [6]
Edexcel
S2
Specimen
Q6
14 marks
Standard +0.3
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.
- 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
- no sheep, [1]
- more than 2 sheep. [4]
A sheepdog has been sent into the field to round up the sheep.
- 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.
- Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]
Edexcel
S3
2015
June
Q3
11 marks
Standard +0.3
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 |
- 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 | \(s\) |
- Find the value of \(r\) and the value of \(s\), giving your answers to 2 decimal places. [3]
- Stating your hypotheses clearly, use a 10\% level of significance to test the motorway supervisor's belief. Show your working clearly. [7]
Edexcel
S3
2011
June
Q5
13 marks
Standard +0.3
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
- Write down two assumptions that will support modelling the number of hurricanes per year by a Poisson distribution.
[2]
- Show that the mean number of hurricanes per year from Table 1 is 4.4875
[2]
- 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
- Test, at the 5\% level of significance, whether or not the data can be modelled by a Poisson distribution. State your hypotheses clearly.
[6]
AQA
S2
2010
June
Q5
13 marks
Standard +0.3
The number of telephone calls received, during an \(8\)-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of \(7\).
- Determine the probability that, during a given \(8\)-hour period, the number of telephone calls received that request an urgent visit by an engineer is:
- at most \(5\); [1 mark]
- exactly \(7\); [2 marks]
- at least \(5\) but fewer than \(10\). [3 marks]
- Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer. [1 mark]
- The IT company has \(4\) engineers available for urgent visits and it may be assumed that each of these engineers takes exactly \(1\) hour for each such visit.
At \(10\)am on a particular day, all \(4\) engineers are available for urgent visits.
- State the maximum possible number of telephone calls received between \(10\)am and \(11\)am that request an urgent visit and for which an engineer is immediately available. [1 mark]
- Calculate the probability that at \(11\)am an engineer will not be immediately available to make an urgent visit. [4 marks]
- Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer. [1 mark]
Edexcel
S2
Q3
10 marks
Standard +0.3
A secretarial agency carefully assesses the work of a new recruit, with the following results after 150 pages:
| No of errors | 0 | 1 | 2 | 3 | 4 | 5 | 6 | | No of pages | 16 | 38 | 41 | 29 | 17 | 7 | 2 |
- Find the mean and variance of the number of errors per page. [4 marks]
- Explain how these results support the idea that the number of errors per page follows a Poisson distribution. [1 mark]
- After two weeks at the agency, the secretary types a fresh piece of work, six pages long, which is found to contain 15 errors.
The director suspects that the secretary was trying especially hard during the early period and that she is now less conscientious. Using a Poisson distribution with the mean found in part (a), test this hypothesis at the 5% significance level. [5 marks]
Edexcel
S2
Q5
12 marks
Moderate -0.3
In a survey of 22 families, the number of children, \(X\), in each family was given by the following
table, where \(f\) denotes the frequency:
- Find the mean and variance of \(X\). [4 marks]
- Explain why these results suggest that \(X\) may follow a Poisson distribution. [1 mark]
- State another feature of the data that suggests a Poisson distribution. [1 mark]
It is sometimes suggested that the number of children in a family follows a Poisson distribution
with mean 2·4. Assuming that this is correct,
- find the probability that a family has less than two children. [3 marks]
- Use this result to find the probability that, in a random sample of 22 families, exactly 11 of
the families have less than two children. [3 marks]
Edexcel
S2
Q4
11 marks
Standard +0.3
A Geiger counter is observed in the presence of a radioactive source.
In 100 one-minute intervals, the number of counts recorded are as follows:
| No of counts, \(X\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | | Frequency | 10 | 24 | 29 | 16 | 12 | 6 | 3 |
- Find the mean and variance of this data, and show that it supports the idea that the random variable \(X\) is following a Poisson distribution. [5 marks]
- Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute. [4 marks]
- Find the number of one-minute intervals, in the sample of 100, in which more than 6 counts would be expected. [2 marks]
OCR
S2
2016
June
Q6
12 marks
Moderate -0.3
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 \(\lambda\).
- State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [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]
- At a busy time of the day, \(\lambda = 30\).
- Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
- Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]
OCR MEI
S2
2007
January
Q3
18 marks
Standard +0.3
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.
| Number of repairs | 0 | 1 | 2 | 3 | \(>3\) | | Frequency | 53 | 20 | 6 | 1 | 0 |
- Show that the sample mean is 0.4375. [1]
- 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.
- Find P\((X = 1)\). Comment on your answer in relation to the data in the table. [4]
- 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]
- 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.
- Assuming that free repairs are required independently, find the probability that
- the two appliances need a total of 3 free repairs between them,
- each appliance needs exactly one free repair.
[5]
Edexcel
S2
Q3
10 marks
Moderate -0.3
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.
- 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]
- Find the probability that on one particular day he repairs
- no CD players,
- more than 6 CD players. [3 marks]
- Find the probability that over 10 working days he will repair more than 6 CD players on exactly 3 of the days. [3 marks]
Edexcel
S2
Q6
12 marks
Moderate -0.3
A shoe shop sells on average 4 pairs of shoes per hour on a weekday morning.
- 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]
- Explain why this model might have to be modified for modelling the number of sales made per hour on a Saturday morning. [1 mark]
- Find the probability that on a weekday morning the shop sells
- more than 4 pairs in a one-hour period,
- no pairs in a half-hour period,
- 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.
- 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]
Edexcel
S2
Q2
9 marks
Moderate -0.8
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.
- Suggest a suitable distribution for modelling the number of species that visit a bird table meeting these criteria. [1 mark]
- Explain why the parameter used with this model may need to be changed if
- 50 g of nuts are used instead of breadcrumbs,
- 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
- exactly 6 different species visit the table, [2 marks]
- more than 2 different species visit the table. [4 marks]
Edexcel
S2
Q4
13 marks
Standard +0.3
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
- there will be no failures in a one-hour period, [1 mark]
- 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
- less than 60 failures, [5 marks]
- exactly 72 failures. [4 marks]
Edexcel
S2
Q2
9 marks
Moderate -0.3
The manager of a supermarket receives an average of 6 complaints per day from customers.
Find the probability that on one day she receives
- 3 complaints, [3 marks]
- 10 or more complaints. [2 marks]
The supermarket is open on six days each week.
- Find the probability that the manager receives 10 or more complaints on no more than one day in a week. [4 marks]
Edexcel
S2
Q4
10 marks
Standard +0.3
A rugby player scores an average of 0.4 tries per match in which he plays.
- 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.
- 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]
Edexcel
S2
Q6
13 marks
Moderate -0.8
A shop receives weekly deliveries of 120 eggs from a local farm. The proportion of eggs received from the farm that are broken is 0.008
- Explain why it is reasonable to use the binomial distribution to model the number of eggs that are broken in each delivery. [3 marks]
- Use the binomial distribution to calculate the probability that at most one egg in a delivery will be broken. [4 marks]
- State the conditions under which the binomial distribution can be approximated by the Poisson distribution. [1 mark]
- Using the Poisson approximation to the binomial, find the probability that at most one egg in a delivery will be broken. Comment on your answer. [5 marks]
OCR
Further Statistics AS
Specimen
Q6
13 marks
Moderate -0.3
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.
- State these two assumptions. [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 \(\mathrm{Po}(0.8)\).
- Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
- Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
- Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [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]
SPS
SPS ASFM Statistics
2021
May
Q7
Moderate -0.3
A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
- Find the probability of exactly 4 faults in a 15 metre length of cloth.
(2)
- Find the probability of more than 10 faults in 60 metres of cloth.
(3)
A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
- Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures.
(4)
The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of £1.50 on any pieces that do contain faults.
- Find the retailer's expected profit.
(4)
|