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
FP2
2012
June
Q8
9 marks
Standard +0.3
The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by
$$\text{F}(x) = \begin{cases}
1 - e^{-0.016x} & x \geq 0, \\
0 & x < 0,
\end{cases}$$
and deduce that \(X\) has a negative exponential distribution, stating its mean. [4]
Find
- the median distance between two successive flaws, [3]
- the probability that there is a distance of at least 50 metres between two successive flaws. [2]
CAIE
FP2
2012
June
Q8
9 marks
Standard +0.3
The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by
$$\text{F}(x) = \begin{cases}
1 - e^{-0.016x} & x \geqslant 0, \\
0 & x < 0,
\end{cases}$$
and deduce that \(X\) has a negative exponential distribution, stating its mean.
[4]
Find
- the median distance between two successive flaws, [3]
- the probability that there is a distance of at least 50 metres between two successive flaws. [2]
CAIE
FP2
2017
June
Q10
12 marks
Standard +0.3
Roberto owns a small hotel and offers accommodation to guests. Over a period of \(100\) nights, the numbers of rooms, \(x\), that are occupied each night at Roberto's hotel and the corresponding frequencies are shown in the following table.
| Number of rooms occupied \((x)\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) | | Number of nights | 4 | 9 | 18 | 26 | 20 | 16 | 7 | 0 |
- Show that the mean number of rooms that are occupied each night is \(3.25\). [1]
The following table shows most of the corresponding expected frequencies, correct to \(2\) decimal places, using a Poisson distribution with mean \(3.25\).
| Number of rooms occupied \((x)\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | \(\geqslant 7\) | | Observed frequency | 4 | 9 | 18 | 26 | 20 | 16 | 7 | 0 | | Expected frequency | 3.88 | 12.60 | 20.48 | 22.18 | 18.02 | 11.72 | | |
- Show how the expected value of \(22.18\), for \(x = 3\), is obtained and find the expected values for \(x = 6\) and for \(x \geqslant 7\). [4]
- Use a goodness-of-fit test at the \(5\%\) significance level to determine whether the Poisson distribution is a suitable model for the number of rooms occupied each night at Roberto's hotel. [7]
CAIE
S2
2021
June
Q1
4 marks
Standard +0.3
Accidents at two factories occur randomly and independently. On average, the numbers of accidents per month are 3.1 at factory \(A\) and 1.7 at factory \(B\).
Find the probability that the total number of accidents in the two factories during a 2-month period is more than 3. [4]
CAIE
S2
2021
June
Q5
7 marks
Standard +0.3
On average, 1 in 75000 adults has a certain genetic disorder.
- Use a suitable approximating distribution to find the probability that, in a random sample of 10000 people, at least 1 has the genetic disorder. [3]
- In a random sample of \(n\) people, where \(n\) is large, the probability that no-one has the genetic disorder is more than 0.9.
Find the largest possible value of \(n\). [4]
CAIE
S2
2022
November
Q3
6 marks
Moderate -0.8
1.6% of adults in a certain town ride a bicycle. A random sample of 200 adults from this town is selected.
- Use a suitable approximating distribution to find the probability that more than 3 of these adults ride a bicycle. [4]
- Justify your approximating distribution. [2]
CAIE
S2
2023
November
Q3
10 marks
Standard +0.3
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.
- Assume that the owner is correct.
- Find the probability that there will be at least 4 hits during a 10-minute period. [3]
- Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period. [4]
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 (9.00am to 9.00pm) is usually about twice the number of hits during the night-time (9.00pm to 9.00am).
- Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution. [1]
- Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time. [2]
CAIE
S2
2024
November
Q6
9 marks
Standard +0.3
The numbers of customers arriving at service desks \(A\) and \(B\) during a \(10\)-minute period have the independent distributions \(\text{Po}(1.8)\) and \(\text{Po}(2.1)\) respectively.
- Find the probability that during a randomly chosen \(15\)-minute period more than \(2\) customers will arrive at desk \(A\). [2]
- Find the probability that during a randomly chosen \(5\)-minute period the total number of customers arriving at both desks is less than \(4\). [3]
- An inspector waits at desk \(B\). She wants to wait long enough to be \(90\%\) certain of seeing at least one customer arrive at the desk.
Find the minimum time for which she should wait, giving your answer correct to the nearest minute. [4]
CAIE
S2
2011
June
Q3
7 marks
Moderate -0.3
The number of goals scored per match by Everly Rovers is represented by the random variable \(X\) which has mean 1.8.
- State two conditions for \(X\) to be modelled by a Poisson distribution. [2]
Assume now that \(X \sim \text{Po}(1.8)\).
- Find \(\text{P}(2 < X < 6)\). [2]
- The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus. [3]
CAIE
S2
2011
June
Q5
9 marks
Standard +0.3
The number of adult customers arriving in a shop during a 5-minute period is modelled by a random variable with distribution \(\text{Po}(6)\). The number of child customers arriving in the same shop during a 10-minute period is modelled by an independent random variable with distribution \(\text{Po}(4.5)\).
- Find the probability that during a randomly chosen 2-minute period, the total number of adult and child customers who arrive in the shop is less than 3. [3]
- During a sale, the manager claims that more adult customers are arriving than usual. In a randomly selected 30-minute period during the sale, 49 adult customers arrive. Test the manager's claim at the 2.5\% significance level. [6]
CAIE
S2
2002
November
Q4
7 marks
Standard +0.3
The number of accidents per month at a certain road junction has a Poisson distribution with mean 4.8. A new road sign is introduced warning drivers of the danger ahead, and in a subsequent month 2 accidents occurred.
- A hypothesis test at the 10% level is used to determine whether there were fewer accidents after the new road sign was introduced. Find the critical region for this test and carry out the test. [5]
- Find the probability of a Type I error. [2]
CAIE
S2
2002
November
Q5
8 marks
Standard +0.3
\(X\) and \(Y\) are independent random variables each having a Poisson distribution. \(X\) has mean 2.5 and \(Y\) has mean 3.1.
- Find P\((X + Y > 3)\). [4]
- A random sample of 80 values of \(X\) is taken. Find the probability that the sample mean is less than 2.4. [4]
CAIE
S2
2011
November
Q7
11 marks
Standard +0.8
The numbers of men and women who visit a clinic each hour are independent Poisson variables with means 2.4 and 2.8 respectively.
- Find the probability that, in a half-hour period,
- 2 or more men and 1 or more women will visit the clinic, [4]
- a total of 3 or more people will visit the clinic. [3]
- Find the probability that, in a 10-hour period, a total of more than 60 people will visit the clinic. [4]
CAIE
S2
2020
Specimen
Q3
5 marks
Moderate -0.3
The number of calls received at a small call centre has a Poisson distribution with mean 2 calls per 5 minute period.
- Find the probability exactly 4 calls in a 10 minute period. [2]
- Find the probability at least 3 calls in a 3 minute period. [3]
Edexcel
S2
2016
January
Q3
11 marks
Moderate -0.3
Left-handed people make up 10\% of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.
- Write down an expression for the exact value of \(\mathrm{P}(Y \leq 1)\)
- Evaluate your expression, giving your answer to 3 significant figures. [3]
- Using a Poisson approximation, estimate \(\mathrm{P}(Y \leq 1)\) [2]
- Using a normal approximation, estimate \(\mathrm{P}(Y \leq 1)\) [5]
- Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm{P}(Y \leq 1)\) [1]
Edexcel
S2
2016
January
Q5
10 marks
Standard +0.3
The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1
- Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods. [2]
- Find the probability that this volcano does not erupt in a randomly selected 20 year period. [2]
The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.
- Use the tables to find the value of \(w\) [3]
A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1
She selects a 100 year period at random in order to test her claim.
- State the null hypothesis for this test. [1]
- Determine the critical region for the test at the 5\% level of significance. [2]
Edexcel
S2
2016
January
Q7
12 marks
Standard +0.3
A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution.
The fisherman takes 5 fishing trips each lasting 1 hour.
- Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. [6]
The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour.
Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,
- carry out the test at the 5\% level of significance. State your hypotheses clearly. [6]
Edexcel
S2
Q2
7 marks
Moderate -0.8
On a stretch of motorway accidents occur at a rate of 0.9 per month.
- Show that the probability of no accidents in the next month is 0.407, to 3 significant figures. [1]
Find the probability of
- exactly 2 accidents in the next 6 month period, [3]
- no accidents in exactly 2 of the next 4 months. [3]
Edexcel
S2
Q5
12 marks
Standard +0.3
The maintenance department of a college receives requests for replacement light bulbs at a rate of 2 per week.
Find the probability that in a randomly chosen week the number of requests for replacement light bulbs is
- exactly 4, [2]
- more than 5. [2]
Three weeks before the end of term the maintenance department discovers that there are only 5 light bulbs left.
- Find the probability that the department can meet all requests for replacement light bulbs before the end of term. [3]
The following term the principal of the college announces a package of new measures to reduce the amount of damage to college property. In the first 4 weeks following this announcement, 3 requests for replacement light bulbs are received.
- Stating your hypotheses clearly test, at the 5\% level of significance, whether or not there is evidence that the rate of requests for replacement light bulbs has decreased. [5]
Edexcel
S2
Q2
7 marks
Standard +0.3
The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2.5. The firm appoints a new salesman and wants to find out whether or not house sales increase as a result. After the appointment of the salesman, the number of house sales in a randomly chosen 4-week period is 14.
Stating your hypotheses clearly test, at the 5\% level of significance, whether or not the new salesman has increased house sales. [7]
Edexcel
S2
Q5
13 marks
Moderate -0.3
An Internet service provider has a large number of users regularly connecting to its computers. On average only 3 users every hour fail to connect to the Internet at their first attempt.
- Give 2 reasons why a Poisson distribution might be a suitable model for the number of failed connections every hour. [2]
Find the probability that in a randomly chosen hour
- all Internet users connect at their first attempt, [2]
- more than 4 users fail to connect at their first attempt. [2]
- Write down the distribution of the number of users failing to connect at their first attempt in an 8-hour period. [1]
- Using a suitable approximation, find the probability that 12 or more users fail to connect at their first attempt in a randomly chosen 8-hour period. [6]
Edexcel
S2
Q6
14 marks
Standard +0.3
From past records, a manufacturer of twine knows that faults occur in the twine at random and at a rate of 1.5 per 25 m.
- Find the probability that in a randomly chosen 25 m length of twine there will be exactly 4 faults. [2]
The twine is usually sold in balls of length 100 m. A customer buys three balls of twine.
- Find the probability that only one of them will have fewer than 6 faults. [6]
As a special order a ball of twine containing 500 m is produced.
- Using a suitable approximation, find the probability that it will contain between 23 and 33 faults inclusive. [6]
Edexcel
S2
Q3
12 marks
Moderate -0.3
A botanist suggests that the number of a particular variety of weed growing in a meadow can be modelled by a Poisson distribution.
- Write down two conditions that must apply for this model to be applicable. [2]
Assuming this model and a mean of 0.7 weeds per m², find
- the probability that in a randomly chosen plot of size 4 m² there will be fewer than 3 of these weeds, [4]
- Using a suitable approximation, find the probability that in a plot of 100 m² there will be more than 66 of these weeds. [6]
Edexcel
S2
Q2
7 marks
Moderate -0.8
- Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]
The random variable Y ~ Po(30).
- Estimate P(Y > 28). [6]
|