OCR
GCSE
Further Mathematics A
Y542/01: Statistics
*1920068128*
A Level
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined page at the end of the Printed Answer
Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is
needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [J/508/5510] DC (PQ) 357273/3
OCR is an exempt Charity
Turn over
1 A set of 16 observations of the bivariate data (X, Y ) are summarised as follows.
n = 16 /x = 136 /y = 352 /x2 = 1496 /y2 = 9104 /xy = 3642
Determine an estimate of the value of y corresponding to x = 8 5. .
, 2
2 The number of trees of a particular species found in 1km2 of a forest can be
modelled by the distribution Po( )m .
The forest consists of two regions, A and B.
In the region A, m= 4. The number of trees of this species in a randomly chosen area
of 3km2 in region A is denoted by X.
(a) Find P(16 1 1X20). [3]
In region B, m= 8. The number of trees of this species in a randomly chosen area of
3km2 in region B is denoted by Y.
The random variable Z is defined by Z = +X Y . It may be assumed that
independent.
(b) Write down a formula for P(Z = z), in terms of z. [2]
(c) The average numbers of trees of this species in one square kilometre are
not the same in regions A and B.
Explain why the associated modelling assumption for the distribution of Z
nevertheless
valid. [1]
(d) Show that Y-X does not have a Poisson distribution. [1]
3 The length, in cm, of sand lizards when fully grown is a random variable which is
known to have the distribution N(13.2, 3 4. 2).
Some scientists discover a colony of what they believe to be sand lizards in a
remote location. The scientists measure the lengths of a random sample of 50 fully
grown lizards from this remote location. The mean of this random sample is 14.02
cm. You may assume that the variance of the length of lizards on the remote island
is 3 4. 2cm2.
(a) Test, at the 5% significance level, whether the fully grown lizards in this
location have a
mean length which is different from 13.2 cm.
(b) Now suppose it is not known that the lengths are normally distributed.
(i) Explain why the Central Limit Theorem can be used in this context.
[1]
(ii) State where in your test the Central Limit Theorem would be used.
[1]
4 In this question you must show the parameters of any distributions you use.
© OCR 2025 Y542/01 Jun25
, 3
The continuous random variable V has the distribution N(42, 32).
The continuous random variable W has the distribution N(48, 42).
The random variable X is the sum of 5 independent observations of V.
The random variable Y is the sum of 7 independent observations of W.
(a) Determine P(X+Y 2 560). [3]
(b) Determine the probability that X is less than 60% of Y. [4]
Turn over
5 For many years a coach company has used a make of tyre that has an average
driving lifespan of 1 52. #105km. The coach company decides to fit a new make of
tyre to 12 of its coaches, and records the lifespan of one of the tyres, randomly
chosen, for each of these 12 coaches. The distances, in multiples of 105km, are
given below.
1.35 1.46 1.48 1.53 1.54 1.57
1.62 1.74 1.76 1.82 1.88 1.91
The coach company wishes to test whether the new make of tyres has improved the
driving lifespan.
(a) Explain why a hypothesis test based on a normal distribution may not be
valid. [1]
(b) Carry out a suitable Wilcoxon test at the 5% significance level. [7]
© OCR 2025 Y542/01 Jun25
, 4
(c) Explain why a Wilcoxon test, if it is valid, is preferable to a sign test in
this context. [1]
6 The editor of a mathematical journal investigated whether there was association
between the types of articles published in the journal and the academic backgrounds
of the authors.
The results for a random sample of 64 articles are shown in the table.
Pure maths Applied maths Other
Schools and colleges 6 4 5
Universities 30 12 7
The editor carries out a test at the 10% significance level.
(a) Explain why it is not possible to combine rows of the table before
carrying out the test. [1]
(b) Show that it is necessary to combine columns before carrying out the test.
[2]
(c) Carry out the test. [6] 7 (a) A large number of competitors take part
in two races. A random sample of 6 competitors who take part in both races is
obtained. The value of Spearman’s rank correlation coefficient for the
finishing positions of these 6 competitors is 0.771, correct to 3 significant
figures.
Test, at the 5% significance level, whether there is agreement between the
finishing positions in the two races. [4]
(b) Show that, when only 3 competitors are selected at random, the value of Spearman’s
rank
correlation coefficient can take only 4 different values. You should state what
these values are. [5]
8 The proportion X of charge in a phone battery is measured on a scale from 0
(uncharged) to 1 (fully charged).
For a certain model of phone, X can be modelled as a continuous random variable
with the following probability density function.
f( )x
=*(k+1)
xk 0 G
© OCR 2025 Y542/01 Jun25
GCSE
Further Mathematics A
Y542/01: Statistics
*1920068128*
A Level
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined page at the end of the Printed Answer
Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is
needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [J/508/5510] DC (PQ) 357273/3
OCR is an exempt Charity
Turn over
1 A set of 16 observations of the bivariate data (X, Y ) are summarised as follows.
n = 16 /x = 136 /y = 352 /x2 = 1496 /y2 = 9104 /xy = 3642
Determine an estimate of the value of y corresponding to x = 8 5. .
, 2
2 The number of trees of a particular species found in 1km2 of a forest can be
modelled by the distribution Po( )m .
The forest consists of two regions, A and B.
In the region A, m= 4. The number of trees of this species in a randomly chosen area
of 3km2 in region A is denoted by X.
(a) Find P(16 1 1X20). [3]
In region B, m= 8. The number of trees of this species in a randomly chosen area of
3km2 in region B is denoted by Y.
The random variable Z is defined by Z = +X Y . It may be assumed that
independent.
(b) Write down a formula for P(Z = z), in terms of z. [2]
(c) The average numbers of trees of this species in one square kilometre are
not the same in regions A and B.
Explain why the associated modelling assumption for the distribution of Z
nevertheless
valid. [1]
(d) Show that Y-X does not have a Poisson distribution. [1]
3 The length, in cm, of sand lizards when fully grown is a random variable which is
known to have the distribution N(13.2, 3 4. 2).
Some scientists discover a colony of what they believe to be sand lizards in a
remote location. The scientists measure the lengths of a random sample of 50 fully
grown lizards from this remote location. The mean of this random sample is 14.02
cm. You may assume that the variance of the length of lizards on the remote island
is 3 4. 2cm2.
(a) Test, at the 5% significance level, whether the fully grown lizards in this
location have a
mean length which is different from 13.2 cm.
(b) Now suppose it is not known that the lengths are normally distributed.
(i) Explain why the Central Limit Theorem can be used in this context.
[1]
(ii) State where in your test the Central Limit Theorem would be used.
[1]
4 In this question you must show the parameters of any distributions you use.
© OCR 2025 Y542/01 Jun25
, 3
The continuous random variable V has the distribution N(42, 32).
The continuous random variable W has the distribution N(48, 42).
The random variable X is the sum of 5 independent observations of V.
The random variable Y is the sum of 7 independent observations of W.
(a) Determine P(X+Y 2 560). [3]
(b) Determine the probability that X is less than 60% of Y. [4]
Turn over
5 For many years a coach company has used a make of tyre that has an average
driving lifespan of 1 52. #105km. The coach company decides to fit a new make of
tyre to 12 of its coaches, and records the lifespan of one of the tyres, randomly
chosen, for each of these 12 coaches. The distances, in multiples of 105km, are
given below.
1.35 1.46 1.48 1.53 1.54 1.57
1.62 1.74 1.76 1.82 1.88 1.91
The coach company wishes to test whether the new make of tyres has improved the
driving lifespan.
(a) Explain why a hypothesis test based on a normal distribution may not be
valid. [1]
(b) Carry out a suitable Wilcoxon test at the 5% significance level. [7]
© OCR 2025 Y542/01 Jun25
, 4
(c) Explain why a Wilcoxon test, if it is valid, is preferable to a sign test in
this context. [1]
6 The editor of a mathematical journal investigated whether there was association
between the types of articles published in the journal and the academic backgrounds
of the authors.
The results for a random sample of 64 articles are shown in the table.
Pure maths Applied maths Other
Schools and colleges 6 4 5
Universities 30 12 7
The editor carries out a test at the 10% significance level.
(a) Explain why it is not possible to combine rows of the table before
carrying out the test. [1]
(b) Show that it is necessary to combine columns before carrying out the test.
[2]
(c) Carry out the test. [6] 7 (a) A large number of competitors take part
in two races. A random sample of 6 competitors who take part in both races is
obtained. The value of Spearman’s rank correlation coefficient for the
finishing positions of these 6 competitors is 0.771, correct to 3 significant
figures.
Test, at the 5% significance level, whether there is agreement between the
finishing positions in the two races. [4]
(b) Show that, when only 3 competitors are selected at random, the value of Spearman’s
rank
correlation coefficient can take only 4 different values. You should state what
these values are. [5]
8 The proportion X of charge in a phone battery is measured on a scale from 0
(uncharged) to 1 (fully charged).
For a certain model of phone, X can be modelled as a continuous random variable
with the following probability density function.
f( )x
=*(k+1)
xk 0 G
© OCR 2025 Y542/01 Jun25
