Oxford Cambridge and
RSA Examinations GCE
Further Mathematics
AY542/01: Statistics
A Level question paper
with marking scheme
(merged)
, Oxford Cambridge and RSA
Wednesday 14 June 2023 – Afternoon
A Level Further Mathematics A
Y542/01 Statistics
Time allowed: 1 hour 30 minutes
* 9 9 7 6 8 5 4 6 5 2 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
Mathematics A
• a scientific or graphical calculator
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 pages 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 2023 [J/508/5510] OCR is an exempt Charity
DC (LK) 322893/4 Turn over
, 2
1 A certain section of a library contains several thousand books. A lecturer is looking for a book that
refers to a particular topic. The lecturer believes that one-twentieth of the books in that section
of the library contain a reference to that topic. However, the lecturer does not know which books
they might be, so the lecturer looks in each book in turn for a reference to the topic. The first book
the lecturer finds that refers to the topic is the X th book in which the lecturer looks.
(a) A student says, “There is a maximum value of X as there is only a finite number of books. So
a geometric distribution cannot be a good model for X.”
Explain whether you agree with the student. [1]
(b) (i) State one modelling assumption (not involving the total number of books) needed for X
to be modelled by a geometric distribution in this context. [1]
(ii) Suggest a reason why this assumption may not be valid in this context. [1]
Assume now that X can be well modelled by the distribution Geo(0.05).
(c) The probability that the lecturer needs to look in no more than n books is greater than 0.9.
Find the smallest possible value of n. [3]
(d) The lecturer needs to find four different books that refer to the topic.
Find the probability that the lecturer wants to look in exactly 40 books. [2]
© OCR 2023 Y542/01 Jun23
, 3
2 The director of a concert hall wishes to investigate if the price of the most expensive concert
tickets affects attendance. The director collects data about the price, £P, of the most expensive
tickets and the number of people in the audience, H hundred (rounded to the nearest hundred), for
20 concerts. For each price there are several different concerts. The results are shown in the table.
P (£) 75 65 55 45 35
H (hundred) 27 27 27 26 15
27 27 20 21 12
22 18 16 9
19 18 13
12 16 9
n = 20 / p = 1080 / h = 381 / p 2 = 61300 / h 2 = 8011 / ph = 21535
(a) Calculate the equation of the regression line of h on p. [2]
(b) State what change, if any, there would be to your answer to part (a) if H had been measured
in thousands (to 1 decimal place) rather than in hundreds. [1]
For a special charity concert, the most expensive tickets cost £50.
(c) Use your answer to part (b) to estimate the expected size of the audience for this concert.
Give your answer correct to 1 decimal place. [1]
(d) Comment on the reliability of your answer to part (c). You should refer to
• the value of the product-moment correlation coefficient for the data, which is 0.642
• the value of £50
• any one other relevant factor that should be taken into account. [4]
3 The discrete random variable W has the distribution U(11). The independent discrete random
variable V has the distribution U(5).
(a) It is given that, for constants m and n, with m 2 0 ,
E (mW + nV ) = 0 and Var (mW + nV ) = 1.
Determine the exact values of m and n. [5]
The random variable T is the mean of three independent observations of W.
(b) Explain whether the Central Limit Theorem can be used to say that the distribution of T is
approximately normal. [1]
© OCR 2023 Y542/01 Jun23 Turn over
RSA Examinations GCE
Further Mathematics
AY542/01: Statistics
A Level question paper
with marking scheme
(merged)
, Oxford Cambridge and RSA
Wednesday 14 June 2023 – Afternoon
A Level Further Mathematics A
Y542/01 Statistics
Time allowed: 1 hour 30 minutes
* 9 9 7 6 8 5 4 6 5 2 *
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for A Level Further
QP
Mathematics A
• a scientific or graphical calculator
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 pages 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 2023 [J/508/5510] OCR is an exempt Charity
DC (LK) 322893/4 Turn over
, 2
1 A certain section of a library contains several thousand books. A lecturer is looking for a book that
refers to a particular topic. The lecturer believes that one-twentieth of the books in that section
of the library contain a reference to that topic. However, the lecturer does not know which books
they might be, so the lecturer looks in each book in turn for a reference to the topic. The first book
the lecturer finds that refers to the topic is the X th book in which the lecturer looks.
(a) A student says, “There is a maximum value of X as there is only a finite number of books. So
a geometric distribution cannot be a good model for X.”
Explain whether you agree with the student. [1]
(b) (i) State one modelling assumption (not involving the total number of books) needed for X
to be modelled by a geometric distribution in this context. [1]
(ii) Suggest a reason why this assumption may not be valid in this context. [1]
Assume now that X can be well modelled by the distribution Geo(0.05).
(c) The probability that the lecturer needs to look in no more than n books is greater than 0.9.
Find the smallest possible value of n. [3]
(d) The lecturer needs to find four different books that refer to the topic.
Find the probability that the lecturer wants to look in exactly 40 books. [2]
© OCR 2023 Y542/01 Jun23
, 3
2 The director of a concert hall wishes to investigate if the price of the most expensive concert
tickets affects attendance. The director collects data about the price, £P, of the most expensive
tickets and the number of people in the audience, H hundred (rounded to the nearest hundred), for
20 concerts. For each price there are several different concerts. The results are shown in the table.
P (£) 75 65 55 45 35
H (hundred) 27 27 27 26 15
27 27 20 21 12
22 18 16 9
19 18 13
12 16 9
n = 20 / p = 1080 / h = 381 / p 2 = 61300 / h 2 = 8011 / ph = 21535
(a) Calculate the equation of the regression line of h on p. [2]
(b) State what change, if any, there would be to your answer to part (a) if H had been measured
in thousands (to 1 decimal place) rather than in hundreds. [1]
For a special charity concert, the most expensive tickets cost £50.
(c) Use your answer to part (b) to estimate the expected size of the audience for this concert.
Give your answer correct to 1 decimal place. [1]
(d) Comment on the reliability of your answer to part (c). You should refer to
• the value of the product-moment correlation coefficient for the data, which is 0.642
• the value of £50
• any one other relevant factor that should be taken into account. [4]
3 The discrete random variable W has the distribution U(11). The independent discrete random
variable V has the distribution U(5).
(a) It is given that, for constants m and n, with m 2 0 ,
E (mW + nV ) = 0 and Var (mW + nV ) = 1.
Determine the exact values of m and n. [5]
The random variable T is the mean of three independent observations of W.
(b) Explain whether the Central Limit Theorem can be used to say that the distribution of T is
approximately normal. [1]
© OCR 2023 Y542/01 Jun23 Turn over
