Ocr 2024
OXFORD CAMBRDGE
AND RSA 2024
[Type the document subtitle]
OCR 2024
2024
[Type the company address]
, Oxford Cambridge and RSA
Friday 7 June 2024 – AfternoonAS Level
Further Mathematics B (MEI) Y412/01 Statistics a
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• 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 AnswerBooklet. 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 begiven
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• 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 is an exempt Charity
Turn over
, 3
1 The probability distribution for a discrete random variable X is given in the table below.
x 0 1 2 3
P(X = x) 2c 3c 0.5 - c c
(a) Find the value of c. [2]
(b) Find the value of each of the following.
• E(X )
• Var(X ) [3]
The random variable Y is defined by Y = 2X - 3.
(c) Find the value of each of the following.
• E(Y )
• Var(Y ) [2]
2 In a game of chance there are 32 slots, numbered 1 to 32, and on each turn a ball lands in one
ofthem. You may assume that the process is completely random.
You are given that X is the random variable denoting the number of the slot that the ball lands in
on a given turn.
(a) Suggest a suitable distribution to model X. You should state the value(s) of any parameter(s).
[2]
(b) Write down P(X = 7). [1]
Players of the game start with a score of 0. On each turn a player may choose to play the game
byselecting a number. If the ball lands in the slot with that number then 15 is added to the
player’s score. Otherwise, the player’s score is reduced by 1. A player’s score may become
negative.
A player decides to play the game, selecting the number 7 on each turn, until the ball lands in
theslot numbered 7.
You are given that Y is the random variable denoting the number of turns up to and including
theturn in which the ball lands in the slot numbered 7.
(c) Determine P(Y G 15). [3]
(d) Determine the player’s expected final score. [3]
© OCR 2024 Y412/01 Jun24 Turn over
, 2
3 A glassware factory produces a large number of ornaments each week. Just before they
leave the factory, all the ornaments are checked and some may be found to be defective. The
QualityAssurance Manager of the factory wishes to model the number of defective
ornaments that arefound each week using a Poisson distribution.
The numbers of defective ornaments found each week in a period of 40 weeks are shown in
Table 3.1.
Table 3.1
No. of defective ornaments in a week, r 0 1 2 3 4 5 6 H7
No. of weeks with r defective ornaments, f 2 14 13 5 3 1 2 0
You are given that summary statistics for the data are / f = 40 , /rf = 84 and /r2f = 256.
(a) By using the summary statistics to determine estimates for the mean and variance of the
number of defective ornaments produced by the factory each week, explain how the data
support the suggestion that the number of defective ornaments produced each week can be
modelled using a Poisson distribution. [3]
The Quality Assurance Manager is asked by the head office to carry out a chi-squared hypothesis
test for goodness of fit based on a Po(2) distribution.
(b) Table 3.2, which is incomplete, gives observed frequency, probability, expected
frequencyand chi-squared contribution.
Table 3.2
No. of defective Observed Probability Expected Chi-squared
ornaments in a frequenc frequenc contributio
week, r y y n
0 2 0.13534 5.4134 2.15232
1 14
2 13 0.27067 0.43620
3 5 7.2179
H4 6 0.14288 0.01421
(i) Complete the copy of the table in the Printed Answer Booklet. [4]
(ii) Carry out the test at the 10% significance level. [6]
(c) On one occasion a fork-lift truck in the factory drops a crate containing eight ornaments
andall of them are subsequently found to be defective.
© OCR 2024 Y412/01 Jun24
OXFORD CAMBRDGE
AND RSA 2024
[Type the document subtitle]
OCR 2024
2024
[Type the company address]
, Oxford Cambridge and RSA
Friday 7 June 2024 – AfternoonAS Level
Further Mathematics B (MEI) Y412/01 Statistics a
Time allowed: 1 hour 15 minutes
You must have:
• the Printed Answer Booklet
• the Formulae Booklet for Further Mathematics B
QP
(MEI)
• 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 AnswerBooklet. 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 begiven
for using a correct method, even if your answer is wrong.
• Give your final answers to a degree of accuracy that is appropriate to the context.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 60.
• 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 is an exempt Charity
Turn over
, 3
1 The probability distribution for a discrete random variable X is given in the table below.
x 0 1 2 3
P(X = x) 2c 3c 0.5 - c c
(a) Find the value of c. [2]
(b) Find the value of each of the following.
• E(X )
• Var(X ) [3]
The random variable Y is defined by Y = 2X - 3.
(c) Find the value of each of the following.
• E(Y )
• Var(Y ) [2]
2 In a game of chance there are 32 slots, numbered 1 to 32, and on each turn a ball lands in one
ofthem. You may assume that the process is completely random.
You are given that X is the random variable denoting the number of the slot that the ball lands in
on a given turn.
(a) Suggest a suitable distribution to model X. You should state the value(s) of any parameter(s).
[2]
(b) Write down P(X = 7). [1]
Players of the game start with a score of 0. On each turn a player may choose to play the game
byselecting a number. If the ball lands in the slot with that number then 15 is added to the
player’s score. Otherwise, the player’s score is reduced by 1. A player’s score may become
negative.
A player decides to play the game, selecting the number 7 on each turn, until the ball lands in
theslot numbered 7.
You are given that Y is the random variable denoting the number of turns up to and including
theturn in which the ball lands in the slot numbered 7.
(c) Determine P(Y G 15). [3]
(d) Determine the player’s expected final score. [3]
© OCR 2024 Y412/01 Jun24 Turn over
, 2
3 A glassware factory produces a large number of ornaments each week. Just before they
leave the factory, all the ornaments are checked and some may be found to be defective. The
QualityAssurance Manager of the factory wishes to model the number of defective
ornaments that arefound each week using a Poisson distribution.
The numbers of defective ornaments found each week in a period of 40 weeks are shown in
Table 3.1.
Table 3.1
No. of defective ornaments in a week, r 0 1 2 3 4 5 6 H7
No. of weeks with r defective ornaments, f 2 14 13 5 3 1 2 0
You are given that summary statistics for the data are / f = 40 , /rf = 84 and /r2f = 256.
(a) By using the summary statistics to determine estimates for the mean and variance of the
number of defective ornaments produced by the factory each week, explain how the data
support the suggestion that the number of defective ornaments produced each week can be
modelled using a Poisson distribution. [3]
The Quality Assurance Manager is asked by the head office to carry out a chi-squared hypothesis
test for goodness of fit based on a Po(2) distribution.
(b) Table 3.2, which is incomplete, gives observed frequency, probability, expected
frequencyand chi-squared contribution.
Table 3.2
No. of defective Observed Probability Expected Chi-squared
ornaments in a frequenc frequenc contributio
week, r y y n
0 2 0.13534 5.4134 2.15232
1 14
2 13 0.27067 0.43620
3 5 7.2179
H4 6 0.14288 0.01421
(i) Complete the copy of the table in the Printed Answer Booklet. [4]
(ii) Carry out the test at the 10% significance level. [6]
(c) On one occasion a fork-lift truck in the factory drops a crate containing eight ornaments
andall of them are subsequently found to be defective.
© OCR 2024 Y412/01 Jun24
