Cambridge International AS & A Level
* 5 5 4 8 6 1 2 3 5 1 *
FURTHER MATHEMATICS 9231/41
Paper 4 Further Probability & Statistics May/June 2023
1 hour 30 minutes
You must answer on the question paper.
You will need: List of formulae (MF19)
INSTRUCTIONS
● Answer all questions.
● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
● Write your name, centre number and candidate number in the boxes at the top of the page.
● Write your answer to each question in the space provided.
● Do not use an erasable pen or correction fluid.
● Do not write on any bar codes.
● If additional space is needed, you should use the lined page at the end of this booklet; the question
number or numbers must be clearly shown.
● You should use a calculator where appropriate.
● You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
INFORMATION
● The total mark for this paper is 50.
● The number of marks for each question or part question is shown in brackets [ ].
This document has 12 pages.
DC (LO) 327443
© UCLES 2023 [Turn over
, 2
1 The lengths of the leaves of a particular type of tree are normally distributed with mean μ cm. The
lengths, x cm, of a random sample of 12 leaves of this type are recorded. The results are summarised as
follows.
/ x = 91.2 / x 2 = 695.8
Find a 95% confidence interval for μ. [4]
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© UCLES 2023 9231/41/M/J/23
, 3
2 The children at two large schools, P and Q, are all given the same puzzle to solve. A random sample
of size 10 is taken from the children at school P. Their individual times to complete the puzzle give a
sample mean of 9.12 minutes and an unbiased variance estimate of 2.16 minutes2. A random sample of
size 12 is taken from the children at school Q. Their individual times, x minutes, to complete the puzzle
are summarised by
/ x = 99.6 / (x - x ) 2 = 21.5,
where x is the sample mean. Times to complete the puzzle are assumed to be normally distributed with
the same population variance.
Test at the 5% significance level whether the population mean time taken to complete the puzzle by
children at school P is greater than the population mean time taken to complete the puzzle by children
at school Q. [8]
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© UCLES 2023 9231/41/M/J/23 [Turn over
* 5 5 4 8 6 1 2 3 5 1 *
FURTHER MATHEMATICS 9231/41
Paper 4 Further Probability & Statistics May/June 2023
1 hour 30 minutes
You must answer on the question paper.
You will need: List of formulae (MF19)
INSTRUCTIONS
● Answer all questions.
● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
● Write your name, centre number and candidate number in the boxes at the top of the page.
● Write your answer to each question in the space provided.
● Do not use an erasable pen or correction fluid.
● Do not write on any bar codes.
● If additional space is needed, you should use the lined page at the end of this booklet; the question
number or numbers must be clearly shown.
● You should use a calculator where appropriate.
● You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.
INFORMATION
● The total mark for this paper is 50.
● The number of marks for each question or part question is shown in brackets [ ].
This document has 12 pages.
DC (LO) 327443
© UCLES 2023 [Turn over
, 2
1 The lengths of the leaves of a particular type of tree are normally distributed with mean μ cm. The
lengths, x cm, of a random sample of 12 leaves of this type are recorded. The results are summarised as
follows.
/ x = 91.2 / x 2 = 695.8
Find a 95% confidence interval for μ. [4]
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© UCLES 2023 9231/41/M/J/23
, 3
2 The children at two large schools, P and Q, are all given the same puzzle to solve. A random sample
of size 10 is taken from the children at school P. Their individual times to complete the puzzle give a
sample mean of 9.12 minutes and an unbiased variance estimate of 2.16 minutes2. A random sample of
size 12 is taken from the children at school Q. Their individual times, x minutes, to complete the puzzle
are summarised by
/ x = 99.6 / (x - x ) 2 = 21.5,
where x is the sample mean. Times to complete the puzzle are assumed to be normally distributed with
the same population variance.
Test at the 5% significance level whether the population mean time taken to complete the puzzle by
children at school P is greater than the population mean time taken to complete the puzzle by children
at school Q. [8]
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© UCLES 2023 9231/41/M/J/23 [Turn over
