Cambridge A Level Mathematics - Further 9231 — 2025 Oct/Nov Paper 4 · Variant 4

This document has 12 pages. Any blank pages are indicated. [Turn over * 8 5 7 7 1 3 9 7 4 8 * Cambridge International AS & A Level FURTHER MATHEMATICS 9231/44 Paper 4 Further Probability & Statistics October/November 2025 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 [ ]. DC (DE) 360449/1 © UCLES 2025 ,










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3 9231/44/O/N/25 © UCLES 2025 [Turn over 1 Local residents are concerned about the speed of cars travelling through their village. They record the speed, in kmh 1 - , of a random sample of 13 cars travelling through their village. The recorded speeds are as follows. 40 53 59 42 43 48 62 67 46 82 66 45 70 (a) Construct a 90% confidence interval for the population mean speed of cars passing through the village. [5] … … … … … … … … … … … … … … … … … … … … (b) State an assumption that is necessary for the confidence interval found in 1(a) to be valid. [1] … … … * 0000800000003 * ,










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4 9231/44/O/N/25 © UCLES 2025 2 The continuous random variable X has probability density function f given by . ( ) f x = ( ) ( ) , , x x x x 1 2 0 1 1 2 1 G G G + - 9 4 2 otherwise 0 * (a) Find the cumulative distribution function of X. [3] … … … … … … … … … … … … … … … … … (b) Find the exact value of the median of X. [2] … … … … … … * 0000800000004 * ,










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5 9231/44/O/N/25 © UCLES 2025 [Turn over The random variable Y is defined by X Y = . (c) Find the cumulative distribution function of Y. [3] … … … … … … … … … … … … (d) Determine whether the median of Y is greater than, or less than, the median of X. [2] … … … … … … … … … … … … … * 0000800000005 * ,










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6 9231/44/O/N/25 © UCLES 2025 3 Two different types of juice extractor, machine X and machine Y, are being compared. The manufacturer claims that machine Y extracts more juice per orange on average than machine X. A random sample of 20 oranges is selected and each carefully measured. Oranges of a similar size and mass are then paired and numbered from 1 to 10. One orange from each pair is randomly allocated to machine X with the other allocated to machine Y. The amount of juice, in ml, extracted from each orange is recorded in the following table. pair 1 2 3 4 5 6 7 8 9 10 machine X 65 73 58 61 72 79 64 65 69 71 machine Y 68 72 64 63 75 82 63 63 72 74 (a) Use a t-test to test the manufacturer’s claim at the 1% significance level. [7] … … … … … … … … … … … … … … … … … … … * 0000800000006 * ,










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7 9231/44/O/N/25 © UCLES 2025 [Turn over (b) State an assumption required for the test in 3(a) to be valid. [1] … … … … The manufacturer notices that the amount of juice extracted from the oranges in pair 4 were recorded incorrectly. In fact, the amount of juice extracted from machine X was 60 ml and from machine Y was 62 ml. (c) Explain, with justification, whether the conclusion of the test in 3(a) remains the same. [2] … … … … … … … … … … … … … … … … … … … … * 0000800000007 * ,










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8 9231/44/O/N/25 © UCLES 2025 4 The random variable X takes values 1 and 2 with probabilities 5 2 and 5 3 respectively. (a) Write down the probability generating function ( )t GX of X. [1] … … … The random variable Y is the sum of four independent observations of X. (b) Find the probability generating function ( )t GY of Y. Give your answer in the form ( )t ( ) at b ct GY m n = + , where a, b, c, m and n are constants to be determined. [2] … … … (c) Use ( )t GY to find ( ) Y 6 = P . [2] … … … … … … … … … … … … … … … … … * 0000800000008 * ,










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9 9231/44/O/N/25 © UCLES 2025 [Turn over (d) Find Var(Y ). [5] … … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000009 * ,










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10 9231/44/O/N/25 © UCLES 2025 5 A driving instructor believes that the performance (pass or fail) of students when taking a driving test is associated with their age. The following table summarises the number of students who pass and who fail, and the ages in years of the students taking the test, over a period of three years. age of student under 20 20 – 40 over 40 total pass 34 41 6 81 fail 16 39 9 64 total 50 80 15 145 Test, at the 10% significance level, whether performance is independent of age of student. [7] … … … … … … … … … … … … … … … … … … … … * 0000800000010 * ,










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11 9231/44/O/N/25 © UCLES 2025 6 Students of the same age from two schools, school A and school B, take a large number of quizzes throughout the year and are each awarded a mark out of 1000. The marks of 123 students in school A and 147 students in school B are ranked from lowest (rank 1) to highest (rank 270). The sum of the ranks of the students from school A is 15 355. (a) Carry out a Wilcoxon rank-sum test at the 5% significance level to investigate whether there is a difference in average marks between the students in school A and school B. [6] … … … … … … … … … … … … … … … … … … … … (b) State an assumption that is required for the Wilcoxon rank-sum test to be valid. [1] … … … * 0000800000011 * ,









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12 9231/44/O/N/25 © UCLES 2025 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge. Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. … … … … … … … … … … … … … … … … … … … … … … * 0000800000012 * ,









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This document consists of 17 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge International AS & A Level FURTHER MATHEMATICS 9231/44 Paper 4 Further Probability & Statistics October/November 2025 MARK SCHEME Maximum Mark: 50 Published This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the October/November 2025 series for most Cambridge IGCSE, Cambridge International A and AS Level components, and some Cambridge O Level components.

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 2 of 17 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptions for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently, e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 3 of 17 Mathematics Specific Marking Principles 1 Unless a particular method has been specified in the question, full marks may be awarded for any correct method. However, if a calculation is required then no marks will be awarded for a scale drawing. 2 Unless specified in the question, non-integer answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the degree of accuracy is not affected. 3 Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points. 4 Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (ISW). 5 Where a candidate has misread a number or sign in the question and used that value consistently throughout, provided that number or sign does not alter the difficulty or the method required, award all marks earned and deduct just 1 A or B mark for the misread. 6 Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear.

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 4 of 17 Annotations guidance for centres Examiners use a system of annotations as a shorthand for communicating their marking decisions to one another. Examiners are trained during the standardisation process on how and when to use annotations. The purpose of annotations is to inform the standardisation and monitoring processes and guide the supervising examiners when they are checking the work of examiners within their team. The meaning of annotations and how they are used is specific to each component and is understood by all examiners who mark the component. We publish annotations in our mark schemes to help centres understand the annotations they may see on copies of scripts. Note that there may not be a direct correlation between the number of annotations on a script and the mark awarded. Similarly, the use of an annotation may not be an indication of the quality of the response. The annotations listed below were available to examiners marking this component in this series. Annotations Annotation Meaning More information required Accuracy mark awarded zero Accuracy mark awarded one Independent accuracy mark awarded zero Independent accuracy mark awarded one Independent accuracy mark awarded two Benefit of the doubt Blank Page Incorrect Dep Used to indicate DM0 or DM1

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 5 of 17 Annotation Meaning DM1 Dependent on the previous M1 mark(s) Follow through Indicate working that is right or wrong Highlighter Highlight a key point in the working Ignore subsequent work Judgement Judgement Method mark awarded zero Method mark awarded one Method mark awarded two Misread Omission or Other solution Off-page comment Allows comments to be entered at the bottom of the RM marking window and then displayed when the associated question item is navigated to. On-page comment Allows comments to be entered in speech bubbles on the candidate response. Judgment made by the PE Premature approximation Special case

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 6 of 17 Annotation Meaning Indicates that work/page has been seen Error in number of significant figures Correct Transcription error Correct answer from incorrect working

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 7 of 17 Mark Scheme Notes The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Mark for a correct result or statement independent of method marks. DM or DB When a part of a question has two or more ‘method’ steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly, when there are several B marks allocated. The notation DM or DB is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. FT Implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. • A or B marks are given for correct work only (not for results obtained from incorrect working) unless follow through is allowed (see abbreviation FT above). • For a numerical answer, allow the A or B mark if the answer is correct to 3 significant figures or would be correct to 3 significant figures if rounded (1 decimal place for angles in degrees). • The total number of marks available for each question is shown at the bottom of the Marks column. • Wrong or missing units in an answer should not result in loss of marks unless the guidance indicates otherwise. • Square brackets [ ] around text or numbers show extra information not needed for the mark to be awarded.

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 8 of 17 Abbreviations AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) CAO Correct Answer Only (emphasising that no ‘follow through’ from a previous error is allowed) CWO Correct Working Only ISW Ignore Subsequent Working SOI Seen Or Implied SC Special Case (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance) WWW Without Wrong Working AWRT Answer Which Rounds To

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 9 of 17 Question Answer Marks Guidance 1(a) 2 723, 42261 x x  =  = B1 723 x = implied by AWRT 55.6 for x . 2 42261 x  = implied by AWRT 171 for 2s (or 158 if biased). ( ) 2 2 '723' 1 2222 12 13 13 '42261' 170.9231 s   = − = =   M1 Correct form, must be unbiased, 13.07 s = . Implied by AWRT 171. 1.782 B1 1.782 or 1.78 seen. '723' '170.9231' '1.782' 13 13  M1 Correct form, must be a t-value.   49.2, 62.1 A1 Accept with inequality signs or open brackets. Condone   62.1,49.2 . Do not accept 55.6 6.5  . 5 1(b) The population (of car speeds) is normally distributed. OR Underlying distribution is normal. B1 Population mean normally distributed earns B0. (Underlying) data is normal earns B0. It is normal distribution earns B0. 1

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 10 of 17 Question Answer Marks Guidance 2(a) ( ) ( ) ( ) 2 2 4 9 9 3 1 3 0 0 0 1 F 1 1 1 2 2 2 x x x x x x x x    +   =  − +    ≤ ≤ ≤ M1 Attempt to integrate both parts, condone missing c , ignore domains. A1 Both parts correct with correct c , ignore domains. B1 Correct domains. Must cover all reals, may overlap at boundaries. Must have 0 for 0 x  and 1 for 2 x  . 3 2(b) 2 2 4 1 9 9 2 m m + = M1 Identifies correct interval for their piecewise ( ) F x to form an equation of the form equation ( ) 1 2 F m = OE. ( ) 1 2 2 13 m = −+ A1 OE, must clearly state one answer. 2 2(c) ( ) ( ) ( ) 4 2 2 4 9 9 3 2 1 3 0 0 0 1 G 2 1 2 2 1 1 y y y y y y y y    +   =  − +     ≤ ≤ ≤ M1 Substitute 2 x y = into both parts of their ( ) F x , ignore domains. A1 Correct functions in terms of ,y AEF. B1 Correct domain in terms of y . Domain may overlap at boundaries. Condone missing 0 for 0 y  but must have 1 for 2 y  . 3

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 11 of 17 Question Answer Marks Guidance 2(d) ( ) ( ) 1 2 G 2 13 0.379 −+ = M1 Evaluates ( ) G their m . 0.379 0.5  so the median of Y is greater than the median of X . A1 Alternative method for question 2(d) 2 2 13 2 y m −+ = M1 Expresses y m in terms of x m . 2 1 y x m m =  and hence y x m m  . A1 Uses the fact squaring a number between 0 and 1 makes it smaller. Alternative method for question 2(d) 4 2 2 4 1 9 9 2 Solves y y + = . M1 Identifies correct interval for their piecewise ( ) G y to form equation of the form ( ) 1 2 G y = OE. 0.896 0.803 y x m m =  = A1 0.803 may be seen in part 2(b). A0 for ( ) 1 2 2 13 0.896 −+  without ( ) 1 2 2 13 −+ evaluated. 2

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 12 of 17 Question Answer Marks Guidance 3(a) 0 1 H : H : Y X Y X     =  B1 If in words, must have population means. Allow H0: 1 0, H : 0 d d   =  if d defined or consistent with working. Differences: 3, –1, 6, 2, 3, 3, –1, –2, 3, 3 M1 Attempt at signed differences, SOI. ( )   2 2 '19' 1 9 10 '91' 6.1 ds = − = M1 Correct form, unbiased, implied by 2 6.1 ds = or 2.47 ds = seen. '19' 10 '6.1' 10 2.43 = M1 Correct form using their 2 ds . Allow M1 for   '696' '677' 10 10 1 1 10 10 0.666 ' ' ps − = + from 2-sample t-test. A1 AWRT 2.43. '2.43' 2.821,  accept H0 / do not reject H0 / not significant. M1 Compare their 2.43 with 2.821 and appropriate result (may be in terms of H1). Allow M1 for comparing their 0.666 with 2.552 and appropriate result from 2-sample t-test. Test result may be implied by an attempt at an appropriate conclusion in context ignoring hypotheses. Their 2.43 must come from an attempt at standardising. Insufficient evidence to support the manufacturer’s claim. OR Insufficient evidence to support that (on average) machine Y extracts more juice than machine X. A1 Correct conclusion from correct working ignoring hypotheses. In context. Level of uncertainty in language is used (for example, not ‘prove’). Allow ‘Not enough evidence / no sufficient evidence that / to show / conclude that…’. Do not accept ‘No evidence…’. Do not accept statements such as ‘there is sufficient evidence to suggest…’. 7

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 13 of 17 Question Answer Marks Guidance 3(b) Population of differences is normally distributed. OR Underlying distribution of differences is normal. B1 Population mean normally distributed earns B0. (Underlying) data is normal earns B0. Must mention population/distribution and differences. Following 2-sample test in part 3(a), allow either of the following: Populations/underlying distributions are normal. Populations have equal variances. Must refer to both populations. 1 3(c) The difference remains the same for pair 4. The set of differences has not changed. OR Redo the test: 2.43 2.821  *B1 Comment that recognises that the differences remain unchanged. OR Compare their 2.43 from part 3(a) with 2.821. Following 2-sample test in part 3(a), compare their 0.653 with 2.552. The conclusion remains the same. DB1 2

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 14 of 17 Question Answer Marks Guidance 4(a) ( ) 2 3 2 5 5 G X t t t = + B1 CAO 1 4(b) ( ) ( ) 4 4 1 625 G 2 3 Y t t t = + ( ) 4 4 3 2 5 5 t t   = +     M1 Attempt at 4th power of their ( ) G X t . A1 2 4(c) Attempt at coefficient of ( ) 6 in GY t their t . M1 Identify required term, coefficient must a non-zero probability. May be seen in part 4(b). ( ) 216 625 P 6 Y = = A1 AWRT 0.346. 2

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 15 of 17 Question Answer Marks Guidance 4(d) ( ) ( ) ( ) 4 3 3 4 4 12 625 625 ' G 2 3 2 3 Y t t t t t = + + + OR ( ) ( ) 3 4 5 6 7 1 625 ' G 64 480 1296 1512 648 Y t t t t t t = + + + + M1 Attempt at differentiating their ( ) GY t from part 4(b). ( ) ( ) ( ) ( ) 4 3 2 2 3 4 96 108 12 625 625 625 '' G 2 3 2 3 2 3 Y t t t t t t t = + + + + + OR ( ) ( ) 2 3 4 5 6 1 625 '' G 192 1920 6480 9072 4536 Y t t t t t t = + + + + M1 Attempt at differentiating their ( ) ' GY t . ( ) ( ) 32 888 5 25 ' '' G 6.4, G 35.52 1 1 Y Y = = = = B1 WWW ( ) ( ) 2 888 32 32 25 5 5 Var Y = + − M1 Use correct formula with their values. ( )   24 25 Var 0.96 Y = = A1 CWO Alternative method for question 4(d) ( ) ( ) 3 8 6 2 2 5 5 5 5 5 ' E 2 or use GX X t t = + = = + . M1 ( ) ( ) 2 3 6 2 14 5 5 5 5 '' E 4 or use GX X t = +  = = M1 ( ) ( ) ( ) ( ) 2 2 8 6 8 8 14 5 5 5 5 5 Var or Var X X = − = + − M1 ( ) 6 25 Var X = A1 ( ) ( ) 24 25 Var 4Var Y X = = A1 CWO 5

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 16 of 17 Question Answer Marks Guidance 5 H0: performance is independent of age. H1: performance is not independent of age. OR H0: passing test is independent of age. H1: passing test is not independent of age. B1 Allow dependent, association or relationship for not independent. For example: H0: No association/relationship between… H1: Association/relationship between… Must refer to “performance” and “age” in H0. 34 (27.93) 41 (44.69) 6 (8.38) 16 (22.07) 39 (35.31) 9 (6.62) M1 At least 3 correct to 3sf (must be seen). A1 All correct to 3sf. ( ) ( ) ( ) 2 2 2 34 27.93 41 44.69 9 6.62 27.93 44.69 6.62 − − − + ++   1.319 0.305 0.676 1.669 0.386 0.856 + + + + + OR 2 2 2 34 41 9 145 27.93 44.69 6.62   + ++ −     M1 At least 2 expressions (not including 145) of correct form seen. May see individual terms not summed. May be implied by 2 contributions correct to 3sf seen. May be implied by AWRT 5.21. 5.21 A1 AWRT 5.21. '5.21' 4.605  , reject H0 / significant. M1 Compare their 5.21 with 4.605 and appropriate result (may be in terms of H1). Test result may be implied by an attempt at an appropriate conclusion in context ignoring hypotheses. Sufficient evidence to suggest that performance and age are not independent. OR Sufficient evidence to suggest that performance and age are dependent. OR Sufficient evidence to suggest that performance and age are associated. OR Sufficient evidence to support the (driving) instructor’s belief. A1 Correct conclusion from correct working ignoring hypotheses. Must refer to both performance and age. In context. Level of uncertainty in language is used (for example, not ‘prove’). Do not accept statements such as ‘there is insufficient evidence to suggest…’. 7

9231/44 Cambridge International AS & A Level – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 17 of 17 Question Answer Marks Guidance 6(a) H0: population median mark for school A is equal to population median mark for school B. H1: population median mark for school A is not equal to population median mark for school B. B1 Allow m , but not . If m defined, must see population. ( )   1 2 123 123 147 1 16666.5  + + = ( )   1 12 123 147 123 147 1 408329.25   + + = B1 Both expressions correct. Implied by 16666.5 and 408329.25 seen. 15355 0.5 '16666.5' '408329.25' + − M1 OR attempts to find critical value or p-value using their mean and variance. Condone continuity correction omitted or incorrect. 2.05 − A1 AWRT 2.05 − or CV = 15414 or 0.0201 p = . Condone continuity correction omitted or incorrect. 2.05 1.96 − − OR 0.980 0.975  OR 0.020 0.025  OR 15355 0.5 15414 +  , reject H0 / significant. M1 Compare their 2.05  with 1.96  (consistent signs) or their 0.980 with 0.975, or their 0.020/0.040 with 0.025/0.05 or their 15355.5 with 15414 (condone continuity correction omitted or incorrect). Consistent signs and appropriate result (may be in terms of H1). Test result can be implied by an attempt at an appropriate conclusion in context. Sufficient evidence to suggest that there is a difference in average marks between the students in school A and B. OR Sufficient evidence to suggest that, on average, the test scores for students at the two schools are not equal. A1 Correct conclusion from correct working, ignoring hypotheses. Must consider continuity correction e.g. 15355.5 seen. In context. Level of uncertainty in language is used (for example, not ‘prove’). Do not accept statements such as ‘there is insufficient evidence to suggest…’. 6 6(b) The (underlying) distributions are symmetrical/similar shape. OR The populations are symmetrical/similar shape. OR No repeated marks. B1 1