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Oxford Cambridge and RSA

Level 3 CertificateQuantitative Problem Solving (MEI)H867/02 Statistical Problem Solving

Insert

Wednesday 23 May 2018 – MorningTime allowed: 2 hours

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18.1

210

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0014

223

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dia

Sout

h A

sia

1236

3446

312

67.8

162

19.8

986

4000

168

237

Mal

dive

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uth

Asi

a39

3595

176

75.1

598

15.5

912

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8N

epal

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3098

6975

4167

.19

164

21.0

779

1500

203

239

Paki

stan

Sout

h A

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1961

7438

06

67.0

516

623

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7131

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uth

Asi

a21

8664

4556

76.3

581

16.2

412

265

0014

5

12

H867/02/I Jun18© OCR 2018


Copyright Information

OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.

If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.

For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.

OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

*7021607587*

INSTRUCTIONS• The Insert will be found inside this document.• Use black ink. You may use an HB pencil for graphs and diagrams.• Complete the boxes above with your name, centre number and candidate number.• Answer all the questions.• Write your answer to each question in the space provided. If additional space is

required, you should use the lined page(s) at the end of this booklet. The question number(s) must be clearly shown.

• Do not write in the barcodes.• You are advised that an answer may receive no marks unless you show sufficient detail

of the working to indicate that a correct method is being used.

INFORMATION• The total mark for this paper is 60.• The marks for each question are shown in brackets [ ].• This document consists of 20 pages.• Final answers should be given to a degree of accuracy appropriate to the context.

© OCR 2018 [601/4782/9]DC (LK/CT) 154646/2

Last name

First name

Candidatenumber

Centrenumber


Level 3 CertificateQuantitative Problem Solving (MEI)H867/02 Statistical Problem Solving

Wednesday 23 May 2018 – MorningTime allowed: 2 hours

You must have:• the Insert (inserted)• the Statistical Tables (ST1) (inserted)

You may use:• a scientific or graphical calculator

OCR is an exempt CharityTurn over

* H 8 6 7 0 2 *

2

© OCR 2018

Answer all the questions.

Section A (30 marks)

1 A house building company applies for planning permission to build houses on a flood plain just outside a small town. Elaine is a journalist with the local newspaper. She wants to write a fair and informed report about the proposed development.

She commissions 5 assistants. In order to obtain the views of 100 people, each of them is to ask a sample of 20 people, 10 male and 10 female, two questions.

• Do you support the proposed housing development?• What is the main reason for your view?

(i) Which of the following terms describes the sample best?

Opportunity, Simple Random, Stratified, Quota, Cluster, Self-selected. [1]

The results for the first question are summarised in the table below.

Male Female

Interviewer Yes No Don’t know Yes No Don’t know

A 6 3 1 4 4 2

B 5 5 0 3 5 2

C 4 6 0 5 2 3

D 3 5 2 3 3 4

E 5 4 1 3 4 3

Total 23 23 4 18 18 14

(ii) State two general points that Elaine can conclude from the figures in the table. [2]

The most common reasons given are:

For: We need more housing in this town; there’s nowhere for young people to live (except with their parents).

Against: Building on a flood plain means that places further down the river are more likely to be flooded.

(iii) For each of these reasons, make one suggestion as to what further data Elaine should try to collect.

In each case say how she might obtain the data. [4]

3

Turn over© OCR 2018

1 (i)

1 (ii)

1 (iii) For

Against

4

© OCR 2018

2 Miranda is an administrator in a company that develops and hosts online tests. The company develops a game that measures reaction times. When it is nearly ready, Miranda, who has not been involved in its design, is asked to trial it.

Her first attempt gave her reaction time to be 520 milliseconds. Her next four attempts, in order, gave times of 415, 352, 242 and 268 ms.

Miranda then tries the game 200 times more. Her times for these 200 attempts are recorded and displayed as the frequency chart below. It is suggested that they can be modelled by a Normal distribution.

100 200 300 4000

10

20

30

40

Frequency

Reaction time (ms)

(i) Show that for a Normal distribution with mean 250 and standard deviation 50, the probability of an observation being between 250 and 275 is 0.1915. [3]

2 (i)

5


(ii) The equivalent figures for some other intervals are given in the table below. Complete this table. Give two reasons why the information in this table and the frequency chart on the previous page

indicates that this Normal distribution is indeed a good model for Miranda’s times. [3]

2 (ii)Interval Probability Normal model

frequencyMiranda’s frequency

250 – 275 0.1915 38.30 39

275 – 300 0.1499 29.98 29

300 – 325 0.0918 18.37 20

325 – 350 0.0441 8.81 8

Over 350 5

6

© OCR 2018

Miranda talks about the test to two friends drinking wine at a pub. After closing time they go to her house and play the game. In total they do 40 tests; their times are shown on the box and whisker plot below.

350 ms 450 ms 560 ms 618 ms 830 ms

(iii) Compare these times with Miranda’s times. Give a possible explanation for any difference. [2]

2 (iii)

7


(iv) Miranda’s company plan to keep a record of all the times of people playing the game and to show the overall distribution in a diagram. The graph below is their prediction of what it will look like when the data come in.

(A) Identify its main features. [2] (B) Give one possible explanation for the shape of the curve. [1]

100 200 300 400 500 600

Time (ms)

2 (iv)(A)

2 (iv)(B)

8

© OCR 2018

3 Salim is a doctor. One of his patients gives him a bottle of home-made medicine made from wild plants using an old family recipe.

Sometimes the patient gives himself a dose of the medicine. He says his legs are a bit stiff the next day but

otherwise he feels fine.

Salim obtains permission to do a pilot study on possible side effects of the medicine. He invites everyone who works at his surgery to take part and 10 healthy people volunteer.

• Before they take the medicine, they are timed running 100 m. These times, in seconds, are denoted by t1.

• A few days later, they are given measured doses of the medicine and timed again the following day. These times are denoted by t2.

The results are given in Table 3.1.

Volunteer A B C D E F G H I J

Dose (ml) 1.00 2.00 0.00 1.50 0.50 1.75 0.25 0.75 1.25 2.25

t1 (s) 12.3 11.6 12.8 13.4 15.1 11.2 12.3 17.5 16.3 14.4

t2 (s) 12.0 13.0 12.8 14.1 14.5 12.4 12.2 17.1 16.9 14.9

Table 3.1

(i) Which of the following terms describes the sample best?

Opportunity, Simple Random, Stratified, Quota, Cluster, Self-selected. [1]

3 (i)

Salim wants to investigate whether there is any relationship between the dose and the change in times between the first and second runs. He uses the figures in Table 3.1 to carry out a test based on Spearman’s Rank Correlation Coefficient at the 5% significance level.

(ii) State the null and alternative hypotheses for this test.

Complete Table 3.2 and carry out the test. State the result. [8]

9


3 (ii)

Volunteer Dose Dose rank t2 – t1 (t2 – t1) rank d d2

A 1.00 5 –0.3

B 2.00 1.4 10

C 0.00 1 0.0

D 1.50 0.7

E 0.50 –0.6 1

F 1.75 1.2

G 0.25 –0.1

H 0.75 –0.4

I 1.25 0.6

J 2.25 10 6 16

Σ

Table 3.2

10

© OCR 2018

A week later the volunteers are timed on a third 100 m run. These times are denoted by t3. Their three times are given in Table 3.3.

Volunteer A B C D E F G H I J

t1 (s) 12.3 11.6 12.8 13.4 15.1 11.2 12.3 17.5 16.3 14.4

t2 (s) 12.0 13.0 12.8 14.1 14.5 12.4 12.2 17.1 16.9 14.9

t3 (s) 12.3 11.5 12.9 13.6 15.0 11.2 12.3 17.7 16.2 14.4

Table 3.3

(iii) Salim has to write a short report for his practice manager, commenting on the pilot study.

Give three points that the report might contain. [3]

3 (iii)

11


Section B (30 marks)

The questions in this section are based on the pre-release data. A hard copy of this is provided with this examination paper.

4 (i) The GDP per capita is given for 228 countries in the pre-release data.

Find the median value and show that Ecuador, Macedonia and Azerbaijan are the countries with GDP per capita closest to it. [3]

(ii) Find the figure half way between the lowest and highest values of the GDP per capita and compare it with the median value. What does this tell you about the distribution of wealth in the world? [3]

4 (i)

4 (ii)

12

© OCR 2018

5 (i) Using relevant figures from the pre-release data set, estimate the number of babies born in a year in Argentina. [3]

A different measure of birth rate is considered. It is the number of babies born in a year per female aged between 15 and 54 (inclusive). In Argentina there are 11 692 613 females in this age range.

(ii) Calculate the new measure. [2]

(iii) The range 15 to 54 covers 40 years. Multiply your answer to part (ii) by 40. What information does this give you? [2]

5 (i)

13


5 (ii)

5 (iii)

14

© OCR 2018

6 Dipali wants to know if it is more healthy to live on an island or in a mainland country. To investigate this she starts by selecting 20 islands at random (from the pre-release data) and 30 mainland countries.

She classifies these countries according to whether their life expectancy is Low, Medium or High, using her own scale.

The results are shown in Table 6.1.

fo Low Medium High Total

Islands 1 9 10 20

Mainland countries 12 11 7 30

Total 13 20 17 50

Table 6.1

Dipali uses the data in Table 6.1 to carry out a χ2 test at the 5% significance level.

(i) State the null and alternative hypotheses. [1]

(ii) Complete Table 6.2 and carry out the test, showing that the result is significant. [7]

6 (i)

15


6 (ii)Expected frequency, fe Low Medium High Total

Islands 5.2 8 20

Mainland countries 30

Total 13 20 17 50

Table 6.2

16

© OCR 2018

Dipali is encouraged by this result and decides to continue her investigation using all the countries covered by the pre-release data, except for the 17 countries for which the life expectancy is not given. She starts by working out the means of the life expectancies for the 68 islands and for the mainland countries.

(iii) The sum of the life expectancies for the islands is 5149.36 years. The sum of the life expectancies for the mainland countries is 10 781.86 years.

Calculate the mean of the life expectancies for the islands and the mean of the life expectancies for the mainland countries. [2]

6 (iii)

17


(iv) Dipali then works out the equivalent weighted means, taking the populations of the countries into account. Table 6.3 contains relevant parts of a spreadsheet to work out the weighted mean of the life expectancies of three islands. The answer for the weighted mean should appear in cell T25.

Fill in the missing numbers in Table 6.3. [4]

6 (iv)P Q R T

20 Island Population Life expectancy Q × R

21 Barbados 289 690 74.99

22 Comoros 766 875 63.48

23 New Zealand 4 401 916 80.93

24 Total

25

Table 6.3

18

© OCR 2018

Dipali uses her spreadsheet correctly to find that the weighted mean life expectancy for all 68 islands is 75.62 years and that for all the mainland countries is 70.65 years.

(v) Dipali has used three different techniques in her investigation:

• a χ2 test; • comparing the simple means of the life expectancies of the two groups; • comparing the weighted means.

State which of these techniques you consider to be the most appropriate and give a reason why it is better than each of the other two. The two reasons which you give should be different. [3]

6 (v)

END OF QUESTION PAPER

19

© OCR 2018

ADDITIONAL ANSWER SPACE

If additional space is required, you should use the following lined page(s). The question number(s) must be clearly shown in the margin(s).

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© OCR 2018


Copyright Information

OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.

If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.

For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.

OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

....................................................................................................................................................................................

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Oxford Cambridge and RSA Examinations

Level 3 Certificate

Quantitative Problem Solving (MEI)

Unit H867/02 Statistical Problem Solving

OCR Level 3 Certificate

Mark Schemes for June 2018

OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of candidates of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT, business, languages, teaching/training, administration and secretarial skills. It is also responsible for developing new specifications to meet national requirements and the needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is invested back into the establishment to help towards the development of qualifications and support, which keep pace with the changing needs of today’s society. This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which marks were awarded by examiners. It does not indicate the details of the discussions which took place at an examiners’ meeting before marking commenced. All examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes should be read in conjunction with the published question papers and the report on the examination. © OCR 2018

H867/02 Mark Scheme June 2018

3

Annotations and abbreviations

Annotation in scoris Meaning

and

BOD Benefit of doubt

FT Follow through

ISW Ignore subsequent working

M0, M1 Method mark awarded 0, 1

A0, A1 Accuracy mark awarded 0, 1

B0, B1 Independent mark awarded 0, 1

SC Special case

^ Omission sign

MR Misread

Highlighting

Other abbreviations in mark scheme

Meaning

E1 Mark for explaining

U1 Mark for correct units

G1 Mark for a correct feature on a graph

M1 dep* Method mark dependent on a previous mark, indicated by *

cao Correct answer only

oe Or equivalent

rot Rounded or truncated

soi Seen or implied

www Without wrong working


4

Subject-specific Marking Instructions

a Annotations should be used whenever appropriate during your marking.

The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded.

b An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, award marks according to the spirit of the basic scheme; if you are in any doubt whatsoever (especially if several marks or candidates are involved) you should contact your Team Leader.

c The following types of marks are available. M A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually 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, eg by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. 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). Therefore M0 A1 cannot ever be awarded. B Mark for a correct result or statement independent of Method marks.


5

E A given result is to be established or a result has to be explained. This usually requires more working or explanation than the establishment of an unknown result. Unless otherwise indicated, marks once gained cannot subsequently be lost, eg wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.

d When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation ‘dep *’ is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.

e The abbreviation ft implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only — differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, exactly what is acceptable will be detailed in the mark scheme rationale. If this is not the case please consult your Team Leader. Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be ‘follow through’. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question.

f Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. Candidates are expected to give numerical answers to an appropriate degree of accuracy, with 3 significant figures often being the norm. Small variations in the degree of accuracy to which an answer is given (e.g. 2 or 4 significant figures where 3 is expected) should not normally be penalised, while answers which are grossly over- or under-specified should normally result in the loss of a mark. The situation regarding any particular cases where the accuracy of the answer may be a marking issue should be detailed in the mark scheme rationale. If in doubt, contact your Team Leader.


6

g Rules for replaced work If a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests.

If there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.

h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate’s data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A mark in the question. Note that a miscopy of the candidate’s own working is not a misread but an accuracy error.

i Anything in the mark scheme which is in square brackets […] is not required for the mark to be earned, but if present it must be correct.


7

Question Answer Mark Guidance

1 (i) Quota B1 Not stratified

[1]

1 (ii) There is no difference between the proportions of males and females answering Yes

and No. B1 Other sensible answers possible

More females say "Don't know" B1

[2]

1 (iii) For Data about the housing B1 Allow any reasonable description of relevant data,

and any reasonable data collection method (which

would enable collection of required data).

Survey of households B1

Against Data relating to the risk of flooding B1

Historical records B1

[4]

2 (i) For 275,

275 2500.5

50z

M1

A1

M1: standardisation of 275 attempted (ignore

wrong sign or √ errors)

A1: z=0.5 correct

For 250, z = 0 B1 B1: 0.6915 – 0.5 seen

(0.5) (0) 0.6915 0.5

0.1915

SC: 0.6915 – 0.5 seen (no z-values) – gets full

marks

SC: Calculator answers must show the detail of

what was entered. Full marks available.

[3]


8


2 (ii)

Interval Probability Normal model

frequency

Miranda's

frequency

250 - 275 0.1915 38.30 39

275 - 300 0.1499 29.98 29

300 - 325 0.0918 18.37 20

325 - 350 0.0441 8.81 8

Over 350 0.0228 4.56 5

Table 2.2

B1

The figures have been obtained from a calculator.

Accept slightly different figures from candidates

using the given figures or tables.

All Miranda's frequencies are close to those for the Normal model

B1 One comment comparing the frequencies in the

table

Fig 2.1 is close to being symmetrical about the mean (250 ms) B1 One comment about the shape of the chart

allow ‘bell shaped’

SC B1: ‘data fits normal curve / is bell shaped’

[3]


9


2 (iii) Most of their times are more than 3 standard deviations from Miranda's mean so their

reactions are slow M1 Accept “Miranda’s times are generally much lower

than these times”

Can be implied by the explanation.

Maybe this is the result of the wine they have drunk / they did less practice

A1

Their explanation must relate to their description.

SC B1 if there are both right and wrong full

answers

[2]

2 (iv) (A) It is nearly a Normal distribution (with the same mode as Miranda's times)

B1 one comment related to the unimodality

allow ‘single peak’ or ‘approximately bell shaped’

Allow a correct comment about 250.

but it is skewed (to the right)

B1

one comment related to skew

allow a reasonable description of ‘skewed’(?)

Allow a correct comment about the range.

(B) This is because of some people who can be expected to take longer, for example

beginners and those who have been drinking B1

The explanation must relate to skew

[3]

3 (i) Self-selected B1 Accept Opportunity

[1]


10


3 (ii) H0: There is no association/correlation between dose and change in running

time not ‘relationship’ or ‘dependence’

Condone “increase” instead of “change”

H1: There is association/correlation between dose and change in running time B1 not ‘positive association’ in H1

Volunteer Dose Dose rank t2 - t1 (t2 - t1) rank d d2

A 1.0 5 -0.3 3 -2 4

B 2.0 9 1.4 10 1 1

C 0.0 1 0.0 5 4 16

D 1.5 7 0.7 8 1 1

E 0.5 3 -0.6 1 -2 4

F 1.75 8 1.2 9 1 1

G 0.25 2 -0.1 4 2 4

H 0.75 4 -0.4 2 -2 4

I 1.25 6 0.6 7 1 1

J 2.25 10 0.5 6 -4 16

Σ 0 52

B1

B1

B1

One fully correct row

One fully correct ‘rank’ column

Correct Σd2

2

s 2

6 6 521 1

10 991

dr

n n

M1 Attempt to use correct formula

s 0.6848r A1 cao, art 0.68 or 0.685 (not f/t their table)

Critical value for n=10, 2-tailed, 5% significance level, is 0.6485 B1 0.6485 seen


11


Since 0.6848 > 0.6485, Ho is rejected. (The evidence does not support H0.) A1 Dependent on M1

f/t their rs and cv if both clearly seen.

This mark may be awarded for a context based

comment.

[8]

3 (iii) The Spearman test suggests a side effect is possible. B1 One mark for each (different) sensible point

The times for the 3rd test are almost the same as those for the 1st suggesting that any

effects are short term B1

The data for the first two tests suggest that lower doses decrease times (people run

faster) but higher doses increase times (people run slower). B1

A random sample is required. / A (much) larger study is required. Comments about sample size or nature can score at

most B1 in total.

[3]


12


4 (i) The median is midway between those ranked 114 and 115.

B1 not 114th

give if 114.5 seen

Macedonia and Azerbaijan are ranked equal 113 with GDP per capita of $10 800.

Ecuador is ranked 115 with GDP per capita of $10 600. B1 All correct countries, ranks and GDP values seen.

So the median is $10 700. B1 cao

[3]

4 (ii) The country with the greatest GDP per capita is Qatar at $102 100 B1 Both correct GDPs

The country with the least GDP per Capita is Congo DDR at $400

$102 100 $400$51 250

2

M1 Finding the average of their two numbers

This is very much greater than the median. It suggests that a large amount of wealth is

concentrated in a small number of countries.

A1 Needs an explicit comparison with the median and

a comment concerning distribution of wealth. (does

not need the correct median in (i))

Not ‘there is a large difference between richest and

poorest countries’

[3]


13


5 (i) Population of Argentina is 43 024 374, birth rate is 16.88 per thousand B1 both correct numbers seen

Number of babies per year is

16.8843 024 374 726 251 ( 726 000)

1000

M1

A1

using their numbers

Allow 2 to 6 s.f., art 730 000

[3]

5 (ii) Babies 726 2510.0621... ( 0.0621)

Females 11 692 613

M1

A1

using their answer from (i)

f/t their (i), allow 2 to 4 s.f.

[2]

5 (iii) 0.0621 × 40 = 2.48 B1 f/t their (ii)

Average number of children per female B1 Only give if “2.48” is between 1 and 10

[2]


14


6 (i) H0: The proportion of countries with Low, Medium and High life expectancy is

independent of whether the countries are islands or mainland countries

OR ‘life expectancy and location are

Independent’

H1: The proportion of countries with Low, Medium and High life expectancy

depends on whether the countries are islands or mainland countries

B1 Both hypotheses correct

not ‘correlation’ or similar

[1]

6 (ii) Expected

frequency, fe

Low Medium High Total

Islands 5.2 8 6.8 20

Mainland

countries

7.8 12 10.2 30

Total 13 20 17 50

M1

A1

At least one correct cell

All cells correct

2

25.2 1

...5.2

X

M1 Evidence of correct calculation

2 3.3923 0.125 1.5059 2.2615 0.0833 1.0039 8.37X A1 X2 value, cao, at least 2 s.f.

3 1 2 1 2 M1

Critical value = 5.991 A1

8.37 > 5.991 so reject H0 (the result is significant) A1 Needs explicit comparison, correct X2 and c.v.

[7]


15


6 (iii) Islands

5149.3675.725(8)...

68

B1 Accept 2 to 5 s.f.

Mainland countries

10781.86 10781.8670.012...

239 17 68 154

B1 Accept 2 to 5 s.f.

ignore any incorrect statements/working

[2]

6 (iv) P Q R T

20 Island Population Life expectancy Q × R

21 Barbados 289690 74.99 21723103

22 Comoros 766875 63.48 48680590

23 New Zealand 4401916 80.93 356247062

24 Total 5458481 426650755

25 78.16

B1

B1

B1

B1

One correct Q × R

Correct Q24

Correct T24 (allow if in T25)

Correct T25, 3 s.f. or better

[4]


16


6 (v) The weighted mean is the best. B1

The χ2 test uses a sample not the whole population.

It does not tell you which group has the greater life expectancy

B1 Explicit comparison with χ2 test

The simple mean does not take the territories' populations into account B1 Explicit comparison with simple mean

[3]

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Oxford Cambridge and RSA Examinations

Level 3 Certificate

Quantitative Problem Solving (MEI) OCR Level 3 Certificate Quantitative Problem Solving H867

OCR Report to Centres June 2018

OCR Report to Centres – June 2018

9

H867/02 Statistical Problem Solving

1. General Comments: The overall quality of responses was high this year, with candidates clearly engaging with the questions and very few blank answer spaces. The calculation aspect of most topics was done particularly well, one exception being the calculation of Normal probabilities. Future candidates would benefit from more practice in using Normal probability tables or calculators, and also understanding the difference between calculating probabilities and estimating them using the ‘rule of thumb’ percentage points, which are introduced in Paper 1. Most candidates attempted to interpret results of tests and calculations and there were many well thought-through and sensible answers. As in previous years, some wrote general comments rather than relating their interpretation explicitly to their calculations, but this was much less common than last year. There has also been an improvement in candidates considering the whole question when writing their final interpretation. The use of technical language remains a stumbling block for many candidates, particularly when stating hypotheses and describing shapes of distributions. Terms such as ‘association’, ‘dependence’, ‘average’, ‘skew’ and ‘Normal’ are often used colloquially rather than with their precise technical meanings. Centres should encourage correct vocabulary in discussions throughout the course. Future candidates would also benefit from spending more time discussing which tests and calculations can and cannot be used to answer various statistical questions. They should be encouraged to devise their own investigations and measures, compare the results and discuss which are most appropriate in a given situation. 2. Comments on Individual Questions: Question No. 1 This question was about sampling and data collection methods. In part (i) most candidates correctly identified quota sampling, but some opted for stratified sampling, possibly because the categories were ‘male’ and ‘female’. The latter is not correct since the required number for each category was specified with no reference to actual population proportions. Part (ii) required some observations to be made about the data and most candidates were able to provide two sensible comments. Best answers made it clear whether they were referring to the sample data or the whole population. For example, in the sample the same number of people answered ‘Yes’ and ‘No’, but this only suggests that the proportions are approximately equal in the population. Candidates should be reminded to write in full sentences so that it is clear, for example, whether they are referring to males, females or the whole sample.


10

Part (iii) invited candidates to suggest additional information that could be collected to support each of the opposing views. There were many good answers, but some suggested data that could not be reasonably collected for the purposes of writing a newspaper article (for example, ‘record the level of rainfall in the next ten years’). To gain full marks candidates needed to make a clear description of the information to be collected, and suggest a plausible method of obtaining it. Question No. 2 This question was about using Normal distribution to model reaction times. Part (i) proved to be one of the most challenging questions on the paper, with most candidates unable to demonstrate the use of Normal probability tables. Many tried to estimate the probability from the standard percentage points, which they know from Paper 1. Some realised that they needed to find a z-value but did not know how to continue from there. Some sketched and shaded normal distribution diagrams, which is a useful first step. Centres should remember that the use of calculators is permitted and, indeed, encouraged in this specification, so Normal probabilities can be found from a calculator rather than the tables. Many candidates were able to fill in the missing numbers in the table in part (ii) and provide at least one correct comment comparing those expected frequencies to the observed ones. Two comments were required, one about the chart and one about the table. Candidates needed to state clearly, which one they were referring to. For example, saying that ‘the data in the table is bell shaped’ is not correct, as the table only shows one half of the distribution. Almost all candidates missed the subtle point that the question was referring to ‘this Normal distribution’ rather than ‘a Normal distribution’. This means that it is not enough to note that the chart is approximately bell-shaped, but also that its peak is around 250ms. However, the marking on this occasion was lenient in this respect. Part (iii) required at least one comparison and an explanation for that particular feature. Most candidates opted to comment on the difference in the average; although some incorrectly referred to the mean of 560 ms, (the box plot shows the median). Some also successfully commented on the spread of the data, correctly noting that a possible explanation for the larger spread is that the data comes from three different people. The description in part (iv) proved more challenging, with most candidates seeming to lack the technical vocabulary to describe the skewness of the distribution. The strongest candidates realised that the skew meant that some people were a lot slower than average and were able to give sensible justification for why this might be. As in part (ii) many used ‘symmetrical’ when they presumably mean ‘bell shaped’. Learners would benefit from seeing examples of many different distributions, as many seem to be equating ‘symmetrical’ with ‘Normal’. Question No. 3 This question was centred on conducting and interpreting a Spearman’s rank test for correlation. The calculations for the test were generally carried out accurately and the comparison with the critical value led to the correct conclusion. More attention should be given to the correct phrasing of the hypotheses and interpreting the conclusion. The hypotheses for this test should mention ‘correlation’ or ‘association’ (which describe an increasing or decreasing relationship) rather than ‘relationship’ or ‘dependence’. Some class time should be given to the discussion of these distinctions and to looking at examples of different questions and phrasing of appropriate hypotheses to answer them.


11

Similar discussion would improve the quality of conclusions drawn from the tests. In this question, the test result was statistically significant, suggesting that the medicine seems to have possible side effects. The dose in fact seems to have an effect on the change in the running time, rather than on the time itself, with a larger dose being associated with a larger increase in time; smaller doses seem to in fact lead to a decrease in running time. Some candidates expressed these subtleties very well, while others struggled to phrase their answers clearly. Most candidates picked up on the fact that the third set of times being similar to the first meant that any side effects were temporary. Some also commented on the fact that the medicine may have a different effect on different people (as the correlation is not perfect, so there are other factors involved). As in previous years, an insignificant minority of candidates included comments of a general nature rather than relating their conclusions to the test itself; for example, they talked about participants getting tired, or commented on possible inaccuracies in the measurement procedures. One mark in part (iii) was available for comments concerning possible improvements to the experimental set up, but the main focus of the question was on interpretation of the Spearman’s rank hypothesis test. A small number of candidates answered the question in general rather than specific terms, for example saying that ‘Simon could include information about the results of his hypothesis test’ rather that stating the actual result. Such responses gained no credit. Questions 4, 5 and 6 were based on the pre-release data set. Candidates showed good familiarity with the data set and were generally able to find the required information. Answers were almost always given to a sensible degree of accuracy. Question No. 4 The calculation of the median in part (i) presented difficulties for a majority of candidates. There are 228 countries, so the median is the mean of the 114th and 115th value. This needed to be clearly stated to gain the first mark. It is not correct to claim, as many candidates did, that ‘there is no country number 114, so the median is between 113th and 115th value’; if the countries were written out in a list in order of GDP, the 114th place would be occupied by one of Macedonia or Azerbaijan. Many candidates were still able to obtain the correct median using this erroneous reasoning, but in order to gain the remaining two marks they needed to state the three countries, their ranks and GDPs. Part (ii) produced some good answers, although many candidates made general comments about ‘unequal distribution of the wealth in the world’ rather than referring to their calculation explicitly. The correct answer is that the mid-range being greater than the median tells us that there are a small number of countries with a very large GDP, and a large number of countries with a small GDP. Question No. 5 This question was about calculating and interpreting various measures of birth rates. Most candidates showed a good understanding of ‘births per 1000’ and were able to calculate the number of babies after one year correctly. Some misinterpreted the measure as giving the number of births per 1000 females and got half the required answer. The most common mistake was dividing rather than multiplying by 16.88.


12

The alternative measure introduced in part (ii) proved more difficult to calculate and interpret, with very few correct responses to part (iii) seen. Future candidates would benefit from seeing different examples of measures, devising their own and reflecting on how different measures can be used to draw different conclusions about the data. Question No. 6 This looked at a different way to answer the question ‘Is it healthier to live on an island or a mainland country?’. Parts (i) and (ii) required a completion of a chi-squared test to see whether there is any association between the type of country and life expectancy. The calculations were again carried out well, with sufficient detail usually shown. The hypotheses often used terms ‘correlation’ or ‘association’ which are not correct in this context; the correct null hypothesis is that the life expectancy is independent of whether the country is mainland or island. Part (iii) was done very well, although a sizeable minority of candidates miscalculated the number of mainland countries. Part (iv) also produced many fully correct answers, with the final calculation for cell T25 proving too challenging for a lot of candidates. We have adjusted the mark scheme to allow for a minor issue with this question. The final part of this question asked candidates to compare the three measures they have worked with. They needed to state their preferred measure and explain why they consider it better than the other two. Thus two different comparisons were required, rather than just a list of some positives about their chosen measure. The expected answer was that the weighted mean is the most appropriate: it is better than the simple mean because it is not affected by the size of the countries, and it is better than the chi-squared test which only tells us that the life expectancy is dependent on the type of the country, but not which type of country is better. It is also correct to say that Dipali’s chi-squared test only used a sample, while the calculation of the means used the whole data set. Some candidates mistakenly claimed that the weighted means were only calculated for three countries, which was not the case. It is possible to claim that the chi-squared test is most appropriate because its result would not be affected by a few countries having an usually large or unusually small life expectancy (since the countries are only categorised as ‘low’, ‘medium’ and ‘high’). This answer was given credit, but it was difficult to find two different reasons for the two comparisons in this case.

Published: 15 August 2018 Version 1.0 1

Level 3 Certificate, Level 3 Extended Project and FSMQ raw mark grade boundaries June 2018 series

Level 3 Certificate Mathematics - Quantitative Methods (MEI)Max Mark a b c d e u

G244 A 01 Introduction to Quantitative Methods with Coursework (WrittenPaper) Raw 72 58 50 43 36 28 0

G244 A 02 Introduction to Quantitative Methods with Coursework(Coursework) Raw 18 14 12 10 8 7 0

UMS 100 80 70 60 50 40 0Overall 90 72 62 53 44 35 0

Level 3 Certificate Mathematics - Quantitative Reasoning (MEI)Max Mark a b c d e u

H866 01 Introduction to quantitative reasoning Raw 72 56 49 42 35 28 0H866 02 Critical maths Raw 60 44 39 34 29 24 0

*To create the overall boundaries, component 02 is weighted to give marks out of 72 Overall 144 109 96 83 70 57 0

Level 3 Certificate Mathematics - Quantitative Problem Solving (MEI)Max Mark a b c d e u

H867 01 Introduction to quantitative reasoning Raw 72 56 49 42 35 28 0H867 02 Statistical problem solving Raw 60 40 36 32 28 24 0

*To create the overall boundaries, component 02 is weighted to give marks out of 72 Overall 144 104 92 80 69 57 0

Advanced Free Standing Mathematics Qualification (FSMQ)Max Mark a b c d e u

6993 01 Additional Mathematics Raw 100 56 50 44 38 33 0

Intermediate Free Standing Mathematics Qualification (FSMQ)Max Mark a b c d e u

6989 01 Foundations of Advanced Mathematics (MEI) Raw 40 35 30 25 20 16 0

For more information about results and grade calculations, see https://www.ocr.org.uk/students/getting-your-results/

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