statistics 4040 2013 - 2014

This document consists of 19 printed pages and 1 blank page.

DC (NH/CGW) 66916/2 UCLES 2013 [Turn over

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STATISTICS 4040/12Paper 1 October/November 2013

2 hours 15 minutesCandidates answer on the question paper.Additional Materials: Pair of compasses

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READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use a soft pencil for any diagrams or graphs.Do not use staples, paper clips, highlighters, glue or correction fluid.DO NOT WRITE IN ANY BARCODES.

Answer all questions in Section A and not more than four questions from Section B.If working is needed for any question it must be shown below that question.The use of an electronic calculator is expected in this paper.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or part question.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONSGeneral Certificate of Education Ordinary Level

24040/12/O/N/13 UCLES 2013

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Section A [36 marks]

Answer all of the questions 1 to 6.

1 Seven statistical measures are

mean,median,mode,range,interquartile range,variance

and standard deviation.

In each of the following situations, one of these measures is to be found by the person described. State the appropriate measure in each case.

(i) A doctor finds the most common age of her patients.

................................................... [1]

(ii) An athlete who competes in the 100 metres sprint finds the difference between his slowest and quickest practice times.

................................................... [1]

(iii) A graduate who seeks employment with a company finds a measure of central tendency for the salaries of the companys employees. The company has twenty employees, of whom three are managers earning salaries very much higher than the other employees.

................................................... [1]

(iv) A teacher finds a measure of dispersion for the scores of her pupils in a test, in which no pupil scored an exceptionally high mark, and no pupil scored an exceptionally low mark.

................................................... [1]

(v) A biologist finds a measure of dispersion for the growth of twelve plants over a period of three months. Two plants have been attacked by insects and have grown very much less than the others.

................................................... [1]

(vi) A sociologist finds a measure of central tendency for the first names given to the male babies born in a hospital over a period of six months.

................................................... [1]

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2 A large keep fit class for women is held at a sports club once every week. The manager of the club asks the class instructor to select a sample of size 10 from the class.

(i) State the method of sampling used if the instructor decides to select

(a) the first 10 women to arrive at the class,

................................................... [1]

(b) women at regular intervals from the class register.

................................................... [1]

The sample is required to obtain responses to a proposal to change the time of the class from Monday evening to Monday afternoon. For class members the only items of data presently available to the instructor are name and age.

(ii) State, and justify, two other items of data relating to class members which the instructor needs to know when selecting the sample in order to avoid bias in responses. You are not required to describe how the sample is selected.

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

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

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

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

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

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

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

...................................................................................................................................... [4]

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3 In a photographic equipment store a record was kept of the number of cameras sold each day. The values, for eleven consecutive days, were as follows.

6 0 8 2 1 6 0 9 6 4 1

For these values find

(i) the mode,

................................................... [1]

(ii) the mean, correct to one decimal place,

................................................... [2]

(iii) the median.

................................................... [2]

The values recorded for the next three days were x, x + 1 and x + 2.

(iv) If the median for the entire fourteen-day period was the same as the median for the first eleven days, find x.

x = .................................................. [1]


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4 The diagram below shows the number of actors at a film festival who have worked in one or more of the cities Mumbai, Los Angeles and Rome.

Mumbai Los Angeles

Rome

8

4

5

63

13 9

(i) Find the number of actors who have worked in Mumbai.

................................................... [1]

(ii) Interpret the value 6 in the diagram.

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

...................................................................................................................................... [1]

A journalist selects one of these actors at random for interview.

Find the probability of selecting an actor who has worked in

(iii) Mumbai or Los Angeles or both,

................................................... [2]

(iv) Los Angeles and Rome,

................................................... [1]

(v) Rome, given that the actor has worked in Mumbai and Los Angeles.

................................................... [1]

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5 In this question you are not required to draw any charts.

A charity, Camfam, classifies the income it receives under the headings Special Events, Donations, Grants, and Other Sources. In Camfams report for 2010, the following percentage bar chart was given, which represents a total income of $80 million.

1009080706050Percentage

Special Events

2010

403020100

Donations Grants Other Sources

(i) Find the income which Camfam received in 2010 from Grants.

$ ................................................... [2]

(ii) If a pie chart were to be drawn to represent this information, find the angle which would represent the sector for Special Events.

................................................... [2]

Camfams total income in 2011 was $60 million.Two pie charts, one for 2010 and one for 2011, are to be presented together in a new report.

(iii) Find, in its simplest terms, the ratio of the area of the chart representing 2010 to the area of the chart representing 2011.

................................................... [2]


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6 The following table is to show the distance, in kilometres, between any two of the five towns A, B, C, D and E.

42 B

20

36

39

18 25

A

C

D

E

For example, the distance between B and D is 39 km.

(i) Complete the table using the following information.

(a) The distance between B and C is 10 km more than the distance between D and E.

[1]

(b) The distance between A and C is two thirds of the distance between A and E.

[1]

(c) The distance between A and B is twice the distance between C and E.

[1]

(d) C is 19 km further from D than B is from E.

[1]

Dimitri lives in town A, but has one friend in each of the towns D and E. He makes a journey in which he leaves his home, visits each of these friends once, and then returns home.

(ii) Find the distance which Dimitri travels to complete the journey.

............................................. km [2]

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Section B [64 marks]

Answer not more than four of the questions 7 to 11.

Each question in this section carries 16 marks.

7 In this question the fertility rate of a population is defined as the number of births per 1000 females.

The table below gives information about the female population and age group fertility rates in a particular city for the year 2012, together with the standard population of the area in which the city is situated.

Age group of females Births

Population of females in age group

Age group fertility rate

Standard population of females (%)

Under 20 2900 50 18

20 29 4500 184 22

30 39 5250 136 25

Over 39 5800 15 35

(i) Calculate, to 1 decimal place, the standardised fertility rate for the city.

................................................... [4]

(ii) Calculate the number of births for each age group and insert the values in the table above.

[2]


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(iii) Calculate, to 1 decimal place, the crude fertility rate for the city.

................................................... [4]

There are equal numbers of males and females in the city and in the standard population.The standardised and crude death rates for the city in 2012 were 8.5 and 7.8 per thousand of the population respectively.

(iv) Using one of these values, and any other appropriate values from parts (i), (ii) and (iii), find the increase in the population of the city in 2012 due to births and deaths.

................................................... [5]

It is not possible to obtain an accurate measure of population increase or decrease in a city from information on births and deaths alone.

(v) State what additional information is required.

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

...................................................................................................................................... [1]

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8 In a large residential building there are 120 apartments, of which 50 are private apartments (owned by the residents) and 70 are company apartments (owned by the company which constructed the building).

If two apartments are chosen at random, find the probability of choosing

(i) two private apartments,

................................................... [2]

(ii) at least one company apartment.

................................................... [2]

The weekly rents, in dollars, charged on the company apartments are represented in the histogram below, from which one rectangle, representing the $400 to under $500 class, has been omitted.

0

5

200 250 300Weekly rent ($)

350 400 450 500

10

15Number

ofapartments

per $50

20

25

11


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Use the histogram to find the number of company apartments for which the weekly rent was

(iii) from $250 to under $400,

................................................... [2]

(iv) from $225 to under $250.

................................................... [2]

There were 10 company apartments for which the weekly rent was from $400 to under $500.

(v) Complete the histogram by drawing on the grid the rectangle representingthe $400 to under $500 class.

[1]

(vi) Write down the term used to describe the $300 to under $350 class.

................................................... [1]

The private apartments are of three different sizes. There are 24 apartments with three rooms, 14 with four rooms, and 12 with five rooms.A safety expert, conducting a survey on the use of smoke detectors, chooses three private apartments at random.

(vii) If the apartments chosen have 12 rooms in total, find the probability that the apartments are all of the same size.

................................................... [6]

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9 The mid-day temperature at a particular location in a city was measured every day throughout the year 2010. The following table summarises the results obtained.

Temperature (C) Number of days Cumulative frequency0 under 5 8

5 under 10 25

10 under 15 52

15 under 20 81

20 under 25 79

25 under 30 68

30 under 35 37

35 under 40 15

(i) Complete the cumulative frequency column in the above table. [2]

(ii) Plot the cumulative frequencies on the grid opposite, joining the points by a smooth curve. [3]

(iii) Use your graph to estimate

(a) the median of these temperatures,

.............................................. C [1]

(b) the interquartile range of these temperatures.

.............................................. C [4]

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0

50

50 10 15Temperature (C)

20 25 30 35 40

100

150

Cum

ula

tive fr

eque

ncy

(days

)

200

250

300

350

400

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When the results were obtained, a scientist predicted that, because of climate change, temperatures in the city would increase at the rate of 0.5 C every ten years.Assume that this prediction is accurate.

For this particular location,

(iv) use your answers to part (iii) to estimate, for the year 2050,

(a) the median of the mid-day temperatures,

........................................C [2]

(b) the interquartile range of the mid-day temperatures,

........................................C [1]

(v) use your graph to estimate, for the period 2010 to 2050, the increase in the number of days with a mid-day temperature of more than 36 C.

................................................... [3]

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10 Emilie, a student teacher, conducted research on the number of pupils and the number of teachers in the schools in the town of Astra, where she lives. The schools supplied the following data.

School A B C D E F G H

Number of pupils, x 760 1219 927 470 1361 628 381 1085

Number of teachers, y 29 44 33 34 52 24 16 40

(i) Plot these data on the grid below.

0

10

2000 400 600

Number of pupils

Numberof

teachers

800 1000 1200 1400 x

y

20

30

40

50

60

[2]

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The data have an overall mean of (853.875, 34) and an upper semi-average of (1148, 42.25).

(ii) Show how the value 1148 is calculated.

[2]

(iii) Find the lower semi-average.

................................................... [2]

(iv) Without plotting the averages, and without drawing the line, find the equation of the line of best fit in the form y = mx + c.

................................................... [3]

(v) Explain briefly why the value of c which you have found in part (iv) might give you cause for concern.

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

...................................................................................................................................... [1]

Emilie discovered later that the data supplied by one of the schools gave, incorrectly, the total number of people employed by the school, and not the number of teachers.

(vi) Ignoring the point representing the school which supplied incorrect data, draw, by eye, on the grid in part (i), a line of best fit through the remaining seven points. [1]

(vii) Use the line you have drawn in part (vi) to find its equation in the form y = mx + c.

................................................... [3]

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Emilie repeated the research for schools in the nearby town of Belport, for which she found the equation of the line of best fit to be y = 0.0431x + 1.72 .

(viii) Using this equation, and your answer to part (vii), state in which of the two towns a pupil might choose to be educated, if free to choose. Explain your answer briefly.

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

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

...................................................................................................................................... [2]

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11 (i) Give one advantage and one disadvantage of forming a large set of data into a grouped frequency distribution.

Advantage .........................................................................................................................

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

Disadvantage ....................................................................................................................

...................................................................................................................................... [2]

The presenter of a radio programme, in which recordings of popular songs are played, plans his programme. For each song chosen he writes down the song length, in terms of time, in minutes, taken to play the song. The following table summarises the song lengths.

Song length (minutes)

Number of songs

2.8 under 3.2 3

3.2 under 3.4 5

3.4 under 3.6 9

3.6 under 3.8 8

3.8 under 4.0 7

4.0 under 4.2 4

4.2 under 4.6 2

(ii) Estimate, in minutes, the mean and standard deviation of these song lengths. Give your answers to 3 significant figures.

Mean = ......................................................

Standard deviation = .................................................. [8]

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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.

University of Cambridge International Examinations 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.

Information about five of the presenters earlier programmes is shown below.

Programme Number of songs playedMean of song lengths

(minutes)Standard deviation of

song lengths (minutes)P 38 3.70 0.339

Q 39 3.52 0.328R 42 3.69 0.294

S 37 3.83 0.305

T 38 3.74 0.291

(iii) State in which of the programmes P, Q, R, S or T, songs were generally

(a) shortest in length,

................................................... [1]

(b) most similar in length.

................................................... [1]

All the presenters programmes are three hours in duration. Songs are not played continuously throughout each programme; for some of the time the presenter talks about the songs and the singers.

A listener switched on programme P at a random time during its transmission.

(iv) Find the probability that a song was not being played at that moment.

................................................... [4]

This document consists of 19 printed pages and 1 blank page.

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24040/13/O/N/13 UCLES 2013

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1 A survey was carried out to discover whether the quantity of traffic on a busy road was sufficient to justify the installation of a pedestrian crossing. At intervals throughout one day an investigator recorded the number of vehicles passing the proposed location in periods of 30 seconds duration.

The numbers he recorded were:

12 51 64 55 51 61 31 22 20 15 34 14 69 35

When his record sheet was examined the number shown here as was illegible, but it was certainly a single-digit number.

Although this number is unknown, name, but do not calculate,

(i) two measures of central tendency (average) which can still be found,

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

................................................... [2]

(ii) one measure of dispersion which can still be found,

................................................... [1]

(iii) one measure of central tendency (average) which cannot be found,

................................................... [1]

(iv) two measures of dispersion which cannot be found.

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

................................................... [2]


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2 The pie chart below illustrates the distribution by location of the total net profit of $787 million earned by an international company in the year 2011.

AsiaNorth

America

Europe

Restof the

World

(i) Measure, to the nearest degree, the sector angles of the pie chart, and insert them in the appropriate places on the chart.

[2]

(ii) Calculate, to the nearest $million, the net profit of the company in Asia.

$ ...................................... million [1]

(iii) Measure and state the radius, in centimetres, of the above pie chart.

............................................. cm [1]

The total net profit of the same company in the year 2005 was $523 million.

(iv) Calculate, correct to 2 significant figures, the radius, in centimetres, of the comparable pie chart for 2005.

............................................ cm [2]

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3 A factory employs both male and female staff in each of the three categories managerial, inspection and production.

There are altogether 3500 employees, of whom 2150 are male. There are a total of 660 managerial staff, 540 male inspection staff and 785 female production staff.

(i) Insert these values in the appropriate places in the following table.

Managerial Inspection Production TOTAL

Male

Female

TOTAL[1]

Two thirds of the managerial staff are female.

(ii) Use this further information to complete the table.

[5]


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4 There are 50 girls in their final year at a school. The diagram below illustrates the number of the girls who play each of the sports badminton, volleyball and handball.

115 6

9

3

7

Badminton Volleyball

Handballx

8

(i) Calculate the value of x, and state what it represents.

x = ......................................................

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

...................................................................................................................................... [2]

(ii) Find

(a) how many more girls play volleyball than play handball,

................................................... [1]

(b) how many more girls play exactly two sports than play exactly one sport.

................................................... [1]

Half of the girls who play volleyball and two thirds of the girls who play only handball say they intend to continue playing sport after they have left school.

(iii) Find the number of girls who intend to continue playing sport after they have left school.

................................................... [2]

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5 In answering this question you are not required to draw a histogram.

The times taken, in minutes, by 174 people to complete an aptitude test are summarised in the following table.

Time (minutes) Number of peopleHeight of rectangle

(units)10 under 30 28

30 under 40 36 18

40 under 45 40

45 under 50 32

50 under 75 20

75 under 120 18

TOTAL 174

The times are to be illustrated by a histogram, in which the 30 under 40 class is represented by a rectangle of height 18 units.

(i) Calculate the height of the rectangle representing the 40 under 45 class, and insert the value in the table.

[1]

(ii) Calculate the heights of the rectangles representing the remaining four classes, and insert the values in the table.

[3]

(iii) If the final two classes were combined into a single 50 under 120 class, calculate, to 2 decimal places, the height of the rectangle which would represent the combined class.

................................................... [2]


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6 (a) (i) Describe the situation which can lead to the method of systematic sampling producing a biased sample.

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

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

.............................................................................................................................. [1]

(ii) There are 380 students at a college. It is proposed to take a systematic sample of 20 of the students. Explain briefly how this could be achieved.

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

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

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

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

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

.............................................................................................................................. [3]

(b) Briefly explain how a population could be stratified, prior to taking a stratified sample, in order to ascertain the views of members of the public on

(i) a proposed increase in the tax on tobacco products,

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

.............................................................................................................................. [1]

(ii) aircraft noise.

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

.............................................................................................................................. [1]

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7 (a) A test for a particular disease has a 95% chance of correctly giving a positive result for a person who has the disease, but a 10% chance of incorrectly giving a positive result for a person who does not have the disease.

(i) Find the chance that the test gives a negative result for a person who has the disease, and insert it in the following table.

Person has the disease

Person does not have the disease

P(test result positive) 0.95P(test result negative)

[1]

(ii) Complete the table.

[1]

15% of the people who are tested are believed to have the disease.

A person is chosen at random and tested.

(iii) Calculate the probability that the test gives a correct result for this person.

................................................... [4]


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(b) Give all probabilities in this part of the question as fractions.

The following diagram classifies the members of a tennis club as to whether they are male or female, left-handed or right-handed, and whether or not they have represented the club in matches.

Left-handed Right-handed

Male 5

0

1

3

5

4

7

8Female

represented clubRepresented club

A member of the club is chosen at random.

(i) Calculate the probability that this member has represented the club in matches.

................................................... [1]

A female member is chosen at random.

(ii) Calculate the probability that she is right-handed.

................................................... [2]

A member who has represented the club in matches is chosen at random.

(iii) Calculate the probability that this member is left-handed.

................................................... [2]

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(c) Laura walks to school. On her route she passes two shops, A and B. The probability that she will go into shop A on any morning is 0.2, and into shop B is 0.7.Her decision of whether to go into one of the shops is independent of whether she goes into the other shop. If she goes into either or both shops the probability that she will be late for school is 0.09.

(i) Calculate the probability that on any morning she will go into exactly one shop and be late for school.

................................................... [3]

Laura has been told that she must aim to be late on no more than 5% of the schooldays on which she goes into exactly one shop.

(ii) State, with a reason, whether she is likely to achieve this target.

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

.............................................................................................................................. [2]

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8 (a) The table below summarises information about the number of GCE O Level subjects passed by different numbers of pupils at a school in the year 2011.

Number of subjects (x) 0 1 2 3 4 5 6Number of pupils (frequency) 1 3 6 12 15 10 7

Cumulative frequency 1 4 10 22 37 47 54

An appropriate cumulative frequency graph is to be drawn to represent these data.

(i) On the grid below, draw and label two axes, the horizontal axis representing the number of subjects passed and the vertical axis representing cumulative frequency.

[2]

(ii) Draw an appropriate cumulative frequency graph to represent these data.

[4]

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(b) The cumulative frequency graph below illustrates the lengths of journey times, in minutes, to their homes of a number of students at a college at the end of one particular day.

0

40

0 10 20 30Journey time (minutes)

40 50 60

80

120

Cumulativefrequency

160

200

Use the graph to estimate

(i) the median journey time,

.................................... minutes [1]

(ii) the lower quartile time,

..................................... minutes [1]

(iii) the 90th percentile time,

..................................... minutes [1]

(iv) the number of students whose journey time was longer than 23 minutes,

................................................... [3]

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(v) the percentile corresponding to a journey time of 17 minutes.

................................................... [2]

On the next day, due to bad weather, the journey time of all students was 5 minutes longer than the original times illustrated in the graph.

Compared with the original times, state, without further calculation, the effect which the bad weather had on

(vi) the upper quartile journey time,

.............................................................................................................................. [1]

(vii) the interquartile range of journey times.

.............................................................................................................................. [1]

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9 The following table gives information about the populations and deaths in two towns, A and B, during the course of one year, together with the standard population of the area in which both towns are situated.

AgeTown A Town B

Standard populationPopulation Deaths Death rate(per thousand) Population Deaths

0 under 15 5000 45 p = 6000 66 400

15 under 45 3750 15 4 27000 54 300

45 under 65 2500 25 10 15000 60 200

65 and over 1250 q = 32 2000 30 100

(i) For town A, calculate the values of p and of q and insert them in the table.

[2]

(ii) Calculate the crude death rate of town B.

................................................... [4]

(iii) Calculate the standardised death rate of town B.

................................................... [4]

(iv) Use the population figures given in the table to state why the crude death rate and the standardised death rate of town A are equal.

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

...................................................................................................................................... [2]

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The table shows that far more deaths occurred in town B than in town A during the year, and yet the standardised death rate for town B is much lower than that for town A.

(v) Give two reasons why this situation has occurred.

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

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

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

.............................................................................................................................. [2]

It was subsequently discovered that a small number of inhabitants of town B, none of whom had died during the year, had been misclassified by being included incorrectly in the 45 under 65 class, when in fact they were all 65 and over.

(vi) State, with a reason, the effect, if any, which correcting this error would have on the crude death rate of town B.

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

.............................................................................................................................. [2]

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10 The time, in minutes, taken by each of 6 children to walk 1 kilometre, is given in the following table.

Child A B C D E F

Age in years (x) 13 8 7 15 12 9Time in minutes (y) 12 23 25 11 18 23

(i) Plot these data on the grid below.

00 2 4 6Age (years)

8 10 12 14 16x

10

Time(minutes)

20

y30

[2] (ii) Calculate the overall mean and the two semi-averages of the data, and plot them on

your graph.

[5]

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(iii) Use your plotted averages to draw a line of best fit. [1]

(iv) Using any valid method, obtain the equation of your line of best fit, and write it in the form y = mx + c.

................................................... [3]

(v) Use your equation to estimate, to the nearest minute, the time taken to walk 1 kilometre by a child aged 14 years.

................................................... [1]

(vi) (a) Comment on how well your line of best fit matches the data points.

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

.............................................................................................................................. [1]

(b) From the graph, identify the child for whom your line of best fit most overestimates the time taken.

................................................... [1]

(vii) State, with a reason, whether it would be valid to use your line of best fit to estimate the time taken to walk 1 kilometre by a person whose age is outside the range of values given in the table.

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

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

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

.............................................................................................................................. [2]

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11 The following table summarises the increase, in dollars, of the annual income of a sample of 200 people between the years 2006 and 2011 (a negative value indicates a decrease).

Increase in annual income ($x)

Class mid-point (m) y =

m 750250

Frequency(f ) fy fy

2

2500 under 0 14

0 under 1500 99

1500 under 2500 39

2500 under 5000 25

5000 under 10000 23

TOTAL 200

(i) Obtain the mid-point, m, for each of the five classes and insert the values in the table.

[1]

(ii) For each class, obtain the value of the scaled variable, y, where

y = m 750250

,

and insert the values of y in the table.

[2]

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(iii) Obtain the values of fy and fy 2 and use them to estimate the values of the mean of y and the variance of y.

Mean = ......................................................

Variance = .................................................. [7]

(iv) Use your results from part (iii) to estimate

(a) the mean of x,

................................................... [2]

(b) the variance of x.

................................................... [3]

(v) State the units of the variance of x.

................................................... [1]

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1 Events A, B, C and D are four of the possible outcomes of an experiment such that

P(A) = 0.15 , P(B) = 0.2 , P(C) = 0.4 and P(D) = 0.24 .

(i) If events A and B are independent, find

(a) P(A B),

................................................... [2]

(b) P(A B).

................................................... [2]

(ii) If events C and D are mutually exclusive, find

(a) P(C D),

................................................... [1]

(b) P(C D).

................................................... [1]


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2 (i) The annual salaries of the employees at a company have a mean of $m and a standard deviation of $s, where s 0.

A new employee arrives at the company and is paid an annual salary of $m. The mean and standard deviation of the salaries of the employees are now recalculated

to include the salary of the new employee.

For each of the mean and the standard deviation, state whether it will increase, decrease, or stay the same when this new employees salary is included.

Mean .......................................................

Standard deviation ................................................... [2]

(ii) At another company, at the end of 2011, the employees annual salaries had a mean of $12 000 and a standard deviation of $1000.

During 2012, each of the employees salaries increased by 5%. At the end of that year they each also received an annual bonus of $200.

Calculate the mean and standard deviation of the annual incomes (salaries plus bonuses) of the employees at the end of 2012.

Mean $ .......................................................

Standard deviation $ ................................................... [4]

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3 Ariana and Bella are playing a game.They each have 4 cards, which are numbered 1, 2, 3 and 4.Each shuffles her own cards and turns one over at random.

(i) If the cards show the same number, Ariana wins and Bella must pay Ariana $3. If the cards show different numbers, Bella wins and Ariana must pay Bella $1.

By finding the probabilities of Ariana and Bella winning, show whether or not the game is fair.

[3]

(ii) In a second game the numbers shown on the cards are added together. If the total is 4 or less, Ariana wins and Bella must pay Ariana $5. If the total is 5 or more, Bella wins.

If the game is to be fair, how much should Ariana pay Bella if Bella wins?

$ ................................................... [3]


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4 The pupils in a class should arrive for registration at 9.00 am. On one particular day, 25 pupils were early, with a mean arrival time of 8.51 am. On the same day, 9 pupils were late with a mean arrival time of 9.21 am, and 2 pupils arrived at 9.00 am exactly.

If x represents the number of minutes a pupil was late (a pupil who was early would have a negative value of x),

(i) find x, and hence find the mean arrival time for all 36 pupils.

x = .......................................................Mean = ................................................... [3]

If x 2 = 5096 for the 36 pupils, (ii) find the standard deviation of x, correct to one decimal place.

................................................... [3]

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5 The change in a countrys annual production (in millions of tonnes) of 4 commodities between 2011 and 2012 is shown in the change chart below.

3 2.5 2 1.5 1 0.5 0

Change in annual production between 2011 and 2012 (in millions of tonnes)

0.5 1 1.5 2 2.5

3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5

Maize

Wheat

Cotton

Rice

The quantity produced (in millions of tonnes) of the 4 commodities in 2011 in this country is shown in the table below.

Commodity Quantity produced in 2011(millions of tonnes)Quantity produced in 2012

(millions of tonnes)Wheat 78.6

Rice 99.2

Cotton 22.6

Maize 17.3

(i) Use these data and the change chart to find the quantities of the commodities produced in 2012 and complete the table. [2]


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(ii) On the grid below, draw a dual bar chart to show the quantities produced in 2011 and 2012 of each of the 4 commodities.

[3]

(iii) State one advantage of a dual bar chart over a change chart.

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

...................................................................................................................................... [1]

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6 (a) For each of the following state whether the variable is discrete or continuous and whether it is qualitative or quantitative.

Discrete or Continuous Qualitative or Quantitative (i) the heights of the players

in a football competition [1]

(ii) the towns of birth of the players in a football competition [1]

(b) A football team used the diagram below to illustrate the number of goals it had scored per match in a season in both the league and cup competitions.

20181614

1210Number ofmatches

864

0 1 2Number of goals

matches played in the cupmatches played in the league

3 4

2

0

(i) State the full name given to this type of diagram.

................................................... [1]

(ii) Explain why the above diagram is more appropriate than a histogram to illustrate these data.

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

.............................................................................................................................. [1]

(iii) Find the proportion of matches played in the cup in which the team scored 2 or more goals.

................................................... [2]


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7 (a) The total number of visitors at a tourist attraction has been recorded for every quarter over a three-year period.

(i) Explain why it might be appropriate to calculate moving average values when establishing the trend in the number of visitors.

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

.............................................................................................................................. [1]

(ii) If an n-point moving average is to be calculated, state an appropriate value for n.

................................................... [1]

(iii) State, with a reason, whether centring would be necessary in this case.

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

.............................................................................................................................. [2]

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(b) A hospital records the number of patients admitted at two-monthly intervals over a period of two years and the results are shown in the table below, together with the 6-point moving average values for these data.

Number of patients

6-point total

6-point moving average value

Centred movingaverage value

2010

Jan Feb 241

Mar Apr 208

May Jun x =

1272 212

Jul Aug 185

1290 215

Sep Oct 209

1290 215

Nov Dec 261

1296 216

2011

Jan Feb 259

y = z =

Mar Apr 208

1323 220.5

May Jun 174

1332 222

Jul Aug 197

Sep Oct 224

Nov Dec 270

(i) Calculate the values of x, y and z and insert them in the table.

[3]

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(ii) Calculate the centred moving average values and insert them in the appropriate places in the table.

[3]

(iii) Plot the centred moving average values on the grid below and draw a trend line through the points.

215

JanFe

b

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

D

ec

JanFe

b

Mar

Apr

May

Jun

JanFe

b

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

D

ec2010 2011 2012

225

235

Num

ber o

f pat

ient

s

210

220

230

[3]

(iv) Explain what the trend line you have drawn tells you.

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

.............................................................................................................................. [1]

The seasonal component for Mar Apr is 11.25 .

(v) Estimate the number of patients admitted to the hospital during the period Mar Apr 2012.

................................................... [2]

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8 The students at a college take one of three programmes of study: Physics, Chemistry and Mathematics (PCM) or Physics, Chemistry and Biology (PCB) or Economics, Geography and Mathematics (EGM). The numbers of students who study each programme are shown in the table below.

PCM PCB EGM TOTAL

Male 60 40 40 140

Female 40 90 30 160

TOTAL 100 130 70 300

(i) Find the probability that a student chosen at random

(a) is a male studying PCM,

................................................... [1]

(b) is female,

................................................... [1]

(c) is studying Physics as part of their programme,

................................................... [1]

(d) is studying PCB, given that they are male.

................................................... [1]

(ii) If two different students are chosen at random, find the probability that they are taking the same programme of study.

................................................... [3]

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(iii) If three different students are chosen at random, find the probability that they are each taking a different programme of study.

................................................... [3]

Students are required to buy textbooks for each subject that they study: one textbook for each of Physics, Chemistry and Biology and two textbooks for each of Mathematics, Economics and Geography.

(iv) Find how many textbooks a student taking each programme of study must buy, and complete the table below.

Course PCM PCB EGM

Number oftextbooks [1]

(v) If one of the textbooks owned by a student at the college is lost at random, find the probability that it

(a) belongs to a student on the PCM programme,

................................................... [3]

(b) is a Mathematics textbook.

................................................... [2]

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9 (a) The values of a variable are formed into a grouped frequency distribution, with one of the classes stated as 50 60 . State the true class limits of this class if the variable is

Lower class limit Upper class limit

(i) the ages of the residents in a block of flats, [1]

(ii) the lengths of some rods, measured in mm, to the nearest mm, [1]

(iii) the lengths of some rods, measured in mm, to the nearest 10 mm. [1]

(b) A fisherman recorded, in grams (g), to the nearest 100 grams, the masses of 100 fish he had caught in river A.

Mass of fish (grams) Number of fish Cumulative frequency100 200 12

300 400 31

500 700 29

800 1000 14

1100 1400 8

1500 2000 4

2100 3000 2

(i) State, with a reason, which of the mean or the median would be the more appropriate measure of central tendency to use in this case.

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

.............................................................................................................................. [2]

(ii) Find the cumulative frequencies and complete the table above. [1]

(iii) Without drawing a graph, calculate an estimate of the interquartile range of the masses of the fish.

................................................... [6]

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(iv) The fisherman also recorded the masses of 100 fish caught in river B and found the interquartile range of the masses of these fish to be 352 g. Explain what this tells you about the masses of the fish caught in river B compared to those caught in river A.

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

.............................................................................................................................. [1]

(v) Without drawing a graph, calculate an estimate of the percentage of fish in river A with a mass of less than 650 g.

................................................... [3]

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10 A hairdresser classifies the expenditure on her business into three categories: Rent, Equipment and Wages.

The cost of Rent has increased from $240 per month in 2010 to $256 per month in 2012.The price relative of Equipment in 2012 is 110, taking 2010 as base year.The hourly rate of the Wages of her employees has decreased by 2% between 2010 and 2012.

(i) (a) Calculate the price relative, to the nearest whole number, of Rent for 2012, taking 2010 as base year.

................................................... [2]

(b) Explain what the price relative of 110 for Equipment indicates.

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

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

.............................................................................................................................. [3]

(c) State the price relative of Wages for 2012, taking 2010 as base year.

................................................... [1]

(d) Present the price relatives for 2010 and 2012 for each of Rent, Equipment and Wages in a suitable table.

[2]

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The hairdresser wishes to calculate a weighted aggregate cost index, using weights calculated in 2010, for the three categories.

(ii) (a) Briefly describe how these weights could be calculated.

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

.............................................................................................................................. [1]

The weights in 2010 for Rent, Equipment and Wages were calculated as 7, 2 and 5 respectively.

(b) Calculate, to the nearest integer, a weighted aggregate cost index for 2012, taking 2010 as base year.

................................................... [3]

(c) Her total expenditure on the hairdressing business in 2010 came to $5760. Use your answer to part (b) to estimate, to the nearest dollar, her total expenditure on the business in 2012.

................................................... [2]

(d) Give two possible reasons why this estimate might be very inaccurate.

Reason 1 ...................................................................................................................

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

Reason 2 ...................................................................................................................

.............................................................................................................................. [2]

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11 A small village has a population of 60 people aged 10 and over.A group of researchers wish to find out what the people of the village think about proposed changes to the timetable for the buses that pass through the village. Each researcher has a list of the population and thinks of a different way to select a sample.

(i) The first researcher plans to stand at the village bus stop at 7 am on a Monday morning and ask the first six people from the population who come to wait for a bus. Explain why this might not produce a reliable sample.

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

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

...................................................................................................................................... [2]

(ii) A second researcher decides to take a simple random sample of size six from the population of 60 people.

(a) Explain what the researcher would need to do with the population list before being able to select the sample from a random number table.

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

.............................................................................................................................. [2]

(b) Use the random number table below, starting at the beginning of the first row and working along the row, to select a simple random sample of size six from the population of 60 people, ensuring that no one is selected more than once.

RANDOM NUMBER TABLE

15 08 73 00 60 15 31 52 86 47 82 99 04 33

23 05 65 27 46 13 81 50 49 34 29 08 94 72

.............................................................................................................................. [2]

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(iii) A third researcher decides to take a systematic sample of size six from the population.

(a) Explain clearly how they should use a random number table to select the first value for such a sample.

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

.............................................................................................................................. [1]

(b) Use the random number table below, starting at the beginning of the first row and working along the row, to select a systematic sample of size six.

RANDOM NUMBER TABLE

36 04 85 06 63 22 16 64 12 51 25 92 74 43

35 75 21 44 56 20 83 59 98 35 27 08 14 69

.............................................................................................................................. [3]

[Question 11 continues on the next page]

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The table below shows the population, split into three different age groups.

10 18years

19 65years

66 yearsand over TOTAL

Numberof people 20 30 10 60

(iv) A fourth researcher decides to take a random sample of size six, stratified by age group.

(a) State how many people from each age group would be needed for such a sample.

10 18 years .......................................................

19 65 years .......................................................

66 years and over ................................................... [1]

(b) Explain clearly what the researcher would need to do before selecting the random sample, stratified by age group, from a random number table.

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

.............................................................................................................................. [2]

(c) Use the random number table below, starting at the beginning of the first row and working along the row, to select a random sample of size six, stratified by age group, ensuring that no one is selected more than once. Use every number if the age group to which it relates has not yet been fully sampled.

RANDOM NUMBER TABLE

17 55 82 25 07 16 35 42 89 37 91 98 24 38

77 29 38 02 47 19 80 53 16 40 28 07 94 73

.............................................................................................................................. [2]

(d) Explain why a random sample, stratified by age group, might be a good idea in this situation.

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

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

.............................................................................................................................. [1]

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1 State, for each of the following variables, whether it is discrete or continuous:

(i) the number of items of mail delivered each day to a particular address;

................................................... [1]

(ii) the distances run by a number of athletes during 1 hour.

................................................... [1]

The variables described above are each grouped into classes labelled 0 4, 5 9,10 14 etc.

State the true lower and upper class limits for the 5 9 class for

(iii) the variable described in (i),

...................................................................................................................................... [2]

(iv) the variable described in (ii), after the distances have been rounded to the nearest integer.

...................................................................................................................................... [2]


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2 Give a brief explanation of the meaning of each of the following terms when used in the calculation of index numbers:

(i) base year;

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

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

...................................................................................................................................... [2]

(ii) weight;

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

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

...................................................................................................................................... [2]

(iii) price relative.

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

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

...................................................................................................................................... [2]

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3 The body lengths (including the tail) of a sample of 45 white-footed Texas mice were measured in millimetres. 25 of the mice were found to be male and 20 female. The following table summarises the data obtained on mouse length.

Number of mice Sum of lengths Sum of squares of lengths

Male 25 4325 748 369

Female 20 3060 468 252

(i) Explain why the mean length of the total sample of 45 mice is not just given by (mean length of male mice + mean length of female mice) / 2.

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

...................................................................................................................................... [1]

(ii) Calculate, to 1 decimal place, the mean and the standard deviation of the lengths of the total sample of 45 mice.

Mean = ..................................................

Standard deviation = .................................................. [5]


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4 Values of experimental readings taken by different people are to be scaled for purposes of comparison. The readings have a mean of 37 and a standard deviation of 5. The scaled values are to have a mean of 100 and a standard deviation of 10.

Calculate

(i) the scaled value corresponding to a reading of 55,

................................................... [2]

(ii) the reading corresponding to a scaled value of 87.5,

................................................... [2]

(iii) the reading which is unaltered when scaled.

................................................... [2]

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5

School A School B

The bar chart above is intended to illustrate information about how many boys and girls attend each of two schools, A and B.

(i) The bar chart is incomplete. List three items of detail which are missing.

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

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

................................................... [2]

(ii) State the name of this type of bar chart.

................................................... [1]

(iii) Explain how you know that the bar chart illustrates the actual number of boys and girls, and not percentages.

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

...................................................................................................................................... [1]

(iv) Another type of diagram which could be used to illustrate the data is a pictogram. State a disadvantage of pictograms, compared with bar charts, when illustrating frequencies such as the number of pupils at a school.

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

...................................................................................................................................... [1]

(v) Give a reason why a change chart could not be used to illustrate these data.

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

...................................................................................................................................... [1]


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6 A farmer classifies the expenditure in running his farm under four headings: Animal Feed, Labour, Fuel and Professional Services (e.g. veterinary services). The price relatives for each of these headings for the year 2011, taking 2006 as base year, and the weight allocated by the farmer to each heading are given in the following table.

Price relative Weight

Animal Feed 104 14

Labour 110 6

Fuel 107 4

Professional Services 102 3

(i) Calculate, correct to 2 decimal places, the overall percentage increase in the farmers weighted cost index from 2006 to 2011.

................................................... [4]

(ii) In 2011 the farmers income was 7% greater than it had been in 2006. State, with a reason, whether or not the farm was more profitable than it had been five years earlier.

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

...................................................................................................................................... [2]

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7 This question must be answered by calculation. An answer using a graphical method will not be awarded any marks.

The following table summarises the heights, in centimetres, of a sample of 8585 adult males in the United Kingdom.

Height (cm) Frequency Cumulative frequency150 under 160 144

160 under 165 1232

165 under 170 2213

170 under 175 2559

175 under 180 1709

180 under 190 705

190 under 200 23

(i) Calculate the cumulative frequencies and insert them in the table. [2]

(ii) (a) State the class in which the median height lies.

................................................... [1]

(b) Estimate, to 1 decimal place, the median height.

............................................. cm [3]

(iii) (a) State the class in which the lower quartile height lies.

................................................... [1]

(b) Estimate, to 1 decimal place, the lower quartile height.

............................................. cm [3]


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The upper quartile height, correct to 1 decimal place, is 175.9 cm.

(iv) (a) Estimate the interquartile range of the heights.

............................................. cm [1]

(b) Compare the distances of the quartiles from the median, and comment on whether this is what you would expect in a distribution of the heights of a large number of adult males.

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

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

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

.............................................................................................................................. [3]

(v) If a cumulative frequency curve were drawn to illustrate this distribution, state, with a reason, in which part of the graph the curve would be at its steepest.

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

...................................................................................................................................... [2]

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8 In this question give all answers as fractions in their lowest terms.

Two identical bags each contain a number of coloured balls.Bag X contains 4 white and 7 blue balls. Bag Y contains 3 blue and 8 red balls.

(i) A bag is chosen at random and a ball selected at random from it. Find the probability that the selected ball is blue.

................................................... [3]

(ii) Two balls are chosen at random from bag Y. Find the probability that they are of the same colour.

................................................... [3]

(iii) One ball is chosen at random from each bag. Find the probability that the chosen balls are of the same colour.

................................................... [4]

(iv) A bag is chosen at random and two balls are selected at random from it. Find the probability that both selected balls are white.

................................................... [3]

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(v) All the balls from both bags are emptied into a third bag, bag Z. Two balls are then chosen at random from bag Z. Find the probability that both selected balls are white.

................................................... [2]

(vi) Explain briefly why the answer to part (iv) is greater than the answer to part (v).

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

...................................................................................................................................... [1]

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9 Three unbiased six-sided dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled simultaneously.

Find the probability that the numbers on the uppermost faces will be

(i) three 1s,

................................................... [1]

(ii) three of the same number except 1,

................................................... [1]

(iii) exactly two 1s and some other number.

................................................... [3]

A game, in which three such dice are rolled simultaneously and for which the entry fee is $1, is organised. Prizes are paid for certain outcomes on the uppermost faces, as given in the following table.

Outcome Prize paid ($)Three 1s 6

Three of the same number except 1 4

Exactly two 1s andsome other number 3

(iv) Calculate, to the nearest cent, the organisers expected profit each time the game is played.

................................................... [3]

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In another game, a contestant chooses three cards at random from a set of ten. The numbers on the cards are 1, 1, 1, 2, 2, 2, 3, 4, 5 and 6. Prizes are again paid as given in the previous table.

(v) By first calculating the appropriate probabilities, calculate, to the nearest cent, the entry fee which should be charged to make this a fair game.

................................................... [8]

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10 (a) A large housing estate contains approximately equal numbers of three types of dwelling: detached houses (D), semi-detached houses (S) and bungalows (B). A research organisation wishes initially to get some idea of how many occupants there tend to be in each type of dwelling. It has instructed an interviewer to call at four of each type of dwelling to ask how many people live there but the choice of exactly which dwellings is up to the interviewer.

(i) State the name of the method of sampling being used.

................................................... [1]

(ii) Give a reason why the research organisation could not just simply use a list of registered voters for the estate.

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

.............................................................................................................................. [1]

The interviewer labelled his chosen dwellings 1 to 12, and the following is a copy of the notes he made during a number of visits to the estate:

ad = adult(s) ch = child(ren)

1 B 2ad 2 S 3ch 2ad 3 S 2ad 4ch 4 D 7ch 2ad 5 B call again later 6 D 2ad 5ch 7 D 2ad 5ch 8 B no reply 9 S 4ch 2ad 10 S no reply 11 D 2ad 12 B 1ch 1ad 5 call again 8 still no reply 10 2ad 5 2ad 8 still no reply 8 2ad

(iii) For the twelve dwellings chosen, find the total number of

(a) adults,

................................................... [1]

(b) bungalows with no children.

................................................... [1]

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(iv) Draw up and complete a table showing the number of dwellings, classified by their type and by the number of children who live in them.

[3](v) The research organisation is to carry out a survey on behalf of a manufacturer of

childrens clothes. If it only has sufficient funding to investigate the expenditure on such clothes by the inhabitants of one type of dwelling, state, with a reason, which type it should choose.

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

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

.............................................................................................................................. [2]

[Question 10 continues on the next page]

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(b) A group of 60 people are each allocated a different two-digit random number in the range 01 to 60. The 20 men are numbered 01 to 20 and the 40 women are numbered 21 to 60.A sample of size six is to be selected by different sampling methods using the following random number table, starting at the beginning of the row for each sample. No person may be selected more than once in any one sample.

RANDOM NUMBER TABLE

21 32 07 42 98 81 21 57 81 59 31 17 36

Select

(i) a simple random sample, .............................................................................................................................. [2]

(ii) a systematic sample,

.............................................................................................................................. [3] (iii) a sample stratified by gender, using every number if the gender to which it relates

has not yet been fully sampled.

.............................................................................................................................. [2]

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11 (a) (i) A companys sales are recorded every month over a period of several years. Use this example to explain briefly the meaning of the term

(a) trend,

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

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

....................................................................................................................... [1]

(b) seasonal variation,

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

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

....................................................................................................................... [1]

(c) cyclic variation.

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

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

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

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