Cambridge International Examinations Cambridge Ordinary Level

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1 Cambridge International Examinations Cambridge Ordinary Level * * STATISTICS 4040/12 Paper 1 October/November 2015 Candidates answer on the Question Paper. Additional Materials: Pair of compasses Protractor 2 hours 15 minutes 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 an HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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. This document consists of 19 printed pages and 1 blank page. DC (ST/FD) /3 [Turn over

2 2 Section A [36 marks] Answer all of the questions 1 to 6. 1 Four types of sample which may be obtained from a population by a researcher are: simple random, stratified, quota, and systematic. State for which of these types of sample (i) the individual items are selected at regular intervals from a sampling frame, the choice of which individual items are selected is left to the researcher, (iii) the sample is selected so that the proportions of different categories in the sample correspond with those of the population. 2 The frequency distribution of a discrete variable, X, is given in the table below. x or more Frequency, f For this distribution, (i) name a measure of central tendency which cannot be found exactly, name, but do not find, a measure of central tendency which can be found exactly, (iii) name a measure of dispersion which cannot be found exactly, (iv) name, and find, a measure of dispersion which can be found exactly. Name......[3] 4040/12/O/N/15

3 3 3 Randa s friend Sonia claims that, of the two drinks tea and coffee, males generally prefer tea, whilst females generally prefer coffee. To investigate this Randa asks her friends and relatives their preferences. She records whether the person asked is male (M) or female (F), and whether they prefer tea (T) or coffee (C), or express no preference (X). Her raw data is as follows: FC MT FC MX MC FC FT MC FC FC FX MC FT FC MC FT FC MT FX MC For example, the first person asked was female and preferred coffee. (i) Summarise the data in a two-way table. [4] Explain whether or not Randa s survey has shown Sonia s claim to be correct [2] 4040/12/O/N/15 [Turn over

4 4 4 A man owns three shops in a town. For a trial period of 12 days he offers for sale in each shop a new type of chocolate bar. He records the number of these bars sold each day in each shop over this period, and calculates the measures shown in the table below for the daily sales in each shop. Shop Mean Median Standard deviation A B C (i) State in which one of the shops, A, B or C, daily sales were (a) generally highest, (b) generally most similar, (c) definitely fewer than 4 bars on six of the days. Find the total number of these chocolate bars sold in all three shops combined over this period....[3] 4040/12/O/N/15

5 5 5 In a game, a turn consists of throwing three unbiased six-sided dice, each with faces numbered 1, 2, 3, 4, 5 and 6. The score for a turn is the sum of any numbers which appear more than once. For example, if 5, 2, 5 appear, the score is 10; if 5, 2, 1 appear, the score is zero. (i) Write down four integers between 0 and 18 which it is impossible to score in one turn....[2] Find the probability of obtaining a score of 12 in one turn....[5] 4040/12/O/N/15 [Turn over

6 6 6 At a town centre car park the electronic barrier records the length of stay of all vehicles parked there. When a vehicle leaves the car park, it is not allowed to return the same day. The lengths of stay for the 120 vehicles using the car park on one particular day are summarised in the graph below Cumulative frequency (vehicles) Length of stay (hours) (i) Write down the name of this type of graph.... Use the graph to estimate, in hours, the median length of stay, (iii) the 85th percentile length of stay....[2] 4040/12/O/N/15

7 7 For the first two hours parking is free. For stays lasting from 2 hours up to 5 hours the charge is $6, from 5 hours up to 8 hours it is $9, and from 8 hours up to 10 hours it is $12. (iv) Use the graph to estimate the total amount paid in parking charges on this particular day....[4] 4040/12/O/N/15 [Turn over

8 8 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 a school science experiment, a beaker of hot water is allowed to cool, and its temperature is measured every 5 minutes. The results are shown in the following table. Time, x (minutes) Temperature, y ( C) (i) Calculate the overall mean and the two semi-averages of these data [5] Use the values obtained in part (i) to find the equation of the line of best fit to these data in the form y = mx + c....[3] (iii) Use your equation to estimate the temperature of the water after 30 minutes, giving your answer to the nearest degree....[2] 4040/12/O/N/15

9 9 (iv) On the grid below, plot the data given in the table at the start of the question Temperature ( C) Time (minutes) [2] (v) Draw on the grid the line whose equation you found in part for times between 0 and 30 minutes. [2] (vi) By inspecting the points plotted, explain briefly why it can be considered that it was inappropriate to find a line of best fit in the form y = mx + c in this case (vii) State how the actual water temperature after 30 minutes will compare with the value calculated in part (iii) /12/O/N/15 [Turn over

10 10 8 At a college, students are enrolled into one of four departments: Arts, Languages, Science, or Technology. The ages of students in these departments, for the year 2014, are shown in the table below. Students aged 25 under 30 are classed as mature students. Age (years) Department Arts Languages Science Technology 18 under under under under under TOTAL (i) Find the number of students at the college who are under 20 years of age. Of all the students, show that the percentage who are enrolled in Science is 32.5%, correct to 3 significant figures. [1] (iii) Of all the students, find the percentage who are mature students....[2] 4040/12/O/N/15

11 11 (iv) On the grid below draw a histogram to illustrate the ages of students enrolled in Languages. The rectangles representing the 18 under 19 class and the 19 under 20 class have already been drawn for you Number of students per 1 year Age (years) [4] 4040/12/O/N/15 [Turn over

12 12 At the end of the year, of all the students, 144 obtained distinctions in their examinations. The departments in which these 144 students were enrolled are represented by the following pie chart, which is drawn to scale. (v) Find the number of students in Arts who obtained distinctions....[2] (vi) Of all the students enrolled in Science, find the percentage who obtained distinctions. Half of the distinctions in Technology were earned by mature students....[3] (vii) Of all the mature students, find the percentage who obtained distinctions in Technology....[3] 4040/12/O/N/15

13 13 9 At an international sporting event, the team from a particular country includes swimmers and track athletes. (i) The diagram below shows the number of swimmers who are specialists in one or more of the styles breaststroke, freestyle and backstroke. Use this information to find the number of these swimmers who are specialists in (a) backstroke, (b) breaststroke and freestyle, (c) breaststroke and freestyle but not backstroke, (d) breaststroke or backstroke or both. One of these swimmers is chosen at random to appear on television. Find the probability of choosing a specialist in (e) exactly two of these styles, (f) freestyle, given that the swimmer is a specialist in breaststroke. 4040/12/O/N/15 [Turn over

14 14 The diagram below shows the number of track athletes who enter one or more of the events 100 metres, 200 metres and 400 metres. Use this information to find the number of these track athletes who enter (a) only the 200 metres, (b) the 100 metres, (c) the 100 metres or the 400 metres, (d) the 100 metres and the 400 metres. (iii) One of the swimmers in part (i) and one of the track athletes in part are chosen at random to undergo blood tests. Find the probability that the swimmer is a specialist in freestyle and the track athlete enters more than one of the given events....[4] 4040/12/O/N/15

15 15 (iv) Later, one of the track athletes in part, who currently enters both the 100 metres and the 200 metres, decides to enter also the 400 metres. Draw and label a new Venn diagram to represent the track athletes in part after this change has been made. [2] 4040/12/O/N/15 [Turn over

16 16 10 A postman wears a pedometer, with which he measures the daily distance he walks when delivering mail. The following table summarises the data he collected over 50 working days. Daily distance walked, x (km) Number of days, f 0 under under under under under under 20 2 (i) State the modal class. Estimate, in kilometres, the mean and standard deviation of the daily distance walked. Give your answers correct to 3 significant figures. Mean =... Standard deviation =...[7] (iii) State the units in which the variance of the daily distance walked would be measured. (iv) From the data in the table, possible values for the range, r, are given by a r b. Find a and b. a =... b =...[2] 4040/12/O/N/15

17 17 Later the postman has to deliver mail to a new apartment building. The mail boxes are inside the building, and to gain access he must enter a four-digit security code on a keypad outside the building The postman has forgotten the exact code, but he remembers, correctly, that the first digit is 4, and the other digits are odd numbers which are different from each other. He uses this knowledge, but otherwise randomly guesses. Find the probability that he enters the correct code (v) on the first attempt,...[3] (vi) on the second attempt....[2] 4040/12/O/N/15 [Turn over

18 18 11 In this question calculate all mortality rates as deaths per thousand admissions. Where values do not work out exactly give your answers to two decimal places. At Northshore hospital, the medical condition of patients admitted is recorded as one of non-urgent, stable, serious, or extremely serious. The table below gives information on the number of admissions and mortality (number of deaths) at the hospital for the year 2014, together with the standard population of admissions for hospitals in the area. Medical condition Mortality Admissions Medical condition mortality rate Standard population of admissions (%) Non-urgent Stable Serious Extremely serious (i) Calculate the crude mortality rate for Northshore hospital....[4] Calculate the mortality rate for each medical condition and insert the values in the table above. [2] (iii) Calculate the standardised mortality rate for Northshore hospital....[4] 4040/12/O/N/15

19 19 The table below gives mortality rate information about Southshore hospital, which is situated in the same area as Northshore hospital, also for the year Medical condition Medical condition mortality rate (deaths per thousand admissions) Admissions Non-urgent Stable Serious Extremely serious (iv) Calculate the standardised mortality rate for Southshore hospital in the year 2014, using the same standard population as for Northshore hospital....[2] (v) Find how many fewer deaths there were at Southshore hospital than at Northshore hospital in [2] The local government of the area where Northshore and Southshore hospitals are situated has sufficient funds available to improve medical care in one of the hospitals only. (vi) State, with a reason, to which of these two hospitals the funds should be allocated [2] 4040/12/O/N/15

20 20 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. 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. 4040/12/O/N/15

21 Cambridge International Examinations Cambridge Ordinary Level * * STATISTICS 4040/13 Paper 1 October/November 2015 Candidates answer on the Question Paper. Additional Materials: Pair of compasses Protractor 2 hours 15 minutes 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 an HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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. This document consists of 17 printed pages and 3 blank pages. DC (ST/FD) /3 [Turn over

22 2 Section A [36 marks] Answer all of the questions 1 to 6. 1 A teacher gives her pupils a test consisting of 5 questions, in which each question is worth 1 mark. The test scores of the pupils are shown in the following chart. 8 Number of pupils Test score (i) State the number of pupils whose test score is 1 mark. Find the number of pupils taking the test....[2] (iii) State the modal test score. (iv) Find the median test score....[2] 4040/13/O/N/15

23 3 2 A bird protection society is concerned by the declining numbers of certain birds, particularly blackbirds and sparrows. To monitor the situation, members of the public were asked to count the number of birds in their gardens during a 1-hour period last Sunday. Andy took part in the survey and altogether he counted 212 birds. His results are shown in the pie chart below, which has a radius of 3 cm. Blackbird Other Sparrow (i) Measure the angle for blackbirds. Give your answer correct to the nearest degree. Calculate the number of blackbirds that Andy saw. Katrina also took part in the survey and counted 137 birds in total. She wants to show this information in a comparative pie chart....[2] (iii) Calculate the radius of Katrina s pie chart, correct to 2 decimal places....[2] Katrina saw the same number of sparrows as Andy. (iv) Explain why the angle for sparrows on Katrina s pie chart will be larger than the angle for sparrows on Andy s pie chart. You are not required to find either of these angles /13/O/N/15 [Turn over

24 4 3 Rishi and Rakhi are conducting a survey about the meals at the school canteen. They decide to interview pupils aged 12, 13 and 14. The table below shows the number of pupils of each age. Age Number of pupils TOTAL 175 Rather than asking all the pupils they decide to take a sample. (i) Explain how they should select a systematic sample of size [3] Alternatively, they could take a stratified sample of size 25. If they choose this method, calculate the number of pupils aged 14 that would be interviewed....[2] After discussion they each conduct their own survey independently. They both take a systematic sample of size 25. When they compare their results, Rishi notices that he has interviewed one more student aged 12 than Rakhi has. (iii) Explain why this is possible /13/O/N/15

25 5 4 Students at a local college have the option of taking a 1-year vocational Computing course. At the end of the course the students achieve either a PASS or a FAIL. Information about the numbers taking the course is shown in the table below. Year Number of students taking the Computing course Number of PASSES Boys Girls Boys Girls Total number of students (i) State the number of girls who took the Computing course in Of the boys who took the Computing course in 2010, calculate the percentage who achieved a PASS....[2] (iii) Of all the students at the college in 2013, calculate the percentage who took the Computing course and achieved a PASS....[2] (iv) Identify one trend in the number of students taking the Computing course from 2010 to /13/O/N/15 [Turn over

26 6 5 A bag contains 10 counters, of which 3 are red. Basil selects counters from the bag one at a time, at random, without replacement. He stops if he selects a red counter or if he has selected a total of 4 counters. Let X be the number of counters selected. (i) Find P(X = 1). Show that P(X = 2) = 7/30. [2] (iii) Complete the following table: x Probability 7/30 [3] 4040/13/O/N/15

27 7 6 A child s toy consists of a box of coloured pieces. Each piece is green or blue, a square or a triangle, and made of wood or plastic. (i) Write down a variable of the pieces which is (a) qualitative, (b) quantitative and discrete, (c) quantitative and continuous. The diagram below shows the number of pieces which have one or more of the properties square, made of wood, and green. Square Made of wood Green Find the number of pieces which are (a) green and made of wood, (b) squares which are made of plastic, (c) green plastic triangles. 4040/13/O/N/15 [Turn over

28 8 Section B [64 marks] Answer not more than four of the questions 7 to 11. Each question in this section carries 16 marks. 7 (a) Gary drives to work every day. On his journey he has to pass through two sets of traffic lights. The probability that he has to stop at the first set of traffic lights is The probability that he has to stop at the second set is These events are independent. Find the probability that on any particular journey to work he has to stop at at least one set of traffic lights....[3] (b) Alex and Beatrice play a series of games. In each game they throw, alternately, a dart at a dart board. The first player to hit the bull s-eye (centre) wins the game. They play a number of games until one of them wins the series by winning three games. In any game, if Alex throws first the probability that he wins the game is If Beatrice throws first the probability that she wins the game is To decide who starts the first game in the series an unbiased coin is tossed. Subsequent games in the series are started by the winner of the previous game. Find the probability that (i) Beatrice wins the toss and wins the first game,...[2] Alex wins the toss and Beatrice wins the first game,...[2] 4040/13/O/N/15

29 9 (iii) Alex wins the first game,...[2] (iv) Beatrice wins the series by three games to zero....[2] They start a new series. Beatrice wins the first game and Alex wins the second game. (v) Find the probability that Beatrice wins the series....[5] 4040/13/O/N/15 [Turn over

30 10 8 Konrad is conducting an experiment. He attaches a spring to a stand, then attaches a mass to the bottom of the spring, and then measures the length of the spring. He repeats this experiment for eight different masses. His results are shown in the table below. Mass, x (g) Length of spring, y (cm) (i) Plot these data on the grid below Length of spring (cm) Mass (g) [2] 4040/13/O/N/15

31 11 The overall mean is (47.5, 38.7). Calculate the lower semi-average and the upper semi-average [3] (iii) Plot these three averages on your graph and hence draw the line of best fit. [2] (iv) Calculate the equation of the line of best fit in the form y = mx + c....[4] (v) State what the value of c represents.... (vi) Estimate the length of the spring for a mass of (a) 42 g, (b) 75 g. (vii) Which of your estimates in part (vi) is likely to be more reliable? Give a reason for your answer [2] 4040/13/O/N/15 [Turn over

32 people applied to be contestants in a quiz show. As part of the selection process they were given 60 seconds to solve a set of simple puzzles. The times taken by those who completed the puzzles are summarised in the cumulative frequency graph below Cumulative frequency (people) Completion time (seconds) (i) State the number of people who failed to complete the puzzles within the allotted time. 4040/13/O/N/15

33 13 Find, for all 140 people, (a) the median completion time, (b) the interquartile range of the completion times,...[4] (c) the completion time of the quickest person. The people were graded for speed. Those who took less than 40 seconds were graded A. Those who took 40 seconds or more, but less than 49 seconds, were graded B. Those who took 49 seconds or more were graded C. (iii) Find the number of people who were graded (a) A,...[2] (b) B....[2] (iv) Find the percentile of the quickest grade C person....[2] The people who completed the puzzles within the allotted time were also graded A, B or C for accuracy. The table shows the cumulative percentages of these people graded A, B or C. Grade for accuracy A A or B A or B or C Percentage (v) Find the maximum number of people that could have been graded B for both speed and accuracy. 4040/13/O/N/15...[3] [Turn over

34 14 10 Dirota is a keen gardener. She likes to grow tomatoes in her greenhouse. She sows a packet of seeds in a tray of compost and after five weeks she measures the heights of the plants. The results are summarised in the histogram below Number of 12 plants per 1 cm of plant height Plant height (cm) (i) Find an estimate for the mode of the heights....[2] Estimate the number of plants which are (a) more than 6.5 cm in height,...[2] (b) more than 5 cm in height....[2] 4040/13/O/N/15

35 15 Each plant is transferred to its own pot. Dirota puts the plants in order of size, starting with the shortest, and numbers the pots. The pot which contains the shortest plant is numbered 1; the pot with the second shortest plant is numbered 2 etc. (iii) Estimate the height of the plant in pot number [3] After 8 weeks she has 60 surviving plants. She measures the height, x cm, of each of the plants and finds that Σx = 443 and Σx 2 = (iv) Calculate the mean and standard deviation of x. Mean... Standard deviation...[5] Dirota sees an advertisement for a new plant food, which adds 2 cm to plant growth during the first 8 weeks after sowing. (v) Write down what the mean and standard deviation of the heights of Dirota s plants would have been after 8 weeks if she had used this new plant food. Mean... Standard deviation...[2] 4040/13/O/N/15 [Turn over

36 16 11 The table below gives information about the population and deaths in the town of Ashville for the year 2012, together with the standard population of the area in which Ashville is situated. Age group Number of deaths Population in age group Standard population (%) 0 under 25 a under b under and over c (i) The death rate for the 0 under 25 age group was 6 per thousand. Show that a = 21. The death rate for the 25 under 45 age group was 2.5 per thousand. Find the value of b. [1]...[2] (iii) Calculate the death rates per thousand for the other two age groups. 45 under 65 group and over group...[2] (iv) Calculate the crude death rate per thousand for Ashville, correct to 2 decimal places....[4] (v) Write down the value of c. 4040/13/O/N/15

37 17 (vi) Calculate the standardised death rate per thousand for Ashville....[4] Birchville is a town in the same area as Ashville. For Birchville, the crude death rate is 5.21 per thousand and the standardised death rate is 3.44 per thousand. (vii) State, with a reason, which of the two towns appears to have the healthier environment [2] 4040/13/O/N/15

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40 20 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. 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. 4040/13/O/N/15

41 Cambridge International Examinations Cambridge Ordinary Level * * STATISTICS 4040/22 Paper 2 October/November 2015 Candidates answer on the Question Paper. Additional Materials: Pair of compasses Protractor 2 hours 15 minutes 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 an HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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. This document consists of 20 printed pages. DC (ST/FD) /2 [Turn over

42 2 Section A [36 marks] Answer all of the questions 1 to 6. 1 Two of the units used to measure temperature are degrees Fahrenheit ( F) and degrees Celsius ( C). The table below shows the mean and standard deviation of the daily mid-day temperatures in the town of Weatherville for 1970 and Mean Standard deviation F 6.6 F... C... C C 3.4 C 5 The formula y = ( x - 32) converts x F to y C. 9 (i) Find the mean and standard deviation, in C, for the daily mid-day temperatures in the town in 1970, and complete the table above. [2] Explain what the figures in the table tell you about the differences between the daily mid-day temperatures in Weatherville in 1970 and [2] 4040/22/O/N/15

43 3 2 (a) A and B are two possible outcomes of the same experiment. State, in each case below, whether the two events are definitely mutually exclusive, definitely not mutually exclusive, or whether there is insufficient information to decide. P(A) = 0.3 and P(B) = 0.3 P(A) = 0.3 and P(B) = 0.7 P(A) = 0.3 and P(B) = 0.8 [2] (b) C and D are two independent events where P(C) = 0.4 and P(D) = 0.3. Find (i) P(C D),...[2] P(C D)....[2] 4040/22/O/N/15 [Turn over

44 4 3 The total petrol consumption, in thousands of barrels per day, for the years 2011 and 2012 in three countries is shown in the table below New Zealand Chile Austria (i) Show that the percentage change in petrol consumption from 2011 to 2012 for New Zealand was 2.3%, correct to two significant figures. [1] Find also the percentage change in petrol consumption from 2011 to 2012 for Chile and Austria. Chile... Austria...[2] (iii) On the grid below draw a change chart to show the percentage changes in petrol consumption for the three countries. [3] 4040/22/O/N/15

45 5 4 A class of 30 pupils took a Statistics test. For the whole class, the mean mark was 58.1 and the standard deviation was 8.1. (i) The marks are to be scaled to a mean of 50 and a standard deviation of 10. (a) Find, correct to one decimal place, the scaled mark for a pupil whose original mark was 48. (b) Find the mark which would remain unchanged by the scaling process....[2] The 23 boys in the class had an original mean mark of 56. Find the original mean mark for the 7 girls in the class....[2]...[3] 4040/22/O/N/15 [Turn over

46 6 5 The average daily sales, in dollars, for a shop were recorded for each of the quarters of three consecutive years. Using these figures, values of an appropriate moving average were calculated. These were then used to estimate quarterly components, and to draw the trend line from which the quarterly components had been removed. Quarter I II III IV Quarterly component q Average daily sales ($) I II III IV I II III IV I II III IV I II Year 1 Year 2 Year 3 Year 4 (i) Use the above information to estimate the sales figures for quarter I and quarter II of year 4. Quarter I... Quarter II...[3] Explain what the value of the quarterly component for quarter III tells you about the original data for that quarter /22/O/N/15

47 7 (iii) Explain what this trend line tells you (iv) Find the value of q. 4040/22/O/N/15 [Turn over

48 8 6 The table below gives information about the masses, measured to the nearest gram, of the 180 apples in a crop. Mass (g) Number of apples Cumulative frequency (i) The largest 25% of the apples from this crop are to be classed as extra-large. Use linear interpolation to calculate an estimate of the minimum mass of an extra-large apple, correct to one decimal place....[4] Apples under 116 g are to be classed as small. Use linear interpolation to calculate an estimate of the number of apples from this crop which will be classed as small....[3] 4040/22/O/N/15

49 9 Section B [64 marks] Answer not more than four of the questions 7 to 11. Each question in this section carries 16 marks. 7 (a) There are 600 employees at a company. A sample of the employees is to be given a questionnaire. (i) Give one advantage and one disadvantage of taking a sample rather than asking all 600 employees. Advantage Disadvantage......[2] The 600 employees are numbered 001 to 600. Use the random number table below, starting at the beginning of the table, to help you select a systematic sample of size six [3] (iii) Explain what is meant by an unbiased sampling method /22/O/N/15 [Turn over

50 10 (b) At another company there are 60 employees numbered 00 to 59. Table 1 shows the number of employees of each job type and gender. Table 2 shows the numbers they have been assigned. Table 1 Clerical Technical Executive Male Female Table 2 Clerical Technical Executive Male Female (i) Use the random number table below, starting at the beginning of the table, to select a sample of size 6, stratified by job type, showing your working clearly [5] Explain, with a reason, how well the sample you have found in part (b)(i) represents the genders [3] (iii) A sample of the employees is to be given a questionnaire about how much they enjoy their work. Explain, with a reason, whether it would be more appropriate to take a sample stratified by job type or a sample stratified by gender for this questionnaire [2] 4040/22/O/N/15

51 11 8 (a) Students at a Technical College must choose to specialise in two out of the three options Plumbing, Carpentry and Building. (i) State whether the variable the options chosen by the students is quantitative or qualitative. Give a reason for your answer [2] The chart below shows the percentage of students taking each combination in both 2012 and Plumbing and Building Percentage Carpentry and Building Plumbing and Carpentry It is known that 33 students studied both Plumbing and Carpentry in Find the total number of students studying each of the three subjects Plumbing, Carpentry and Building at the college in Plumbing... Carpentry /22/O/N/15 Building...[5] [Turn over

52 12 (iii) For each of the following statements, state whether it is definitely true, definitely false or whether there is insufficient information to decide. Give a reason for each of your answers. (a) In 2013 more students at the college were studying Plumbing than Carpentry [2] (b) More students at the college were studying the combination Plumbing and Building in 2013 than in [2] (b) The members of two cycling clubs, Pedal Powers and Speedy Wheelers, measured the distances they could cycle in one hour. (i) State whether the variable the distance travelled by each cyclist is discrete or continuous. Give a reason for your answer [2] They decided to record the distances by counting the number of whole laps of a particular track they had completed in one hour. (For example, a cyclist who had completed laps would record their distance as 24 laps.) The data were grouped with classes stated as 20 22, 23 25, etc. State the lower and upper class boundaries of the class and find the class width. Lower class boundary... Upper class boundary... Class width...[2] 4040/22/O/N/15

53 13 The frequency polygons below show the distances cycled by the members at each club. 8 Number of cyclists 6 4 Pedal Powers Speedy Wheelers (iii) Distance (laps) Compare the distances cycled by the members of each club /22/O/N/15 [Turn over

54 14 9 The results for the candidates who took their driving test on a particular day at a driving test centre are shown in the table. Male Female Pass 4 3 Fail 6 5 (i) Find the probability that a candidate chosen at random is (a) female, (b) a male who passed their test, (c) someone who passed, given that they are female. If two candidates are chosen at random, find the probability that one has passed and one has failed the test....[4] (iii) If four candidates are chosen at random, one at a time, find the probability that the 4th candidate is the first to be both female and to have passed the test....[3] 4040/22/O/N/15

55 15 (iv) If two males and two females are chosen at random, find the probability that the four candidates consist of one who has passed and three who have failed the test....[4] Of those that failed, 1/3 of the males and 2/5 of the females retook their test within 3 months. (v) Find the probability that a candidate chosen at random, from those who had failed, retook their test within 3 months....[2] 4040/22/O/N/15 [Turn over

56 16 10 A farmer classified the expenditure on his farm into four categories: Animal food, Labour, Fuel and Veterinary services. Taking 2011 as base year, the price relatives for each of these categories for the years 2012 and 2013 are shown in the table below. Price relative Animal food Labour x 103 Fuel Veterinary services The average rate of pay for the labourers on the farm increased from $7.96 per hour in 2011 to $8.52 per hour in (i) Show that x, the price relative for Labour in 2012, is 107, correct to the nearest whole number. [2] Find, correct to the nearest cent, the average rate of pay per hour for the labourers on the farm in [2] (iii) Explain what the price relative of 97 for Veterinary services tells you [2] (iv) For each category find the price relative in 2013, taking 2012 as base year, correct to the nearest whole number. Animal food... Labour... Fuel... Veterinary services...[3] 4040/22/O/N/15

57 17 Using his expenditure in 2012, the farmer assigned weights to each category as shown in the table below. Weight Animal food 12 Labour 9 Fuel 4 Veterinary services 2 (v) Calculate, correct to one decimal place, a weighted aggregate cost index for the farmer s expenses in 2013, taking 2012 as base year....[3] (vi) If the total expenditure on these items in 2012 was $ , estimate, correct to 3 significant figures, the expenditure on these items in [2] It was later discovered that, although all the price relatives used were correct, this estimate was very inaccurate. The farmer considered four possible explanations for this: A B C D The price of animal food had increased The number of employees had changed More animals had become ill The price of fuel had changed (vii) State which two of these are not possible explanations for the estimate being inaccurate and explain your answer [2] 4040/22/O/N/15 [Turn over

58 A game consists of spinning an arrow. The arrow can point to one of the sectors marked 1, 2, 3, 4 or 5 and the probability of the arrow pointing to each is shown in the table. x P(x) p (i) Use the values in the table to find the value of p. It costs $2.40 to play the game once and a player wins a prize of a number of dollars equal to the number in the sector where the arrow stops. Find the expected profit or loss for someone playing the game....[3] 4040/22/O/N/15

59 19 (iii) In another game the arrow is spun twice and the numbers obtained are added together. Again it costs $2.40 to play. This time the prize is $y if they have a total of 3 or less, otherwise they get nothing. (a) Find the probability of a total of 3 or less in this game. (b) Find the value of y if it is a fair game....[2]...[2] (c) 100 people are to play this game, and the prize awarded is $9 for a total of 3 or less. By finding the number of people expected to win, calculate the expected profit or loss for the game owner....[3] [Question 11 continues on the next page] 4040/22/O/N/15 [Turn over

60 20 Another spinner is as shown in the picture below. The arrow is equally likely to point in any direction when it stops. The arrow is spun once and $11 is charged to play (iv) If it is to be a fair game and the prizes are to be proportional to the numbers of the sectors, find how much the prizes should be. Prize for sector 1... Prize for sector 2... Prize for sector 3...[5] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series. 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. 4040/22/O/N/15

61 Cambridge International Examinations Cambridge Ordinary Level * * STATISTICS 4040/23 Paper 2 October/November 2015 Candidates answer on the Question Paper. Additional Materials: Pair of compasses Protractor 2 hours 15 minutes 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 an HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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. This document consists of 17 printed pages and 3 blank pages. DC (ST/FD) [Turn over

62 2 Section A [36 marks] Answer all of the questions 1 to 6. 1 The ages, in completed years, of the children attending two swimming clubs, Fun Swim and Aqua Splash, are shown in the table below. Age (years) Fun Swim Aqua Splash (i) State the lower and upper class boundaries of the 9 11 class. Lower class boundary... Upper class boundary...[1] On the grid below, draw frequency polygons to show the children s ages at each swimming club Number of children Age (years) [4] (iii) Use the frequency polygons to compare the children s ages at each swimming club /23/O/N/15

63 3 2 Each student in a class took a Physics test, a Chemistry test and a Biology test. The scores for the class are summarised in the table below. Physics Chemistry Biology Mean Standard deviation a The students scores on each test were then scaled to a mean of 50 and a standard deviation of 10. (i) Salma had a scaled mark of 45 in the Chemistry test. Find her actual score in the Chemistry test....[2] Peter had the same scaled mark in Physics and Chemistry. He had an actual score of 82 in the Physics test. Find his actual score in the Chemistry test....[2] (iii) In the Biology test Kumar had an actual score of 39 and a scaled mark of Find a, the standard deviation of the actual scores in the Biology test....[2] 4040/23/O/N/15 [Turn over

64 3 (a) (i) Given that P(A' ) = 0.7, P(B) = 0.6 and P(A B) = 0.7, find P(A B). 4...[3] Hence explain whether or not A and B are independent events [2] (b) Four cards are numbered 1, 2, 3 and 4. The following are possible events when one card is selected at random. C : an even number is chosen D : an odd number is chosen E : the 3 or the 4 is chosen F : the 4 is chosen From the above, list all the pairs of mutually exclusive events /23/O/N/15

65 5 4 The areas in km 2 of various types of land in two countries A and B are shown in the table below. The types of land have been categorised as being Urban, Farmland or Other (including forest, desert, lakes etc.). Area (km 2 ) Country A Country B Urban Farmland Other TOTAL (i) Display the data in a percentage sectional (component) bar chart to allow the proportions of the area of each type of land in each country to be compared. [4] Use the table and your percentage sectional bar chart to make two statements comparing the area of urban land in the two countries [2] 4040/23/O/N/15 [Turn over

66 6 5 (a) Give one advantage and one disadvantage of taking a sample rather than obtaining data from the whole population. Advantage Disadvantage......[2] (b) For each of the following statements, state whether it is always true, sometimes true, or never true. Give a reason for each of your answers. (i) A representative sample of people should contain equal numbers of males and females [2] A sampling method which produces a sample consisting of the people numbered 000 to 029 from a population listing numbered 000 to 599 is biased [2] 4040/23/O/N/15

67 7 6 A museum monitored the number of visitors it had. In a particular week, for the six days from Monday to Saturday, the mean daily attendance was 38 and the variance was 71. When the visitor numbers for the Sunday of that week were also included, the mean for all seven days became 39. Find (i) the number of visitors to the museum on the Sunday of that week,...[2] the variance of the number of visitors for all seven days of that week, correct to 1 decimal place....[4] 4040/23/O/N/15 [Turn over

68 8 Section B [64 marks] Answer not more than four of the questions 7 to 11. Each question in this section carries 16 marks. 7 A number of bags each contain 2 white beads and 3 black beads. (i) Two such bags are taken and a bead is selected at random from each bag. Find the probability of selecting one bead of each colour....[4] Three such bags are taken and a bead is selected at random from each bag. Find the probability of selecting at least two black beads....[5] 4040/23/O/N/15

69 9 (iii) One such bag is chosen and three beads are selected at random, without replacement. Find the probability of selecting both white beads....[4] (iv) Another such bag is chosen. All the beads are selected one at a time and threaded onto a string. Find the probability that the beads on the string alternate in colour....[3] 4040/23/O/N/15 [Turn over

70 10 8 A small village has 120 residents. The table below summarises their ages. Age, x years Number of people Cumulative frequency 0 x x x x x x x (i) Calculate the cumulative frequencies of the data and insert them in the final column of the table. [1] Use linear interpolation to calculate an estimate of the median age of the residents, correct to one decimal place....[4] (iii) 20% of the residents are aged p years or more. Use linear interpolation to calculate an estimate of p....[4] 4040/23/O/N/15

71 11 (iv) Use linear interpolation to calculate an estimate of the percentage of people in the village aged 18 and over but less than [5] (v) State why your answers to parts, (iii) and (iv) are only estimates and explain what assumption you are making when calculating these estimates [2] 4040/23/O/N/15 [Turn over

72 12 9 A company employs three grades of worker: skilled, semi-skilled and unskilled. The manager of the company wishes to calculate a weighted aggregate cost index for the total wage bill of the company. The incomplete table 1 below is to show the rates of pay per hour for each of these grades for three consecutive years. The incomplete table 2 below is to show the price relatives of the rates of hourly pay, for these three years, taking 2011 as base year. Table 1 Rate of pay per hour ($) Skilled Semi-skilled 7.50 Unskilled Table 2 Price relative Skilled Semi-skilled Unskilled (i) Use the information provided in the tables to find the rates of pay missing from table 1. Insert them in the appropriate places in table 1. [2] Use the rates of pay from table 1 and the fact that 2011 is the base year to find the price relatives missing from table 2, correct to the nearest whole number. Insert them in the appropriate places in table 2. [4] 4040/23/O/N/15

73 13 (iii) The table below shows the number of workers at each grade at the company in Number of workers Skilled 10 Semi-skilled 6 Unskilled 5 By considering the total cost of employing each grade of worker in 2011, show that weights of 9, 5 and 4 should be assigned to the skilled, semi-skilled and unskilled workers respectively. State what assumption you have made in this calculation [3] (iv) When the earnings of the manager for 2011 are included, the ratio of the weights becomes 9 : 5 : 4 : 2 for the workers who are skilled, semi-skilled, unskilled and the manager respectively. The price relative for the manager in 2013, taking 2011 as base, is 108. Calculate a weighted aggregate cost index for the total wage bill in 2013 with 2011 as base year....[4] (v) Explain what this figure tells you. State what additional assumption you have made in giving this answer [3] 4040/23/O/N/15 [Turn over

74 14 10 The table below shows the number of marriages (in thousands) in a particular country each quarter for a period of 3 years. Year Quarter Number of marriages (thousands) Four-quarter total Centred total Centred moving average value I II III c = IV I II III a = IV I b =... II III 94.8 IV 42.1 (i) Give a reason for finding moving average values and explain why it is useful to do this [2] Explain clearly why it is necessary to centre the moving average values in this table [2] 4040/23/O/N/15

75 15 (iii) Calculate the values of a, b and c and insert them in the table. [3] (iv) Use the number of marriages and centred moving average values for quarter II of 2012 and 2013 to find an estimate of the seasonal component for quarter II....[3] (v) Plot all the centred moving average values on the grid below and draw the trend line Number of marriages 56 (thousands) I II III IV I II III IV I II III IV I II III [3] (vi) Comment on what the trend line tells you (vii) Use your trend line and answer to part (iv) to estimate the number of marriages in quarter II of [2] 4040/23/O/N/15 [Turn over