• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy isspecified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
• Read each question carefully and make sure you know what you have to do before starting your answer.
• You are reminded of the need for clear presentation in your answers.
This document consists of 6 printed pages and 2 blank pages.
(i) Calculate the equation of the regression line of y on x. [4]
(ii) Use your equation to estimate the temperature after 12 minutes. [2]
(iii) It is given that the value of the product moment correlation coefficient is close to +1. Commenton the reliability of using your equation to estimate y when
6 A coin is biased so that the probability that it will show heads on any throw is 23. The coin is thrown
repeatedly.
The number of throws up to and including the first head is denoted by X. Find
(i) P(X = 4), [3]
(ii) P(X < 4), [3]
(iii) E(X). [2]
7 A bag contains three 1p coins and seven 2p coins. Coins are removed at random one at a time, withoutreplacement, until the total value of the coins removed is at least 3p. Then no more coins are removed.
(i) Copy and complete the probability tree diagram. [5]
Find the probability that
(ii) exactly two coins are removed, [3]
(iii) the total value of the coins removed is 4p. [3]
8 In the 2001 census, the household size (the number of people living in each household) was recorded.The percentages of households of different sizes were then calculated. The table shows the percentagesfor two wards, Withington and Old Moat, in Manchester.
Household size
1 2 3 4 5 6 7 or more
Withington 34.1 26.1 12.7 12.8 8.2 4.0 2.1
Old Moat 35.1 27.1 14.7 11.4 7.6 2.8 1.3
(i) Calculate the median and interquartile range of the household size for Withington. [3]
(ii) Making an appropriate assumption for the last class, which should be stated, calculate the meanand standard deviation of the household size for Withington. Give your answers to an appropriatedegree of accuracy. [6]
The corresponding results for Old Moat are as follows.
Median Interquartile Mean Standardrange deviation
2 2 2.4 1.5
(iii) State one advantage of using the median rather than the mean as a measure of the averagehousehold size. [1]
(iv) By comparing the values for Withington with those for Old Moat, explain briefly why theinterquartile range may be less suitable than the standard deviation as a measure of the variationin household size. [1]
(v) For one of the above wards, the value of Spearman’s rank correlation coefficient betweenhousehold size and percentage is −1. Without any calculation, state which ward this is. Explainyour answer. [2]
9 A variable X has the distribution B(11, p).(i) Given that p = 3
4, find P(X = 5). [2]
(ii) Given that P(X = 0) = 0.05, find p. [4]
(iii) Given that Var(X) = 1.76, find the two possible values of p. [5]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher willbe pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),which is itself a department of the University of Cambridge.
• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy isspecified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
• Read each question carefully and make sure you know what you have to do before starting your answer.
• You are reminded of the need for clear presentation in your answers.
This document consists of 6 printed pages and 2 blank pages.
1 The table shows the probability distribution for a random variable X.
x 0 1 2 3
P(X = x) 0.1 0.2 0.3 0.4
Calculate E(X) and Var(X). [5]
2 Two judges each placed skaters from five countries in rank order.
Position 1st 2nd 3rd 4th 5th
Judge 1 UK France Russia Poland Canada
Judge 2 Russia Canada France UK Poland
Calculate Spearman’s rank correlation coefficient, rs, for the two judges’ rankings. [5]
3 (i) How many different teams of 7 people can be chosen, without regard to order, from a squadof 15? [2]
(ii) The squad consists of 6 forwards and 9 defenders. How many different teams containing3 forwards and 4 defenders can be chosen? [2]
4 A bag contains 6 white discs and 4 blue discs. Discs are removed at random, one at a time, withoutreplacement.
(i) Find the probability that
(a) the second disc is blue, given that the first disc was blue, [1]
(b) the second disc is blue, [3]
(c) the third disc is blue, given that the first disc was blue. [3]
(ii) The random variable X is the number of discs which are removed up to and including the firstblue disc. State whether the variable X has a geometric distribution. Explain your answer briefly.
5 The numbers of births, in thousands, to mothers of different ages in England and Wales, in 1991 and2001 are illustrated by the cumulative frequency curves.
(i) In which of these two years were there more births? How many more births were there in thisyear? [2]
(ii) The following quantities were estimated from the diagram.
Median age Interquartile Proportion of mothers Proportion of mothersYear (years) range (years) giving birth aged below 25 giving birth aged 35 or above
1991 27.5 7.3 33% 9%
2001 18%
(a) Find the values missing from the table. [5]
(b) Did the women who gave birth in 2001 tend to be younger or older or about the same age asthe women who gave birth in 1991? Using the table and your values from part (a), give tworeasons for your answer. [3]
6 A machine with artificial intelligence is designed to improve its efficiency rating with practice. Thetable shows the values of the efficiency rating, y, after the machine has carried out its task variousnumbers of times, x.
These data are illustrated in the scatter diagram.
(i) (a) Calculate the value of r, the product moment correlation coefficient. [3]
(b) Without calculation, state with a reason the value of rs, Spearman’s rank correlationcoefficient. [2]
(ii) A researcher suggests that the data for x = 0 and x = 1 should be ignored. Without calculation,state with a reason what effect this would have on the value of
(a) r, [2]
(b) rs. [2]
(iii) Use the diagram to estimate the value of y when x = 29. [1]
(iv) Jack finds the equation of the regression line of y on x for all the data, and uses it to estimate thevalue of y when x = 29. Without calculation, state with a reason whether this estimate or the onefound in part (iii) will be the more reliable. [2]
7 On average, 25% of the packets of a certain kind of soup contain a voucher. Kim buys one packet ofsoup each week for 12 weeks. The number of vouchers she obtains is denoted by X.
(i) State two conditions needed for X to be modelled by the distribution B(12, 0.25). [2]
In the rest of this question you should assume that these conditions are satisfied.
(ii) Find P(X ≤ 6). [2]
In order to claim a free gift, 7 vouchers are needed.
(iii) Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.[1]
(iv) Find the probability that Kim will be able to claim a free gift in the 12th week but not before.[4]
8 (i) A biased coin is thrown twice. The probability that it shows heads both times is 0.04. Find theprobability that it shows tails both times. [3]
(ii) Another coin is biased so that the probability that it shows heads on any throw is p. The probabilitythat the coin shows heads exactly once in two throws is 0.42. Find the two possible values of p.
[5]
9 (i) A random variable X has the distribution Geo�15�. Find
(a) E(X), [2]
(b) P(X = 4), [2]
(c) P(X > 4). [2]
(ii) A random variable Y has the distribution Geo(p), and q = 1 − p.
(a) Show that P(Y is odd) = p + q2p + q4p + . . . . [1]
(b) Use the formula for the sum to infinity of a geometric progression to show that
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),which is itself a department of the University of Cambridge.
Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education
MATHEMATICS 4732Probability & Statistics 1
Tuesday 18 JANUARY 2005 Afternoon 1 hour 30 minutes
Additional materials:Answer bookletGraph paperList of Formulae (MF1)
TIME 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree ofaccuracy is specified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carryinglarger numbers of marks later in the paper.
• You are reminded of the need for clear presentation in your answers.
3 Two commentators gave ratings out of 100 for seven sports personalities. The ratings are shown inthe table below.
Personality A B C D E F G
Commentator I 73 76 78 65 86 82 91Commentator II 77 78 79 80 86 89 95
(i) Calculate Spearman’s rank correlation coefficient for these ratings. [5]
(ii) State what your answer tells you about the ratings given by the two commentators. [1]
4 The table below shows the probability distribution of the random variable X.
x −2 −1 0 1 2
P(X = x) 14
15
k 25
110
(i) Find the value of the constant k. [2]
(ii) Calculate the values of E(X) and Var(X). [5]
5 On average 1 in 20 members of the population of this country has a particular DNA feature. Membersof the population are selected at random until one is found who has this feature.
(i) Find the probability that the first person to have this feature is
(a) the sixth person selected, [3]
(b) not among the first 10 people selected. [3]
(ii) Find the expected number of people selected. [2]
6 Louise and Marie play a series of tennis matches. It is given that, in any match, the probability thatLouise wins the first two sets is 3
8.
(i) Find the probability that, in 5 randomly chosen matches, Louise wins the first two sets in exactly2 of the matches. [3]
It is also given that Louise and Marie are equally likely to win the first set.
(ii) Show that P(Louise wins the second set, given that she won the first set) = 34. [2]
(iii) The probability that Marie wins the first two sets is 13. Find
P(Marie wins the second set, given that she won the first set). [2]
7 It is known that, on average, one match box in 10 contains fewer than 42 matches. Eight boxes areselected, and the number of boxes that contain fewer than 42 matches is denoted by Y .
(i) State two conditions needed to model Y by a binomial distribution. [2]
Assume now that a binomial model is valid.
(ii) Find
(a) P(Y = 0), [2]
(b) P(Y ≥ 2). [2]
(iii) On Wednesday 8 boxes are selected, and on Thursday another 8 boxes are selected. Find theprobability that on one of these days the number of boxes containing fewer than 42 matches is 0,and that on the other day the number is 2 or more. [3]
8 An examination paper consists of 8 questions, of which one is on geometric distributions and one ison binomial distributions.
(i) If the 8 questions are arranged in a random order, find the probability that the question ongeometric distributions is next to the question on binomial distributions. [3]
Four of the questions, including the one on geometric distributions, are worth 7 marks each, and theremaining four questions, including the one on binomial distributions, are worth 9 marks each. The7-mark questions are the first four questions on the paper, but are arranged in random order. The9-mark questions are the last four questions, but are arranged in random order. Find the probabilitythat
(ii) the questions on geometric distributions and on binomial distributions are next to one another,[3]
(iii) the questions on geometric distributions and on binomial distributions are separated by at least 2other questions. [4]
9 Five observations of bivariate data produce the following results, denoted as (xi, y
(i) Show that the regression line of y on x has gradient −0.06, and find its equation in the formy = a + bx. [4]
(ii) The regression line is used to estimate the value of y corresponding to x = 20, but the value x = 20is accurate only to the nearest whole number. Calculate the difference between the largest andthe smallest values that the estimated value of y could take. [3]
The numbers e1, e2, e3, e4, e5 are defined by
ei= a + bx
i− y
ifor i = 1, 2, 3, 4, 5.
(iii) The values of e1, e2 and e3 are 0.6, −0.7 and 0.2 respectively. Calculate the values of e4 and e5.[2]
(iv) Calculate the value of e21 + e2
2 + e23 + e2
4 + e25 and explain the relevance of this quantity to the
regression line found in part (i). [2]
(v) Find the mean and the variance of e1, e2, e3, e4, e5. [4]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonablee�ort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local ExaminationsSyndicate (UCLES), which is itself a department of the University of Cambridge.
Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education
MATHEMATICS 4732Probability & Statistics 1
Thursday 9 JUNE 2005 Morning 1 hour 30 minutes
Additional materials:Answer bookletGraph paperList of Formulae (MF1)
TIME 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree ofaccuracy is specified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carryinglarger numbers of marks later in the paper.
• You are reminded of the need for clear presentation in your answers.
This question paper consists of 5 printed pages and 3 blank pages.
1 (i) Calculate the value of Spearman’s rank correlation coefficient between the two sets of rankings,A and B, shown in Table 1. [4]
A 1 2 3 4 5
B 4 1 3 2 5
Table 1
(ii) The value of Spearman’s rank correlation coefficient between the set of rankings B and a thirdset of rankings, C, is known to be −1. Copy and complete Table 2 showing the set of rankings C.
[2]
B 4 1 3 2 5
C
Table 2
2 The probability that a certain sample of radioactive material emits an alpha-particle in one unit of timeis 0.14. In one unit of time no more than one alpha-particle can be emitted. The number of units oftime up to and including the first in which an alpha-particle is emitted is denoted by T .
(i) Find the value of
(a) P(T = 5), [3]
(b) P(T < 8). [3]
(ii) State the value of E(T). [2]
3 In a supermarket the proportion of shoppers who buy washing powder is denoted by p. 16 shoppersare selected at random.
(i) Given that p = 0.35, use tables to find the probability that the number of shoppers who buywashing powder is
(a) at least 8, [3]
(b) between 4 and 9 inclusive. [2]
(ii) Given instead that p = 0.38, find the probability that the number of shoppers who buy washingpowder is exactly 6. [3]
4 The table shows the latitude, x (in degrees correct to 3 significant figures), and the average rainfall y(in cm correct to 3 significant figures) of five European cities.
(i) Calculate the product moment correlation coefficient. [3]
(ii) The values of y in the table were in fact obtained from measurements in inches and converted intocentimetres by multiplying by 2.54. State what effect it would have had on the value of the productmoment correlation coefficient if it had been calculated using inches instead of centimetres. [1]
(iii) It is required to estimate the annual rainfall at Bergen, where x = 60.4. Calculate the equationof an appropriate line of regression, giving your answer in simplified form, and use it to find therequired estimate. [5]
5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph,where the data points are joined by a smooth curve.
Use the curve to estimate
(i) the interquartile range of the marks, [3]
(ii) x, if 40% of the candidates scored more than x marks, [3]
(iii) the number of candidates who scored more than 68 marks. [2]
Five of the candidates are selected at random, with replacement.
(iv) Estimate the probability that all five scored more than 68 marks. [3]
It is subsequently discovered that the candidates’ marks in the range 35 to 55 were evenly distributed— that is, roughly equal numbers of candidates scored 35, 36, 37, …, 55.
(v) What does this information suggest about the estimate of the interquartile range found in part (i)?[2]
6 Two bags contain coloured discs. At first, bag P contains 2 red discs and 2 green discs, and bag Qcontains 3 red discs and 1 green disc. A disc is chosen at random from bag P, its colour is noted andit is placed in bag Q. A disc is then chosen at random from bag Q, its colour is noted and it is placedin bag P. A disc is then chosen at random from bag P.
The tree diagram shows the different combinations of three coloured discs chosen.
(i) Write down the values of a, b, c, d, e and f . [4]
The total number of red discs chosen, out of 3, is denoted by R. The table shows the probabilitydistribution of R.
r 0 1 2 3
P(R = r) 110
k 920
15
(ii) Show how to obtain the value P(R = 2) = 920
. [3]
(iii) Find the value of k. [2]
(iv) Calculate the mean and variance of R. [5]
7 A committee of 7 people is to be chosen at random from 18 volunteers.
(i) In how many different ways can the committee be chosen? [2]
The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. Thecommittee is to be chosen randomly. Find the probability that the committee will
(ii) consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester, [4]
(iii) include exactly 5 people from Worcester, [4]
(iv) include at least 2 people from each of the three cities. [4]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonablee�ort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local ExaminationsSyndicate (UCLES), which is itself a department of the University of Cambridge.
• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.
• Read each question carefully and make sure you know what you have to do before startingyour answer.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree ofaccuracy is specified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• You are reminded of the need for clear presentation in your answers.
This document consists of 6 printed pages and 2 blank pages.
1 (i) The letters A, B, C, D and E are arranged in a straight line.
(a) How many different arrangements are possible? [2]
(b) In how many of these arrangements are the letters A and B next to each other? [3]
(ii) From the letters A, B, C, D and E, two different letters are selected at random. Find the probabilitythat these two letters are A and B. [2]
2 A random variable T has the distribution Geo(15). Find
(i) P(T = 4), [2]
(ii) P(T > 4), [2]
(iii) E(T). [1]
3 A sample of bivariate data was taken and the results were summarised as follows.
n = 5 Σ x = 24 Σ x2 = 130 Σ y = 39 Σ y2 = 361 Σ xy = 212
(i) Show that the value of the product moment correlation coefficient r is 0.855, correct to 3 significantfigures. [2]
(ii) The ranks of the data were found. One student calculated Spearman’s rank correlation coefficientrs, and found that rs = 0.7. Another student calculated the product moment coefficient, R, ofthese ranks. State which one of the following statements is true, and explain your answer briefly.
(A) R = 0.855(B) R = 0.7(C) It is impossible to give the value of R without carrying out a calculation using the original
data.[2]
(iii) All the values of x are now multiplied by a scaling factor of 2. State the new values of r and rs.[2]
4 A supermarket has a large stock of eggs. 40% of the stock are from a firm called Eggzact. 12% of thestock are brown eggs from Eggzact.
An egg is chosen at random from the stock. Calculate the probability that
(i) this egg is brown, given that it is from Eggzact, [2]
(ii) this egg is from Eggzact and is not brown. [2]
5 (i) 20% of people in the large town of Carnley support the Residents’ Party. 12 people from Carnleyare selected at random. Out of these 12 people, the number who support the Residents’ Party isdenoted by U.
Find
(a) P(U ≤ 5), [2]
(b) P(U ≥ 3). [3]
(ii) 30% of people in Carnley support the Commerce Party. 15 people from Carnley are selected atrandom. Out of these 15 people, the number who support the Commerce Party is denoted by V .
Find P(V = 4). [3]
6 The probability distribution for a random variable Y is shown in the table.
y 1 2 3
P(Y = y) 0.2 0.3 0.5
(i) Calculate E(Y) and Var(Y). [5]
Another random variable, Z, is independent of Y . The probability distribution for Z is shown in thetable.
� 1 2 3
P(Z = �) 0.1 0.25 0.65
One value of Y and one value of Z are chosen at random. Find the probability that
(ii) Y + Z = 3, [3]
(iii) Y × Z is even. [3]
7 (i) Andrew plays 10 tennis matches. In each match he either wins or loses.
(a) State, in this context, two conditions needed for a binomial distribution to arise. [2]
(b) Assuming these conditions are satisfied, define a variable in this context which has a binomialdistribution. [1]
(ii) The random variable X has the distribution B(21, p), where 0 < p < 1.
Given that P(X = 10) = P(X = 9), find the value of p. [5]
Key: 1 4 0 represents a male aged 41 and a female aged 40.
(i) Find the median and interquartile range for the males. [3]
(ii) The median and interquartile range for the females are 27 and 15 respectively. Make twocomparisons between the ages of the males and the ages of the females. [2]
(iii) The mean age of the males is 30.7 and the mean age of the females is 27.5, each correct to1 decimal place. Give one advantage of using the median rather than the mean to compare theages of the males with the ages of the females. [1]
A record was kept of the number of hours, X, spent by each member at the club in a year. The resultswere summarised by
n = 49, Σ(x − 200) = 245, Σ(x − 200)2 = 9849.
(iv) Calculate the mean and standard deviation of X. [6]
9 It is thought that the pH value of sand (a measure of the sand’s acidity) may affect the extent to whicha particular species of plant will grow in that sand. A botanist wished to determine whether there wasany correlation between the pH value of the sand on certain sand dunes, and the amount of each of twoplant species growing there. She chose random sections of equal area on each of eight sand dunes andmeasured the pH values. She then measured the area within each section that was covered by each ofthe two species. The results were as follows.
Dune A B C D E F G H
pH value, x 8.5 8.5 9.5 8.5 6.5 7.5 8.5 9.0
Area, y cm2,covered
Species P 150 150 575 330 45 15 340 330
Species Q 170 15 80 230 75 25 0 0
The results for species P can be summarised by
n = 8, Σ x = 66.5, Σ x2 = 558.75, Σ y = 1935, Σ y2 = 711 275, Σ xy = 17 082.5.
(i) Give a reason why it might be appropriate to calculate the equation of the regression line of y onx rather than x on y in this situation. [1]
(ii) Calculate the equation of the regression line of y on x for species P, in the form y = a + bx, givingthe values of a and b correct to 3 significant figures. [4]
(iii) Estimate the value of y for species P on sand where the pH value is 7.0. [2]
The values of the product moment correlation coefficient between x and y for species P and Q arerP = 0.828 and rQ = 0.0302.
(iv) Describe the relationship between the area covered by species Q and the pH value. [1]
(v) State, with a reason, whether the regression line of y on x for species P will provide a reliableestimate of the value of y when the pH value is
(a) 8, [1]
(b) 4. [1]
(vi) Assume that the equation of the regression line of y on x for species Q is also known. State, witha reason, whether this line will provide a reliable estimate of the value of y when the pH valueis 8. [1]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),which is itself a department of the University of Cambridge.
Additional materials (required):Answer Booklet (8 pages)List of Formulae (MF1)
INSTRUCTIONS TO CANDIDATES
• Write your name in capital letters, your Centre Number and Candidate Number in the spacesprovided on the Answer Booklet.
• Read each question carefully and make sure you know what you have to do before startingyour answer.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree ofaccuracy is specified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• You are reminded of the need for clear presentation in your answers.
This document consists of 6 printed pages and 2 blank pages.
1 (i) State the value of the product moment correlation coefficient for each of the following scatterdiagrams. [2]
x x
y y
(a) (b)
(ii) Calculate the value of Spearman’s rank correlation coefficient for the following data. [5]
x 3.8 4.1 4.5 5.3
y 1.4 0.8 0.7 1.2
2 A class consists of 7 students from Ashville and 8 from Bewton. A committee of 5 students is chosenat random from the class.
(i) Find the probability that 2 students from Ashville and 3 from Bewton are chosen. [3]
(ii) In fact 2 students from Ashville and 3 from Bewton are chosen. In order to watch a video, all5 committee members sit in a row. In how many different orders can they sit if no two studentsfrom Bewton sit next to each other? [2]
3 (i) A random variable X has the distribution B(8, 0.55). Find
(a) P(X < 7), [1]
(b) P(X = 5), [2]
(c) P(3 ≤ X < 6). [3]
(ii) A random variable Y has the distribution B(10, 512
4 At a fairground stall, on each turn a player receives prize money with the following probabilities.
Prize money £0.00 £0.50 £5.00
Probability 1720
110
120
(i) Find the probability that a player who has two turns will receive a total of £5.50 in prize money.[3]
(ii) The stall-holder wishes to make a profit of 20p per turn on average. Calculate the amount thestall-holder should charge for each turn. [4]
5 (i) A bag contains 12 red discs and 10 black discs. Two discs are removed at random, withoutreplacement. Find the probability that both discs are red. [2]
(ii) Another bag contains 7 green discs and 8 blue discs. Three discs are removed at random, withoutreplacement. Find the probability that exactly two of these discs are green. [3]
(iii) A third bag contains 45 discs, each of which is either yellow or brown. Two discs are removed atrandom, without replacement. The probability that both discs are yellow is 1
15. Find the number
of yellow discs which were in the bag at first. [4]
6 (i) The numbers of males and females in Year 12 at a school are illustrated in the pie chart. Thenumber of males in Year 12 is 128.
Males
Females
120°
Year 12
(a) Find the number of females in Year 12. [1]
(b) On a corresponding pie chart for Year 13, the angle of the sector representing males is 150◦.Explain why this does not necessarily mean that the number of males in Year 13 is morethan 128. [1]
(ii) All the Year 12 students took a General Studies examination. The results are illustrated in thebox-and-whisker plots.
Mark
0 10 20 30 40 50 60 70 80 90 100
Year 12 Females
Year 12 Males
(a) One student said “The Year 12 pie chart shows that there are more females than males, butthe box-and-whisker plots show that there are more males than females.”
Comment on this statement. [1]
(b) Give two comparisons between the overall performance of the females and the males in theGeneral Studies examination. [2]
(c) Give one advantage and one disadvantage of using box-and-whisker plots rather thanhistograms to display the results. [2]
(iii) The mean mark for 102 of the male students was 51. The mean mark for the remaining 26 malestudents was 59. Calculate the mean mark for all 128 male students. [3]
7 Once each year, Paula enters a lottery for a place in an annual marathon. Each time she enters thelottery, the probability of her obtaining a place is 0.3. Find the probability that
(i) the first time she obtains a place is on her 4th attempt, [3]
(ii) she does not obtain a place on any of her first 6 attempts, [2]
(iii) she needs fewer than 10 attempts to obtain a place, [3]
(iv) she obtains a place exactly twice in her first 5 attempts. [3]
8 A city council attempted to reduce traffic congestion by introducing a congestion charge. The chargewas set at £4.00 for the first year and was then increased by £2.00 each year. For each of the firsteight years, the council recorded the average number of vehicles entering the city centre per day. Theresults are shown in the table.
Charge, £x 4 6 8 10 12 14 16 18
Average number of vehiclesper day, y million 2.4 2.5 2.2 2.3 2.0 1.8 1.7 1.5
(i) Calculate the product moment correlation coefficient for these data. [3]
(ii) Explain why x is the independent variable. [1]
(iii) Calculate the equation of the regression line of y on x. [4]
(iv) (a) Use your equation to estimate the average number of vehicles which will enter the city centreper day when the congestion charge is raised to £20.00. [2]
(b) Comment on the reliability of your estimate. [2]
(v) The council wishes to estimate the congestion charge required to reduce the average number ofvehicles entering the city per day to 1.0 million. Assuming that a reliable estimate can be madeby extrapolation, state whether they should use the regression line of y on x or the regression lineof x on y. Give a reason for your answer. [2]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES),which is itself a department of the University of Cambridge.
Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education
MATHEMATICS 4732Probability & Statistics 1
Wednesday 24 MAY 2006 Afternoon 1 hour 30 minutes
Additional materials:8 page answer bookletGraph paperList of Formulae (MF1)
TIME 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree ofaccuracy is specified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carryinglarger numbers of marks later in the paper.
• You are reminded of the need for clear presentation in your answers.
1 Some observations of bivariate data were made and the equations of the two regression lines werefound to be as follows.
y on x : y = −0.6x + 13.0x on y : x = −1.6y + 21.0
(i) State, with a reason, whether the correlation between x and y is negative or positive. [1]
(ii) Neither variable is controlled. Calculate an estimate of the value of x when y = 7.0. [2]
(iii) Find the values of x and y. [3]
2 A bag contains 5 black discs and 3 red discs. A disc is selected at random from the bag. If it is red itis replaced in the bag. If it is black, it is not replaced. A second disc is now selected at random fromthe bag.
Find the probability that
(i) the second disc is black, given that the first disc was black, [1]
(ii) the second disc is black, [3]
(iii) the two discs are of different colours. [3]
3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in arow.
(i) How many different arrangements of the letters are possible? [3]
(ii) In how many of these arrangements are all three Ds together? [2]
The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
(iii) Find the probability that at least one of these 2 cards has D printed on it. [3]
4 (i) The random variable X has the distribution B(25, 0.2). Using the tables of cumulative binomialprobabilities, or otherwise, find P(X ≥ 5). [2]
(ii) The random variable Y has the distribution B(10, 0.27). Find P(Y = 3). [2]
(iii) The random variable Z has the distribution B(n, 0.27). Find the smallest value of n such thatP(Z ≥ 1) > 0.95. [3]
5 The probability distribution of a discrete random variable, X, is given in the table.
6 The table shows the total distance travelled, in thousands of miles, and the amount of commissionearned, in thousands of pounds, by each of seven sales agents in 2005.
Agent A B C D E F G
Distance travelled 18 15 12 14 16 24 13
Commission earned 18 45 19 24 27 22 23
(i) (a) Calculate Spearman’s rank correlation coefficient, rs, for these data. [5]
(b) Comment briefly on your value of rs with reference to this context. [1]
(c) After these data were collected, agent A found that he had made a mistake. He had actuallytravelled 19 000 miles in 2005. State, with a reason, but without further calculation, whetherthe value of Spearman’s rank correlation coefficient will increase, decrease or stay the same.
[2]
The agents were asked to indicate their level of job satisfaction during 2005. A score of 0 representedno job satisfaction, and a score of 10 represented high job satisfaction. Their scores, y, together withthe data for distance travelled, x, are illustrated in the scatter diagram below.
(ii) For this scatter diagram, what can you say about the value of
(a) Spearman’s rank correlation coefficient, [1]
(b) the product moment correlation coefficient? [1]
7 In a UK government survey in 2000, smokers were asked to estimate the time between their wakingand their having the first cigarette of the day. For heavy smokers, the results were as follows.
Time between waking 1 to 4 5 to 14 15 to 29 30 to 59 At least 60and first cigarette minutes minutes minutes minutes minutes
Percentage of smokers 31 27 19 14 9
Times are given correct to the nearest minute.
(i) Assuming that ‘At least 60 minutes’ means ‘At least 60 minutes but less than 240 minutes’,calculate estimates for the mean and standard deviation of the time between waking and firstcigarette for these smokers. [6]
(ii) Find an estimate for the interquartile range of the time between waking and first cigarette forthese smokers. Give your answer correct to the nearest minute. [4]
(iii) The meaning of ‘At least 60 minutes’ is now changed to ‘At least 60 minutes but less than480 minutes’. Without further calculation, state whether this would cause an increase, a decreaseor no change in the estimated value of
(a) the mean, [1]
(b) the standard deviation, [1]
(c) the interquartile range. [1]
8 Henry makes repeated attempts to light his gas fire. He makes the modelling assumption that theprobability that the fire will light on any attempt is 1
3.
Let X be the number of attempts at lighting the fire, up to and including the successful attempt.
(i) Name the distribution of X, stating a further modelling assumption needed. [2]
In the rest of this question, you should use the distribution named in part (i).
(ii) Calculate
(a) P(X = 4), [3]
(b) P(X < 4). [3]
(iii) State the value of E(X). [1]
(iv) Henry has to light the fire once a day, starting on March 1st. Calculate the probability that thefirst day on which fewer than 4 attempts are needed to light the fire is March 3rd. [3]
Advanced Subsidiary General Certificate of EducationAdvanced General Certificate of Education
MATHEMATICS 4732Probability & Statistics 1
Thursday 12 JANUARY 2006 Afternoon 1 hour 30 minutes
Additional materials:8 page answer bookletGraph paperList of Formulae (MF1)
TIME 1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answerbooklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree ofaccuracy is specified in the question or is clearly appropriate.
• You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carryinglarger numbers of marks later in the paper.
• You are reminded of the need for clear presentation in your answers.
This question paper consists of 5 printed pages and 3 blank pages.
1 Jenny and John are each allowed two attempts to pass an examination.
(i) Jenny estimates that her chances of success are as follows.
• The probability that she will pass on her first attempt is 23.
• If she fails on her first attempt, the probability that she will pass on her second attempt is 34.
Calculate the probability that Jenny will pass. [3]
(ii) John estimates that his chances of success are as follows.
• The probability that he will pass on his first attempt is 23.
• Overall, the probability that he will pass is 56.
Calculate the probability that if John fails on his first attempt, he will pass on his second attempt.[3]
2 For each of 50 plants, the height, h cm, was measured and the value of (h − 100) was recorded. Themean and standard deviation of (h − 100) were found to be 24.5 and 4.8 respectively.
(i) Write down the mean and standard deviation of h. [2]
The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and5.1 cm respectively.
(ii) Describe briefly how the heights of the second group of plants compare with the first. [2]
(iii) Calculate the mean height of all 150 plants. [2]
3 In Mr Kendall’s cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately hisdaughter has removed all the labels for a school project and so the tins are identical in appearance.Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he hasopened the two tins of pineapples.
Let X be the number of tins that Mr Kendall opens.
(i) Show that P(X = 3) = 15. [4]
(ii) The probability distribution of X is given in the table below.
4 Each day, the Research Department of a retail firm records the firm’s daily income, to be used forstatistical analysis. The results are summarised by recording the number of days on which the dailyincome is within certain ranges.
(i)
The histogram shows the results for 300 days. By considering the total area of the histogram,
(a) find the number of days on which the daily income was between £4000 and £6000, [4]
(b) calculate an estimate of the number of days on which the daily income was between £2700and £3200. [3]
(ii) The Research Department offers to provide any of the following statistical diagrams: histogram,frequency polygon, box-and-whisker plot, cumulative frequency graph, stem-and-leaf diagramand pie chart.
Which one of these statistical diagrams would most easily enable managers to
(a) read off the median and quartile values of the daily income, [1]
(b) find the range of the top 10% of values of the daily income? [1]
5 Andrea practises shots at goal. For each shot the probability of her scoring a goal is 25. Each shot is
independent of other shots.
(i) Find the probability that she scores her first goal
(a) on her 5th shot, [2]
(b) before her 5th shot. [3]
(ii) (a) Find the probability that she scores exactly 1 goal in her first 5 shots. [3]
(b) Hence find the probability that she scores her second goal on her 6th shot. [2]
6 An examination paper consists of two parts. Section A contains questions A1, A2, A3 and A4.Section B contains questions B1, B2, B3, B4, B5, B6 and B7.
Candidates must choose three questions from section A and four questions from section B. The orderin which they choose the questions does not matter.
(i) In how many ways can the seven questions be chosen? [3]
(ii) Assuming that all selections are equally likely, find the probability that a particular candidatechooses question A1 but does not choose question B1. [3]
(iii) Following a change of syllabus, the form of the examination remains the same except thatcandidates who choose question A1 are not allowed to choose question B1. In how many wayscan the seven questions now be chosen? [3]
7 Past experience has shown that when seeds of a certain type are planted, on average 90% will germinate.A gardener plants 10 of these seeds in a tray and waits to see how many will germinate.
(i) Name an appropriate distribution with which to model the number of seeds that germinate, givingthe value(s) of any parameters. State any assumption(s) needed for the model to be valid. [4]
(ii) Use your model to find the probability that fewer than 8 seeds germinate. [2]
Later the gardener plants 20 trays of seeds, with 10 seeds in each tray.
(iii) Calculate the probability that there are at least 19 trays in each of which at least 8 seeds germinate.[4]
(b) Explain what your answer indicates about the populations of these countries and their capitalcities. [1]
(ii) Calculate the product moment correlation coefficient, r. [2]
The data are illustrated in the scatter diagram.
(iii) By considering the diagram, state the effect on the value of the product moment correlationcoefficient, r, if the data for France and the United Kingdom were removed from the calculation.
[1]
(iv) In a certain country in Africa, most people live in remote areas and hence the population of thecountry is unknown. However, the population of the capital city is known to be approximately1 million. An official suggests that the population of this country could be estimated by using aregression line drawn on the above scatter diagram.
(a) State, with a reason, whether the regression line of y on x or the regression line of x on ywould need to be used. [2]
(b) Comment on the reliability of such an estimate in this situation. [2]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonablee�ort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will bepleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local ExaminationsSyndicate (UCLES), which is itself a department of the University of Cambridge.
