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OCR Further Statistics AS 2019 June Q6
9 marks Moderate -0.3
6 A bag contains a mixture of blue and green beads, in unknown proportions. The proportion of green beads in the bag is denoted by \(p\).
  1. Sasha selects 10 beads at random, with replacement. Write down an expression, in terms of \(p\), for the variance of the number of green beads Sasha selects. Freda selects one bead at random from the bag, notes its colour, and replaces it in the bag. She continues to select beads in this way until a green bead is selected. The first green bead is the \(X\) th bead that Freda selects.
  2. Assume that \(p = 0.3\). Find
    1. \(\mathrm { P } ( X \geqslant 5 )\),
    2. \(\operatorname { Var } ( X )\).
  3. In fact, on the basis of a large number of observations of \(X\), it is found that \(\mathrm { P } ( X = 3 ) = \frac { 4 } { 25 } \times \mathrm { P } ( X = 1 )\). Estimate the value of \(p\).
OCR Further Statistics AS 2019 June Q7
9 marks Standard +0.3
7 In a standard model from genetic theory, the ratios of types \(a , b , c\) and \(d\) of a characteristic from a genetic cross are predicted to be 9:3:3:1. Andrei collects 120 specimens from such a cross, and the numbers corresponding to each type of the characteristic are given in the table.
Type\(a\)\(b\)\(c\)\(d\)
Frequency5133306
Andrei tests, at the 1\% significance level, whether the observed frequencies are consistent with the standard model.
  1. State appropriate hypotheses for the test.
  2. Carry out the test.
  3. State with a reason which one of the frequencies is least consistent with the standard model.
  4. Suggest a different, improved model by changing exactly two of the ratio values.
OCR Further Statistics AS 2019 June Q8
7 marks Challenging +1.8
8 Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to \(n\) in an order of Jane's own choice and Alex tells Jane what order he believes was chosen. They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round. They use a value of \(n\) such that, if Alex chooses the order of the integers at random, the probability that Alex’s claim will be accepted is less than \(1 \%\). Determine the smallest possible value of \(n\). \section*{END OF QUESTION PAPER}
OCR Further Statistics AS 2022 June Q1
8 marks Moderate -0.3
1 A geography student chose a certain point in a stream and took measurements of the speed of flow, \(v \mathrm {~ms} ^ { - 1 }\), of water at various depths, \(d \mathrm {~m}\), below the surface at that point. The results are shown in the table.
\(d\)0.10.150.20.250.30.350.40.450.5
\(v\)0.80.50.71.21.11.31.61.40.4
\(n = 9 \quad \sum d = 2.7 \quad \sum v = 9.0 \quad \sum d ^ { 2 } = 0.96 \quad \sum v ^ { 2 } = 10.4 \quad \sum \mathrm {~d} v = 2.85\)
    1. Explain why \(d\) is an example of an independent, controlled variable.
    2. Use two relevant terms to describe the variable \(v\) in a similar way. A statistician believes that the point ( \(0.5,0.4\) ) may be an anomaly.
  2. Calculate the equation of the least squares regression line of \(v\) on \(d\) for all the points in the table apart from ( \(0.5,0.4\) ).
  3. Use the equation of the line found in part (b) to estimate the value of \(v\) when \(d = 0.5\).
  4. Use your answer to part (c) to comment on the statistician’s belief.
  5. Use the diagram in the Printed Answer Booklet (which does not illustrate the data in this question) to explain what is meant by "least squares regression line".
OCR Further Statistics AS 2022 June Q2
7 marks Standard +0.3
2 Eight runners took part in two races. The positions in which the runners finished in the two races are shown in the table.
RunnerABCDEFGH
First race31562874
Second race43872561
Test at the 5\% significance level whether those runners who do better in one race tend to do better in the other.
OCR Further Statistics AS 2022 June Q3
9 marks Standard +0.8
3 A discrete random variable \(X\) has the following probability distribution.
\(x\)1234
\(\mathrm { P } ( X = x )\)\(p\)0.310.3\(p ^ { 2 }\)
  1. Determine the value of \(p\).
  2. It is given that \(\mathrm { E } ( a X + b ) = \operatorname { Var } ( a X + b ) = 23.19\), where \(a\) and \(b\) are positive constants. Determine the value of \(a\) and the value of \(b\).
OCR Further Statistics AS 2022 June Q4
7 marks Standard +0.3
4 A school pupil keeps a note of whether her journeys to school and from school are delayed. The results for a random sample of journeys are shown in the table.
\cline { 2 - 3 } \multicolumn{1}{c|}{}Direction of journey
\cline { 2 - 3 } \multicolumn{1}{c|}{}To schoolFrom school
Delayed6456
Not delayed74106
Test at the 10\% significance level whether there is association between delays and the direction of the journey.
OCR Further Statistics AS 2022 June Q5
9 marks Standard +0.3
5 The manager of an emergency response hotline believes that calls are made to the hotline independently and at constant average rate throughout the day. From a small random sample of the population, the manager finds that the mean number of calls made in a 1-hour period is 14.4. Let \(R\) denote the number of calls made in a randomly chosen 1-hour period.
  1. Using evidence from the small sample, state a suitable distribution with which to model \(R\). You should give the value(s) of any parameter(s).
  2. In this part of the question, use the distribution and value(s) of the parameter(s) from your answer to part (a).
    1. Find \(\mathrm { P } ( R > 20 )\).
    2. Given that \(\mathrm { P } ( \mathrm { R } = \mathrm { r } ) > \mathrm { P } ( \mathrm { R } = \mathrm { r } + 1 )\), show algebraically that \(r > 13.4\).
    3. Hence write down the mode of the distribution. The manager also finds, from records over many years, that the modal value of \(R\) is 10 .
  3. Use this result to comment on the validity of the distribution used in part (b).
  4. Assume now that the type of distribution used in part (b) is valid. Find the range(s) of values of the parameter(s) of this distribution that would correspond to the modal value of \(R\) being 10.
OCR Further Statistics AS 2022 June Q6
10 marks Standard +0.8
6 A teacher has 10 different mathematics books. Of these books, 5 are on Algebra, 3 are on Calculus and 2 are on Trigonometry. The teacher chooses 5 of the books at random.
  1. Find the probability that 3 of the books are on Algebra. The teacher now arranges all 10 books in random order on a shelf.
  2. Find the probability that the Calculus books are next to each other and the Trigonometry books are next to each other. \section*{In this question you must show detailed reasoning.}
  3. Find the probability that 2 of the Calculus books are next to each other but the third Calculus book is separated from the other 2 by at least 1 other book.
OCR Further Statistics AS 2022 June Q7
10 marks Standard +0.8
7 Each of three students, \(\mathrm { X } , \mathrm { Y }\) and Z , was given an identical pack of 48 cards, of which 12 cards were red and 36 were blue. They were each told to carry out a different experiment, as follows: Student X: Choose a card from the pack, at random, 20 times altogether, with replacement. Record how many times you obtain a red card. Student Y: Choose a card from the pack, at random, 20 times altogether, without replacement. Record how many times you obtain a red card. Student Z: Choose single cards from the pack at random, with replacement, until you obtain the first red card. Record how many cards you have chosen, including the first red card.
  1. Find the probability that student Z has to choose more than 8 cards in order to obtain the first red card. Each student carries out their experiment 30 times. The frequencies of the results recorded by each student are shown in the following table, but not necessarily with the rows in the order \(\mathrm { X } , \mathrm { Y } , \mathrm { Z }\) :
    Number recorded012345678\(\geqslant 9\)Observed MeanObserved Variance
    \multirow{3}{*}{Observed Frequencies}Student 100137864105.031.97
    Student 208542332124.0311.57
    Student 301254653404.973.70
    \section*{(b) In this question you must show detailed reasoning.} Two other students make the following statements about the results. For each of the statements, explain whether you agree with the statement. Do not carry out any hypothesis tests, but in each case you should give two justifications for your answer.
    1. "The second row is a good match with the expected results for student Z ."
    2. "The third row is definitely student X 's results."
OCR Further Statistics AS 2023 June Q1
7 marks Moderate -0.8
1 A radar device is used to detect flaws in motorway roads before they become dangerous. The number of flaws in a 1 km stretch of motorway is denoted by \(X\). It may be assumed that flaws occur randomly.
  1. State two further assumptions that are necessary for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be modelled by distribution \(\operatorname { Po } ( 5.7 )\).
  2. Determine the probability that in a randomly chosen stretch of motorway, of length 1 km , there are between 8 and 11 flaws, inclusive.
  3. Determine the probability that in two randomly chosen, non-overlapping, stretches of motorway, each of length 5 km , there are at least 30 flaws in one stretch and fewer than 30 flaws in the other stretch.
OCR Further Statistics AS 2023 June Q2
7 marks Standard +0.8
2 A music lover has 30 CDs arranged in a random order in a line on a shelf. Of these CDs, 7 are classed as Baroque, 10 as Classical and 13 as Romantic.
  1. Determine the probability that all 7 Baroque CDs are next to each other.
  2. Determine the probability that, of the 10 CDs furthest to the left on the shelf, at least 6 are Baroque.
OCR Further Statistics AS 2023 June Q3
8 marks Standard +0.3
3 An insurance company collected data concerning the age, \(x\) years, of policy holders and the average size of claim, \(\pounds y\) thousand. The data is summarised as follows. \(n = 32 \quad \sum x = 1340 \quad \sum y = 612 \quad \sum x ^ { 2 } = 64282 \quad \sum y ^ { 2 } = 13418 \quad \sum x y = 27794\)
  1. Find the variance of \(x\).
  2. Find the equation of the regression line of \(y\) on \(x\).
  3. Hence estimate the expected size of claim from a policy holder of age 48. Tom is aged 48. He claims that the range of the data probably does not include people of his age because the mean age for the data is 41.875 , and 48 is not close to this.
  4. Use your answer to part (a) to determine how likely it is that Tom's claim is correct.
  5. Comment on the reliability of your estimate in part (c). You should refer to the value of the product-moment correlation coefficient for the data, which is 0.579 correct to 3 significant figures.
OCR Further Statistics AS 2023 June Q4
7 marks Standard +0.3
4 A discrete random variable \(W\) has the probability distribution shown in the following table, in which \(a\) and \(b\) are constants.
\(w\)585960616263
\(\mathrm { P } ( W = w )\)\(a\)\(b\)0.20.20.10.1
It is given that \(\mathrm { E } ( W - 60 ) = 0.15\). Determine the value of \(\operatorname { Var } ( 4 W - 60 )\).
OCR Further Statistics AS 2023 June Q5
9 marks Standard +0.3
5 A psychologist investigates the relationship between 'openness' and 'creativity' in adults. Each member of a random sample of 15 adults is given two tests, one on openness and one on creativity. Each test has a maximum score of 75 . The results are given in the table.
AdultABCDEFGHIJKLMNO
Openness, \(x\)393429204035203655314143333033
Creativity, \(y\)593417294946455460384635435634
\(n = 15 \quad \sum x = 519 \quad \sum y = 645 \quad \sum x ^ { 2 } = 19033 \quad \sum y ^ { 2 } = 29751 \quad \sum x y = 23034\)
  1. Use Pearson's product-moment correlation coefficient to test, at the \(5 \%\) significance level, whether there is positive association between openness and creativity.
  2. State what the value of Pearson's product-moment correlation coefficient shows about a scatter diagram illustrating the data.
  3. A student suggests that there is a way to obtain a more accurate measure of the correlation. Before carrying out the test it would be better to standardise the test scores so that they have the same mean and variance. Explain whether you agree with this suggestion.
OCR Further Statistics AS 2023 June Q6
12 marks Standard +0.3
6 A machine is used to toss a coin repeatedly. Rosa believes that the outcome of each toss made by the machine is not independent of the previous toss. Rosa gets the machine to toss a coin 6 times and record the number of heads, \(X\), obtained. After recording the number of heads obtained, Rosa resets the machine and gets it to toss the coin 6 more times. Rosa again records the number of heads obtained and she repeats this procedure until she has recorded 88 independent values of \(X\).
  1. The sample mean and sample variance of \(X\) are 3.35 and 3.392 respectively. Explain what these results suggest about the validity of a binomial model \(\mathrm { B } ( 6 , p )\) for the data. Rosa uses a computer spreadsheet to work out the probabilities for a more sophisticated model in which the outcome of each toss is dependent on the outcome of the previous toss. Her model suggests that the probabilities \(\mathrm { P } ( X = x )\), for \(x = 0,1,2,3,4,5,6\), are approximately in the ratio \(5 : 6 : 7 : 8 : 7 : 6 : 5\). She carries out a \(\chi ^ { 2 }\) test to investigate whether this model is a good fit for the data. The following table shows the full results of the experiments, together with some of the calculations needed for the test.
    \(x\)0123456Total
    Observed frequency710161515111488
    Expected frequency
    Contribution to \(\chi ^ { 2 }\) statistic0.90.33330.28570.06250.0714
  2. In the Printed Answer Booklet, complete the table.
  3. Carry out the test, using a 10\% significance level.
  4. Rosa says that the results definitely show that one of the two proposed models is correct. Comment on this statement.
OCR Further Statistics AS 2023 June Q7
10 marks Standard +0.8
7 A town council is planning to introduce a new set of parking regulations. An interviewer contacts randomly chosen people in the town and asks them whether they are in favour of the proposal. The first person who is not in favour of the regulation is the \(R\) th person interviewed. It can be assumed that the probability that any randomly chosen person is not in favour of the proposal is a constant \(p\), and that \(p\) does not equal 0 or 1 . Assume first that \(\mathrm { E } ( R ) = 10\).
  1. Determine \(\mathrm { P } ( R \geqslant 14 )\). Now, without the assumption that \(\mathrm { E } ( R ) = 10\), consider a general value of \(p\).
    It is given that \(\mathrm { P } ( R = 3 ) - 0.4 \times \mathrm { P } ( R = 2 ) - a \times \mathrm { P } ( R = 1 ) = 0\), where \(a\) is a positive constant.
  2. Determine the range of possible values of \(a\).
OCR Further Statistics AS 2024 June Q1
8 marks Moderate -0.8
1 The random variable \(W\) can take values 1,2 or 3 and has a discrete uniform distribution.
  1. Write down the value of \(\mathrm { E } ( 2 W )\).
  2. Find the value of \(\operatorname { Var } ( 2 W )\).
  3. Determine the value of the constant \(k\) for which \(\mathrm { E } ( 2 \mathrm {~W} + \mathrm { k } ) = \operatorname { Var } ( 2 \mathrm {~W} + \mathrm { k } )\). The random variable \(S\) has the probability distribution shown in the following table.
    \(S\)23456
    \(P ( S = S )\)\(\frac { 2 } { 9 }\)\(\frac { 1 } { 9 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 9 }\)\(\frac { 2 } { 9 }\)
  4. Calculate \(\operatorname { Var } ( S )\).
OCR Further Statistics AS 2024 June Q2
8 marks Standard +0.3
2 For a random sample of 160 employees of a large company, the principal method of transport for getting to work, arranged according to grade of employee, is shown in the table.
GradeWalk or cyclePrivate motorised transportPublic transport
A9136
B164341
C11813
A test is carried out at the \(5 \%\) significance level of whether there is association between grade of employee and method of transport.
  1. State appropriate hypotheses for the test. The contributions to the test statistic are shown in the following table, correct to 3 decimal places.
    GradeWalk or cyclePrivate motorised transportPublic transport
    A1.1570.2891.929
    B1.8780.2250.327
    C2.0061.8000.083
  2. Show how the value 0.225 is obtained.
  3. Complete the test, stating the conclusion.
  4. Which combination of grade of employee and method of transport most strongly suggests association? Justify your answer.
OCR Further Statistics AS 2024 June Q3
11 marks Standard +0.3
3 The ages, \(x\) years, and the reaction time, \(t\) seconds, in an experiment carried out on a sample of 15 volunteers are summarised as follows. \(n = 15 \quad \sum x = 762 \quad \sum t = 8.7 \quad \sum x ^ { 2 } = 44204 \quad \sum t ^ { 2 } = 5.65 \quad \sum x t = 490.1\)
  1. Calculate the value of the product moment correlation coefficient between \(x\) and \(t\).
  2. Calculate the equation of the line of regression of \(t\) on \(x\). Give your answer in the form \(\mathrm { t } = \mathrm { a } + \mathrm { bx }\) where \(a\) and \(b\) are constants to be determined.
  3. Explain the relevance of the quantity \(\sum ( t - a - b x ) ^ { 2 }\) to your answer to part (b).
  4. Estimate the reaction time, in seconds, for a volunteer aged 42. It is subsequently decided to measure the reaction time in tenths of a second rather than in seconds (so, for example, a time of 0.6 seconds would now be recorded as 6 ).
    1. State what effect, if any, this change would have on your answer to part (a).
    2. State what effect, if any, this change would have on your answer to part (b). It is known that the sample of 15 volunteers consisted almost entirely of students and retired people.
  6. Using this information, and the value of the product moment correlation coefficient, comment on the reliability of your estimate in part (d).
OCR Further Statistics AS 2024 June Q5
9 marks Standard +0.3
5 In a fashion competition, two judges gave marks to a large number of contestants. The value of Spearman's rank correlation coefficient, \(\mathrm { r } _ { \mathrm { s } }\), between the marks given to 7 randomly chosen contestants is \(\frac { 27 } { 28 }\).
  1. An excerpt from the table of critical values of \(\mathrm { r } _ { \mathrm { s } }\) is shown below. \section*{Critical values of Spearman's rank correlation coefficient}
    1-tail test5\%2.5\%1\%0.5\%
    2-tail test10\%5\%2\%1\%
    \multirow{3}{*}{\(n\)}60.82860.88570.94291.0000
    70.71430.78570.89290.9286
    80.64290.73810.83330.8810
    Test whether there is evidence, at the 1\% significance level, that the judges agree with each another. The marks given by the two judges to the 7 randomly chosen contestants were as follows, where \(x\) is an integer.
    ContestantABCD\(E\)\(F\)G
    Judge 164656778798086
    Judge 2616378808190\(x\)
  2. Use the value \(\mathrm { r } _ { \mathrm { s } } = \frac { 27 } { 28 }\) to determine the range of possible values of \(x\).
  3. Give a reason why it might be preferable to use the product moment correlation coefficient rather than Spearman's rank correlation coefficient in this context.
OCR Further Statistics AS 2024 June Q6
12 marks Standard +0.3
6 Anika walks along a street that contains parked cars. The number of cars that Anika passes, up to and including the first car that is white, is denoted by \(X\).
  1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Geo } ( p )\), where \(0 < p < 1\).
  2. For \(p = 0.1\), find \(\mathrm { P } ( X > 6 )\). The number of cars that Anika passes, up to but not including the first car that is white, is denoted by \(Y\).
  3. For a general value of \(p\), determine a simplified expression for \(\mathrm { E } ( Y ) \div \operatorname { Var } ( Y )\), in terms of \(p\). Ben walks along a different street that also contains parked cars. The number of cars that Ben passes, up to and including the first white car on which the last digit of the number plate is even is denoted by \(Z\). It may be assumed that \(Z\) can be well modelled by the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } p \right)\), where \(p\) is the parameter of the distribution of \(X\). It is given that \(\mathrm { P } ( \mathrm { Z } = 3 ) = \mathrm { kP } ( \mathrm { X } = 3 )\), where \(k\) is a positive constant.
  4. Determine the range of possible values of \(k\).
OCR Further Statistics AS 2020 November Q1
5 marks Moderate -0.3
1 Five observations of bivariate data \(( x , y )\) are given in the table.
\(x\)781264
\(y\)201671723
  1. Find the value of Pearson's product-moment correlation coefficient.
  2. State what your answer to part (a) tells you about a scatter diagram representing the data.
  3. A new variable \(a\) is defined by \(\mathrm { a } = 3 \mathrm { x } + 4\). Dee says "The value of Pearson's product-moment correlation coefficient between \(a\) and \(y\) will not be the same as the answer to part (a)." State with a reason whether you agree with Dee.
OCR Further Statistics AS 2020 November Q2
8 marks Moderate -0.8
2 Every time a spinner is spun, the probability that it shows the number 4 is 0.2 , independently of all other spins.
  1. A pupil spins the spinner repeatedly until it shows the number 4. Find the mean of the number of spins required.
  2. Calculate the probability that the number of spins required is between 3 and 10 inclusive.
  3. Each pupil in a class of 30 spins the spinner until it shows the number 4. Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\).
OCR Further Statistics AS 2020 November Q3
9 marks Moderate -0.3
3 An investor obtains data about the profits of 8 randomly chosen investment accounts over two one-year periods. The profit in the first year for each account is \(p \%\) and the profit in the second year for each account is \(q \%\). The results are shown in the table and in the scatter diagram.
AccountABCDEFGH
\(p\)1.62.12.42.72.83.35.28.4
\(q\)1.62.32.22.23.12.97.64.8
\(n = 8 \quad \sum \mathrm { p } = 28.5 \quad \sum \mathrm { q } = 26.7 \quad \sum \mathrm { p } ^ { 2 } = 136.35 \quad \sum \mathrm { q } ^ { 2 } = 116.35 \quad \sum \mathrm { pq } = 116.70\) \includegraphics[max width=\textwidth, alt={}, center]{bf1468d1-e02e-47d2-bf41-5bc8f5b4d7c4-3_782_1280_998_242}
  1. State which, if either, of the variables \(p\) and \(q\) is independent.
  2. Calculate the equation of the regression line of \(q\) on \(p\).
    1. Use the regression line to estimate the value of \(q\) for an investment account for which \(p = 2.5\).
    2. Give two reasons why this estimate could be considered reliable.
  4. Comment on the reliability of using the regression line to predict the value of \(q\) when \(p = 7.0\).