Questions — OCR Further Statistics AS (58 questions)

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OCR Further Statistics AS 2018 June Q1
1 A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is 0.1 . The first weekday in September on which he receives a delivery of books to review is the Xth weekday of September.
  1. State an assumption needed for \(X\) to be well modelled by a geometric distribution.
  2. Find \(\mathrm { P } ( X = 11 )\).
  3. Find \(\mathrm { P } ( X \leqslant 8 )\).
  4. Find \(\operatorname { Var } ( X )\).
  5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review.
OCR Further Statistics AS 2018 June Q2
2 The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(\mathrm { P } ( W = w )\)0.250.36\(x\)\(x ^ { 2 }\)
  1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
  2. Find \(\operatorname { Var } ( 3 W + 6 )\).
OCR Further Statistics AS 2018 June Q3
3 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).
  1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
  2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
  3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4.
OCR Further Statistics AS 2018 June Q4
4 Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \sum x = 3600 , \sum x ^ { 2 } = 660500 , \sum y = 1440 , \sum y ^ { 2 } = 105280 , \sum x y = 260990$$
  1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
  2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
  3. Assume now that the assumption in part (ii) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
  4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
OCR Further Statistics AS 2018 June Q5
5
  1. A team of 9 is chosen at random from a class consisting of 8 boys and 12 girls.
    Find the probability that the team contains no more than 3 girls.
  2. A group of \(n\) people, including Mr and Mrs Laplace, are arranged at random in a line. The probability that Mr and Mrs Laplace are placed next to each other is less than 0.1 . Find the smallest possible value of \(n\).
OCR Further Statistics AS 2018 June Q7
7 An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.
\(c\)1.941.781.621.511.521.4
\(m\)6.57.27.47.68.39.7
  1. State which, if either, of \(m\) and \(c\) is an independent variable.
  2. Calculate the equation of the least squares regression line of \(c\) on \(m\).
  3. State what effect, if any, there would be on your answer to part (ii) if the masses of the fish had been recorded in kilograms rather than pounds. ( \(1 \mathrm {~kg} \approx 2.2\) pounds.)
  4. The data is illustrated in the scatter diagram. Explain what is meant by 'least squares', illustrating your answer using the copy of this diagram in the Printed Answer Booklet.
    \includegraphics[max width=\textwidth, alt={}, center]{708e125e-43a8-40d8-94db-0ed80337d273-4_719_1043_961_513}
OCR Further Statistics AS 2018 June Q8
8 The table shows the results of a random sample drawn from a population which is thought to have the distribution \(\mathrm { U } ( 20 )\).
Range\(1 \leqslant x \leqslant 8\)\(9 \leqslant x \leqslant 12\)\(13 \leqslant x \leqslant 20\)
Observed frequency12\(y\)\(28 - y\)
Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}
OCR Further Statistics AS 2019 June Q1
1 When a spinner is spun, the outcome is equally likely to be 1,2 or 3 . In a competition, the spinner is spun twice and the outcomes are added to give a total score \(T\).
  1. Show that the expectation of \(T\) is 4 .
  2. Find the variance of \(T\). A competitor pays \(\pounds 1.50\) to enter the competition and receives \(\pounds X\), where \(X = 0.3 T\).
    1. Find the expectation of the competitor's profit.
    2. Find the variance of the competitor's profit.
OCR Further Statistics AS 2019 June Q2
2 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).
  1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
  2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
  3. State a necessary assumption for the validity of your calculation in part (b).
OCR Further Statistics AS 2019 June Q3
3
  1. Shula calculates the value of Spearman's rank correlation coefficient \(r _ { s }\) for 9 pairs of rankings.
    Find the largest possible value of \(r _ { s }\) that Shula can obtain that is less than 1 .
  2. A set of bivariate data consists of 5 pairs of values. It is known that for this data the value of Spearman's rank correlation coefficient is - 1 but the value of Pearson's product-moment correlation coefficient is not - 1 . Sketch a possible scatter diagram illustrating the data.
OCR Further Statistics AS 2019 June Q4
4 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.
  1. Find the probability that all the men are next to each other.
  2. Find the probability that no two men are next to one another.
OCR Further Statistics AS 2019 June Q5
5 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.
  1. For all sixteen candidates, the value of the product moment correlation coefficient \(r\) for the marks on the two papers was 0.701 correct to 3 significant figures. Test whether there is evidence, at the \(5 \%\) significance level, of association between the marks on the two papers.
  2. A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by
    \(n = 10 , \sum x = 750 , \sum y = 690 , \sum x ^ { 2 } = 57690 , \sum y ^ { 2 } = 49676 , \sum x y = 50829\).
    1. Calculate the value of \(r\) for these 10 candidates.
    2. What do the two values of \(r\), in parts (a) and (b)(i), tell you about the scores of the sixteen candidates?
OCR Further Statistics AS 2019 June Q6
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
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
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
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
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
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
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
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
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
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
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
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
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.