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OCR C4 2016 June Q10
12 marks Standard +0.8
10
  1. Express \(\frac { 16 + 5 x - 2 x ^ { 2 } } { ( x + 1 ) ^ { 2 } ( x + 4 ) }\) in partial fractions.
  2. It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( 16 + 5 x - 2 x ^ { 2 } \right) y } { ( x + 1 ) ^ { 2 } ( x + 4 ) }$$ and that \(y = \frac { 1 } { 256 }\) when \(x = 0\). Find the exact value of \(y\) when \(x = 2\). Give your answer in the form \(A \mathrm { e } ^ { n }\).
OCR S1 2009 January Q1
8 marks Easy -1.2
1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table.
NumberProbability
00.7
10.2
20.1
The spinner is spun twice. The total of the two numbers on which it lands is denoted by \(X\).
  1. Show that \(\mathrm { P } ( X = 2 ) = 0.18\). The probability distribution of \(X\) is given in the table.
    \(x\)01234
    \(\mathrm { P } ( X = x )\)0.490.280.180.040.01
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2009 January Q2
8 marks Moderate -0.8
2 The table shows the age, \(x\) years, and the mean diameter, \(y \mathrm {~cm}\), of the trunk of each of seven randomly selected trees of a certain species.
Age \(( x\) years \()\)11122028354551
Mean trunk diameter \(( y \mathrm {~cm} )\)12.216.026.439.239.651.360.6
$$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$
  1. (a) Use an appropriate formula to show that the gradient of the regression line of \(y\) on \(x\) is 1.13 , correct to 2 decimal places.
    (b) Find the equation of the regression line of \(y\) on \(x\).
  2. Use your equation to estimate the mean trunk diameter of a tree of this species with age
    (a) 30 years,
    (b) 100 years. It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places.
  3. Comment on the reliability of each of your two estimates.
OCR S1 2009 January Q3
10 marks Moderate -0.8
3 Erika is a birdwatcher. The probability that she will see a woodpecker on any given day is \(\frac { 1 } { 8 }\). It is assumed that this probability is unaffected by whether she has seen a woodpecker on any other day.
  1. Calculate the probability that Erika first sees a woodpecker
    1. on the third day,
    2. after the third day.
    3. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
    4. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.
OCR S1 2009 January Q4
7 marks Moderate -0.3
4 Three tutors each marked the coursework of five students. The marks are given in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)
Tutor 17367604839
Tutor 26250617665
Tutor 34250635471
  1. Calculate Spearman's rank correlation coefficient, \(r _ { \mathrm { s } }\), between the marks for tutors 1 and 2 .
  2. The values of \(r _ { \mathrm { s } }\) for the other pairs of tutors, are as follows. $$\begin{array} { c c } \text { Tutors } 1 \text { and 3: } & r _ { \mathrm { s } } = - 0.9 \\ \text { Tutors } 2 \text { and 3: } & r _ { \mathrm { s } } = 0.3 \end{array}$$ State which two tutors differ most widely in their judgements. Give your reason.
OCR S1 2009 January Q5
8 marks Easy -1.3
5 The stem-and-leaf diagram shows the masses, in grams, of 23 plums, measured correct to the nearest gram.
5567889
61235689
700245678
80
97
9
\(\quad\) Key \(: 6 \mid 2\) means 62
  1. Find the median and interquartile range of these masses.
  2. State one advantage of using the interquartile range rather than the standard deviation as a measure of the variation in these masses.
  3. State one advantage and one disadvantage of using a stem-and-leaf diagram rather than a box-and-whisker plot to represent data.
  4. James wished to calculate the mean and standard deviation of the given data. He first subtracted 5 from each of the digits to the left of the line in the stem-and-leaf diagram, giving the following.
    0567889
    11235689
    200245678
    30
    47
    The mean and standard deviation of the data in this diagram are 18.1 and 9.7 respectively, correct to 1 decimal place. Write down the mean and standard deviation of the data in the original diagram.
OCR S1 2009 January Q6
12 marks Standard +0.3
6 A test consists of 4 algebra questions, A, B, C and D, and 4 geometry questions, G, H, I and J.
The examiner plans to arrange all 8 questions in a random order, regardless of topic.
  1. (a) How many different arrangements are possible?
    (b) Find the probability that no two Algebra questions are next to each other and no two Geometry questions are next to each other. Later, the examiner decides that the questions should be arranged in two sections, Algebra followed by Geometry, with the questions in each section arranged in a random order.
  2. (a) How many different arrangements are possible?
    (b) Find the probability that questions A and H are next to each other.
    (c) Find the probability that questions B and J are separated by more than four other questions.
OCR S1 2009 January Q7
12 marks Moderate -0.8
7 At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).
  1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
  2. Find
    1. \(\mathrm { P } ( X = 3 )\),
    2. \(\mathrm { P } ( X \geqslant 1 )\).
    3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .
OCR S1 2009 January Q8
7 marks Moderate -0.3
8 A game uses an unbiased die with faces numbered 1 to 6 . The die is thrown once. If it shows 4 or 5 or 6 then this number is the final score. If it shows 1 or 2 or 3 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
  1. Find the probability that the final score is 4 .
  2. Given that the die is thrown only once, find the probability that the final score is 4 .
  3. Given that the die is thrown twice, find the probability that the final score is 4 .
OCR S1 2011 January Q2
11 marks Moderate -0.8
2 The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.2 )\). Find
  1. \(\mathrm { P } ( X = 3 )\),
  2. \(\mathrm { P } ( 3 \leqslant X \leqslant 5 )\),
  3. \(\mathrm { P } ( X > 4 )\). Two independent values of \(X\) are found.
  4. Find the probability that the total of these two values is 3 .
OCR S1 2011 January Q3
12 marks Moderate -0.8
3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, \(\pounds x\) thousand, and the annual profit, \(\pounds y\) thousand. A summary of the figures for 12 years is as follows. $$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$
  1. Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .
  2. Comment briefly on this value in this context.
  3. A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.
  4. Calculate the equation of the regression line of \(y\) on \(x\).
  5. Estimate the annual profit during a year when \(\pounds 7400\) was spent on advertising.
OCR S1 2011 January Q4
7 marks Moderate -0.8
4 Jenny and Omar are each allowed two attempts at a high jump.
  1. The probability that Jenny will succeed on her first attempt is 0.6 . If she fails on her first attempt, the probability that she will succeed on her second attempt is 0.7 . Calculate the probability that Jenny will succeed.
  2. The probability that Omar will succeed on his first attempt is \(p\). If he fails on his first attempt, the probability that he will succeed on his second attempt is also \(p\). The probability that he succeeds is 0.51 . Find \(p\). \(530 \%\) of packets of Natural Crunch Crisps contain a free gift. Jan buys 5 packets each week.
OCR S1 2011 January Q6
10 marks Moderate -0.8
6
  1. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number.
    1333559
    How many different 7 -digit numbers can be formed using these cards?
  2. The diagram below shows 5 white cards and 10 grey cards, each with a letter printed on it. \includegraphics[max width=\textwidth, alt={}, center]{98ac515d-fd47-4864-afd6-321e9848d6cb-04_398_801_596_632} From these cards, 3 white cards and 4 grey cards are selected at random without regard to order.
    1. How many selections of seven cards are possible?
    2. Find the probability that the seven cards include exactly one card showing the letter A .
OCR S1 2011 January Q7
5 marks Easy -1.2
7 The probability distribution of a discrete random variable, \(X\), is shown below.
\(x\)02
\(\mathrm { P } ( X = x )\)\(a\)\(1 - a\)
  1. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  2. Show that \(\operatorname { Var } ( X ) = 4 a ( 1 - a )\).
OCR S1 2012 January Q1
4 marks Easy -1.3
1 The probability distribution of a random variable \(X\) is shown in the table.
\(x\)1234
\(\mathrm { P } ( X = x )\)0.10.3\(2 p\)\(p\)
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\).
OCR S1 2012 January Q2
10 marks Easy -1.8
2 In an experiment, the percentage sand content, \(y\), of soil in a given region was measured at nine different depths, \(x \mathrm {~cm}\), taken at intervals of 6 cm from 0 cm to 48 cm . The results are summarised below. $$n = 9 \quad \Sigma x = 216 \quad \Sigma x ^ { 2 } = 7344 \quad \Sigma y = 512.4 \quad \Sigma y ^ { 2 } = 30595 \quad \Sigma x y = 10674$$
  1. State, with a reason, which variable is the independent variable.
  2. Calculate the product moment correlation coefficient between \(x\) and \(y\).
  3. (a) Calculate the equation of the appropriate regression line.
    (b) This regression line is used to estimate the percentage sand content at depths of 25 cm and 100 cm . Comment on the reliability of each of these estimates. You are not asked to find the estimates.
OCR S1 2012 January Q3
6 marks Standard +0.3
3 A random variable \(X\) has the distribution \(\mathrm { B } ( 13,0.12 )\).
  1. Find \(\mathrm { P } ( X < 2 )\). Two independent values of \(X\) are found.
  2. Find the probability that exactly one of these values is equal to 2 .
OCR S1 2012 January Q4
8 marks Standard +0.8
4
  1. The table gives the heights and masses of 5 people.
    Person\(A\)\(B\)\(C\)\(D\)\(E\)
    Height (m)1.721.631.771.681.74
    Mass (kg)7562646070
    Calculate Spearman's rank correlation coefficient.
  2. In an art competition the value of Spearman's rank correlation coefficient, \(r _ { s }\), calculated from two judges' rankings was 0.75 . A late entry for the competition was received and both judges ranked this entry lower than all the others. By considering the formula for \(r _ { s }\), explain whether the new value of \(r _ { s }\) will be less than 0.75 , equal to 0.75 , or greater than 0.75 .
OCR S1 2012 January Q5
11 marks Moderate -0.8
5 At a certain resort the number of hours of sunshine, measured to the nearest hour, was recorded on each of 21 days. The results are summarised in the table.
Hours of sunshine0\(1 - 3\)\(4 - 6\)\(7 - 9\)\(10 - 15\)
Number of days06942
The diagram shows part of a histogram to illustrate the data. The scale on the frequency density axis is 2 cm to 1 unit. \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-3_944_1778_699_148}
  1. (a) Calculate the frequency density of the \(1 - 3\) class.
    (b) Fred wishes to draw the block for the 10 - 15 class on the same diagram. Calculate the height, in centimetres, of this block.
  2. A cumulative frequency graph is to be drawn. Write down the coordinates of the first two points that should be plotted. You are not asked to draw the graph.
  3. (a) Calculate estimates of the mean and standard deviation of the number of hours of sunshine.
    (b) Explain why your answers are only estimates.
OCR S1 2012 January Q6
5 marks Moderate -0.8
6 The diagrams illustrate all or part of the probability distributions of the discrete random variables \(V , W , X , Y\) and \(Z\). \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_365_370_296} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_376_370_838} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_362_370_1400} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_421_359_879_580} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_355_881_1142}
  1. One of these variables has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } \right)\). State, with a reason, which variable this is.
  2. One of these variables has the distribution \(\mathrm { B } \left( 4 , \frac { 1 } { 2 } \right)\). State, with reasons, which variable this is. \(760 \%\) of the voters at a certain polling station are women. Voters enter the polling station one at a time. The number of voters who enter, up to and including the first woman, is denoted by \(X\).
OCR S1 2012 January Q8
8 marks Moderate -0.8
8 On average, half the plants of a particular variety produce red flowers and the rest produce blue flowers.
  1. Ann chooses 8 plants of this variety at random. Find the probability that more than 6 plants produce red flowers.
  2. Karim chooses 22 plants of this variety at random.
    1. Find the probability that the number of these plants that produce blue flowers is equal to the number that produce red flowers.
    2. Hence find the probability that the number of these plants that produce blue flowers is greater than the number that produce red flowers.
OCR S1 2012 January Q9
12 marks Moderate -0.3
9 A bag contains 9 discs numbered 1, 2, 3, 4, 5, 6, 7, 8, 9 .
  1. Andrea chooses 4 discs at random, without replacement, and places them in a row.
    1. How many different 4 -digit numbers can be made?
    2. How many different odd 4-digit numbers can be made?
    3. Andrea's 4 discs are put back in the bag. Martin then chooses 4 discs at random, without replacement. Find the probability that
      (a) the 4 digits include at least 3 odd digits,
      (b) the 4 digits add up to 28 .
OCR S1 2011 June Q1
7 marks Moderate -0.8
1 Five salesmen from a certain firm were selected at random for a survey. For each salesman, the annual income, \(x\) thousand pounds, and the distance driven last year, \(y\) thousand miles, were recorded. The results were summarised as follows. $$n = 5 \quad \Sigma x = 251 \quad \Sigma x ^ { 2 } = 14323 \quad \Sigma y = 65 \quad \Sigma y ^ { 2 } = 855 \quad \Sigma x y = 3247$$
  1. (a) Show that the product moment correlation coefficient, \(r\), between \(x\) and \(y\) is - 0.122 , correct to 3 significant figures.
    (b) State what this value of \(r\) shows about the relationship between annual income and distance driven last year for these five salesmen.
    (c) It was decided to recalculate \(r\) with the distances measured in kilometres instead of miles. State what effect, if any, this would have on the value of \(r\).
  2. Another salesman from the firm is selected at random. His annual income is known to be \(\pounds 52000\), but the distance that he drove last year is unknown. In order to estimate this distance, a regression line based on the above data is used. Comment on the reliability of such an estimate.
OCR S1 2011 June Q2
5 marks Easy -1.2
2 The orders in which 4 contestants, \(P , Q , R\) and \(S\), were placed in two competitions are shown in the table.
Position1st2nd3rd4th
Competition 1\(Q\)\(R\)\(S\)\(P\)
Competition 2\(Q\)\(P\)\(R\)\(S\)
Calculate Spearman's rank correlation coefficient between these two orders.
OCR S1 2011 June Q3
10 marks Moderate -0.3
3
  1. A random variable, \(X\), has the distribution \(\mathrm { B } ( 12,0.85 )\). Find
    1. \(\mathrm { P } ( X > 10 )\),
    2. \(\mathrm { P } ( X = 10 )\),
    3. \(\operatorname { Var } ( X )\).
    4. A random variable, \(Y\), has the distribution \(\mathrm { B } \left( 2 , \frac { 1 } { 4 } \right)\). Two independent values of \(Y\) are found. Find the probability that the sum of these two values is 1 .