Questions — OCR S1 (160 questions)

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OCR S1 2009 January Q2
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
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
    (a) on the third day,
    (b) after the third day.
  2. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
  3. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.
OCR S1 2009 January Q4
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
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
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
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
    (a) \(\mathrm { P } ( X = 3 )\),
    (b) \(\mathrm { P } ( X \geqslant 1 )\).
  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
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 2010 January Q1
1 Andy makes repeated attempts to thread a needle. The number of attempts up to and including his first success is denoted by \(X\).
  1. State two conditions necessary for \(X\) to have a geometric distribution.
  2. Assuming that \(X\) has the distribution \(\operatorname { Geo } ( 0.3 )\), find
    (a) \(\mathrm { P } ( X = 5 )\),
    (b) \(\mathrm { P } ( X > 5 )\).
  3. Suggest a reason why one of the conditions you have given in part (i) might not be satisfied in this context. 240 people were asked to guess the length of a certain road. Each person gave their guess, \(l \mathrm {~km}\), correct to the nearest kilometre. The results are summarised below.
    \(l\)\(10 - 12\)\(13 - 15\)\(16 - 20\)\(21 - 30\)
    Frequency113206
  4. (a) Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\).
    (b) Explain why your answers are only estimates.
  5. A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class.
  6. Explain which class contains the median value of \(l\).
  7. Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km . Without calculation state whether the following will increase, decrease or remain the same:
    (a) the mean of \(l\),
    (b) the standard deviation of \(l\).
OCR S1 2010 January Q3
3 The heights, \(h \mathrm {~m}\), and weights, \(m \mathrm {~kg}\), of five men were measured. The results are plotted on the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5c25d6cf-2c23-4b49-88fb-e4abe6c281e4-3_738_956_386_593} The results are summarised as follows. $$n = 5 \quad \Sigma h = 9.02 \quad \Sigma m = 377.7 \quad \Sigma h ^ { 2 } = 16.382 \quad \Sigma m ^ { 2 } = 28558.67 \quad \Sigma h m = 681.612$$
  1. Use the summarised data to calculate the value of the product moment correlation coefficient, \(r\).
  2. Comment on your value of \(r\) in relation to the diagram.
  3. It was decided to re-calculate the value of \(r\) after converting the heights to feet and the masses to pounds. State what effect, if any, this will have on the value of \(r\).
  4. One of the men had height 1.63 m and mass 78.4 kg . The data for this man were removed and the value of \(r\) was re-calculated using the original data for the remaining four men. State in general terms what effect, if any, this will have on the value of \(r\).
OCR S1 2010 January Q4
4 A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 8 }\)
  1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). The die is thrown 10 times.
  2. Find the probability that there are not more than 4 throws on which the score is 1 .
  3. Find the probability that there are exactly 4 throws on which the score is 2 .
OCR S1 2010 January Q5
5 A washing-up bowl contains 6 spoons, 5 forks and 3 knives. Three of these 14 items are removed at random, without replacement. Find the probability that
  1. all three items are of different kinds,
  2. all three items are of the same kind.
OCR S1 2010 January Q6
6
  1. A student calculated the values of the product moment correlation coefficient, \(r\), and Spearman's rank correlation coefficient, \(r _ { s }\), for two sets of bivariate data, \(A\) and \(B\). His results are given below. $$\begin{array} { l l } A : & r = 0.9 \text { and } r _ { s } = 1
    B : & r = 1 \quad \text { and } r _ { s } = 0.9 \end{array}$$ With the aid of a diagram where appropriate, explain why the student's results for \(A\) could both be correct but his results for \(B\) cannot both be correct.
  2. An old research paper has been partially destroyed. The surviving part of the paper contains the following incomplete information about some bivariate data from an experiment.
    \includegraphics[max width=\textwidth, alt={}, center]{5c25d6cf-2c23-4b49-88fb-e4abe6c281e4-4_339_1200_1117_511} Calculate the missing constant at the end of the equation of the second regression line.
OCR S1 2010 January Q7
7 The table shows the numbers of male and female members of a vintage car club who own either a Jaguar or a Bentley. No member owns both makes of car.
\cline { 2 - 3 } \multicolumn{1}{c|}{}MaleFemale
Jaguar2515
Bentley128
One member is chosen at random from these 60 members.
  1. Given that this member is male, find the probability that he owns a Jaguar. Now two members are chosen at random from the 60 members. They are chosen one at a time, without replacement.
  2. Given that the first one of these members is female, find the probability that both own Jaguars.
OCR S1 2010 January Q8
8 The five letters of the word NEVER are arranged in random order in a straight line.
  1. How many different orders of the letters are possible?
  2. In how many of the possible orders are the two Es next to each other?
  3. Find the probability that the first two letters in the order include exactly one letter E.
    \(9 R\) and \(S\) are independent random variables each having the distribution \(\operatorname { Geo } ( p )\).
  4. Find \(\mathrm { P } ( R = 1\) and \(S = 1 )\) in terms of \(p\).
  5. Show that \(\mathrm { P } ( R = 3\) and \(S = 3 ) = p ^ { 2 } q ^ { 4 }\), where \(q = 1 - p\).
  6. Use the formula for the sum to infinity of a geometric series to show that $$\mathrm { P } ( R = S ) = \frac { p } { 2 - p }$$
OCR S1 2011 January Q2
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
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
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
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.
    (a) How many selections of seven cards are possible?
    (b) Find the probability that the seven cards include exactly one card showing the letter A .
OCR S1 2011 January Q7
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
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
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
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
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
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
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 \(X\).