Questions — OCR S1 (169 questions)

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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 .
OCR S1 2011 June Q4
16 marks Moderate -0.8
4 The table shows information about the time, \(t\) minutes correct to the nearest minute, taken by 50 people to complete a race.
Time (minutes)\(t \leqslant 27\)\(28 \leqslant t \leqslant 30\)\(31 \leqslant t \leqslant 35\)\(36 \leqslant t \leqslant 45\)\(46 \leqslant t \leqslant 60\)\(t \geqslant 61\)
Number of people04281440
  1. In a histogram illustrating the data, the height of the block for the \(31 \leqslant t \leqslant 35\) class is 5.6 cm . Find the height of the block for the \(28 \leqslant t \leqslant 30\) class. (There is no need to draw the histogram.)
  2. The data in the table are used to estimate the median time. State, with a reason, whether the estimated median time is more than 33 minutes, less than 33 minutes or equal to 33 minutes.
  3. Calculate estimates of the mean and standard deviation of the data.
  4. It was found that the winner's time had been incorrectly recorded and that it was actually less than 27 minutes 30 seconds. State whether each of the following will increase, decrease or remain the same:
    1. the mean,
    2. the standard deviation,
    3. the median,
    4. the interquartile range.
OCR S1 2011 June Q5
9 marks Moderate -0.8
5 A bag contains 4 blue discs and 6 red discs. Chloe takes a disc from the bag. If this disc is red, she takes 2 more discs. If not, she takes 1 more disc. Each disc is taken at random and no discs are replaced.
  1. Complete the probability tree diagram in your Answer Book, showing all the probabilities. \includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-4_730_1203_529_511} The total number of blue discs that Chloe takes is denoted by \(X\).
  2. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 5 }\). The complete probability distribution of \(X\) is given below.
    \(x\)012
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 6 }\)\(\frac { 3 } { 5 }\)\(\frac { 7 } { 30 }\)
  3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2011 June Q6
9 marks Moderate -0.3
6 A group of 7 students sit in random order on a bench.
  1. (a) Find the number of orders in which they can sit.
    (b) The 7 students include Tom and Jerry. Find the probability that Tom and Jerry sit next to each other.
  2. The students consist of 3 girls and 4 boys. Find the probability that
    (a) no two boys sit next to each other,
    (b) all three girls sit next to each other.
OCR S1 2011 June Q7
6 marks Moderate -0.8
7 The diagram shows the results of an experiment involving some bivariate data. The least squares regression line of \(y\) on \(x\) for these results is also shown. \includegraphics[max width=\textwidth, alt={}, center]{48ffcd44-d933-40e0-818a-20d6db607298-5_748_919_390_612}
  1. Given that the least squares regression line of \(y\) on \(x\) is used for an estimation, state which of \(x\) or \(y\) is treated as the independent variable.
  2. Use the diagram to explain what is meant by 'least squares'.
  3. State, with a reason, the value of Spearman's rank correlation coefficient for these data.
  4. What can be said about the value of the product moment correlation coefficient for these data?
OCR S1 2011 June Q8
10 marks Standard +0.3
8 Ann, Bill, Chris and Dipak play a game with a fair cubical die. Starting with Ann they take turns, in alphabetical order, to throw the die. This process is repeated as many times as necessary until a player throws a 6 . When this happens, the game stops and this player is the winner. Find the probability that
  1. Chris wins on his first throw,
  2. Dipak wins on his second throw,
  3. Ann gets a third throw,
  4. Bill throws the die exactly three times.
OCR S1 2012 June Q1
9 marks Moderate -0.8
1 For each of the last five years the number of tourists, \(x\) thousands, visiting Sackton, and the average weekly sales, \(\pounds y\) thousands, in Sackton Stores were noted. The table shows the results.
Year20072008200920102011
\(x\)250270264290292
\(y\)4.23.73.23.53.0
  1. Calculate the product moment correlation coefficient \(r\) between \(x\) and \(y\).
  2. It is required to estimate the average weekly sales at Sackton Stores in a year when the number of tourists is 280000 . Calculate the equation of an appropriate regression line, and use it to find this estimate.
  3. Over a longer period the value of \(r\) is - 0.8 . The mayor says, "This shows that having more tourists causes sales at Sackton Stores to decrease." Give a reason why this statement is not correct.
OCR S1 2012 June Q2
6 marks Easy -1.2
2 The masses, \(x \mathrm {~kg}\), of 50 bags of flour were measured and the results were summarised as follows. $$n = 50 \quad \Sigma ( x - 1.5 ) = 1.4 \quad \Sigma ( x - 1.5 ) ^ { 2 } = 0.05$$ Calculate the mean and standard deviation of the masses of these bags of flour.
OCR S1 2012 June Q3
7 marks Easy -1.3
3 The test marks of 14 students are displayed in a stem-and-leaf diagram, as shown below. \includegraphics[max width=\textwidth, alt={}, center]{e23cb28b-49e5-436a-942d-e6320029c634-2_234_261_1425_482} Key: 1 | 6 means 16 marks
  1. Find the lower quartile.
  2. Given that the median is 32 , find the values of \(w\) and \(x\).
  3. Find the possible values of the upper quartile.
  4. State one advantage of a stem-and-leaf diagram over a box-and-whisker plot.
  5. State one advantage of a box-and-whisker plot over a stem-and-leaf diagram.
OCR S1 2012 June Q4
7 marks Easy -1.2
4 A bag contains 5 red discs and 1 black disc. Tina takes two discs from the bag at random without replacement.
  1. The diagram shows part of a tree diagram to illustrate this situation. \section*{First disc}
    \includegraphics[max width=\textwidth, alt={}]{e23cb28b-49e5-436a-942d-e6320029c634-3_264_494_479_550}
    Complete the tree diagram in your Answer Book showing all the probabilities. \section*{Second disc}
  2. Find the probability that exactly one of the two discs is red. All the discs are replaced in the bag. Tony now takes three discs from the bag at random without replacement.
  3. Given that the first disc Tony takes is red, find the probability that the third disc Tony takes is also red.
OCR S1 2012 June Q5
8 marks Easy -1.2
5
  1. Write down the value of Spearman's rank correlation coefficent, \(r _ { s }\), for the following sets of ranks. All the discs are replaced in the bag. Tony now takes three discs from the bag at random without replacement.
  2. Given that the first disc Tony takes is red, find the probability that the third disc Tony takes is also red.
    [0pt] [2
  3. Write down the value of Spearman's rank correlation coefficent, \(r _ { s }\), for the following sets of ranks.
    (b)
    Judge \(A\) ranks1234
    Judge \(C\) ranks4321
    (a)
    (a)
    Judge \(A\) ranks1234
    Judge \(B\) ranks1234
  4. Calculate the value of \(r _ { s }\) for the following ranks.
    Judge \(A\) ranks1234
    Judge \(D\) ranks2413
  5. For each of parts (i)(a), (i)(b) and (ii), describe in everyday terms the relationship between the two judges' opinions.
OCR S1 2012 June Q6
5 marks Moderate -0.3
6 A six-sided die is biased so that the probability of scoring 6 is 0.1 and the probabilities of scoring \(1,2,3,4\), and 5 are all equal. In a game at a fête, contestants pay \(\pounds 3\) to roll this die. If the score is 6 they receive \(\pounds 10\) back. If the score is 5 they receive \(\pounds 5\) back. Otherwise they receive no money back. Find the organiser's expected profit for 100 rolls of the die.
OCR S1 2012 June Q7
9 marks Moderate -0.8
7
  1. 5 of the 7 letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }\) are arranged in a random order in a straight line.
    1. How many different arrangements of 5 letters are possible?
    2. How many of these arrangements end with a vowel (A or E)?
    3. A group of 5 people is to be chosen from a list of 7 people.
      (a) How many different groups of 5 people can be chosen?
      (b) The list of 7 people includes Jill and Jo. A group of 5 people is chosen at random from the list. Given that either Jill and Jo are both chosen or neither of them is chosen, find the probability that both of them are chosen.
OCR S1 2012 June Q8
10 marks Standard +0.8
8
  1. The random variable \(X\) has the distribution \(\mathrm { B } ( 30,0.6 )\). Find \(\mathrm { P } ( X \geqslant 16 )\).
  2. The random variable \(Y\) has the distribution \(\mathrm { B } ( 4,0.7 )\).
    1. Find \(\mathrm { P } ( Y = 2 )\).
    2. Three values of \(Y\) are chosen at random. Find the probability that their total is 10 .
OCR S1 2012 June Q9
11 marks Standard +0.3
9
  1. A clock is designed to chime once each hour, on the hour. The clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 10 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day. Find the probability that the first time it does not chime is
    1. at 0600 on that day,
    2. before 0600 on that day.
    3. Another clock is designed to chime twice each hour: on the hour and at 30 minutes past the hour. This clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 20 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day.
      (a) Find the probability that the first time it does not chime is at either 0030 or 0130 on that day.
      (b) Use the formula for the sum to infinity of a geometric progression to find the probability that the first time it does not chime is at 30 minutes past some hour.