Questions — OCR S1 (169 questions)

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OCR S1 2014 June Q1
7 marks Easy -1.8
1 The stem-and-leaf diagram shows the heights, in metres to the nearest 0.1 m , of a random sample of trees of species \(A\).
5
59
614
6559
72334
7566678
8034
85
means 6.4 m
  1. Find the median and interquartile range of the heights.
  2. The heights, in metres to the nearest 0.1 m , of a random sample of trees of species \(B\) are given below. \(\begin{array} { l l l } 7.6 & 5.2 & 8.5 \end{array}\) 5.2
    6.3
    6.3
    6.8
    7.2
    6.7
    7.3
    5.4
    7.5
    7.4
    6.0
    6.7 In the answer book, complete the back-to-back stem-and-leaf diagram.
  3. Make two comparisons between the heights of the two species of tree.
OCR S1 2014 June Q2
7 marks Moderate -0.8
2
  1. The probability distribution of a random variable \(W\) is shown in the table.
    \(w\)024
    \(\mathrm { P } ( W = w )\)0.30.40.3
    Calculate \(\operatorname { Var } ( W )\).
  2. The random variable \(X\) has probability distribution given by $$\mathrm { P } ( X = x ) = k ( x + 1 ) \quad \text { for } x = 1,2,3,4 .$$
    1. Show that \(k = \frac { 1 } { 14 }\).
    2. Calculate \(\mathrm { E } ( X )\).
OCR S1 2014 June Q3
7 marks Easy -1.2
3 The table shows information about the numbers of people per household in 280900 households in the northwest of England in 2001.
Number of
people
12345 or more
Number of
households
8690092500450003710019400
  1. Taking ' 5 or more' to mean ' 5 or 6 ', calculate estimates of the mean and standard deviation of the number of people per household.
  2. State the values of the median and upper quartile of the number of people per household.
OCR S1 2014 June Q4
10 marks Moderate -0.8
4 Each time Ben attempts to complete a crossword in his daily newspaper, the probability that he succeeds is \(\frac { 2 } { 3 }\). The random variable \(X\) denotes the number of times that Ben succeeds in 9 attempts.
  1. Find
    1. \(\mathrm { P } ( X = 6 )\),
    2. \(\mathrm { P } ( X < 6 )\),
    3. \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). Ben notes three values, \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), of \(X\).
    4. State the total number of attempts to complete a crossword that are needed to obtain three values of \(X\). Hence find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } = 18 \right)\).
OCR S1 2014 June Q5
9 marks Moderate -0.8
5 Tariq collected information about typical prices, \(\pounds y\) million, of four-bedroomed houses at varying distances, \(x\) miles, from a large city. He chose houses at 10 -mile intervals from the city. His results are shown below.
\(x\)1020304050607080
\(y\)1.21.41.20.90.80.50.50.3
$$n = 8 \quad \Sigma x = 360 \quad \Sigma x ^ { 2 } = 20400 \quad \Sigma y = 6.8 \quad \Sigma y ^ { 2 } = 6.88 \quad \Sigma x y = 241$$
  1. Use an appropriate formula to calculate the product moment correlation coefficient, \(r\), showing that \(- 1.0 < r < - 0.9\).
  2. State what this value of \(r\) shows in this context.
  3. Tariq decides to recalculate the value of \(r\) with the house prices measured in hundreds of thousands of pounds, instead of millions of pounds. State what effect, if any, this will have on the value of \(r\).
  4. Calculate the equation of the regression line of \(y\) on \(x\).
  5. Explain why the regression line of \(y\) on \(x\), rather than \(x\) on \(y\), should be used for estimating a value of \(x\) from a given value of \(y\).
OCR S1 2014 June Q6
5 marks Moderate -0.8
6 Fiona and James collected the results for six hockey teams at the end of the season. They then carried out various calculations using Spearman's rank correlation coefficient, \(r _ { s }\).
  1. Fiona calculated the value of \(r _ { s }\) between the number of goals scored FOR each team and the number of goals scored AGAINST each team. She found that \(r _ { s } = - 1\). Complete the table in the answer book showing the ranks.
    TeamABCDEF
    Number of goals FOR (rank)123456
    Number of goals AGAINST (rank)
  2. James calculated the value of \(r _ { s }\) between the number of goals scored and the number of points gained by the 6 teams. He found the value of \(r _ { s }\) to be 1 . He then decided to include the results of another two teams in the calculation of \(r _ { s }\). The table shows the ranks for these two teams.
    TeamGH
    Number of goals scored (rank)78
    Number of points gained (rank)87
    Calculate the value of \(r _ { s }\) for all 8 teams.
OCR S1 2014 June Q7
8 marks Moderate -0.3
7 The table shows the numbers of members of a swimming club in certain categories.
\cline { 2 - 3 } \multicolumn{1}{c|}{}MaleFemale
Adults7845
Children52\(n\)
It is given that \(\frac { 5 } { 8 }\) of the female members are children.
  1. Find the value of \(n\).
  2. Find the probability that a member chosen at random is either female or a child (or both). The table below shows the corresponding numbers for an athletics club.
    \cline { 2 - 3 } \multicolumn{1}{c|}{}MaleFemale
    Adults64
    Children510
  3. Two members of the athletics club are chosen at random for a photograph.
    1. Find the probability that one of these members is a female child and the other is an adult male.
    2. Find the probability that exactly one of these members is female and exactly one is a child.
OCR S1 2014 June Q8
9 marks Moderate -0.3
8 A group of 8 people, including Kathy, David and Harpreet, are planning a theatre trip.
  1. Four of the group are chosen at random, without regard to order, to carry the refreshments. Find the probability that these 4 people include Kathy and David but not Harpreet.
  2. The 8 people sit in a row. Kathy and David sit next to each other and Harpreet sits at the left-hand end of the row. How many different arrangements of the 8 people are possible?
  3. The 8 people stand in a line to queue for the exit. Kathy and David stand next to each other and Harpreet stands next to them. How many different arrangements of the 8 people are possible?
OCR S1 2014 June Q9
10 marks Moderate -0.3
9 Each day Harry makes repeated attempts to light his gas fire. If the fire lights he makes no more attempts. On each attempt, the probability that the fire will light is 0.3 independent of all other attempts. Find the probability that
  1. the fire lights on the 5th attempt,
  2. Harry needs more than 1 attempt but fewer than 5 attempts to light the fire. If the fire does not light on the 6th attempt, Harry stops and the fire remains unlit.
  3. Find the probability that, on a particular day, the fire lights.
  4. Harry's week starts on Monday. Find the probability that, during a certain week, the first day on which the fire lights is Wednesday.
OCR S1 2015 June Q1
6 marks Moderate -0.8
1 For the top 6 clubs in the 2010/11 season of the English Premier League, the table shows the annual salary, \(\pounds x\) million, of the highest paid player and the number of points scored, \(y\).
ClubManchester UnitedManchester CityChelseaArsenalTottenhamLiverpool
\(x\)5.67.46.54.13.66.5
\(y\)807171686258
$$n = 6 \quad \sum x = 33.7 \quad \sum x ^ { 2 } = 200.39 \quad \sum y = 410 \quad \sum y ^ { 2 } = 28314 \quad \sum x y = 2313.9$$
  1. Use a suitable formula to calculate the product moment correlation coefficient, \(r\), between \(x\) and \(y\), showing that \(0 < r < 0.2\).
  2. State what this value of \(r\) shows in this context.
  3. A fan suggests that the data should be used to draw a regression line in order to estimate the number of points that would be scored by another Premier League club, whose highest paid player's salary is \(\pounds 1.7\) million. Give two reasons why such an estimate would be unlikely to be reliable.
OCR S1 2015 June Q2
10 marks Easy -1.3
2 The masses, in grams, of 400 plums were recorded. The masses were then collected into class intervals of width 5 g and a cumulative frequency graph was drawn, as shown below. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-3_1045_1401_358_333}
  1. Find the number of plums with masses in the interval 40 g to 45 g .
  2. Find the percentage of plums with masses greater than 70 g .
  3. Give estimates of the highest and lowest masses in the sample, explaining why their exact values cannot be read from the graph.
  4. On the graph paper in the answer book, draw a box-and-whisker plot to illustrate the masses of the plums in the sample.
  5. Comment briefly on the shape of the distribution of masses.
OCR S1 2015 June Q3
6 marks Moderate -0.8
3 An expert tested the quality of the wines produced by a vineyard in 9 particular years. He placed them in the following order, starting with the best. $$\begin{array} { l l l l l l l l l } 1980 & 1983 & 1981 & 1982 & 1984 & 1985 & 1987 & 1986 & 1988 \end{array}$$
  1. Calculate Spearman's rank correlation coefficient, \(r _ { s }\), between the year of production and the quality of these wines. The years should be ranked from the earliest (1) to the latest (9).
  2. State what this value of \(r _ { s }\) shows in this context.
OCR S1 2015 June Q4
9 marks Moderate -0.3
4 The table shows the load a lorry was carrying, \(x\) tonnes, and the fuel economy, \(y \mathrm {~km}\) per litre, for 8 different journeys. You should assume that neither variable is controlled.
Load
\(( x\) tonnes \()\)
5.15.86.57.17.68.49.510.5
Fuel economy
\(( y \mathrm {~km}\) per litre \()\)
6.26.15.95.65.35.45.35.1
$$n = 8 \quad \sum x = 60.5 \quad \sum y = 44.9 \quad \sum x ^ { 2 } = 481.13 \quad \sum y ^ { 2 } = 253.17 \quad \sum x y = 334.65$$
  1. Calculate the equation of the regression line of \(y\) on \(x\).
  2. Estimate the fuel economy for a load of 9.2 tonnes.
  3. An analyst calculated the equation of the regression line of \(x\) on \(y\). Without calculating this equation, state the coordinates of the point where the two regression lines intersect.
  4. Describe briefly the method required to estimate the load when the fuel economy is 5.8 km per litre.
OCR S1 2015 June Q5
10 marks Standard +0.3
5 Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in any particular year is 0.27 .
  1. Find the probability that the first time Jack wins a ticket is
    1. on his 8th attempt,
    2. after his 8th attempt.
    3. Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.
    4. Find the probability that Jack wins his 3rd ticket on his 9th attempt and his 4th ticket on his 12th attempt.
OCR S1 2015 June Q6
8 marks Moderate -0.8
6
  1. The seven digits \(1,1,2,3,4,5,6\) are arranged in a random order in a line. Find the probability that they form the number 1452163.
  2. Three of the seven digits \(1,1,2,3,4,5,6\) are chosen at random, without regard to order.
    1. How many possible groups of three digits contain two 1s?
    2. How many possible groups of three digits contain exactly one 1?
    3. How many possible groups of three digits can be formed altogether?
OCR S1 2015 June Q7
8 marks Standard +0.3
7 Froox sweets are packed into tubes of 10 sweets, chosen at random. \(25 \%\) of Froox sweets are yellow.
  1. Find the probability that in a randomly selected tube of Froox sweets there are
    1. exactly 3 yellow sweets,
    2. at least 3 yellow sweets.
    3. Find the probability that in a box containing 6 tubes of Froox sweets, there is at least 1 tube that contains at least 3 yellow sweets.
OCR S1 2015 June Q8
9 marks Standard +0.3
8 A game is played with a fair, six-sided die which has 4 red faces and 2 blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or until the die has been thrown 4 times.
  1. In the answer book, complete the probability tree diagram for one turn. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-5_314_302_1000_884}
  2. Find the probability that in one particular turn the die is thrown 4 times.
  3. Adnan and Beryl each have one turn. Find the probability that Adnan throws the die more times than Beryl.
  4. State one change that needs to be made to the rules so that the number of throws in one turn will have a geometric distribution.
OCR S1 2015 June Q9
6 marks Moderate -0.3
9 The random variable \(X\) has probability distribution given by $$\mathrm { P } ( X = x ) = a + b x \quad \text { for } x = 1,2 \text { and } 3 ,$$ where \(a\) and \(b\) are constants.
  1. Show that \(3 a + 6 b = 1\).
  2. Given that \(\mathrm { E } ( X ) = \frac { 5 } { 3 }\), find \(a\) and \(b\).
OCR S1 2011 January Q5
10 marks Moderate -0.8
  1. The number of free gifts that Jan receives in a week is denoted by \(X\). Name a suitable probability distribution with which to model \(X\), giving the value(s) of any parameter(s). State any assumption(s) necessary for the distribution to be a valid model. Assume now that your model is valid.
  2. Find
    1. \(\mathrm { P } ( X \leqslant 2 )\),
    2. \(\mathrm { P } ( X = 2 )\).
    3. Find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts. 6
    4. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number. {}
      3
      2
    5. 2
    6. 2
    7. 2
    8. \multirow[t]{21}{*}{3
    9. }
      10. \multirow[t]{4}{*}{3
    10. }
      11. 3
    11. {}
OCR S1 2012 January Q7
8 marks Moderate -0.8
  1. State a suitable distribution that can be used as a model for \(X\), giving the value(s) of any parameter(s). State also any necessary condition(s) for this distribution to be a good model. Use the distribution stated in part (i) to find
  2. \(\mathrm { P } ( X = 4 )\),
  3. \(\mathrm { P } ( X \geqslant 4 )\).
OCR S1 Q3
8 marks Moderate -0.8
3 In a supermarket the proportion of shoppers who buy washing powder is denoted by \(p\). 16 shoppers are selected at random.
  1. Given that \(p = 0.35\), use tables to find the probability that the number of shoppers who buy washing powder is
    1. at least 8,
    2. between 4 and 9 inclusive.
    3. Given instead that \(p = 0.38\), find the probability that the number of shoppers who buy washing powder is exactly 6 . \section*{June 2005}
OCR S1 Q4
8 marks Moderate -0.3
4 The table shows the latitude, \(x\) (in degrees correct to 3 significant figures), and the average rainfall \(y\) (in cm correct to 3 significant figures) of five European cities.
City\(x\)\(y\)
Berlin52.558.2
Bucharest44.458.7
Moscow55.853.3
St Petersburg60.047.8
Warsaw52.356.6
$$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$
  1. Calculate the product moment correlation coefficient.
  2. The values of \(y\) in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54. State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
  3. It is required to estimate the annual rainfall at Bergen, where \(x = 60.4\). Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate. \section*{June 2005}
OCR S1 Q5
13 marks Moderate -0.8
5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-04_1335_1319_404_413} Use the curve to estimate
  1. the interquartile range of the marks,
  2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
  3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
  4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
  5. What does this information suggest about the estimate of the interquartile range found in part (i)? \section*{June 2005}
OCR S1 Q6
13 marks Standard +0.3
6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-05_858_980_573_585}
  1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).
    \(r\)0123
    \(\mathrm { P } ( R = r )\)\(\frac { 1 } { 10 }\)\(k\)\(\frac { 9 } { 20 }\)\(\frac { 1 } { 5 }\)
  2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
  3. Find the value of \(k\).
  4. Calculate the mean and variance of \(R\).
OCR S1 Q7
14 marks Moderate -0.3
7 A committee of 7 people is to be chosen at random from 18 volunteers.
  1. In how many different ways can the committee be chosen? The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. The committee is to be chosen randomly. Find the probability that the committee will
  2. consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
  3. include exactly 5 people from Worcester,
  4. include at least 2 people from each of the three cities. 1 Jenny and John are each allowed two attempts to pass an examination.
  5. Jenny estimates that her chances of success are as follows.
    • The probability that she will pass on her first attempt is \(\frac { 2 } { 3 }\).
    • If she fails on her first attempt, the probability that she will pass on her second attempt is \(\frac { 3 } { 4 }\). Calculate the probability that Jenny will pass.
    • John estimates that his chances of success are as follows.
    • The probability that he will pass on his first attempt is \(\frac { 2 } { 3 }\).
    • Overall, the probability that he will pass is \(\frac { 5 } { 6 }\).
    Calculate the probability that if John fails on his first attempt, he will pass on his second attempt. 2 For each of 50 plants, the height, \(h \mathrm {~cm}\), was measured and the value of ( \(h - 100\) ) was recorded. The mean and standard deviation of \(( h - 100 )\) were found to be 24.5 and 4.8 respectively.
  6. Write down the mean and standard deviation of \(h\). The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and 5.1 cm respectively.
  7. Describe briefly how the heights of the second group of plants compare with the first.
  8. Calculate the mean height of all 150 plants. 3 In Mr Kendall's cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately his daughter has removed all the labels for a school project and so the tins are identical in appearance. Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he has opened the two tins of pineapples. Let \(X\) be the number of tins that Mr Kendall opens.
  9. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }\).
  10. The probability distribution of \(X\) is given in the table below.
    \(x\)2345
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 10 }\)\(\frac { 1 } { 5 }\)\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)
    Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).