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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 MEI S1 2012 January Q1
7 marks Easy -1.8
1 The mean daily maximum temperatures at a research station over a 12-month period, measured to the nearest degree Celsius, are given below.
JanFebMarAprMayJunJulAugSepOctNovDec
8152529313134363426158
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(0,10 , \ldots\).
  2. Write down the median of these data.
  3. The mean of these data is 24.3 . Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.
OCR MEI S1 2012 January Q2
7 marks Moderate -0.8
2 The hourly wages, \(\pounds x\), of a random sample of 60 employees working for a company are summarised as follows. $$n = 60 \quad \sum x = 759.00 \quad \sum x ^ { 2 } = 11736.59$$
  1. Calculate the mean and standard deviation of \(x\).
  2. The workers are offered a wage increase of \(2 \%\). Use your answers to part (i) to deduce the new mean and standard deviation of the hourly wages after this increase.
  3. As an alternative the workers are offered a wage increase of 25 p per hour. Write down the new mean and standard deviation of the hourly wages after this 25p increase.
OCR MEI S1 2012 January Q3
8 marks Standard +0.3
3 Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is
  • 0.7 if he won the previous set,
  • 0.4 if Alan won the previous set.
It is not possible to draw a set.
  1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets.
  2. Find the probability that Alan wins the match.
  3. Find the probability that the match ends after exactly four sets have been played.
OCR MEI S1 2012 January Q4
6 marks Moderate -0.8
4 In a food survey, a large number of people are asked whether they like tomato soup, mushroom soup, both or neither. One of these people is selected at random.
  • \(T\) is the event that this person likes tomato soup.
  • \(M\) is the event that this person likes mushroom soup.
You are given that \(\mathrm { P } ( T ) = 0.55 , \mathrm { P } ( M ) = 0.33\) and \(\mathrm { P } ( T \mid M ) = 0.80\).
  1. Use this information to show that the events \(T\) and \(M\) are not independent.
  2. Find \(\mathrm { P } ( T \cap M )\).
  3. Draw a Venn diagram showing the events \(T\) and \(M\), and fill in the probability corresponding to each of the four regions of your diagram.
OCR MEI S1 2012 January Q5
8 marks Standard +0.3
5 A couple plan to have at least one child of each sex, after which they will have no more children. However, if they have four children of one sex, they will have no more children. You should assume that each child is equally likely to be of either sex, and that the sexes of the children are independent. The random variable \(X\) represents the total number of girls the couple have.
  1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 11 } { 16 }\). The table shows the probability distribution of \(X\).
    \(r\)01234
    \(\mathrm { P } ( X = r )\)\(\frac { 1 } { 16 }\)\(\frac { 11 } { 16 }\)\(\frac { 1 } { 8 }\)\(\frac { 1 } { 16 }\)\(\frac { 1 } { 16 }\)
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2012 January Q6
17 marks Moderate -0.3
6 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.
  1. (A) Find the probability that there are exactly 4 smokers in the sample.
    (B) Find the probability that there are at least 3 but no more than 6 smokers in the sample.
    (C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Explain why the alternative hypothesis has the form that it does.
  3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  4. In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 2012 January Q7
19 marks Moderate -0.3
7 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_917_1146_367_447}
  1. Estimate the percentage of lambs with birth weight over 6 kg .
  2. Estimate the median and interquartile range of the data.
  3. Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data. The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep. \includegraphics[max width=\textwidth, alt={}, center]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_328_1616_1749_260}
  4. Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
  5. The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.
  6. One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .