Questions S1 (1967 questions)

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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.
    (a) Find the probability that one of these members is a female child and the other is an adult male.
    (b) 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
    (a) on his 8th attempt,
    (b) after his 8th attempt.
  2. Write down an expression for the probability that Jack wins a ticket on exactly 2 of his first 8 attempts, and evaluate this expression.
  3. 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.
    (a) How many possible groups of three digits contain two 1s?
    (b) How many possible groups of three digits contain exactly one 1?
    (c) 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
    (a) exactly 3 yellow sweets,
    (b) at least 3 yellow sweets.
  2. 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 2010 January Q1
8 marks Easy -1.3
1 A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$\begin{array} { l l l l l l l l l l l l l l l } 73 & 67 & 75 & 64 & 52 & 63 & 75 & 81 & 77 & 72 & 68 & 74 & 79 & 72 & 71 \end{array}$$
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(50,60 , \ldots\).
  2. Write down the median and midrange of the data.
  3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer.
OCR MEI S1 2010 January Q2
8 marks Moderate -0.3
2 In her purse, Katharine has two \(\pounds 5\) notes, two \(\pounds 10\) notes and one \(\pounds 20\) note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(\pounds X\).
  1. (A) Show that \(\mathrm { P } ( X = 10 ) = 0.1\).
    (B) Show that \(\mathrm { P } ( X = 30 ) = 0.2\). The table shows the probability distribution of \(X\).
    \(r\)1015202530
    \(\mathrm { P } ( X = r )\)0.10.40.10.20.2
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2010 January Q3
8 marks Easy -1.2
3 In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random.
  • \(G\) is the event that this person goes to the gym.
  • \(R\) is the event that this person goes running.
You are given that \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).
  1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
  2. Determine whether the events \(G\) and \(R\) are independent.
  3. Find \(\mathrm { P } ( R \mid G )\).
OCR MEI S1 2010 January Q4
5 marks Moderate -0.8
4 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.
  1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
  2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.
OCR MEI S1 2010 January Q6
4 marks Moderate -0.8
6 Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if
  1. no student may win more than 1 prize,
  2. no student may win all 3 prizes. Section B (36 marks)
OCR MEI S1 2010 January Q7
19 marks Moderate -0.8
7 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears.
\includegraphics[max width=\textwidth, alt={}, center]{2f39c509-5429-4193-9526-15fb45b18a38-4_837_1651_466_246}
  1. Calculate the number of pears which are between 90 and 100 mm long.
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
  3. Calculate an estimate of the standard deviation.
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
  5. Name the type of skewness of the distribution.
  6. Illustrate the data using a cumulative frequency diagram.
OCR MEI S1 2010 January Q8
17 marks Standard +0.3
8 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.
Pollution levelLowMediumHigh
Probability0.50.350.15
  1. Three days are chosen at random. Find the probability that the pollution level is
    (A) low on all 3 days,
    (B) low on at least one day,
    (C) low on one day, medium on another day, and high on the other day.
  2. Ten days are chosen at random. Find the probability that
    (A) there are no days when the pollution level is high,
    (B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
  3. Carry out a test at the \(5 \%\) level to determine if there is evidence to suggest that she is correct. Use hypotheses \(\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(\mathrm { H } _ { 1 }\) has this form.
OCR MEI S1 2011 January Q1
3 marks Easy -1.3
1 The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas.
839
935666899
100223469
112478
12345
132
Key: 11 | 2 represents 1120 grams
  1. Find the mode and the median of the data.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 2011 January Q2
4 marks Moderate -0.3
2 The table shows all the possible products of the scores on two fair four-sided dice.
\multirow{2}{*}{}Score on second die
1234
\multirow{4}{*}{Score on first die}11234
22468
336912
4481216
  1. Find the probability that the product of the two scores is less than 10 .
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent.
OCR MEI S1 2011 January Q3
6 marks Moderate -0.8
3 There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.
  1. In how many different ways can the three men be selected?
  2. In how many different ways can the whole committee be selected?
  3. A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women.
OCR MEI S1 2011 January Q5
8 marks Easy -1.3
5 Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel.
\includegraphics[max width=\textwidth, alt={}, center]{7df6dcad-790d-4d0e-b1a5-3371103997d9-3_732_1141_792_500} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus,
  2. Andy walks or travels by bike,
  3. the weather is dry given that Andy walks or travels by bike.
OCR MEI S1 2011 January Q6
8 marks Moderate -0.8
6 A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.
  • Reducing car use
  • Insulating their homes
  • Avoiding air travel
The numbers of citizens who have used each of these methods are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7df6dcad-790d-4d0e-b1a5-3371103997d9-4_703_1087_712_529} One of the citizens is selected at random.
  1. Find the probability that this citizen
    (A) has avoided air travel,
    (B) has used at least two of the three methods.
  2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. Three of the citizens are selected at random.
  3. Find the probability that none of them have avoided air travel. Section B (36 marks)
OCR MEI S1 2011 January Q7
19 marks Moderate -0.3
7 The incomes of a sample of 918 households on an island are given in the table below.
Income
\(( x\) thousand pounds \()\)
\(0 \leqslant x \leqslant 20\)\(20 < x \leqslant 40\)\(40 < x \leqslant 60\)\(60 < x \leqslant 100\)\(100 < x \leqslant 200\)
Frequency23836514212845
  1. Draw a histogram to illustrate the data.
  2. Calculate an estimate of the mean income.
  3. Calculate an estimate of the standard deviation of the incomes.
  4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation.
  5. The incomes were converted into another currency using the formula \(y = 1.15 x\). Calculate estimates of the mean and variance of the incomes in the new currency.