Questions — OCR S1 (160 questions)

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OCR S1 2014 June Q7
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
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
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
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
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
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
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
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
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
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
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
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 2013 January Q1
1 When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values \(2,4,6\) and 8 . The spinner is biased so that \(\mathrm { P } ( X = x ) = k x\), where \(k\) is a constant.
  1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 3 } { 10 }\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Kathryn is allowed three attempts at a high jump. If she succeeds on any attempt, she does not jump again. The probability that she succeeds on her first attempt is \(\frac { 3 } { 4 }\). If she fails on her first attempt, the probability that she succeeds on her second attempt is \(\frac { 3 } { 8 }\). If she fails on her first two attempts, the probability that she succeeds on her third attempt is \(\frac { 3 } { 16 }\). Find the probability that she succeeds.
  4. Khaled is allowed two attempts to pass an examination. If he succeeds on his first attempt, he does not make a second attempt. The probability that he passes at the first attempt is 0.4 and the probability that he passes on either the first or second attempt is 0.58 . Find the probability that he passes on the second attempt, given that he failed on the first attempt.
OCR S1 2013 January Q3
3 The Gross Domestic Product per Capita (GDP), \(x\) dollars, and the Infant Mortality Rate per thousand (IMR), \(y\), of 6 African countries were recorded and summarised as follows. $$n = 6 \quad \sum x = 7000 \quad \sum x ^ { 2 } = 8700000 \quad \sum y = 456 \quad \sum y ^ { 2 } = 36262 \quad \sum x y = 509900$$
  1. Calculate the equation of the regression line of \(y\) on \(x\) for these 6 countries. The original data were plotted on a scatter diagram and the regression line of \(y\) on \(x\) was drawn, as shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{13d8d940-fd63-4b62-bd7a-aa7174f6af4b-3_721_1246_680_408}
  2. The GDP for another country, Tanzania, is 1300 dollars. Use the regression line in the diagram to estimate the IMR of Tanzania.
  3. The GDP for Nigeria is 2400 dollars. Give two reasons why the regression line is unlikely to give a reliable estimate for the IMR for Nigeria.
  4. The actual value of the IMR for Tanzania is 96. The data for Tanzania \(( x = 1300 , y = 96 )\) is now included with the original 6 countries. Calculate the value of the product moment correlation coefficient, \(r\), for all 7 countries.
  5. The IMR is now redefined as the infant mortality rate per hundred instead of per thousand, and the value of \(r\) is recalculated for all 7 countries. Without calculation state what effect, if any, this would have on the value of \(r\) found in part (iv).
OCR S1 2013 January Q4
4
  1. How many different 3-digit numbers can be formed using the digits 1, 2 and 3 when
    (a) no repetitions are allowed,
    (b) any repetitions are allowed,
    (c) each digit may be included at most twice?
  2. How many different 4-digit numbers can be formed using the digits 1, 2 and 3 when each digit may be included at most twice?
OCR S1 2013 January Q5
5 A random variable \(X\) has the distribution \(\mathrm { B } \left( 5 , \frac { 1 } { 4 } \right)\).
  1. Find
    (a) \(\mathrm { E } ( X )\),
    (b) \(\mathrm { P } ( X = 2 )\).
  2. Two values of \(X\) are chosen at random. Find the probability that their sum is less than 2 .
  3. 10 values of \(X\) are chosen at random. Use an appropriate formula to find the probability that exactly 3 of these values are 2 s .
OCR S1 2013 January Q6
6 The masses, \(x\) grams, of 800 apples are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{13d8d940-fd63-4b62-bd7a-aa7174f6af4b-4_592_1363_957_351}
  1. On the frequency density axis, 1 cm represents \(a\) units. Find the value of \(a\).
  2. Find an estimate of the median mass of the apples.
  3. Two judges rank \(n\) competitors, where \(n\) is an even number. Judge 2 reverses each consecutive pair of ranks given by Judge 1, as shown.
    Competitor\(C _ { 1 }\)\(C _ { 2 }\)\(C _ { 3 }\)\(C _ { 4 }\)\(C _ { 5 }\)\(C _ { 6 }\)\(\ldots \ldots\)\(C _ { n - 1 }\)\(C _ { n }\)
    Judge 1 rank123456\(\ldots \ldots\)\(n - 1\)\(n\)
    Judge 2 rank214365\(\ldots \ldots\)\(n\)\(n - 1\)
    Given that the value of Spearman's coefficient of rank correlation is \(\frac { 63 } { 65 }\), find \(n\).
  4. An experiment produced some data from a bivariate distribution. The product moment correlation coefficient is denoted by \(r\), and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).
    (a) Explain whether the statement $$r = 1 \Rightarrow r _ { s } = 1$$ is true or false.
    (b) Use a diagram to explain whether the statement $$r \neq 1 \Rightarrow r _ { s } \neq 1$$ is true or false. 8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 .
  5. Find the probability that
    (a) the first time she succeeds is on her 5th attempt,
    (b) the first time she succeeds is after her 5th attempt,
    (c) the second time she succeeds is before her 4th attempt. Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  6. Find the probability that the first person to hit the target is Sandra, on her
    (a) 2nd attempt,
    (b) 10th attempt. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR S1 2013 January Q8
8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 .
  1. Find the probability that
    (a) the first time she succeeds is on her 5th attempt,
    (b) the first time she succeeds is after her 5th attempt,
    (c) the second time she succeeds is before her 4th attempt. Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  2. Find the probability that the first person to hit the target is Sandra, on her
    (a) 2nd attempt,
    (b) 10th attempt. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
    7
OCR S1 2013 June Q1
1 The lengths, in centimetres, of 18 snakes are given below. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 24 & 62 & 20 & 65 & 27 & 67 & 69 & 32 & 40 & 53 & 55 & 47 & 33 & 45 & 55 & 56 & 49 & 58 \end{array}$$
  1. Draw an ordered stem-and-leaf diagram for the data.
  2. Find the mean and median of the lengths of the snakes.
  3. It was found that one of the lengths had been measured incorrectly. After this length was corrected, the median increased by 1 cm . Give two possibilities for the incorrect length and give a corrected value in each case.
OCR S1 2013 June Q2
2
  1. The table shows the times, in minutes, spent by five students revising for a test, and the grades that they achieved in the test.
    StudentAnnBillCazDenEd
    Time revising0603510045
    GradeCDEBA
    Calculate Spearman's rank correlation coefficient.
  2. The table below shows the ranks given by two judges to four competitors.
    CompetitorPQRS
    Judge 1 rank1234
    Judge 2 rank3214
    Spearman's rank correlation coefficient for these ranks is denoted by \(r _ { s }\). With the same set of ranks for Judge 1, write down a different set of ranks for Judge 2 which gives the same value of \(r _ { s }\). There is no need to find the value of \(r _ { s }\).
OCR S1 2013 June Q3
3 The probability distribution of a random variable \(X\) is shown.
\(x\)1357
\(\mathrm { P } ( X = x )\)0.40.30.20.1
  1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. Three independent values of \(X\), denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), are chosen. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 19\), write down all the possible sets of values for \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and hence find \(\mathrm { P } \left( X _ { 1 } = 7 \right)\).
  3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5 s .
OCR S1 2013 June Q4
4 At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m \mathrm {~kg}\), and for each person the value of \(( m - 5 )\) was recorded. The mean and standard deviation of \(( m - 5 )\) were found to be 0.74 and 0.13 respectively.
  1. Write down the mean and standard deviation of \(m\). The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.
  2. Calculate the mean of all 25 estimates.
  3. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg , comment on this claim.
OCR S1 2013 June Q5
5 The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), \(x \%\), and the Consumer Price Index (CPI), \(y \%\), in the United Kingdom from April 2008 to July 2010.
DateApril 2008July 2008October 2008January 2009April 2009July 2009October 2009January 2010April 2010July 2010
UR, \(x \%\)5.25.76.16.87.57.87.87.97.87.7
CPI, \(y \%\)3.04.44.53.02.31.81.53.53.73.1
These data are summarised below. $$n = 10 \quad \sum x = 70.3 \quad \sum x ^ { 2 } = 503.45 \quad \sum y = 30.8 \quad \sum y ^ { 2 } = 103.94 \quad \sum x y = 211.9$$
  1. Calculate the product moment correlation coefficient, \(r\), for the data, showing that \(- 0.6 < r < - 0.5\).
  2. Karen says "The negative value of \(r\) shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement.
  3. (a) Calculate the equation of the regression line of \(x\) on \(y\).
    (b) Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%.
OCR S1 2013 June Q6
6 The diagram shows five cards, each with a letter on it.
\includegraphics[max width=\textwidth, alt={}, center]{d06430a6-7957-4313-beea-bb320fadb282-4_113_743_315_662} The letters A and E are vowels; the letters B, C and D are consonants.
  1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them.
  2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it.
  3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it.
OCR S1 2013 June Q7
7 In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows.
  • If \(X < 2\), the batch is accepted.
  • If \(X > 2\), the batch is rejected.
  • If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted.
It is given that \(5 \%\) of mugs are defective.
  1. (a) Find the probability that the batch is rejected after just the first sample is checked.
    (b) Show that the probability that the batch is rejected is 0.327 , correct to 3 significant figures.
  2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked.
  3. A bag contains 12 black discs, 10 white discs and 5 green discs. Three discs are drawn at random from the bag, without replacement. Find the probability that all three discs are of different colours.
  4. A bag contains 30 red discs and 20 blue discs. A second bag contains 50 discs, each of which is either red or blue. A disc is drawn at random from each bag. The probability that these two discs are of different colours is 0.54 . Find the number of red discs that were in the second bag at the start.