Questions S1 (2020 questions)

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Edexcel S1 2014 June Q7
10 marks Standard +0.3
7. One event at Pentor sports day is throwing a tennis ball. The distance a child throws a tennis ball is modelled by a normal distribution with mean 32 m and standard deviation 12 m . Any child who throws the tennis ball more than 50 m is awarded a gold certificate.
  1. Show that, to 3 significant figures, 6.68\% of children are awarded a gold certificate. A silver certificate is awarded to any child who throws the tennis ball more than \(d\) metres but less than 50 m . Given that 19.1\% of the children are awarded a silver certificate,
  2. find the value of \(d\). Three children are selected at random from those who take part in the throwing a tennis ball event.
  3. Find the probability that 1 is awarded a gold certificate and 2 are awarded silver certificates. Give your answer to 2 significant figures.
Edexcel S1 2015 June Q1
4 marks Moderate -0.8
  1. The discrete random variable \(X\) can only take the values \(1,2,3\) and 4 For these values the cumulative distribution function is defined by
$$\mathrm { F } ( x ) = k x ^ { 2 } \text { for } x = 1,2,3,4$$ where \(k\) is a constant.
  1. Find the value of \(k\).
  2. Find the probability distribution of \(X\).
Edexcel S1 2015 June Q2
13 marks Moderate -0.3
2. Paul believes there is a relationship between the value and the floor size of a house. He takes a random sample of 20 houses and records the value, \(\pounds v\), and the floor size, \(s \mathrm {~m} ^ { 2 }\) The data were coded using \(x = \frac { s - 50 } { 10 }\) and \(y = \frac { v } { 100000 }\) and the following statistics obtained. $$\sum x = 441.5 , \quad \sum y = 59.8 , \quad \sum x ^ { 2 } = 11261.25 , \quad \sum y ^ { 2 } = 196.66 , \quad \sum x y = 1474.1$$
  1. Find the value of \(S _ { x y }\) and the value of \(S _ { x x }\)
  2. Find the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\) The least squares regression line of \(v\) on \(s\) is \(v = c + d s\)
  3. Show that \(d = 1020\) to 3 significant figures and find the value of \(c\)
  4. Estimate the value of a house of floor size \(130 \mathrm {~m} ^ { 2 }\)
  5. Interpret the value \(d\) Paul wants to increase the value of his house. He decides to add an extension to increase the floor size by \(31 \mathrm {~m} ^ { 2 }\)
  6. Estimate the increase in the value of Paul's house after adding the extension.
Edexcel S1 2015 June Q3
8 marks Easy -1.3
  1. A company employs 90 administrators. The length of time that they have been employed by the company and their gender are summarised in the table below.
Length of time employed, \(x\) yearsFemaleMale
\(x < 4\)916
\(4 \leqslant x < 10\)1420
\(10 \leqslant x\)724
One of the 90 administrators is selected at random.
  1. Find the probability that the administrator is female.
  2. Given that the administrator has been employed by the company for less than 4 years, find the probability that this administrator is male.
  3. Given that the administrator has been employed by the company for less than 10 years, find the probability that this administrator is male.
  4. State, with a reason, whether or not the event 'selecting a male' is independent of the event 'selecting an administrator who has been employed by the company for less than 4 years'.
Edexcel S1 2015 June Q4
9 marks Easy -1.3
  1. A bag contains 19 red beads and 1 blue bead only.
Linda selects a bead at random from the bag. She notes its colour and replaces the bead in the bag. She then selects a second bead at random from the bag and notes its colour. Find the probability that
  1. both beads selected are blue,
  2. exactly one bead selected is red. In another bag there are 9 beads, 4 of which are green and the rest are yellow.
    Linda selects 3 beads from this bag at random without replacement.
  3. Find the probability that 2 of these beads are yellow and 1 is green. Linda replaces the 3 beads and then selects another 4 at random without replacement.
  4. Find the probability that at least 1 of the beads is green.
Edexcel S1 2015 June Q5
12 marks Moderate -0.3
Police measure the speed of cars passing a particular point on a motorway. The random variable \(X\) is the speed of a car. \(X\) is modelled by a normal distribution with mean 55 mph (miles per hour).
  1. Draw a sketch to illustrate the distribution of \(X\). Label the mean on your sketch. The speed limit on the motorway is 70 mph . Car drivers can choose to travel faster than the speed limit but risk being caught by the police. The distribution of \(X\) has a standard deviation of 20 mph .
  2. Find the percentage of cars that are travelling faster than the speed limit. The fastest \(1 \%\) of car drivers will be banned from driving.
  3. Show that the lowest speed, correct to 3 significant figures, for a car driver to be banned is 102 mph . Show your working clearly. Car drivers will just be given a caution if they are travelling at a speed \(m\) such that $$\mathrm { P } ( 70 < X < m ) = 0.1315$$
  4. Find the value of \(m\). Show your working clearly.
Edexcel S1 2015 June Q6
9 marks Moderate -0.8
The random variable \(X\) has a discrete uniform distribution and takes the values \(1,2,3,4\) Find
  1. \(\mathrm { F } ( 3 )\), where \(\mathrm { F } ( x )\) is the cumulative distribution function of \(X\),
  2. \(\mathrm { E } ( X )\).
  3. Show that \(\operatorname { Var } ( X ) = \frac { 5 } { 4 }\) The random variable \(Y\) has a discrete uniform distribution and takes the values $$3,3 + k , 3 + 2 k , 3 + 3 k$$ where \(k\) is a constant.
  4. Write down \(\mathrm { P } ( Y = y )\) for \(y = 3,3 + k , 3 + 2 k , 3 + 3 k\) The relationship between \(X\) and \(Y\) may be written in the form \(Y = k X + c\) where \(c\) is a constant.
  5. Find \(\operatorname { Var } ( Y )\) in terms of \(k\).
  6. Express \(c\) in terms of \(k\).
Edexcel S1 2015 June Q7
6 marks Easy -1.8
7. A doctor is investigating the correlation between blood protein, \(p\), and body mass index, \(b\). He takes a random sample of 8 patients and the data are shown in the table below.
Patient\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
\(b\)3236404442212737
\(p\)1821313921121970
  1. Draw a scatter diagram of these data on the axes provided. \includegraphics[max width=\textwidth, alt={}, center]{36cf6341-1957-45b9-9f7d-0914506f5919-13_938_673_785_614} The doctor decides to leave out patient \(H\) from his calculations.
  2. Give a reason for the doctor's decision. For the 7 patients \(A , B , C , D , E , F\) and \(G\), $$S _ { b p } = 369 , \quad S _ { p p } = 490 \text { and } S _ { b b } = 423 \frac { 5 } { 7 }$$
  3. Find the product moment correlation coefficient, \(r\), for these 7 patients.
  4. Without any further calculations, state how \(r\) would differ from your answer in part (c) if it was calculated for all 8 patients. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{36cf6341-1957-45b9-9f7d-0914506f5919-15_1322_1593_207_173} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The histogram in Figure 1 summarises the times, in minutes, that 200 people spent shopping in a supermarket.
    1. Give a reason to justify the use of a histogram to represent these data. Given that 40 people spent between 11 and 21 minutes shopping in the supermarket, estimate
    2. the number of people that spent between 18 and 25 minutes shopping in the supermarket,
    3. the median time spent shopping in the supermarket by these 200 people. The mid-point of each bar is represented by \(x\) and the corresponding frequency by f .
    4. Show that \(\sum \mathrm { f } x = 6390\) Given that \(\sum \mathrm { f } x ^ { 2 } = 238430\)
    5. for the data shown in the histogram, calculate estimates of
      1. the mean,
      2. the standard deviation. A coefficient of skewness is given by \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\)
    6. Calculate this coefficient of skewness for these data. The manager of the supermarket decides to model these data with a normal distribution.
    7. Comment on the manager's decision. Give a justification for your answer.
Edexcel S1 2004 January Q1
13 marks Moderate -0.8
  1. An office has the heating switched on at 7.00 a.m. each morning. On a particular day, the temperature of the office, \(t { } ^ { \circ } \mathrm { C }\), was recorded \(m\) minutes after 7.00 a.m. The results are shown in the table below.
\(m\)01020304050
\(t\)6.08.911.813.515.316.1
  1. Calculate the exact values of \(S _ { m t }\) and \(S _ { m m }\).
  2. Calculate the equation of the regression line of \(t\) on \(m\) in the form \(t = a + b m\).
  3. Use your equation to estimate the value of \(t\) at 7.35 a.m.
  4. State, giving a reason, whether or not you would use the regression equation in (b) to estimate the temperature
    1. at 9.00 a.m. that day,
    2. at 7.15 a.m. one month later.
Edexcel S1 2004 January Q2
7 marks Easy -1.2
2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\).
  1. Write down 3 properties of the distribution of \(X\). Given that \(\mu = 27\) and \(\sigma = 10\)
  2. find \(\mathrm { P } ( 26 < X < 28 )\).
Edexcel S1 2004 January Q3
10 marks Easy -1.3
3. A discrete random variable \(X\) has the probability function shown in the table below.
\(x\)0123
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 12 }\)
Find
  1. \(\mathrm { P } ( 1 < X \leq 3 )\),
  2. \(\mathrm { F } ( 2.6 )\),
  3. \(\mathrm { E } ( X )\),
  4. \(\mathrm { E } ( 2 X - 3 )\),
  5. \(\operatorname { Var } ( X )\)
Edexcel S1 2004 January Q4
11 marks Moderate -0.8
4. \(\quad\) The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 2 } { 5 } , \mathrm { P } ( B ) = \frac { 1 } { 2 }\) and \(\mathrm { P } \left( A \quad B ^ { \prime } \right) = \frac { 4 } { 5 }\).
  1. Find
    1. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
    2. \(\mathrm { P } ( A \cap B )\),
    3. \(\mathrm { P } ( A \cup B )\),
    4. \(\mathrm { P } \left( \begin{array} { l l } A & B \end{array} \right)\).
  2. State, with a reason, whether or \(\operatorname { not } A\) and \(B\) are
    1. mutually exclusive,
    2. independent.
Edexcel S1 2004 January Q5
18 marks Moderate -0.3
5. The values of daily sales, to the nearest \(\pounds\), taken at a newsagents last year are summarised in the table below.
SalesNumber of days
\(1 - 200\)166
\(201 - 400\)100
\(401 - 700\)59
\(701 - 1000\)30
\(1001 - 1500\)5
  1. Draw a histogram to represent these data.
  2. Use interpolation to estimate the median and inter-quartile range of daily sales.
  3. Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
  4. State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
    (2)
Edexcel S1 2004 January Q6
16 marks Moderate -0.3
6. One of the objectives of a computer game is to collect keys. There are three stages to the game. The probability of collecting a key at the first stage is \(\frac { 2 } { 3 }\), at the second stage is \(\frac { 1 } { 2 }\), and at the third stage is \(\frac { 1 } { 4 }\).
  1. Draw a tree diagram to represent the 3 stages of the game.
  2. Find the probability of collecting all 3 keys.
  3. Find the probability of collecting exactly one key in a game.
  4. Calculate the probability that keys are not collected on at least 2 successive stages in a game.
CAIE S1 2020 Specimen Q1
5 marks Easy -1.2
1 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males.
FemalesMales
(4)5200203(1)
(9)98876400021007(3)
(8)8753310022004566(6)
(6)64210023002335677(9)
(6)754000240112556889(10)
(4)9500253457789(7)
(2)5026046(3)
Key: 2 | 20 | 3 means \\(20200for females and \\)20300 for males.
  1. Find the median and the quartiles of the females' salaries.
    You are given that the median salary of the males is \(\\) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(\\) 25300$.
  2. Draw a pair of box-and-whisker plots in a single diagram on the grid below to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-02_994_1589_1736_310}
CAIE S1 2020 Specimen Q2
4 marks Easy -1.2
2 A summary of the speeds, \(x\) kilometres per hour, of 22 cars passing a certain point gave the following information: $$\Sigma ( x - 50 ) = 81.4 \text { and } \Sigma ( x - 50 ) ^ { 2 } = 671.0 .$$ Find the variance of the speeds and hence find the value of \(\Sigma x ^ { 2 }\).
CAIE S1 2020 Specimen Q3
7 marks Moderate -0.5
3 A book club sends 6 paperback and 2 hardback books to Mrs Hunt. She chooses 4 of these books at random to take with her on holiday. The random variable \(X\) represents the number of paperback books she chooses.
  1. Show that the probability that she chooses exactly 2 paperback books is \(\frac { 3 } { 14 }\).
  2. Draw up the probability distribution table for \(X\).
  3. You are given that \(\mathrm { E } ( X ) = 3\). Find \(\operatorname { Var } ( X )\).
CAIE S1 2020 Specimen Q4
10 marks Moderate -0.5
4 A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
  1. Find on how many days of the year (365 days) the daily sales can be expected to exceed 3900 litres.
    The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
  2. Find the value of \(m\).
  3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
CAIE S1 2020 Specimen Q5
7 marks Moderate -0.5
5 A fair six-sided die, with faces marked 1, 2, 3, 4, 5, 6, is thrown 90 times.
  1. Use an approximation to find the probability that a 3 is obtained fewer than 18 times.
  2. Justify your use of the approximation in part (a).
    On another occasion, the same die is thrown repeatedly until a 3 is obtained.
  3. Find the probability that obtaining a 3 requires fewer than 7 throws.
CAIE S1 2020 Specimen Q6
7 marks Standard +0.3
6 A group of 8 friends travels to the airport in two taxis, \(P\) and \(Q\). Each taxi can take 4 passengers.
  1. The 8 friends divide themselves into two groups of 4, one group for taxi \(P\) and one group for taxi \(Q\), with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_272_456_242_461} \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-11_281_455_233_1151} Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi \(P\) and Jon and Sarah sit in the back of taxi \(P\) next to each other.
  2. Find the number of different seating arrangements that are now possible for the 8 friends.
CAIE S1 2020 Specimen Q7
10 marks Standard +0.3
7 Bag \(A\) contains 4 balls numbered 2, 4, 5, 8. Bag \(B\) contains 5 balls numbered 1, 3, 6, 8, 8. Bag \(C\) contains 7 balls numbered \(2,7,8,8,8,8,9\). One ball is selected at random from each bag.
  • Event \(X\) is 'exactly two of the selected balls have the same number'.
  • Event \(Y\) is 'the ball selected from bag \(A\) has number 4'.
    1. Find \(\mathrm { P } ( X )\).
    2. Find \(\mathrm { P } ( X \cap Y )\) and hence determine whether or not events \(X\) and \(Y\) are independent.
    3. Find the probability that two balls are numbered 2, given that exactly two of the selected balls have the same number.
OCR S1 2009 January Q1
8 marks Easy -1.2
1 Each time a certain triangular spinner is spun, it lands on one of the numbers 0,1 and 2 with probabilities as shown in the table.
NumberProbability
00.7
10.2
20.1
The spinner is spun twice. The total of the two numbers on which it lands is denoted by \(X\).
  1. Show that \(\mathrm { P } ( X = 2 ) = 0.18\). The probability distribution of \(X\) is given in the table.
    \(x\)01234
    \(\mathrm { P } ( X = x )\)0.490.280.180.040.01
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2009 January Q2
8 marks Moderate -0.8
2 The table shows the age, \(x\) years, and the mean diameter, \(y \mathrm {~cm}\), of the trunk of each of seven randomly selected trees of a certain species.
Age \(( x\) years \()\)11122028354551
Mean trunk diameter \(( y \mathrm {~cm} )\)12.216.026.439.239.651.360.6
$$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$
  1. (a) Use an appropriate formula to show that the gradient of the regression line of \(y\) on \(x\) is 1.13 , correct to 2 decimal places.
    (b) Find the equation of the regression line of \(y\) on \(x\).
  2. Use your equation to estimate the mean trunk diameter of a tree of this species with age
    (a) 30 years,
    (b) 100 years. It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places.
  3. Comment on the reliability of each of your two estimates.
OCR S1 2009 January Q3
10 marks Moderate -0.8
3 Erika is a birdwatcher. The probability that she will see a woodpecker on any given day is \(\frac { 1 } { 8 }\). It is assumed that this probability is unaffected by whether she has seen a woodpecker on any other day.
  1. Calculate the probability that Erika first sees a woodpecker
    1. on the third day,
    2. after the third day.
    3. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
    4. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.
OCR S1 2009 January Q4
7 marks Moderate -0.3
4 Three tutors each marked the coursework of five students. The marks are given in the table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)
Tutor 17367604839
Tutor 26250617665
Tutor 34250635471
  1. Calculate Spearman's rank correlation coefficient, \(r _ { \mathrm { s } }\), between the marks for tutors 1 and 2 .
  2. The values of \(r _ { \mathrm { s } }\) for the other pairs of tutors, are as follows. $$\begin{array} { c c } \text { Tutors } 1 \text { and 3: } & r _ { \mathrm { s } } = - 0.9 \\ \text { Tutors } 2 \text { and 3: } & r _ { \mathrm { s } } = 0.3 \end{array}$$ State which two tutors differ most widely in their judgements. Give your reason.