Questions — Edexcel S1 (574 questions)

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Edexcel S1 2014 June Q7
7. The heights of adult females are normally distributed with mean 160 cm and standard deviation 8 cm .
  1. Find the probability that a randomly selected adult female has a height greater than 170 cm . Any adult female whose height is greater than 170 cm is defined as tall. An adult female is chosen at random. Given that she is tall,
  2. find the probability that she has a height greater than 180 cm . Half of tall adult females have a height greater than \(h \mathrm {~cm}\).
  3. Find the value of \(h\).
Edexcel S1 2014 June Q8
8. For the events \(A\) and \(B\), $$\mathrm { P } \left( A ^ { \prime } \cap B \right) = 0.22 \text { and } \mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = 0.18$$
  1. Find \(\mathrm { P } ( A )\).
  2. Find \(\mathrm { P } ( A \cup B )\). Given that \(\mathrm { P } ( A \mid B ) = 0.6\)
  3. find \(\mathrm { P } ( A \cap B )\).
  4. Determine whether or not \(A\) and \(B\) are independent.
Edexcel S1 2015 June Q1
  1. Each of 60 students was asked to draw a \(20 ^ { \circ }\) angle without using a protractor. The size of each angle drawn was measured. The results are summarised in the box plot below.
    \includegraphics[max width=\textwidth, alt={}, center]{9626e3ce-35d6-41b5-a0bd-1185f38b9e36-02_371_1040_340_461}
    1. Find the range for these data.
    2. Find the interquartile range for these data.
    The students were then asked to draw a \(70 ^ { \circ }\) angle.
    The results are summarised in the table below.
    Angle, \(\boldsymbol { a }\), (degrees)Number of students
    \(55 \leqslant a < 60\)6
    \(60 \leqslant a < 65\)15
    \(65 \leqslant a < 70\)13
    \(70 \leqslant a < 75\)11
    \(75 \leqslant a < 80\)8
    \(80 \leqslant a < 85\)7
  2. Use linear interpolation to estimate the size of the median angle drawn. Give your answer to 1 decimal place.
  3. Show that the lower quartile is \(63 ^ { \circ }\) For these data, the upper quartile is \(75 ^ { \circ }\), the minimum is \(55 ^ { \circ }\) and the maximum is \(84 ^ { \circ }\) An outlier is an observation that falls either more than \(1.5 \times\) (interquartile range) above the upper quartile or more than \(1.5 \times\) (interquartile range) below the lower quartile.
    1. Show that there are no outliers for these data.
    2. Draw a box plot for these data on the grid on page 3.
  5. State which angle the students were more accurate at drawing. Give reasons for your answer.
    (3) \includegraphics[max width=\textwidth, alt={}, center]{9626e3ce-35d6-41b5-a0bd-1185f38b9e36-03_378_1059_2067_447}
Edexcel S1 2015 June Q2
2. An estate agent recorded the price per square metre, \(p \pounds / \mathrm { m } ^ { 2 }\), for 7 two-bedroom houses. He then coded the data using the coding \(q = \frac { p - a } { b }\), where \(a\) and \(b\) are positive constants. His results are shown in the table below.
\(p\)1840184818301824181918341850
\(q\)4.04.83.02.41.93.45.0
  1. Find the value of \(a\) and the value of \(b\) The estate agent also recorded the distance, \(d \mathrm {~km}\), of each house from the nearest train station. The results are summarised below. $$\mathrm { S } _ { d d } = 1.02 \quad \mathrm {~S} _ { q q } = 8.22 \quad \mathrm {~S} _ { d q } = - 2.17$$
  2. Calculate the product moment correlation coefficient between \(d\) and \(q\)
  3. Write down the value of the product moment correlation coefficient between \(d\) and \(p\) The estate agent records the price and size of 2 additional two-bedroom houses, \(H\) and \(J\).
    HousePrice \(( \pounds )\)Size \(\left( \mathrm { m } ^ { 2 } \right)\)
    \(H\)15640085
    \(J\)17290095
  4. Suggest which house is most likely to be closer to a train station. Justify your answer.
Edexcel S1 2015 June Q3
  1. A college has 80 students in Year 12.
20 students study Biology
28 students study Chemistry
30 students study Physics
7 students study both Biology and Chemistry
11 students study both Chemistry and Physics
5 students study both Physics and Biology
3 students study all 3 of these subjects
  1. Draw a Venn diagram to represent this information. A Year 12 student at the college is selected at random.
  2. Find the probability that the student studies Chemistry but not Biology or Physics.
  3. Find the probability that the student studies Chemistry or Physics or both. Given that the student studies Chemistry or Physics or both,
  4. find the probability that the student does not study Biology.
  5. Determine whether studying Biology and studying Chemistry are statistically independent.
Edexcel S1 2015 June Q4
  1. Statistical models can provide a cheap and quick way to describe a real world situation.
    1. Give two other reasons why statistical models are used.
    A scientist wants to develop a model to describe the relationship between the average daily temperature, \(x ^ { \circ } \mathrm { C }\), and her household's daily energy consumption, \(y \mathrm { kWh }\), in winter. A random sample of the average daily temperature and her household's daily energy consumption are taken from 10 winter days and shown in the table.
    \(x\)- 0.4- 0.20.30.81.11.41.82.12.52.6
    \(y\)28302625262726242221
    $$\text { [You may use } \sum x ^ { 2 } = 24.76 \quad \sum y = 255 \quad \sum x y = 283.8 \quad \mathrm {~S} _ { x x } = 10.36 \text { ] }$$
  2. Find \(\mathrm { S } _ { x y }\) for these data.
  3. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\) Give the value of \(a\) and the value of \(b\) to 3 significant figures.
  4. Give an interpretation of the value of \(a\)
  5. Estimate her household's daily energy consumption when the average daily temperature is \(2 ^ { \circ } \mathrm { C }\) The scientist wants to use the linear regression model to predict her household's energy consumption in the summer.
  6. Discuss the reliability of using this model to predict her household's energy consumption in the summer.
Edexcel S1 2015 June Q5
  1. In a quiz, a team gains 10 points for every question it answers correctly and loses 5 points for every question it does not answer correctly. The probability of answering a question correctly is 0.6 for each question. One round of the quiz consists of 3 questions.
The discrete random variable \(X\) represents the total number of points scored in one round. The table shows the incomplete probability distribution of \(X\)
\(x\)30150- 15
\(\mathrm { P } ( X = x )\)0.2160.064
  1. Show that the probability of scoring 15 points in a round is 0.432
  2. Find the probability of scoring 0 points in a round.
  3. Find the probability of scoring a total of 30 points in 2 rounds.
  4. Find \(\mathrm { E } ( X )\)
  5. Find \(\operatorname { Var } ( X )\) In a bonus round of 3 questions, a team gains 20 points for every question it answers correctly and loses 5 points for every question it does not answer correctly.
  6. Find the expected number of points scored in the bonus round.
Edexcel S1 2015 June Q6
  1. The random variable \(Z \sim \mathrm {~N} ( 0,1 )\)
    \(A\) is the event \(Z > 1.1\)
    \(B\) is the event \(Z > - 1.9\)
    \(C\) is the event \(- 1.5 < Z < 1.5\)
    1. Find
      1. \(\mathrm { P } ( A )\)
      2. \(\mathrm { P } ( B )\)
      3. \(\mathrm { P } ( C )\)
      4. \(\mathrm { P } ( A \cup C )\)
    The random variable \(X\) has a normal distribution with mean 21 and standard deviation 5
  2. Find the value of \(w\) such that \(\mathrm { P } ( X > w \mid X > 28 ) = 0.625\)
Edexcel S1 2016 June Q1
  1. A biologist is studying the behaviour of bees in a hive. Once a bee has located a source of food, it returns to the hive and performs a dance to indicate to the other bees how far away the source of the food is. The dance consists of a series of wiggles. The biologist records the distance, \(d\) metres, of the food source from the hive and the average number of wiggles, \(w\), in the dance.
Distance, \(\boldsymbol { d } \mathbf { m }\)305080100150400500650
Average number
of wiggles, \(\boldsymbol { w }\)
0.7251.2101.7752.2503.5186.3828.1859.555
[You may use \(\sum w = 33.6 \sum d w = 13833 \mathrm {~S} _ { d d } = 394600 \mathrm {~S} _ { w w } = 80.481\) (to 3 decimal places)]
  1. Show that \(\mathrm { S } _ { d w } = 5601\)
  2. State, giving a reason, which is the response variable.
  3. Calculate the product moment correlation coefficient for these data.
  4. Calculate the equation of the regression line of \(w\) on \(d\), giving your answer in the form \(w = a + b d\) A new source of food is located 350 m from the hive.
    1. Use your regression equation to estimate the average number of wiggles in the corresponding dance.
    2. Comment, giving a reason, on the reliability of your estimate.
Edexcel S1 2016 June Q2
2. The discrete random variable \(X\) has the following probability distribution, where \(p\) and \(q\) are constants.
\(x\)- 2- 1\(\frac { 1 } { 2 }\)\(\frac { 3 } { 2 }\)2
\(\mathrm { P } ( X = x )\)\(p\)\(q\)0.20.3\(p\)
  1. Write down an equation in \(p\) and \(q\) Given that \(\mathrm { E } ( X ) = 0.4\)
  2. find the value of \(q\)
  3. hence find the value of \(p\) Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.275\)
  4. find \(\operatorname { Var } ( X )\) Sarah and Rebecca play a game.
    A computer selects a single value of \(X\) using the probability distribution above.
    Sarah's score is given by the random variable \(S = X\) and Rebecca's score is given by the random variable \(R = \frac { 1 } { X }\)
  5. Find \(\mathrm { E } ( R )\) Sarah and Rebecca work out their scores and the person with the higher score is the winner. If the scores are the same, the game is a draw.
  6. Find the probability that
    1. Sarah is the winner,
    2. Rebecca is the winner.
Edexcel S1 2016 June Q3
3. Before going on holiday to Seapron, Tania records the weekly rainfall ( \(x \mathrm {~mm}\) ) at Seapron for 8 weeks during the summer. Her results are summarised as $$\sum x = 86.8 \quad \sum x ^ { 2 } = 985.88$$
  1. Find the standard deviation, \(\sigma _ { x }\), for these data.
    (3) Tania also records the number of hours of sunshine ( \(y\) hours) per week at Seapron for these 8 weeks and obtains the following $$\bar { y } = 58 \quad \sigma _ { y } = 9.461 \text { (correct to } 4 \text { significant figures) } \quad \sum x y = 4900.5$$
  2. Show that \(\mathrm { S } _ { y y } = 716\) (correct to 3 significant figures)
  3. Find \(\mathrm { S } _ { x y }\)
  4. Calculate the product moment correlation coefficient, \(r\), for these data. During Tania's week-long holiday at Seapron there are 14 mm of rain and 70 hours of sunshine.
  5. State, giving a reason, what the effect of adding this information to the above data would be on the value of the product moment correlation coefficient.
Edexcel S1 2016 June Q4
4. The Venn diagram shows the probabilities of customer bookings at Harry’s hotel.
\(R\) is the event that a customer books a room
\(B\) is the event that a customer books breakfast
\(D\) is the event that a customer books dinner
\(u\) and \(t\) are probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{e3b92a5b-c0ad-4176-9b05-cb07a44aa265-08_604_1047_696_450}
  1. Write down the probability that a customer books breakfast but does not book a room. Given that the events \(B\) and \(D\) are independent
  2. find the value of \(t\)
  3. hence find the value of \(u\)
  4. Find
    1. \(\quad\) P( \(D \mid R \cap B\) )
    2. \(\mathrm { P } \left( D \mid R \cap B ^ { \prime } \right)\) A coach load of 77 customers arrive at Harry’s hotel. Of these 77 customers 40 have booked a room and breakfast 37 have booked a room without breakfast
  5. Estimate how many of these 77 customers will book dinner.
Edexcel S1 2016 June Q5
5. A midwife records the weights, in kg , of a sample of 50 babies born at a hospital. Her results are given in the table below.
Weight ( \(\boldsymbol { w } \mathbf { ~ k g }\) )Frequency (f)Weight midpoint (x)
\(0 \leqslant w < 2\)11
\(2 \leqslant w < 3\)82.5
\(3 \leqslant w < 3.5\)173.25
\(3.5 \leqslant w < 4\)173.75
\(4 \leqslant w < 5\)74.5
[You may use \(\sum \mathrm { f } x ^ { 2 } = 611.375\) ] A histogram has been drawn to represent these data. The bar representing the weight \(2 \leqslant w < 3\) has a width of 1 cm and a height of 4 cm .
  1. Calculate the width and height of the bar representing a weight of \(3 \leqslant w < 3.5\)
  2. Use linear interpolation to estimate the median weight of these babies.
    1. Show that an estimate of the mean weight of these babies is 3.43 kg .
    2. Find an estimate of the standard deviation of the weights of these babies. Shyam decides to model the weights of babies born at the hospital, by the random variable \(W\), where \(W \sim \mathrm {~N} \left( 3.43,0.65 ^ { 2 } \right)\)
  4. Find \(\mathrm { P } ( W < 3 )\)
  5. With reference to your answers to (b), (c)(i) and (d) comment on Shyam's decision. A newborn baby weighing 3.43 kg is born at the hospital.
  6. Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the
    1. mean,
    2. standard deviation.
Edexcel S1 2016 June Q6
6. The time, in minutes, taken by men to run a marathon is modelled by a normal distribution with mean 240 minutes and standard deviation 40 minutes.
  1. Find the proportion of men that take longer than 300 minutes to run a marathon.
    (3) Nathaniel is preparing to run a marathon. He aims to finish in the first 20\% of male runners.
  2. Using the above model estimate the longest time that Nathaniel can take to run the marathon and achieve his aim.
    (3) The time, \(W\) minutes, taken by women to run a marathon is modelled by a normal distribution with mean \(\mu\) minutes. Given that \(\mathrm { P } ( W < \mu + 30 ) = 0.82\)
  3. find \(\mathrm { P } ( W < \mu - 30 \mid W < \mu )\)
Edexcel S1 2017 June Q1
  1. A clothes shop manager records the weekly sales figures, \(\pounds s\), and the average weekly temperature, \(t ^ { \circ } \mathrm { C }\), for 6 weeks during the summer. The sales figures were coded so that \(w = \frac { s } { 1000 }\)
The data are summarised as follows $$\mathrm { S } _ { w w } = 50 \quad \sum w t = 784 \quad \sum t ^ { 2 } = 2435 \quad \sum t = 119 \quad \sum w = 42$$
  1. Find \(\mathrm { S } _ { w t }\) and \(\mathrm { S } _ { t t }\)
  2. Write down the value of \(\mathrm { S } _ { s s }\) and the value of \(\mathrm { S } _ { s t }\)
  3. Find the product moment correlation coefficient between \(s\) and \(t\). The manager of the clothes shop believes that a linear regression model may be appropriate to describe these data.
  4. State, giving a reason, whether or not your value of the correlation coefficient supports the manager's belief.
  5. Find the equation of the regression line of \(w\) on \(t\), giving your answer in the form \(w = a + b t\)
  6. Hence find the equation of the regression line of \(s\) on \(t\), giving your answer in the form \(s = c + d t\), where \(c\) and \(d\) are correct to 3 significant figures.
  7. Using your equation in part (f), interpret the effect of a \(1 ^ { \circ } \mathrm { C }\) increase in average weekly temperature on weekly sales during the summer.
Edexcel S1 2017 June Q2
2. An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, \(\pounds x\) per square foot. His results are given in the table below.
Cost (£ \(\boldsymbol { x }\) )Frequency (f)Midpoint (£y)
\(20 \leqslant x < 40\)1230
\(40 \leqslant x < 45\)1342.5
\(45 \leqslant x < 50\)2547.5
\(50 \leqslant x < 60\)3255
\(60 \leqslant x < 80\)870
A histogram is drawn for these data and the bar representing \(50 \leqslant x < 60\) is 2 cm wide and 8 cm high.
  1. Calculate the width and height of the bar representing \(20 \leqslant x < 40\)
  2. Use linear interpolation to estimate the median cost.
  3. Estimate the mean cost of office space for these data.
  4. Estimate the standard deviation for these data.
  5. Describe, giving a reason, the skewness. Rika suggests that the cost of office space in London can be modelled by a normal distribution with mean \(\pounds 50\) and standard deviation \(\pounds 10\)
  6. With reference to your answer to part (e), comment on Rika's suggestion.
  7. Use Rika's model to estimate the 80th percentile of the cost of office space in London.
Edexcel S1 2017 June Q3
  1. The Venn diagram shows three events \(A , B\) and \(C\), where \(p , q , r , s\) and \(t\) are probabilities.
    \includegraphics[max width=\textwidth, alt={}, center]{319667e7-3f8b-4a33-8fc5-ef72154d1421-10_647_972_306_488}
    (b) Find the value of \(r\).
    (c) Hence write down the value of \(s\) and the value of \(t\).
    (d) State, giving a reason, whether or not the events \(A\) and \(B\) are independent.
    (e) Find \(\mathrm { P } ( B \mid A \cup C )\).
    \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } ( C ) = 0.25\) and the events \(B\) and \(C\) are independent.
    (a) Find the value of \(p\) and the value of \(q\).
Edexcel S1 2017 June Q4
4. The discrete random variable \(X\) has probability distribution
\(x\)- 1012
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(b\)\(c\)
The cumulative distribution function of \(X\) is given by
\(x\)- 1012
\(\mathrm {~F} ( x )\)\(\frac { 1 } { 3 }\)\(d\)\(\frac { 5 } { 6 }\)\(e\)
  1. Find the values of \(a , b , c , d\) and \(e\).
  2. Write down the value of \(\mathrm { P } \left( X ^ { 2 } = 1 \right)\).
    \section*{} \section*{
    \includegraphics[max width=\textwidth, alt={}]{image-not-found}
    } \(T\)
Edexcel S1 2017 June Q5
5. Yuto works in the quality control department of a large company. The time, \(T\) minutes, it takes Yuto to analyse a sample is normally distributed with mean 18 minutes and standard deviation 5 minutes.
  1. Find the probability that Yuto takes longer than 20 minutes to analyse the next sample. (3) The company has a large store of samples analysed by Yuto with the time taken for each analysis recorded. Serena is investigating the samples that took Yuto longer than 15 minutes to analyse. She selects, at random, one of the samples that took Yuto longer than 15 minutes to analyse.
  2. Find the probability that this sample took Yuto more than 20 minutes to analyse. Serena can identify, in advance, the samples that Yuto can analyse in under 15 minutes and in future she will assign these to someone else.
  3. Estimate the median time taken by Yuto to analyse samples in future.
Edexcel S1 2017 June Q6
6. The score, \(X\), for a biased spinner is given by the probability distribution
\(x\)036
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 12 }\)\(\frac { 2 } { 3 }\)\(\frac { 1 } { 4 }\)
Find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\) A biased coin has one face labelled 2 and the other face labelled 5 The score, \(Y\), when the coin is spun has $$\mathrm { P } ( Y = 5 ) = p \quad \text { and } \quad \mathrm { E } ( Y ) = 3$$
  3. Form a linear equation in \(p\) and show that \(p = \frac { 1 } { 3 }\)
  4. Write down the probability distribution of \(Y\). Sam plays a game with the spinner and the coin.
    Each is spun once and Sam calculates his score, \(S\), as follows $$\begin{aligned} & \text { if } X = 0 \text { then } S = Y ^ { 2 }
    & \text { if } X \neq 0 \text { then } S = X Y \end{aligned}$$
  5. Show that \(\mathrm { P } ( S = 30 ) = \frac { 1 } { 12 }\)
  6. Find the probability distribution of \(S\).
  7. Find \(\mathrm { E } ( S )\). Charlotte also plays the game with the spinner and the coin.
    Each is spun once and Charlotte ignores the score on the coin and just uses \(X ^ { 2 }\) as her score. Sam and Charlotte each play the game a large number of times.
  8. State, giving a reason, which of Sam and Charlotte should achieve the higher total score.
    END
Edexcel S1 2018 June Q1
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)24710
\(\mathrm { P } ( X = x )\)\(a\)\(b\)0.1\(c\)
where \(a , b\) and \(c\) are probabilities.
The cumulative distribution function of \(X\) is \(\mathrm { F } ( x )\) and \(\mathrm { F } ( 3 ) = 0.2\) and \(\mathrm { F } ( 6 ) = 0.8\)
  1. Find the value of \(a\), the value of \(b\) and the value of \(c\).
  2. Write down the value of \(\mathrm { F } ( 7 )\).
Edexcel S1 2018 June Q2
2. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway one Monday.
Delay (minutes)Number of motorists (f)Delay midpoint (x)
3-6384.5
7-8257.5
9-10189.5
11-151213
16-20718
(You may use \(\sum \mathrm { f } x ^ { 2 } = 8096.25\) ) A histogram has been drawn to represent these data. The bar representing a delay of (3-6) minutes has a width of 2 cm and a height of 9.5 cm .
  1. Calculate the width and the height of the bar representing a delay of (11-15) minutes.
  2. Use linear interpolation to estimate the median delay.
  3. Calculate an estimate of the mean delay.
  4. Calculate an estimate of the standard deviation of the delays. One coefficient of skewness is given by \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\)
  5. Evaluate this coefficient for the above data, giving your answer to 2 significant figures. On the following Friday, the coefficient of skewness for the delays on this stretch of motorway was - 0.22
  6. State, giving a reason, how the delays on this stretch of motorway on Friday are different from the delays on Monday.
Edexcel S1 2018 June Q3
3. The random variable \(Y\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) The \(\mathrm { P } ( Y > 17 ) = 0.4\) Find
  1. \(\mathrm { P } ( \mu < Y < 17 )\)
  2. \(\mathrm { P } ( \mu - \sigma < Y < 17 )\)
Edexcel S1 2018 June Q4
4．A bag contains 64 coloured beads．There are \(r\) red beads，\(y\) yellow beads and 1 green bead and \(r + y + 1 = 64\) Two beads are selected at random，one at a time without replacement．
（a）Find the probability that the green bead is one of the beads selected． The probability that both of the beads are red is \(\frac { 5 } { 84 }\)
（b）Show that \(r\) satisfies the equation \(r ^ { 2 } - r - 240 = 0\)
（c）Hence show that the only possible value of \(r\) is 16
（d）Given that at least one of the beads is red，find the probability that they are both red．
Edexcel S1 2018 June Q5
5. The score when a spinner is spun is given by the discrete random variable \(X\) with the following probability distribution, where \(a\) and \(b\) are probabilities.
\(x\)- 10245
\(\mathrm { P } ( X = x )\)\(b\)\(a\)\(a\)\(a\)\(b\)
  1. Explain why \(\mathrm { E } ( X ) = 2\)
  2. Find a linear equation in \(a\) and \(b\). Given that \(\operatorname { Var } ( X ) = 7.1\)
  3. find a second equation in \(a\) and \(b\) and simplify your answer.
  4. Solve your two equations to find the value of \(a\) and the value of \(b\). The discrete random variable \(Y = 10 - 3 X\)
  5. Find
    1. \(\mathrm { E } ( Y )\)
    2. \(\operatorname { Var } ( Y )\) The spinner is spun once.
  6. Find \(\mathrm { P } ( Y > X )\).