Questions S1 (1967 questions)

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Edexcel S1 2014 June Q2
  1. The discrete random variable \(X\) has probability distribution
$$\mathrm { P } ( X = x ) = \frac { 1 } { 10 } \quad x = 1,2,3 , \ldots 10$$
  1. Write down the name given to this distribution.
  2. Write down the value of
    1. \(\mathrm { P } ( X = 10 )\)
    2. \(\mathrm { P } ( X < 10 )\) The continuous random variable \(Y\) has the normal distribution \(\mathrm { N } \left( 10,2 ^ { 2 } \right)\)
  3. Write down the value of
    1. \(\mathrm { P } ( Y = 10 )\)
    2. \(\mathrm { P } ( Y < 10 )\)
Edexcel S1 2014 June Q3
3. A large company is analysing how much money it spends on paper in its offices every year. The number of employees, \(x\), and the amount of money spent on paper, \(p\) ( \(\pounds\) hundreds), in 8 randomly selected offices are given in the table below.
\(x\)891214731619
\(p\) (£ hundreds)40.536.130.439.432.631.143.445.7
$$\text { (You may use } \sum x ^ { 2 } = 1160 \quad \sum p = 299.2 \quad \sum p ^ { 2 } = 11422 \quad \sum x p = 3449.5 \text { ) }$$
  1. Show that \(S _ { p p } = 231.92\) and find the value of \(S _ { x x }\) and the value of \(S _ { x p }\)
  2. Calculate the product moment correlation coefficient between \(x\) and \(p\). The equation of the regression line of \(p\) on \(x\) is given in the form \(p = a + b x\).
  3. Show that, to 3 significant figures, \(b = 0.824\) and find the value of \(a\).
  4. Estimate the amount of money spent on paper in an office with 10 employees.
  5. Explain the effect each additional employee has on the amount of money spent on paper. Later the company realised it had made a mistake in adding up its costs, \(p\). The true costs were actually half of the values recorded. The product moment correlation coefficient and the equation of the linear regression line are recalculated using this information.
  6. Write down the new value of
    1. the product moment correlation coefficient,
    2. the gradient of the regression line.
Edexcel S1 2014 June Q4
  1. \(\quad A\) and \(B\) are two events such that
$$\mathrm { P } ( B ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \mid B ) = \frac { 2 } { 5 } \quad \mathrm { P } ( A \cup B ) = \frac { 13 } { 20 }$$
  1. Find \(\mathrm { P } ( A \cap B )\).
  2. Draw a Venn diagram to show the events \(A , B\) and all the associated probabilities. Find
  3. \(\mathrm { P } ( A )\)
  4. \(\mathrm { P } ( B \mid A )\)
  5. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\)
Edexcel S1 2014 June Q5
  1. The table shows the time, to the nearest minute, spent waiting for a taxi by each of 80 people one Sunday afternoon.
Waiting time
(in minutes)
Frequency
\(2 - 4\)15
\(5 - 6\)9
76
824
\(9 - 10\)14
\(11 - 15\)12
  1. Write down the upper class boundary for the \(2 - 4\) minute interval. A histogram is drawn to represent these data. The height of the tallest bar is 6 cm .
  2. Calculate the height of the second tallest bar.
  3. Estimate the number of people with a waiting time between 3.5 minutes and 7 minutes.
  4. Use linear interpolation to estimate the median, the lower quartile and the upper quartile of the waiting times.
  5. Describe the skewness of these data, giving a reason for your answer.
Edexcel S1 2014 June Q6
6. The time taken, in minutes, by children to complete a mathematical puzzle is assumed to be normally distributed with mean \(\mu\) and standard deviation \(\sigma\). The puzzle can be completed in less than 24 minutes by \(80 \%\) of the children. For \(5 \%\) of the children it takes more than 28 minutes to complete the puzzle.
  1. Show this information on the Normal curve below.
  2. Write down the percentage of children who take between 24 minutes and 28 minutes to complete the puzzle.
    1. Find two equations in \(\mu\) and \(\sigma\).
    2. Hence find, to 3 significant figures, the value of \(\mu\) and the value of \(\sigma\). A child is selected at random.
  4. Find the probability that the child takes less than 12 minutes to complete the puzzle.
    \includegraphics[max width=\textwidth, alt={}, center]{ca8418eb-4d35-40f4-af40-77503327ae52-11_314_1255_1375_356}
Edexcel S1 2014 June Q7
7. In a large company, 78\% of employees are car owners,
\(30 \%\) of these car owners are also bike owners,
85\% of those who are not car owners are bike owners.
  1. Draw a tree diagram to represent this information. An employee is selected at random.
  2. Find the probability that the employee is a car owner or a bike owner but not both. Another employee is selected at random. Given that this employee is a bike owner,
  3. find the probability that the employee is a car owner. Two employees are selected at random.
  4. Find the probability that only one of them is a bike owner.
Edexcel S1 2014 June Q1
  1. A random sample of 35 homeowners was taken from each of the villages Greenslax and Penville and their ages were recorded. The results are summarised in the back-to-back stem and leaf diagram below.
TotalsGreenslaxPenvilleTotals
(2)8725567889(7)
(3)98731112344569(11)
(4)4440401247(5)
(5)66522500555(5)
(7)865421162566(4)
(8)8664311705(2)
(5)984328(0)
(1)499(1)
Key: 7 | 3 | 1 means 37 years for Greenslax and 31 years for Penville
Some of the quartiles for these two distributions are given in the table below.
GreenslaxPenville
Lower quartile, \(Q _ { 1 }\)\(a\)31
Median, \(Q _ { 2 }\)6439
Upper quartile, \(Q _ { 3 }\)\(b\)55
  1. Find the value of \(a\) and the value of \(b\). An outlier is a value that falls either $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 }
    & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
  2. On the graph paper opposite draw a box plot to represent the data from Penville. Show clearly any outliers.
  3. State the skewness of each distribution. Justify your answers.
    \includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-03_930_1237_1800_367}
Edexcel S1 2014 June Q2
2. The mark, \(x\), scored by each student who sat a statistics examination is coded using $$y = 1.4 x - 20$$ The coded marks have mean 60.8 and standard deviation 6.60 Find the mean and the standard deviation of \(x\).
\includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-04_99_97_2613_1784}
Edexcel S1 2014 June Q3
3. The table shows data on the number of visitors to the UK in a month, \(v\) (1000s), and the amount of money they spent, \(m\) ( \(\pounds\) millions), for each of 8 months.
Number of visitors
\(v ( 1000 \mathrm {~s} )\)
24502480254024202350229024002460
Amount of money spent
\(m ( \pounds\) millions \()\)
13701350140013301270121013301350
You may use
\(S _ { v v } = 42587.5 \quad S _ { v m } = 31512.5 \quad S _ { m m } = 25187.5 \quad \sum v = 19390 \quad \sum m = 10610\)
  1. Find the product moment correlation coefficient between \(m\) and \(v\).
  2. Give a reason to support fitting a regression model of the form \(m = a + b v\) to these data.
  3. Find the value of \(b\) correct to 3 decimal places.
  4. Find the equation of the regression line of \(m\) on \(v\).
  5. Interpret your value of \(b\).
  6. Use your answer to part (d) to estimate the amount of money spent when the number of visitors to the UK in a month is 2500000
  7. Comment on the reliability of your estimate in part (f). Give a reason for your answer.
Edexcel S1 2014 June Q4
  1. In a factory, three machines, \(J , K\) and \(L\), are used to make biscuits.
Machine \(J\) makes \(25 \%\) of the biscuits. Machine \(K\) makes \(45 \%\) of the biscuits. The rest of the biscuits are made by machine \(L\).
It is known that \(2 \%\) of the biscuits made by machine \(J\) are broken, \(3 \%\) of the biscuits made by machine \(K\) are broken and 5\% of the biscuits made by machine \(L\) are broken.
  1. Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. A biscuit is selected at random.
  2. Calculate the probability that the biscuit is made by machine \(J\) and is not broken.
  3. Calculate the probability that the biscuit is broken.
  4. Given that the biscuit is broken, find the probability that it was not made by machine \(K\).
Edexcel S1 2014 June Q5
5. The discrete random variable \(X\) has the probability function $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 2,4,6
k ( x - 2 ) & x = 8
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 1 } { 18 }\)
  2. Find the exact value of \(\mathrm { F } ( 5 )\).
  3. Find the exact value of \(\mathrm { E } ( X )\).
  4. Find the exact value of \(\mathrm { E } \left( X ^ { 2 } \right)\).
  5. Calculate \(\operatorname { Var } ( 3 - 4 X )\) giving your answer to 3 significant figures.
Edexcel S1 2014 June Q6
6. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below.
Time (seconds)Number of customers, \(f\)
0-302
30-6010
60-7017
70-8025
80-10025
100-1506
A histogram was drawn to represent these data. The \(30 - 60\) group was represented by a bar of width 1.5 cm and height 1 cm .
  1. Find the width and the height of the \(70 - 80\) group.
  2. Use linear interpolation to estimate the median of this distribution. Given that \(x\) denotes the midpoint of each group in the table and $$\sum f x = 6460 \quad \sum f x ^ { 2 } = 529400$$
  3. calculate an estimate for
    1. the mean,
    2. the standard deviation,
      for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
  4. Evaluate this coefficient and comment on the skewness of these data.
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.