Questions — Edexcel S1 (606 questions)

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Edexcel S1 2008 January Q5
16 marks Easy -1.2
5. The following shows the results of a wine tasting survey of 100 people. \begin{displayquote} 96 like wine \(A\),
93 like wine \(B\),
96 like wine \(C\),
92 like \(A\) and \(B\),
91 like \(B\) and \(C\),
93 like \(A\) and \(C\),
90 like all three wines.
  1. Draw a Venn Diagram to represent these data. \end{displayquote} Find the probability that a randomly selected person from the survey likes
  2. none of the three wines,
  3. wine \(A\) but not wine \(B\),
  4. any wine in the survey except wine \(C\),
  5. exactly two of the three kinds of wine. Given that a person from the survey likes wine \(A\),
  6. find the probability that the person likes wine \(C\).
Edexcel S1 2008 January Q6
9 marks Moderate -0.8
6. The weights of bags of popcorn are normally distributed with mean of 200 g and \(60 \%\) of all bags weighing between 190 g and 210 g .
  1. Write down the median weight of the bags of popcorn.
  2. Find the standard deviation of the weights of the bags of popcorn. A shopkeeper finds that customers will complain if their bag of popcorn weighs less than 180 g .
  3. Find the probability that a customer will complain.
Edexcel S1 2008 January Q7
14 marks Moderate -0.8
7. Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered \(0,1,2\), and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable \(R\) is the score on the red die and the random variable \(B\) is the score on the blue die.
  1. Find \(\mathrm { P } ( R = 3\) and \(B = 0 )\). The random variable \(T\) is \(R\) multiplied by \(B\).
  2. Complete the diagram below to represent the sample space that shows all the possible values of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-13_732_771_834_621} \section*{Sample space diagram of \(T\)}
  3. The table below represents the probability distribution of the random variable \(T\).
    \(t\)0123469
    \(\mathrm { P } ( T = t )\)\(a\)\(b\)\(1 / 8\)\(1 / 8\)\(c\)\(1 / 8\)\(d\)
    Find the values of \(a , b , c\) and \(d\). Find the values of
  4. \(\mathrm { E } ( T )\),
  5. \(\operatorname { Var } ( T )\).
Edexcel S1 2009 January Q1
11 marks Moderate -0.8
  1. A teacher is monitoring the progress of students using a computer based revision course. The improvement in performance, \(y\) marks, is recorded for each student along with the time, \(x\) hours, that the student spent using the revision course. The results for a random sample of 10 students are recorded below.
\(x\)
hours
1.03.54.01.51.30.51.82.52.33.0
\(y\)
marks
5302710- 3- 5715- 1020
$$\text { [You may use } \sum x = 21.4 , \quad \sum y = 96 , \quad \sum x ^ { 2 } = 57.22 , \quad \sum x y = 313.7 \text { ] }$$
  1. Calculate \(S _ { x x }\) and \(S _ { x y }\).
  2. Find the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\).
  3. Give an interpretation of the gradient of your regression line. Rosemary spends 3.3 hours using the revision course.
  4. Predict her improvement in marks. Lee spends 8 hours using the revision course claiming that this should give him an improvement in performance of over 60 marks.
  5. Comment on Lee's claim.
Edexcel S1 2009 January Q2
8 marks Moderate -0.8
2. A group of office workers were questioned for a health magazine and \(\frac { 2 } { 5 }\) were found to take regular exercise. When questioned about their eating habits \(\frac { 2 } { 3 }\) said they always eat breakfast and, of those who always eat breakfast \(\frac { 9 } { 25 }\) also took regular exercise. Find the probability that a randomly selected member of the group
  1. always eats breakfast and takes regular exercise,
  2. does not always eat breakfast and does not take regular exercise.
  3. Determine, giving your reason, whether or not always eating breakfast and taking regular exercise are statistically independent.
Edexcel S1 2009 January Q3
16 marks Moderate -0.3
3. When Rohit plays a game, the number of points he receives is given by the discrete random variable \(X\) with the following probability distribution.
\(x\)0123
\(\mathrm { P } ( X = x )\)0.40.30.20.1
  1. Find \(\mathrm { E } ( X )\).
  2. Find \(\mathrm { F } ( 1.5 )\).
  3. Show that \(\operatorname { Var } ( X ) = 1\)
  4. Find \(\operatorname { Var } ( 5 - 3 X )\). Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.
  5. Find the probability that Rohit wins the prize.
Edexcel S1 2009 January Q4
14 marks Moderate -0.8
4. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, \(y\) minutes, for the 11 students were $$17,23,35,36,51,53,54,55,60,77,110$$
  1. Find the median and quartiles for these data. A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
  2. Show that 110 is the only outlier.
  3. Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted.
  4. Show that \(S _ { y y }\) for the remaining 10 students is 2966.9 These 10 students were each asked how many text messages, \(x\), they sent in the same week. The values of \(S _ { x x }\) and \(S _ { x y }\) for these 10 students are \(S _ { x x } = 3463.6\) and \(S _ { x y } = - 18.3\).
  5. Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.
  6. Comment on this belief in the light of your calculation in part (e). \includegraphics[max width=\textwidth, alt={}, center]{d5d000c7-de42-461a-ba05-6c8b2c333780-09_611_1593_297_178}
Edexcel S1 2009 January Q5
16 marks Standard +0.3
5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below.
Number of hoursMid-pointFrequency
0-52.7520
6-76.516
8-10918
11-151325
16-2520.515
26-503810
A histogram was drawn and the group ( \(8 - 10\) ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.
  1. Calculate the width and height of the rectangle representing the group (16-25) hours.
  2. Use linear interpolation to estimate the median and interquartile range.
  3. Estimate the mean and standard deviation of the number of hours spent shopping.
  4. State, giving a reason, the skewness of these data.
  5. State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.
Edexcel S1 2009 January Q6
10 marks Moderate -0.8
6. The random variable \(X\) has a normal distribution with mean 30 and standard deviation 5 .
  1. Find \(\mathrm { P } ( X < 39 )\).
  2. Find the value of \(d\) such that \(\mathrm { P } ( X < d ) = 0.1151\)
  3. Find the value of \(e\) such that \(\mathrm { P } ( X > e ) = 0.1151\)
  4. Find \(\mathrm { P } ( d < X < e )\).
Edexcel S1 2011 January Q1
6 marks Easy -1.2
  1. A random sample of 50 salmon was caught by a scientist. He recorded the length \(l \mathrm {~cm}\) and weight \(w \mathrm {~kg}\) of each salmon.
The following summary statistics were calculated from these data. \(\sum l = 4027 \quad \sum l ^ { 2 } = 327754.5 \quad \sum w = 357.1 \quad \sum l w = 29330.5 \quad S _ { w w } = 289.6\)
  1. Find \(S _ { l l }\) and \(S _ { l w }\)
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(l\) and \(w\).
  3. Give an interpretation of your coefficient.
Edexcel S1 2011 January Q2
4 marks Easy -1.8
  1. Keith records the amount of rainfall, in mm , at his school, each day for a week. The results are given below.
    0.0
    0.5
    1.8
    2.8
    2.3
    5.6
    9.4
Jenny then records the amount of rainfall, \(x \mathrm {~mm}\), at the school each day for the following 21 days. The results for the 21 days are summarised below. $$\sum x = 84.6$$
  1. Calculate the mean amount of rainfall during the whole 28 days. Keith realises that he has transposed two of his figures. The number 9.4 should have been 4.9 and the number 0.5 should have been 5.0 Keith corrects these figures.
  2. State, giving your reason, the effect this will have on the mean.
Edexcel S1 2011 January Q3
9 marks Easy -1.2
3. Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below.
Amount received in sales (£1000s)
Two lowest values3,4
Lower quartile7
Median12
Upper quartile14
Two highest values20,25
An outlier is an observation that falls
either \(1.5 \times\) interquartile range above the upper quartile or \(1.5 \times\) interquartile range below the lower quartile.
  1. On the graph paper below, draw a box plot to represent these data, indicating clearly any outliers.
    (5) \includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-05_933_1226_1283_367}
  2. State the skewness of the distribution of the amount of sales received. Justify your answer.
  3. The company claims that for \(75 \%\) of the months, the amount received per month is greater than \(\pounds 10000\). Comment on this claim, giving a reason for your answer.
    (2)
Edexcel S1 2011 January Q4
6 marks Moderate -0.8
  1. A farmer collected data on the annual rainfall, \(x \mathrm {~cm}\), and the annual yield of peas, \(p\) tonnes per acre.
The data for annual rainfall was coded using \(v = \frac { x - 5 } { 10 }\) and the following statistics were found. $$S _ { v v } = 5.753 \quad S _ { p v } = 1.688 \quad S _ { p p } = 1.168 \quad \bar { p } = 3.22 \quad \bar { v } = 4.42$$
  1. Find the equation of the regression line of \(p\) on \(v\) in the form \(p = a + b v\).
  2. Using your regression line estimate the annual yield of peas per acre when the annual rainfall is 85 cm .
Edexcel S1 2011 January Q5
7 marks Moderate -0.8
5. On a randomly chosen day, each of the 32 students in a class recorded the time, \(t\) minutes to the nearest minute, they spent on their homework. The data for the class is summarised in the following table.
Time, \(t\)Number of students
10-192
20-294
30-398
40-4911
50-695
70-792
  1. Use interpolation to estimate the value of the median. Given that $$\sum t = 1414 \quad \text { and } \quad \sum t ^ { 2 } = 69378$$
  2. find the mean and the standard deviation of the times spent by the students on their homework.
  3. Comment on the skewness of the distribution of the times spent by the students on their homework. Give a reason for your answer.
Edexcel S1 2011 January Q6
14 marks Moderate -0.8
  1. The discrete random variable \(X\) has the probability distribution
\(x\)1234
\(\mathrm { P } ( X = x )\)\(k\)\(2 k\)\(3 k\)\(4 k\)
  1. Show that \(k = 0.1\) Find
  2. \(\mathrm { E } ( X )\)
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  4. \(\operatorname { Var } ( 2 - 5 X )\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
  5. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1\)
  6. Complete the probability distribution table for \(X _ { 1 } + X _ { 2 }\)
    \(y\)2345678
    \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)0.010.040.100.250.24
  7. Find \(\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)\)
Edexcel S1 2011 January Q7
17 marks Standard +0.3
  1. The bag \(P\) contains 6 balls of which 3 are red and 3 are yellow.
The bag \(Q\) contains 7 balls of which 4 are red and 3 are yellow.
A ball is drawn at random from bag \(P\) and placed in bag \(Q\). A second ball is drawn at random from bag \(P\) and placed in bag \(Q\).
A third ball is then drawn at random from the 9 balls in bag \(Q\). The event \(A\) occurs when the 2 balls drawn from bag \(P\) are of the same colour. The event \(B\) occurs when the ball drawn from bag \(Q\) is red.
  1. Complete the tree diagram shown below.
    (4) \includegraphics[max width=\textwidth, alt={}, center]{c78ec7b6-dd06-4de1-94c2-052a5577dd10-12_1201_1390_753_269}
  2. Find \(\mathrm { P } ( A )\)
  3. Show that \(\mathrm { P } ( B ) = \frac { 5 } { 9 }\)
  4. Show that \(\mathrm { P } ( A \cap B ) = \frac { 2 } { 9 }\)
  5. Hence find \(\mathrm { P } ( A \cup B )\)
  6. Given that all three balls drawn are the same colour, find the probability that they are all red.
    (3)
Edexcel S1 2011 January Q8
12 marks Standard +0.3
  1. The weight, \(X\) grams, of soup put in a tin by machine \(A\) is normally distributed with a mean of 160 g and a standard deviation of 5 g .
    A tin is selected at random.
    1. Find the probability that this tin contains more than 168 g .
    The weight stated on the tin is \(w\) grams.
  2. Find \(w\) such that \(\mathrm { P } ( X < w ) = 0.01\) The weight, \(Y\) grams, of soup put into a carton by machine \(B\) is normally distributed with mean \(\mu\) grams and standard deviation \(\sigma\) grams.
  3. Given that \(\mathrm { P } ( Y < 160 ) = 0.99\) and \(\mathrm { P } ( Y > 152 ) = 0.90\) find the value of \(\mu\) and the value of \(\sigma\).
Edexcel S1 2012 January Q1
4 marks Easy -1.2
  1. The histogram in Figure 1 shows the time, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bc8ef6c7-a321-4ecf-962d-f469a95fc8c8-02_1312_673_349_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Complete the table.
    Delay (minutes)Number of motorists
    4-66
    7-8
    921
    10-1245
    13-159
    16-20
  2. Estimate the number of motorists who were delayed between 8.5 and 13.5 minutes by the roadworks.
Edexcel S1 2012 January Q2
9 marks Moderate -0.3
  1. (a) State in words the relationship between two events \(R\) and \(S\) when \(\mathrm { P } ( R \cap S ) = 0\)
The events \(A\) and \(B\) are independent with \(\mathrm { P } ( A ) = \frac { 1 } { 4 }\) and \(\mathrm { P } ( A \cup B ) = \frac { 2 } { 3 }\) Find
(b) \(\mathrm { P } ( B )\) (c) \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\) (d) \(\mathrm { P } \left( B ^ { \prime } \mid A \right)\)
Edexcel S1 2012 January Q3
11 marks Moderate -0.8
3. The discrete random variable \(X\) can take only the values \(2,3,4\) or 6 . For these values the probability distribution function is given by
\(x\)2346
\(\mathrm { P } ( X = x )\)\(\frac { 5 } { 21 }\)\(\frac { 2 k } { 21 }\)\(\frac { 7 } { 21 }\)\(\frac { k } { 21 }\)
where \(k\) is a positive integer.
  1. Show that \(k = 3\) Find
  2. \(\mathrm { F } ( 3 )\)
  3. \(\mathrm { E } ( X )\)
  4. \(\mathrm { E } \left( X ^ { 2 } \right)\)
  5. \(\operatorname { Var } ( 7 X - 5 )\)
Edexcel S1 2012 January Q4
13 marks Easy -1.3
  1. The marks, \(x\), of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below.
MarkTotals
36999\(( 3 )\)
40122234\(( 6 )\)
4566668\(( 5 )\)
50233344\(( 6 )\)
55566779\(( 6 )\)
600000013444\(( 9 )\)
65566789\(( 6 )\)
712333\(( 4 )\)
Key(3|6 means 36)
  1. Write down the modal mark of these students.
  2. Find the values of the lower quartile, the median and the upper quartile. For these students \(\sum x = 2497\) and \(\sum x ^ { 2 } = 143369\)
  3. Find the mean and the standard deviation of the marks of these students.
  4. Describe the skewness of the marks of these students, giving a reason for your answer. The mean and standard deviation of the marks of all the students who sat the examination were 55 and 10 respectively. The examiners decided that the total mark of each student should be scaled by subtracting 5 marks and then reducing the mark by a further \(10 \%\).
  5. Find the mean and standard deviation of the scaled marks of all the students.
Edexcel S1 2012 January Q5
15 marks Moderate -0.8
  1. The age, \(t\) years, and weight, \(w\) grams, of each of 10 coins were recorded. These data are summarised below.
$$\sum t ^ { 2 } = 2688 \quad \sum t w = 1760.62 \quad \sum t = 158 \quad \sum w = 111.75 \quad S _ { w w } = 0.16$$
  1. Find \(S _ { t t }\) and \(S _ { t w }\) for these data.
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(t\) and \(w\).
  3. Find the equation of the regression line of \(w\) on \(t\) in the form \(w = a + b t\)
  4. State, with a reason, which variable is the explanatory variable.
  5. Using this model, estimate
    1. the weight of a coin which is 5 years old,
    2. the effect of an increase of 4 years in age on the weight of a coin. It was discovered that a coin in the original sample, which was 5 years old and weighed 20 grams, was a fake.
  6. State, without any further calculations, whether the exclusion of this coin would increase or decrease the value of the product moment correlation coefficient. Give a reason for your answer.
Edexcel S1 2012 January Q6
13 marks Moderate -0.8
  1. The following shows the results of a survey on the types of exercise taken by a group of 100 people.
65 run
48 swim
60 cycle
40 run and swim
30 swim and cycle
35 run and cycle
25 do all three
  1. Draw a Venn Diagram to represent these data. Find the probability that a randomly selected person from the survey
  2. takes none of these types of exercise,
  3. swims but does not run,
  4. takes at least two of these types of exercise. Jason is one of the above group.
    Given that Jason runs,
  5. find the probability that he swims but does not cycle.
Edexcel S1 2012 January Q7
10 marks Moderate -0.3
  1. A manufacturer fills jars with coffee. The weight of coffee, \(W\) grams, in a jar can be modelled by a normal distribution with mean 232 grams and standard deviation 5 grams.
    1. Find \(\mathrm { P } ( W < 224 )\).
    2. Find the value of \(w\) such that \(\mathrm { P } ( 232 < W < w ) = 0.20\)
    Two jars of coffee are selected at random.
  2. Find the probability that only one of the jars contains between 232 grams and \(w\) grams of coffee.
Edexcel S1 2013 January Q1
7 marks Easy -1.2
  1. A teacher asked a random sample of 10 students to record the number of hours of television, \(t\), they watched in the week before their mock exam. She then calculated their grade, \(g\), in their mock exam. The results are summarised as follows.
$$\sum t = 258 \quad \sum t ^ { 2 } = 8702 \quad \sum g = 63.6 \quad \mathrm {~S} _ { g g } = 7.864 \quad \sum g t = 1550.2$$
  1. Find \(\mathrm { S } _ { t t }\) and \(\mathrm { S } _ { g t }\)
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(t\) and \(g\). The teacher also recorded the number of hours of revision, \(v\), these 10 students completed during the week before their mock exam. The correlation coefficient between \(t\) and \(v\) was -0.753
  3. Describe, giving a reason, the nature of the correlation you would expect to find between \(v\) and \(g\).