Questions — Edexcel S1 (574 questions)

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Edexcel S1 2010 January Q6
  1. The blood pressures, \(p\) mmHg, and the ages, \(t\) years, of 7 hospital patients are shown in the table below.
PatientABCDEFG
\(t\)42744835562660
\(p\)981301208818280135
$$\left[ \sum t = 341 , \sum p = 833 , \sum t ^ { 2 } = 18181 , \sum p ^ { 2 } = 106397 , \sum t p = 42948 \right]$$
  1. Find \(S _ { p p } , S _ { t p }\) and \(S _ { t t }\) for these data.
  2. Calculate the product moment correlation coefficient for these data.
  3. Interpret the correlation coefficient.
  4. On the graph paper on page 17, draw the scatter diagram of blood pressure against age for these 7 patients.
  5. Find the equation of the regression line of \(p\) on \(t\).
  6. Plot your regression line on your scatter diagram.
  7. Use your regression line to estimate the blood pressure of a 40 year old patient. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Question 6 continued} \includegraphics[alt={},max width=\textwidth]{a0058e3c-046f-4271-aee4-33a74c719e2a-12_2071_1729_386_157}
    \end{figure}
Edexcel S1 2010 January Q7
  1. The heights of a population of women are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). It is known that \(30 \%\) of the women are taller than 172 cm and \(5 \%\) are shorter than 154 cm .
    1. Sketch a diagram to show the distribution of heights represented by this information.
    2. Show that \(\mu = 154 + 1.6449 \sigma\).
    3. Obtain a second equation and hence find the value of \(\mu\) and the value of \(\sigma\).
    A woman is chosen at random from the population.
  2. Find the probability that she is taller than 160 cm .
Edexcel S1 2011 January Q1
  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
  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
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
  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
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
  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
  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
  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
  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
  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
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
  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
  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
  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
  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
  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 \(g\).
Edexcel S1 2013 January Q2
2. The discrete random variable \(X\) can take only the values 1,2 and 3 . For these values the cumulative distribution function is defined by $$\mathrm { F } ( x ) = \frac { x ^ { 3 } + k } { 40 } \quad x = 1,2,3$$
  1. Show that \(k = 13\)
  2. Find the probability distribution of \(X\). Given that \(\operatorname { Var } ( X ) = \frac { 259 } { 320 }\)
  3. find the exact value of \(\operatorname { Var } ( 4 X - 5 )\).
Edexcel S1 2013 January Q3
3. A biologist is comparing the intervals ( \(m\) seconds) between the mating calls of a certain species of tree frog and the surrounding temperature ( \(t { } ^ { \circ } \mathrm { C }\) ). The following results were obtained.
\(t { } ^ { \circ } \mathrm { C }\)813141515202530
\(m\) secs6.54.5654321
$$\text { (You may use } \sum t m = 469.5 , \quad \mathrm {~S} _ { t t } = 354 , \quad \mathrm {~S} _ { m m } = 25.5 \text { ) }$$
  1. Show that \(\mathrm { S } _ { t m } = - 90.5\)
  2. Find the equation of the regression line of \(m\) on \(t\) giving your answer in the form \(m = a + b t\).
  3. Use your regression line to estimate the time interval between mating calls when the surrounding temperature is \(10 ^ { \circ } \mathrm { C }\).
  4. Comment on the reliability of this estimate, giving a reason for your answer.
Edexcel S1 2013 January Q4
  1. The length of time, \(L\) hours, that a phone will work before it needs charging is normally distributed with a mean of 100 hours and a standard deviation of 15 hours.
    1. Find \(\mathrm { P } ( L > 127 )\).
    2. Find the value of \(d\) such that \(\mathrm { P } ( L < d ) = 0.10\)
    Alice is about to go on a 6 hour journey.
    Given that it is 127 hours since Alice last charged her phone,
  2. find the probability that her phone will not need charging before her journey is completed.
Edexcel S1 2013 January Q5
  1. A survey of 100 households gave the following results for weekly income \(\pounds y\).
Income \(y\) (£)Mid-pointFrequency \(f\)
\(0 \leqslant y < 200\)10012
\(200 \leqslant y < 240\)22028
\(240 \leqslant y < 320\)28022
\(320 \leqslant y < 400\)36018
\(400 \leqslant y < 600\)50012
\(600 \leqslant y < 800\)7008
(You may use \(\sum f y ^ { 2 } = 12452\) 800)
A histogram was drawn and the class \(200 \leqslant y < 240\) was represented by a rectangle of width 2 cm and height 7 cm .
  1. Calculate the width and the height of the rectangle representing the class $$320 \leqslant y < 400$$
  2. Use linear interpolation to estimate the median weekly income to the nearest pound.
  3. Estimate the mean and the standard deviation of the weekly income for these data. One measure of skewness is \(\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }\).
  4. Use this measure to calculate the skewness for these data and describe its value. Katie suggests using the random variable \(X\) which has a normal distribution with mean 320 and standard deviation 150 to model the weekly income for these data.
  5. Find \(\mathrm { P } ( 240 < X < 400 )\).
  6. With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie's suggestion.
Edexcel S1 2013 January Q6
6. A fair blue die has faces numbered \(1,1,3,3,5\) and 5 . The random variable \(B\) represents the score when the blue die is rolled.
  1. Write down the probability distribution for \(B\).
  2. State the name of this probability distribution.
  3. Write down the value of \(\mathrm { E } ( B )\). A second die is red and the random variable \(R\) represents the score when the red die is rolled. The probability distribution of \(R\) is
    \(r\)246
    \(\mathrm { P } ( R = r )\)\(\frac { 2 } { 3 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)
  4. Find \(\mathrm { E } ( R )\).
  5. Find \(\operatorname { Var } ( R )\). Tom invites Avisha to play a game with these dice.
    Tom spins a fair coin with one side labelled 2 and the other side labelled 5 . When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is greater than the number showing on the coin, Avisha wins, otherwise Tom wins. Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.
  6. Find the probability that Avisha wins the game, stating clearly which die she should use in each case.
Edexcel S1 2013 January Q7
  1. Given that
$$\mathrm { P } ( A ) = 0.35 , \quad \mathrm { P } ( B ) = 0.45 \quad \text { and } \quad \mathrm { P } ( A \cap B ) = 0.13$$ find
  1. \(\mathrm { P } ( A \cup B )\)
  2. \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\) The event \(C\) has \(\mathrm { P } ( C ) = 0.20\)
    The events \(A\) and \(C\) are mutually exclusive and the events \(B\) and \(C\) are independent.
  3. Find \(\mathrm { P } ( B \cap C )\)
  4. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\) and the probabilities for each region.
  5. Find \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)
Edexcel S1 2001 June Q1
  1. Each of the 25 students on a computer course recorded the number of minutes \(x\), to the nearest minute, spent surfing the internet during a given day. The results are summarised below.
$$\Sigma x = 1075 , \Sigma x ^ { 2 } = 44625 .$$
  1. Find \(\mu\) and \(\sigma\) for these data. Two other students surfed the internet on the same day for 35 and 51 minutes respectively.
  2. Without further calculation, explain the effect on the mean of including these two students.
    (2)