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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
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. 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
  2. \(\mathrm { P } ( B )\)
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\)
  4. \(\mathrm { P } \left( B ^ { \prime } \mid A \right)\)
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
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.
  3. 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 \(g\).
Edexcel S1 2013 January Q2
8 marks Moderate -0.8
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
10 marks Moderate -0.8
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
10 marks Standard +0.8
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,
  3. find the probability that her phone will not need charging before her journey is completed.
Edexcel S1 2013 January Q5
15 marks Moderate -0.8
  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
13 marks Standard +0.3
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
12 marks Moderate -0.3
  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
5 marks Easy -1.2
  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)
Edexcel S1 2001 June Q2
5 marks Easy -1.2
2. On a particular day in summer 1993 at 0800 hours the height above sea level, \(x\) metres, and the temperature, \(y ^ { \circ } \mathrm { C }\), were recorded in 10 Mediterranean towns. The following summary statistics were calculated from the results. $$\Sigma x = 7300 , \Sigma x ^ { 2 } = 6599600 , S _ { x y } = - 13060 , S _ { y y } = 140.9 .$$
  1. Find \(S _ { x x }\).
  2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(x\) and \(y\).
  3. Give an interpretation of your coefficient.
Edexcel S1 2001 June Q3
8 marks Moderate -0.3
3. The continuous random variable \(Y\) is normally distributed with mean 100 and variance 256 .
  1. Find \(\mathrm { P } ( Y < 80 )\).
  2. Find \(k\) such that \(\mathrm { P } ( 100 - k \leq Y \leq 100 + k ) = 0.516\).
Edexcel S1 2001 June Q4
12 marks Easy -1.8
4. The discrete random variable \(X\) has the probability function shown in the table below.
\(x\)- 2- 10123
\(\mathrm { P } ( X = x )\)0.1\(\alpha\)0.30.20.10.1
Find
  1. \(\alpha\),
  2. \(\mathrm { P } ( - 1 < X \leq 2 )\),
  3. \(\mathrm { F } ( - 0.4 )\),
  4. \(\mathrm { E } ( 3 X + 4 )\),
  5. \(\operatorname { Var } ( 2 X + 3 )\).
Edexcel S1 2001 June Q5
13 marks Easy -1.2
5. A market researcher asked 100 adults which of the three newspapers \(A , B , C\) they read. The results showed that \(30 \operatorname { read } A , 26\) read \(B , 21\) read \(C , 5\) read both \(A\) and \(B , 7\) read both \(B\) and \(C , 6\) read both \(C\) and \(A\) and 2 read all three.
  1. Draw a Venn diagram to represent these data. One of the adults is then selected at random.
    Find the probability that she reads
  2. at least one of the newspapers,
  3. only \(A\),
  4. only one of the newspapers,
  5. \(A\) given that she reads only one newspaper.
Edexcel S1 2001 June Q6
16 marks Easy -1.2
6. Three swimmers Alan, Diane and Gopal record the number of lengths of the swimming pool they swim during each practice session over several weeks. The stem and leaf diagram below shows the results for Alan.
Lengths20 means 20
20122\(( 4 )\)
255667789\(( 7 )\)
3012224\(( 5 )\)
3566679\(( 5 )\)
401333333444\(( 10 )\)
45556667788999\(( 12 )\)
5000\(( 3 )\)
  1. Find the three quartiles for Alan's results. The table below summarises the results for Diane and Gopal.
    DianeGopal
    Smallest value3525
    Lower quartile3734
    Median4242
    Upper quartile5350
    Largest value6557
  2. Using the same scale and on the same sheet of graph paper draw box plots to represent the data for Alan, Diane and Gopal.
  3. Compare and contrast the three box plots.
Edexcel S1 2001 June Q7
16 marks Moderate -0.3
7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the
number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.
\(x\)1215711184693
\(y\)84138181215141216
  1. Plot these data on a scatter diagram.
  2. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). $$\text { (You may use } \left. \Sigma x ^ { 2 } = 746 , \Sigma x y = 749 . \right)$$
  3. Give an interpretation of the slope and the intercept of your regression line.
  4. State whether or not you think the regression model is reasonable
    1. for the range of \(x\)-values given in the table,
    2. for all possible \(x\)-values. In each case justify your answer either by giving a reason for accepting the model or by suggesting an alternative model. END