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Edexcel S1 Q3
11 marks Moderate -0.3
3. A soccer fan collected data on the number of minutes of league football, \(m\), played by each team in the four main divisions before first scoring a goal at the start of a new season. Her results are shown in the table below.
\(m\) (minutes)Number of teams
\(0 \leq m < 40\)36
\(40 \leq m < 80\)28
\(80 \leq m < 120\)10
\(120 \leq m < 160\)4
\(160 \leq m < 200\)5
\(200 \leq m < 300\)4
\(300 \leq m < 400\)2
\(400 \leq m < 600\)3
  1. Calculate estimates of the mean and standard deviation of these data.
  2. Explain why the mean and standard deviation might not be the best summary statistics to use with these data.
  3. Suggest alternative summary statistics that would better represent these data.
Edexcel S1 Q4
13 marks Standard +0.3
4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.
  1. Calculate the probability that on any one day Alan will run for less than 20 minutes.
  2. Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
  3. On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.
Edexcel S1 Q5
14 marks Easy -1.2
5. In a survey unemployed people were asked how many months it had been, to the nearest month, since they were last employed on a full-time basis. The data collected is summarised in this stem and leaf diagram.
Number of months(2 | 1 means 21 months)Totals
011224446779(11)
102355689( )
21568( )
3079( )
45( )
527(2)
63(1)
70(1)
  1. Write down the values needed to complete the totals column on the stem and leaf diagram.
  2. State the mode of these data.
  3. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
  4. determine if there are any outliers in these data,
  5. draw a box plot representing these data on graph paper,
  6. describe the skewness of these data and suggest a reason for it.
Edexcel S1 Q6
17 marks Easy -1.8
6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:
\(a\)123
\(\mathrm { P } ( A = a )\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
  1. State the name of this distribution.
  2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:
    \(b\)123
    \(\mathrm { P } ( B = b )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 4 }\)
  3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
  4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
  5. Find the probability distribution of \(C\).
  6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).
Edexcel S1 Q1
7 marks Moderate -0.8
  1. A shop recorded the number of pairs of gloves, \(n\), that it sold and the average daytime temperature, \(T ^ { \circ } \mathrm { C }\), for each month over a 12-month period.
The data was then summarised as follows: $$\Sigma T = 124 , \quad \Sigma n = 384 , \quad \Sigma T ^ { 2 } = 1802 , \quad \Sigma n ^ { 2 } = 18518 , \quad \Sigma T n = 2583 .$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Comment on what your value shows and suggest a reason for this.
Edexcel S1 Q2
8 marks Standard +0.3
2. Events \(A\) and \(B\) are independent. Given also that $$\mathrm { P } ( A ) = \frac { 3 } { 4 } \quad \text { and } \quad \mathrm { P } \left( A \cap B ^ { \prime } \right) = \frac { 1 } { 4 }$$ Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\mathrm { P } ( B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\).
Edexcel S1 Q3
10 marks Easy -1.2
3. The random variable \(X\) is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of \(a\) and \(b\) for
  1. \(\mathrm { E } ( 2 X + 3 )\),
  2. \(\quad \operatorname { Var } ( 2 X + 3 )\),
  3. \(\mathrm { E } \left( X ^ { 2 } \right)\).
  4. Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$
Edexcel S1 Q4
11 marks Standard +0.3
  1. An engineer tested a new material under extreme conditions in a wind tunnel. He recorded the number of microfractures, \(n\), that formed and the wind speed, \(v\) metres per second, for 8 different values of \(v\) with all other conditions remaining constant. He then coded the data using \(x = v - 700\) and \(y = n - 20\) and calculated the following summary statistics.
$$\Sigma x = 100 , \quad \Sigma y = 23 , \quad \Sigma x ^ { 2 } = 215000 , \quad \Sigma x y = 11600 .$$
  1. Find an equation of the regression line of \(y\) on \(x\).
  2. Hence, find an equation of the regression line of \(n\) on \(v\).
  3. Use your regression line to estimate the number of microfractures that would be formed if the material was tested in a wind speed of 900 metres per second with all other conditions remaining constant.
    (2 marks)
Edexcel S1 Q5
12 marks Moderate -0.3
5. An antiques shop recorded the value of items stolen to the nearest pound during each week for a year giving the data in the table below.
Value of goods stolen (£)Number of weeks
0-19931
200-3996
400-5993
600-7994
800-9995
1000-19992
2000-29991
Letting \(x\) represent the mid-point of each group and using the coding \(y = \frac { x - 699.5 } { 200 }\),
  1. find \(\sum\) fy.
  2. estimate to the nearest pound the mean and standard deviation of the value of the goods stolen each week using your value for \(\sum f y\) and \(\sum f y ^ { 2 } = 424\).
    (6 marks)
    The median for these data is \(\pounds 82\).
  3. Explain why the manager of the shop might be reluctant to use either the mean or the median in summarising these data.
    (3 marks)
Edexcel S1 Q6
12 marks Moderate -0.3
6. At the start of a gameshow there are 10 contestants of which 6 are female. In each round of the game, one contestant is eliminated. All of the contestants have the same chance of progressing to the next round each time.
  1. Show that the probability that the first two contestants to be eliminated are both male is \(\frac { 2 } { 15 }\).
  2. Find the probability that more females than males are eliminated in the first three rounds of the game.
  3. Given that the first contestant to be eliminated is male, find the probability that the next two contestants to be eliminated are both female.
    (3 marks)
Edexcel S1 Q7
15 marks Moderate -0.8
7. A cyber-cafe recorded how long each user stayed during one day giving the following results.
Length of stay
(minutes)
\(0 -\)\(30 -\)\(60 -\)\(90 -\)\(120 -\)\(240 -\)\(360 -\)
Number of users153132231720
  1. Use linear interpolation to estimate the median and quartiles of these data. The results of a previous study had led to the suggestion that the length of time each user stays can be modelled by a normal distribution with a mean of 72 minutes and a standard deviation of 48 minutes.
  2. Find the median and quartiles that this model would predict.
  3. Comment on the suitability of the suggested model in the light of the new results.
AQA S2 2006 January Q1
9 marks Moderate -0.3
1 A study undertaken by Goodhealth Hospital found that the number of patients each month, \(X\), contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .
    1. Calculate \(\mathrm { P } ( X = 2 )\).
    2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
    1. Write down the distribution of \(Y\), the number of patients contracting this superbug in a given 6-month period.
    2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
  3. State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.
AQA S2 2006 January Q2
12 marks Standard +0.3
2 Year 12 students at Newstatus School choose to participate in one of four sports during the Spring term. The students' choices are summarised in the table.
SquashBadmintonArcheryHockeyTotal
Male516301970
Female4203353110
Total9366372180
  1. Use a \(\chi ^ { 2 }\) test, at the \(5 \%\) level of significance, to determine whether the choice of sport is independent of gender.
  2. Interpret your result in part (a) as it relates to students choosing hockey.
AQA S2 2006 January Q3
9 marks Standard +0.3
3 The time, \(T\) minutes, that parents have to wait before seeing a mathematics teacher at a school parents' evening can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). At a recent parents' evening, a random sample of 9 parents was asked to record the times that they waited before seeing a mathematics teacher. The times, in minutes, are $$\begin{array} { l l l l l l l l l } 5 & 12 & 10 & 8 & 7 & 6 & 9 & 7 & 8 \end{array}$$
  1. Construct a \(90 \%\) confidence interval for \(\mu\).
  2. Comment on the headteacher's claim that the mean time that parents have to wait before seeing a mathematics teacher is 5 minutes.
AQA S2 2006 January Q4
11 marks Easy -1.2
4
  1. A random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} k & a < x < b \\ 0 & \text { otherwise } \end{cases}$$
    1. Show that \(k = \frac { 1 } { b - a }\).
    2. Prove, using integration, that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )\).
  2. The error, \(X\) grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: $$f ( x ) = \begin{cases} k & - 2 < x < 4 \\ 0 & \text { otherwise } \end{cases}$$
    1. Write down the value of the mean, \(\mu\), of \(X\).
    2. Evaluate the standard deviation, \(\sigma\), of \(X\).
    3. Hence find \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).
AQA S2 2006 January Q5
6 marks Moderate -0.8
5 The Globe Express agency organises trips to the theatre. The cost, \(\pounds X\), of these trips can be modelled by the following probability distribution:
\(\boldsymbol { x }\)40455574
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.300.240.360.10
  1. Calculate the mean and standard deviation of \(X\).
  2. For special celebrity charity performances, Globe Express increases the cost of the trips to \(\pounds Y\), where $$Y = 10 X + 250$$ Determine the mean and standard deviation of \(Y\).
AQA S2 2006 January Q6
8 marks Moderate -0.3
6 In previous years, the marks obtained in a French test by students attending Topnotch College have been modelled satisfactorily by a normal distribution with a mean of 65 and a standard deviation of 9 . Teachers in the French department at Topnotch College suspect that this year their students are, on average, underachieving. In order to investigate this suspicion, the teachers selected a random sample of 35 students to take the French test and found that their mean score was 61.5.
  1. Investigate, at the \(5 \%\) level of significance, the teachers' suspicion.
  2. Explain, in the context of this question, the meaning of a Type I error.
AQA S2 2006 January Q7
10 marks Standard +0.3
7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, \(T\) hours, is modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(\mathrm { E } ( T ) = \frac { 8 } { 15 }\).
    1. Find the cumulative distribution function, \(\mathrm { F } ( t )\), for \(0 \leqslant t \leqslant 1\).
    2. Hence, or otherwise, for a commuter selected at random, find $$\mathrm { P } ( \text { mean } < T < \text { median } )$$
AQA S2 2006 January Q8
10 marks Moderate -0.3
8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed. In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured. The volumes, in millilitres, of sherry in these bottles are found to be
9961006100999910071003
998101099799610081007
Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the \(5 \%\) level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.
AQA S2 2007 January Q1
5 marks Moderate -0.3
1 Alan's journey time, in minutes, to travel home from work each day is known to be normally distributed with mean \(\mu\). Alan records his journey time, in minutes, on a random sample of 8 days as being $$\begin{array} { l l l l l l l l } 36 & 38 & 39 & 40 & 50 & 35 & 36 & 42 \end{array}$$ Construct a \(95 \%\) confidence interval for \(\mu\).
AQA S2 2007 January Q2
13 marks Moderate -0.3
2 The number of computers, \(A\), bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5. The number of computers, \(B\), bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0 .
    1. Calculate \(\mathrm { P } ( A = 4 )\).
    2. Determine \(\mathrm { P } ( B \leqslant 6 )\).
    3. Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day.
  2. Calculate the probability that a total of fewer than 10 computers is bought from these two stores on at least 4 out of 5 consecutive days.
  3. The numbers of computers bought from the Choicebuy computer store over a 10-day period are recorded as $$\begin{array} { l l l l l l l l l l } 8 & 12 & 6 & 6 & 9 & 15 & 10 & 8 & 6 & 12 \end{array}$$
    1. Calculate the mean and variance of these data.
    2. State, giving a reason based on your results in part (c)(i), whether or not a Poisson distribution provides a suitable model for these data.
AQA S2 2007 January Q3
8 marks Standard +0.3
3 The handicap committee of a golf club has indicated that the mean score achieved by the club's members in the past was 85.9 . A group of members believes that recent changes to the golf course have led to a change in the mean score achieved by the club's members and decides to investigate this belief. A random sample of the scores, \(x\), of 100 club members was taken and is summarised by $$\sum x = 8350 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 15321$$ where \(\bar { x }\) denotes the sample mean.
Test, at the \(5 \%\) level of significance, the group's belief that the mean score of 85.9 has changed.
AQA S2 2007 January Q4
9 marks Easy -1.3
4 The number of fish, \(X\), caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution:
\(\boldsymbol { x }\)123456\(\geqslant 7\)
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.010.050.140.30\(k\)0.120
  1. Find the value of \(k\).
  2. Find:
    1. \(\mathrm { E } ( X )\);
    2. \(\operatorname { Var } ( X )\).
  3. When Pearl sells her fish, she earns a profit, in pounds, given by $$Y = 5 X + 2$$ Find:
    1. \(\mathrm { E } ( Y )\);
    2. the standard deviation of \(Y\).
AQA S2 2007 January Q5
8 marks Standard +0.3
5 Jasmine's French teacher states that a homework assignment should take, on average, 30 minutes to complete. Jasmine believes that he is understating the mean time that the assignment takes to complete and so decides to investigate. She records the times, in minutes, that it takes for a random sample of 10 students to complete the French assignment, with the following results: $$\begin{array} { l l l l l l l l l l } 29 & 33 & 36 & 42 & 30 & 28 & 31 & 34 & 37 & 35 \end{array}$$
  1. Test, at the \(1 \%\) level of significance, Jasmine's belief that her French teacher has understated the mean time that it should take to complete the homework assignment.
  2. State an assumption that you must make in order for the test used in part (a) to be valid.
AQA S2 2007 January Q6
14 marks Standard +0.3
6 The waiting time, \(T\) minutes, before being served at a local newsagents can be modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 8 } t ^ { 2 } & 0 \leqslant t < 1 \\ \frac { 1 } { 16 } ( t + 5 ) & 1 \leqslant t \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  1. Sketch the graph of f.
  2. For a customer selected at random, calculate \(\mathrm { P } ( T \geqslant 1 )\).
    1. Show that the cumulative distribution function for \(1 \leqslant t \leqslant 3\) is given by $$\mathrm { F } ( t ) = \frac { 1 } { 32 } \left( t ^ { 2 } + 10 t - 7 \right)$$
    2. Hence find the median waiting time.