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AQA S1 2014 June Q3
12 marks Easy -1.2
3 The table shows the colour of hair and the colour of eyes of a sample of 750 people from a particular population.
AQA S1 2014 June Q4
7 marks Moderate -0.3
4 Every year, usually during early June, the Isle of Man hosts motorbike races. Each race consists of three consecutive laps of the island's course. To compete in a race, a rider must first complete at least one qualifying lap. The data refer to the lightweight motorbike class in 2012 and show, for each of a random sample of 10 riders, values of $$u = x - 100 \quad \text { and } \quad v = y - 100$$ where \(x\) denotes the average speed, in mph, for the rider's fastest qualifying lap and \(y\) denotes the average speed, in mph, for the rider's three laps of the race.
\cline { 2 - 11 } \multicolumn{1}{c|}{}Rider
\cline { 2 - 11 } \multicolumn{1}{c|}{}\(\mathbf { A }\)\(\mathbf { B }\)\(\mathbf { C }\)\(\mathbf { D }\)\(\mathbf { E }\)\(\mathbf { F }\)\(\mathbf { G }\)\(\mathbf { H }\)\(\mathbf { I }\)\(\mathbf { J }\)
\(\boldsymbol { u }\)7.8813.024.292.886.267.033.6011.7813.1511.69
\(\boldsymbol { v }\)6.6310.163.630.475.708.013.307.3113.0811.82
    1. Calculate the value of \(r _ { u v }\), the product moment correlation coefficient between \(u\) and \(v\).
    2. Hence state the value of \(r _ { x y }\), giving a reason for your answer.
  2. Interpret your value of \(r _ { x y }\) in the context of this question.
AQA S1 2014 June Q5
13 marks Moderate -0.3
5 An analysis of the number of vehicles registered by each household within a city resulted in the following information.
Number of vehicles registered012\(\geqslant 3\)
Percentage of households18472510
  1. A random sample of 30 households within the city is selected. Use a binomial distribution with \(n = 30\), together with relevant information from the table in each case, to find the probability that the sample contains:
    1. exactly 3 households with no registered vehicles;
    2. at most 5 households with three or more registered vehicles;
    3. more than 10 households with at least two registered vehicles;
    4. more than 5 households but fewer than 10 households with exactly two registered vehicles.
  2. If a random sample of \(\mathbf { 1 5 0 }\) households within the city were to be selected, estimate the mean and the variance for the number of households in the sample that would have either one or two registered vehicles.
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-16_1075_1707_1628_153}
AQA S1 2014 June Q6
12 marks Moderate -0.8
6 A rubber seal is fitted to the bottom of a flood barrier. When no pressure is applied, the depth of the seal is 15 cm . When pressure is applied, a watertight seal is created between the flood barrier and the ground. The table shows the pressure, \(x\) kilopascals ( kPa ), applied to the seal and the resultant depth, \(y\) centimetres, of the seal.
\(\boldsymbol { x }\)255075100125150175200250300
\(\boldsymbol { y }\)14.713.412.811.911.010.39.79.07.56.7
    1. State the value that you would expect for \(a\) in the equation of the least squares regression line, \(y = a + b x\).
    2. Calculate the equation of the least squares regression line, \(y = a + b x\).
    3. Interpret, in context, your value for \(b\).
  2. Calculate an estimate of the depth of the seal when it is subjected to a pressure of 225 kPa .
    1. Give a statistical reason as to why your equation is unlikely to give a realistic estimate of the depth of the seal if it were to be subjected to a pressure of 400 kPa .
    2. Give a reason based on the context of this question as to why your equation will not give a realistic estimate of the depth of the seal if it were to be subjected to a pressure of 525 kPa .
      [0pt] [3 marks]
      \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-20_946_1709_1761_153}
      \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-21_2484_1707_221_153}
      \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-23_2484_1707_221_153}
AQA S1 2014 June Q7
10 marks Moderate -0.3
7 The volume of water, \(V\), used by a guest in an en suite shower room at a small guest house may be modelled by a random variable with mean \(\mu\) litres and standard deviation 65 litres. A random sample of 80 guests using this shower room showed a mean usage of 118 litres of water.
    1. Give a numerical justification as to why \(V\) is unlikely to be normally distributed.
    2. Explain why \(\bar { V }\), the mean of a random sample of 80 observations of \(V\), may be assumed to be approximately normally distributed.
    1. Construct a \(98 \%\) confidence interval for \(\mu\).
    2. Hence comment on a claim that \(\mu\) is 140 .
      \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-24_1526_1709_1181_153}
      \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-25_2484_1707_221_153}
      \includegraphics[max width=\textwidth, alt={}, center]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-27_2490_1719_217_150} \includegraphics[max width=\textwidth, alt={}, center]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-28_2486_1728_221_141}
AQA S1 2016 June Q1
5 marks Moderate -0.8
1 The table shows the heights, \(x \mathrm {~cm}\), and the arm spans, \(y \mathrm {~cm}\), of a random sample of 12 men aged between 21 years and 40 years.
\(\boldsymbol { x }\)152166154159179167155168174182161163
\(\boldsymbol { y }\)143154151153168160146163170175155158
  1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
  2. Interpret, in context, your value calculated in part (a).
AQA S1 2016 June Q2
8 marks Moderate -0.8
2 A small chapel was open to visitors for 55 days during the summer of 2015. The table summarises the daily numbers of visitors.
Number of visitorsNumber of days
20 or fewer1
212
223
236
248
2510
2613
277
282
291
30 or more2
Total55
  1. For these data:
    1. state the modal value;
    2. find values for the median and the interquartile range.
  2. Name one measure of average and one measure of spread that cannot be calculated exactly from the data in the table.
    [0pt] [2 marks]
  3. Reference to the raw data revealed that the 3 unknown exact values in the table were 13,37 and 58. Making use of this additional information, together with the data in the table, calculate the value of each of the two measures that you named in part (b).
    [0pt] [3 marks]
AQA S1 2016 June Q3
14 marks Easy -1.2
3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.
\multirow{2}{*}{}Charge band for course of treatment
Band 0Band 1Band 2Band 3Total
\multirow{4}{*}{Age of patient (years)}Under 1932435080
Between 19 and 401762223104
Between 41 and 6528823531176
66 or over1353686140
Total9024013040500
  1. Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
    1. aged between 41 and 65;
    2. aged 66 or over and charged at band 2;
    3. aged between 19 and 40 and charged at most at band 1;
    4. aged 41 or over, given that the patient was charged at band 2;
    5. charged at least at band 2, given that the patient was not aged 66 or over.
  2. Four patients at this dental surgery, not included in the above 500 patients, are selected at random. Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
    [0pt] [5 marks]
AQA S1 2016 June Q4
9 marks Moderate -0.8
4 As part of her science project, a student found the mass, \(y\) grams, of a particular compound that dissolved in 100 ml of water at each of 12 different set temperatures, \(x ^ { \circ } \mathrm { C }\). The results are shown in the table.
\(\boldsymbol { x }\)202530354045505560657075
\(\boldsymbol { y }\)242262269290298310326355359375390412
  1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
  2. Interpret, in context, your value for the gradient of this regression line.
  3. Use your equation to estimate the mass of the compound which will dissolve in 100 ml of water at \(68 ^ { \circ } \mathrm { C }\).
  4. Given that the values of the 12 residuals for the regression line of \(y\) on \(x\) lie between - 7 and + 9 , comment, with justification, on the likely accuracy of your estimate in part (c).
    [0pt] [2 marks]
AQA S1 2016 June Q5
18 marks Moderate -0.3
5 Still mineral water is supplied in 1.5-litre bottles. The actual volume, \(X\) millilitres, in a bottle may be modelled by a normal distribution with mean \(\mu = 1525\) and standard deviation \(\sigma = 9.6\).
  1. Determine the probability that the volume of water in a randomly selected bottle is:
    1. less than 1540 ml ;
    2. more than 1535 ml ;
    3. between 1515 ml and 1540 ml ;
    4. not 1500 ml .
  2. The supplier requires that only 10 per cent of bottles should contain more than 1535 ml of water. Assuming that there has been no change in the value of \(\sigma\), calculate the reduction in the value of \(\mu\) in order to satisfy this requirement. Give your answer to one decimal place.
  3. Sparkling spring water is supplied in packs of six 0.5 -litre bottles. The actual volume in a bottle may be modelled by a normal distribution with mean 508.5 ml and standard deviation 3.5 ml . Stating a necessary assumption, determine the probability that:
    1. the volume of water in each of the 6 bottles from a randomly selected pack is more than 505 ml ;
    2. the mean volume of water in the 6 bottles from a randomly selected pack is more than 505 ml .
      [0pt] [7 marks]
AQA S1 2016 June Q6
12 marks Moderate -0.8
6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.
ColourBlueGreenRedOrangeYellowWhite
Proportion0.250.250.180.120.150.05
  1. A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with \(n = 50\), together with relevant information from the table, to estimate the probability that this sample contains:
    1. exactly 4 red loom bands;
    2. at most 10 yellow loom bands;
    3. at least 30 blue or green loom bands;
    4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
  2. The random variable \(R\) denotes the number of red loom bands in a random sample of \(\mathbf { 3 0 0 }\) loom bands selected from the box. Estimate values for the mean and the variance of \(R\).
    [0pt] [2 marks]
AQA S1 2016 June Q7
9 marks Standard +0.3
7 Customers buying euros ( €) at a travel agency must pay for them in pounds ( \(\pounds\) ). The amounts paid, \(\pounds x\), by a sample of 40 customers were, in ascending order, as follows.
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.