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AQA S2 2013 June Q4
9 marks Standard +0.3
4 Gamma-ray bursts (GRBs) are pulses of gamma rays lasting a few seconds, which are produced by explosions in distant galaxies. They are detected by satellites in orbit around Earth. One particular satellite detects GRBs at a constant average rate of 3.5 per week (7 days). You may assume that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
  1. Find the probability that the satellite detects:
    1. exactly 4 GRBs during one particular week;
    2. at least 2 GRBs on one particular day;
    3. more than 10 GRBs but fewer than 20 GRBs during the 28 days of February 2013.
  2. Give one reason, apart from the constant average rate, why it is likely that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
    (1 mark)
AQA S2 2013 June Q5
13 marks Moderate -0.8
5 In a computer game, players try to collect five treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable \(X\). The probability distribution of \(X\) is defined by $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 2 } & x = 1,2,3,4 \\ k & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$
    1. Show that \(k = \frac { 1 } { 20 }\).
    2. Calculate the value of \(\mathrm { E } ( X )\).
    3. Show that \(\operatorname { Var } ( X ) = 1.5275\).
    4. Find the probability that Isaac collects more than 2 treasures.
  2. The number of points that Isaac scores for collecting treasures is \(Y\) where $$Y = 100 X - 50$$ Calculate the mean and the standard deviation of \(Y\).
AQA S2 2013 June Q6
13 marks Standard +0.3
6 A supermarket buys pears from a local supplier. The supermarket requires the mean weight of the pears to be at least 175 grams. William, the fresh-produce manager at the supermarket, suspects that the latest batch of pears delivered does not meet this requirement.
  1. William weighs a random sample of 6 pears, obtaining the following weights, in grams. $$\begin{array} { l l l l l l } 160.6 & 155.4 & 181.3 & 176.2 & 162.3 & 172.8 \end{array}$$ Previous batches of pears have had weights that could be modelled by a normal distribution with standard deviation 9.4 grams. Assuming that this still applies, show that a hypothesis test at the \(5 \%\) level of significance supports William's suspicion.
    (7 marks)
  2. William then weighs a random sample of 20 pears. The mean of this sample is 169.4 grams and \(s = 11.2\) grams, where \(s ^ { 2 }\) is an unbiased estimate of the population variance. Assuming that the population from which this sample is taken has a normal distribution but with unknown standard deviation, test William's suspicion at the \(\mathbf { 1 \% }\) level of significance.
  3. Give a reason why the probability of a Type I error occurring was smaller when conducting the test in part (b) than when conducting the test in part (a).
AQA S2 2013 June Q7
15 marks Standard +0.3
7 A continuous random variable \(X\) has the probability density function defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 3 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Sketch the graph of f on the axes below.
    1. Find the cumulative distribution function, F , for \(0 \leqslant x \leqslant 1\).
    2. Hence, or otherwise, find the value of the lower quartile of \(X\).
    1. Show that the cumulative distribution function for \(1 \leqslant x \leqslant 2\) is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 3 } \left( 5 x - x ^ { 2 } - 3 \right)$$
    2. Hence, or otherwise, find the value of the upper quartile of \(X\). \includegraphics[max width=\textwidth, alt={}, center]{03c1e107-3377-4b0d-9daf-7f70233c18b5-5_554_1050_1217_424}
AQA S2 2014 June Q1
7 marks Moderate -0.3
1 Vanya collected five samples of air and measured the carbon dioxide content of each sample, in parts per million by volume (ppmv). The results were as follows. $$\begin{array} { l l l l l } 387 & 375 & 382 & 379 & 381 \end{array}$$
  1. Assuming that these data form a random sample from a normal distribution with mean \(\mu\) ppmv, construct a \(90 \%\) confidence interval for \(\mu\).
    [0pt] [6 marks]
  2. Vanya repeated her sampling procedure on each of 30 days and, for each day's results, a \(90 \%\) confidence interval for \(\mu\) was constructed. On how many of these 30 days would you expect \(\mu\) to lie outside that day's confidence interval?
    [0pt] [1 mark]
AQA S2 2014 June Q2
11 marks Moderate -0.3
2 A large multinational company recruits employees from all four countries in the UK. For a sample of 250 recruits, the percentages of males and females from each of the countries are shown in Table 1. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 1}
\cline { 2 - 5 } \multicolumn{1}{c|}{}EnglandScotlandWales
Northern
Ireland
Male22.817.610.86.8
Female15.617.27.61.6
\end{table}
  1. Add the frequencies to the contingency table, Table 2, below.
  2. Carry out a \(\chi ^ { 2 }\)-test at the \(10 \%\) significance level to investigate whether there is an association between country and gender of recruits.
  3. By comparing observed and expected values, make one comment about the distribution of female recruits.
    [0pt] [1 mark] \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 2}
    EnglandScotlandWalesNorthern IrelandTotal
    Male145
    Female105
    Total250
    \end{table}
AQA S2 2014 June Q3
11 marks Moderate -0.3
3 A box contains a large number of pea pods. The number of peas in a pod may be modelled by the random variable \(X\). The probability distribution of \(X\) is tabulated below.
\(\boldsymbol { x }\)2 or fewer345678 or more
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)00.10.2\(a\)0.3\(b\)0
  1. Two pods are picked randomly from the box. Find the probability that the number of peas in each pod is at most 4.
  2. It is given that \(\mathrm { E } ( X ) = 5.1\).
    1. Determine the values of \(a\) and \(b\).
    2. Hence show that \(\operatorname { Var } ( X ) = 1.29\).
    3. Some children play a game with the pods, randomly picking a pod and scoring points depending on the number of peas in the pod. For each pod picked, the number of points scored, \(N\), is found by doubling the number of peas in the pod and then subtracting 5. Find the mean and the standard deviation of \(N\).
      [0pt] [3 marks]
AQA S2 2014 June Q4
11 marks Standard +0.3
4 A continuous random variable \(X\) has a probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(b > a > 0\).
    1. Prove that \(k = b - a\).
    2. Write down the value of \(\mathrm { E } ( X )\).
    3. Show, by integration, that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)\).
    4. Hence derive a simplified formula for \(\operatorname { Var } ( X )\).
  2. Given that \(a = 4\) and \(\operatorname { Var } ( X ) = 3\), find the numerical value of \(\mathrm { E } ( X )\).
    \includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-08_1347_1707_1356_153}
AQA S2 2014 June Q5
14 marks Standard +0.3
5 Peter, a geologist, is studying pebbles on a beach. He uses a frame, called a quadrat, to enclose an area of the beach. He then counts the number of quartz pebbles, \(X\), within the quadrat. He repeats this procedure 40 times, obtaining the following summarised data. $$\sum x = 128 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 126.4$$ Peter believes that the distribution of \(X\) can be modelled by a Poisson distribution with \(\lambda = 3.2\).
  1. Use the summarised data to support Peter's belief.
  2. Using Peter's model, calculate the probability that:
    1. a single placing of the quadrat contains more than 5 quartz pebbles;
    2. a single placing of the quadrat contains at least 3 quartz pebbles but fewer than 8 quartz pebbles;
    3. when the quadrat is placed twice, at least one placing contains no quartz pebbles.
  3. Peter also models the number of flint pebbles enclosed by the quadrat by a Poisson distribution with mean 5 . He assumes that the number of flint pebbles enclosed by the quadrat is independent of the number of quartz pebbles enclosed by the quadrat. Using Peter's models, calculate the probability that a single placing of the quadrat contains a total of either 9 or 10 pebbles which are quartz or flint.
    [0pt] [3 marks]
AQA S2 2014 June Q6
12 marks Standard +0.3
6 South Riding Alarms (SRA) maintains household burglar-alarm systems. The company aims to carry out an annual service of a system in a mean time of 20 minutes.
Technicians who carry out an annual service must record the times at which they start and finish the service.
  1. Gary is employed as a technician by SRA and his manager, Rajul, calculates the times taken for 8 annual services carried out by Gary. The results, in minutes, are as follows. $$\begin{array} { l l l l l l l l } 24 & 25 & 29 & 16 & 18 & 27 & 19 & 23 \end{array}$$ Assume that these times may be regarded as a random sample from a normal distribution. Carry out a hypothesis test, at the \(10 \%\) significance level, to examine whether the mean time for an annual service carried out by Gary is 20 minutes.
    [0pt] [8 marks]
  2. Rajul suspects that Gary may be taking longer than 20 minutes on average to carry out an annual service. Rajul therefore calculates the times taken for 100 annual services carried out by Gary. Assume that these times may also be regarded as a random sample from a normal distribution but with a standard deviation of 4.6 minutes. Find the highest value of the sample mean which would not support Rajul's suspicion at the \(5 \%\) significance level. Give your answer to two decimal places.
    [0pt] [4 marks] \(7 \quad\) A continuous random variable \(X\) has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 4 } { 5 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 20 } ( x - 3 ) ( 3 x - 11 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
    1. Find \(\mathrm { P } ( X < 1 )\).
      1. Show that, for \(1 \leqslant x \leqslant 3\), the cumulative distribution function, \(\mathrm { F } ( x )\), is given by $$\mathrm { F } ( x ) = \frac { 1 } { 20 } \left( x ^ { 3 } - 10 x ^ { 2 } + 33 x - 16 \right)$$
      2. Hence verify that the median value of \(X\) lies between 1.13 and 1.14 .
        [0pt] [3 marks] QUESTION
        PART Answer space for question 7
        REFERENCE REFERENCE
        \includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-20_2290_1707_221_153}
AQA S2 2015 June Q1
9 marks Standard +0.3
1 In a survey of the tideline along a beach, plastic bottles were found at a constant average rate of 280 per kilometre, and drinks cans were found at a constant average rate of 220 per kilometre. It may be assumed that these objects were distributed randomly and independently. Calculate the probability that:
  1. a 10 m length of tideline along this beach contains no more than 5 plastic bottles;
  2. a 20 m length of tideline along this beach contains exactly 2 drinks cans;
  3. a 30 m length of tideline along this beach contains a total of at least 12 but fewer than 18 of these two types of object.
    [0pt] [4 marks]
AQA S2 2015 June Q2
8 marks Moderate -0.3
2 The continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$
  1. Write down, in terms of \(a\) and \(b\), the value of \(k\).
    1. Given that \(\mathrm { E } ( X ) = 1\) and \(\operatorname { Var } ( X ) = 3\), find the values of \(a\) and \(b\).
    2. Four independent values of \(X\) are taken. Find the probability that exactly one of these four values is negative.
      [0pt] [3 marks]
AQA S2 2015 June Q3
10 marks Moderate -0.3
3 A machine fills bags with frozen peas. Measurements taken over several weeks have shown that the standard deviation of the weights of the filled bags of peas has been 2.2 grams. Following maintenance on the machine, a quality control inspector selected 8 bags of peas. The weights, in grams, of the bags were $$\begin{array} { l l l l l l l l } 910.4 & 908.7 & 907.2 & 913.2 & 905.6 & 911.1 & 909.5 & 907.9 \end{array}$$ It may be assumed that the bags constitute a random sample from a normal distribution.
  1. Giving the limits to four significant figures, calculate a 95\% confidence interval for the mean weight of a bag of frozen peas filled by the machine following the maintenance:
    1. assuming that the standard deviation of the weights of the bags of peas is known to be 2.2 grams;
    2. assuming that the standard deviation of the weights of the bags of peas may no longer be 2.2 grams.
  2. The weight printed on the bags of peas is 907 grams. One of the inspector's concerns is that bags should not be underweight. Make two comments about this concern with regard to the data and your calculated confidence intervals.
    [0pt] [2 marks]
AQA S2 2015 June Q4
11 marks Standard +0.3
4 Wellgrove village has a main road running through it that has a 40 mph speed limit. The villagers were concerned that many vehicles travelled too fast through the village, and so they set up a device for measuring the speed of vehicles on this main road. This device indicated that the mean speed of vehicles travelling through Wellgrove was 44.1 mph . In an attempt to reduce the mean speed of vehicles travelling through Wellgrove, life-size photographs of a police officer were erected next to the road on the approaches to the village. The speed, \(X \mathrm { mph }\), of a sample of 100 vehicles was then measured and the following data obtained. $$\sum x = 4327.0 \quad \sum ( x - \bar { x } ) ^ { 2 } = 925.71$$
  1. State an assumption that must be made about the sample in order to carry out a hypothesis test to investigate whether the desired reduction in mean speed had occurred.
  2. Given that the assumption that you stated in part (a) is valid, carry out such a test, using the \(1 \%\) level of significance.
  3. Explain, in the context of this question, the meaning of:
    1. a Type I error;
    2. a Type II error.
      [0pt] [2 marks]
AQA S2 2015 June Q5
10 marks Standard +0.3
5 In a particular town, a survey was conducted on a sample of 200 residents aged 41 years to 50 years. The survey questioned these residents to discover the age at which they had left full-time education and the greatest rate of income tax that they were paying at the time of the survey. The summarised data obtained from the survey are shown in the table.
\multirow{2}{*}{Greatest rate of income tax paid}Age when leaving education (years)\multirow[b]{2}{*}{Total}
16 or less17 or 1819 or more
Zero323439
Basic1021217131
Higher175830
Total1512029200
  1. Use a \(\chi ^ { 2 }\)-test, at the \(5 \%\) level of significance, to investigate whether there is an association between age when leaving education and greatest rate of income tax paid.
  2. It is believed that residents of this town who had left education at a later age were more likely to be paying the higher rate of income tax. Comment on this belief.
    [0pt] [1 mark]
AQA S2 2015 June Q6
12 marks Moderate -0.3
6 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 2 } x - \frac { 1 } { 16 } x ^ { 2 } & 0 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{cases}$$
  1. Find the probability that \(X\) lies between 0.4 and 0.8 .
  2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
    1. Find the value of \(\mathrm { E } ( X )\).
    2. Show that \(\operatorname { Var } ( X ) = \frac { 8 } { 9 }\).
  4. The continuous random variable \(Y\) is defined by $$Y = 3 X - 2$$ Find the values of \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).
AQA S2 2015 June Q7
15 marks Standard +0.3
7 Each week, a newsagent stocks 5 copies of the magazine Statistics Weekly. A regular customer always buys one copy. The demand for additional copies may be modelled by a Poisson distribution with mean 2. The number of copies sold in a week, \(X\), has the probability distribution shown in the table, where probabilities are stated correct to three decimal places.
\(\boldsymbol { x }\)12345
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.1350.2710.271\(a\)\(b\)
  1. Show that, correct to three decimal places, the values of \(a\) and \(b\) are 0.180 and 0.143 respectively.
  2. Find the values of \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\), showing the calculations needed to obtain these values, and hence calculate the standard deviation of \(X\).
  3. The newsagent makes a profit of \(\pounds 1\) on each copy of Statistics Weekly that is sold and loses 50 p on each copy that is not sold. Find the mean weekly profit for the newsagent from sales of this magazine.
  4. Assuming that the weekly demand remains the same, show that the mean weekly profit from sales of Statistics Weekly will be greater if the newsagent stocks only 4 copies.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{6cdf244b-168a-4be5-8ef8-8125daae8608-24_2488_1728_219_141}
AQA S3 2008 June Q1
7 marks Moderate -0.3
1 The best performances of a random sample of 20 junior athletes in the long jump, \(x\) metres, and in the high jump, \(y\) metres, were recorded. The following statistics were calculated from the results. $$S _ { x x } = 7.0036 \quad S _ { y y } = 0.8464 \quad S _ { x y } = 1.3781$$
  1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
    (2 marks)
  2. Assuming that these data come from a bivariate normal distribution, investigate, at the \(1 \%\) level of significance, the claim that for junior athletes there is a positive correlation between \(x\) and \(y\).
  3. Interpret your conclusion in the context of this question.
AQA S3 2008 June Q2
8 marks Moderate -0.3
2 A survey of a random sample of 200 passengers on UK internal flights revealed that 132 of them were on business trips.
  1. Construct an approximate \(98 \%\) confidence interval for the proportion of passengers on UK internal flights that are on business trips.
  2. Hence comment on the claim that more than 60 per cent of passengers on UK internal flights are on business trips.
AQA S3 2008 June Q3
6 marks Standard +0.3
3 Pitted black olives in brine are sold in jars labelled " 340 grams net weight". Two machines, A and B, independently fill these jars with olives before the brine is added. The weight, \(X\) grams, of olives delivered by machine A may be modelled by a normal distribution with mean \(\mu _ { X }\) and standard deviation 4.5. The weight, \(Y\) grams, of olives delivered by machine B may be modelled by a normal distribution with mean \(\mu _ { Y }\) and standard deviation 5.7. The mean weight of olives from a random sample of 10 jars filled by machine A is found to be 157 grams, whereas that from a random sample of 15 jars filled by machine \(B\) is found to be 162 grams. Test, at the \(1 \%\) level of significance, the hypothesis that \(\mu _ { X } = \mu _ { Y }\).
(6 marks)
AQA S3 2008 June Q4
10 marks Moderate -0.8
4 A manufacturer produces three models of washing machine: basic, standard and deluxe. An analysis of warranty records shows that \(25 \%\) of faults are on basic machines, \(60 \%\) are on standard machines and 15\% are on deluxe machines. For basic machines, 30\% of faults reported during the warranty period are electrical, \(50 \%\) are mechanical and \(20 \%\) are water-related. For standard machines, 40\% of faults reported during the warranty period are electrical, \(45 \%\) are mechanical and 15\% are water-related. For deluxe machines, \(55 \%\) of faults reported during the warranty period are electrical, \(35 \%\) are mechanical and \(10 \%\) are water-related.
  1. Draw a tree diagram to represent the above information.
  2. Hence, or otherwise, determine the probability that a fault reported during the warranty period:
    1. is electrical;
    2. is on a deluxe machine, given that it is electrical.
  3. A random sample of 10 electrical faults reported during the warranty period is selected. Calculate the probability that exactly 4 of them are on deluxe machines.
AQA S3 2008 June Q5
7 marks Standard +0.8
5 The daily number of emergency calls received from district A may be modelled by a Poisson distribution with a mean of \(\lambda _ { \mathrm { A } }\). The daily number of emergency calls received from district B may be modelled by a Poisson distribution with a mean of \(\lambda _ { \mathrm { B } }\). During a period of 184 days, the number of emergency calls received from district A was 3312, whilst the number received from district B was 2760.
  1. Construct an approximate \(95 \%\) confidence interval for \(\lambda _ { \mathrm { A } } - \lambda _ { \mathrm { B } }\).
  2. State one assumption that is necessary in order to construct the confidence interval in part (a).
AQA S3 2008 June Q6
18 marks Standard +0.3
6 An aircraft, based at airport A, flies regularly to and from airport B.
The aircraft's flying time, \(X\) minutes, from A to B has a mean of 128 and a variance of 50 .
The aircraft's flying time, \(Y\) minutes, on the return flight from B to A is such that $$\mathrm { E } ( Y ) = 112 , \quad \operatorname { Var } ( Y ) = 50 \quad \text { and } \quad \rho _ { X Y } = - 0.4$$
  1. Given that \(F = X + Y\) :
    1. find the mean of \(F\);
    2. show that the variance of \(F\) is 60 .
  2. At airport B , the stopover time, \(S\) minutes, is independent of \(F\) and has a mean of 75 and a variance of 36 . Find values for the mean and the variance of:
    1. \(T = F + S\);
    2. \(M = F - 3 S\).
  3. Hence, assuming that \(T\) and \(M\) are normally distributed, determine the probability that, on a particular round trip of the aircraft from A to B and back to A :
    1. the time from leaving A to returning to A exceeds 300 minutes;
    2. the stopover time is greater than one third of the total flying time.
AQA S3 2008 June Q7
19 marks Standard +0.8
7
  1. The random variable \(X\) has a Poisson distribution with \(\mathrm { E } ( X ) = \lambda\).
    1. Prove, from first principles, that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\).
    2. Hence deduce that \(\operatorname { Var } ( X ) = \lambda\).
  2. The independent Poisson random variables \(X _ { 1 }\) and \(X _ { 2 }\) are such that \(\mathrm { E } \left( X _ { 1 } \right) = 5\) and \(\mathrm { E } \left( X _ { 2 } \right) = 2\). The random variables \(D\) and \(F\) are defined by $$D = X _ { 1 } - X _ { 2 } \quad \text { and } \quad F = 2 X _ { 1 } + 10$$
    1. Determine the mean and the variance of \(D\).
    2. Determine the mean and the variance of \(F\).
    3. For each of the variables \(D\) and \(F\), give a reason why the distribution is not Poisson.
  3. The daily number of black printer cartridges sold by a shop may be modelled by a Poisson distribution with a mean of 5 . Independently, the daily number of colour printer cartridges sold by the same shop may be modelled by a Poisson distribution with a mean of 2. Use a distributional approximation to estimate the probability that the total number of black and colour printer cartridges sold by the shop during a 4 -week period ( 24 days) exceeds 175.
AQA S3 2009 June Q1
8 marks Standard +0.3
1 An analysis of a random sample of 150 urban dwellings for sale showed that 102 are semi-detached. An analysis of an independent random sample of 80 rural dwellings for sale showed that 36 are semi-detached.
  1. Construct an approximate \(99 \%\) confidence interval for the difference between the proportion of urban dwellings for sale that are semi-detached and the proportion of rural dwellings for sale that are semi-detached.
  2. Hence comment on the claim that there is no difference between these two proportions.