Questions — AQA S3 (74 questions)

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AQA S3 2011 June Q6
6 The weight, \(X\) grams, of a dressed pheasant may be modelled by a normal random variable with a mean of 1000 and a standard deviation of 120 . Pairs of dressed pheasants are selected for packing into boxes. The total weight of a pair, \(Y = X _ { 1 } + X _ { 2 }\) grams, may be modelled by a normal distribution with a mean of 2000 and a standard deviation of 140 .
    1. Show that \(\operatorname { Cov } \left( X _ { 1 } , X _ { 2 } \right) = - 4600\).
    2. Given that \(X _ { 1 } - X _ { 2 }\) may be assumed to be normally distributed, determine the probability that the difference between the weights of a selected pair of dressed pheasants exceeds 250 grams.
  2. The weight of a box is independent of the total weight of a pair of dressed pheasants, and is normally distributed with a mean of 500 grams and a standard deviation of 40 grams. Determine the probability that a box containing a pair of dressed pheasants weighs less than 2750 grams.
    \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-16_2486_1714_221_153}
AQA S3 2011 June Q7
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 \(\operatorname { Var } ( X ) = \mathrm { E } ( X )\).
  2. The random variable \(Y\) has a Poisson distribution with \(\mathrm { E } ( Y ) = 2.5\). Given that \(Z = 4 Y + 30\) :
    1. show that \(\operatorname { Var } ( Z ) = \mathrm { E } ( Z )\);
    2. give a reason why the distribution of \(Z\) is not Poisson.
      \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-19_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-20_2486_1714_221_153}
AQA S3 2011 June Q8
8 The tensile strength of rope is measured in kilograms. The standard deviation of the tensile strength of a particular design of 10 mm diameter rope is known to be 285 kilograms. A retail organisation, which buys such rope from two manufacturers, A and B , wishes to compare their ropes for mean tensile strength. The mean tensile strength, \(\bar { x }\), of a random sample of 80 lengths from manufacturer A was 3770 kilograms. The mean tensile strength, \(\bar { y }\), of a random sample of 120 lengths from manufacturer B was 3695 kilograms.
    1. Test, at the \(5 \%\) level of significance, the hypothesis that there is no difference between the mean tensile strength of rope from manufacturer A and that of rope from manufacturer B.
    2. Why was it not necessary to know the distributions of tensile strength in order for your test in part (a)(i) to be valid?
    1. Deduce that, for your test in part (a)(i), the critical values of \(( \bar { x } - \bar { y } )\) are \(\pm 80.63\), correct to two decimal places.
    2. In fact, the mean tensile strength of rope from manufacturer A exceeds that of rope from manufacturer B by 125 kilograms. Determine the probability of a Type II error for a test of the hypothesis in part (a)(i) at the \(5 \%\) level of significance, based upon a random sample of 80 lengths from manufacturer A and a random sample of 120 lengths from manufacturer B. (4 marks)
AQA S3 2012 June Q1
1 A wildlife expert measured the neck lengths, \(x\) metres, and the tail lengths, \(y\) metres, of a sample of 12 mature male giraffes as part of a study into their physical characteristics. The results are shown in the table.
AQA S3 2012 June Q2
2 As part of a comparison of two varieties of cucumber, Fanfare and Marketmore, random samples of harvested cucumbers of each variety were selected and their lengths measured, in centimetres. The results are summarised in the table.
\multirow{2}{*}{}\multirow[b]{2}{*}{Sample size}Length (cm)
Sample meanSample standard deviation
\multirow{2}{*}{Cucumber variety}Fanfare5022.01.31
Marketmore7521.60.702
  1. Test, at the \(1 \%\) level of significance, the hypothesis that there is no difference between the mean length of harvested Fanfare cucumbers and that of harvested Marketmore cucumbers.
  2. In addition to length, name one other characteristic of cucumbers that could be used for comparative purposes.
AQA S3 2012 June Q3
3 A hotel has three types of room: double, twin and suite. The percentage of rooms in the hotel of each type is 40,45 and 15 respectively. Each room in the hotel may be occupied by \(0,1,2\), or 3 or more people. The proportional occupancy of each type of room is shown in the table.
AQA S3 2012 June Q4
4 The manager of a medical centre suspects that patients using repeat prescriptions were requesting, on average, more items during 2011 than during 2010. The mean number of items on a repeat prescription during 2010 was 2.6.
An analysis of a random sample of 250 repeat prescriptions during 2011 showed a total of 688 items requested. The number of items requested on a repeat prescription may be modelled by a Poisson distribution. Use a distributional approximation to investigate, at the \(5 \%\) level of significance, the manager's suspicion.
AQA S3 2012 June Q5
5 A random sample of 125 people was selected from a council's electoral roll. Of these, 68 were in favour of a proposed local building plan.
  1. Construct an approximate 98\% confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal.
  2. Calculate, to the nearest 5, an estimate of the minimum sample size necessary in order that an approximate \(98 \%\) confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal has a width of at most 10 per cent.
AQA S3 2012 June Q6
6 Alyssa lives in the country but works in a city centre.
Her journey to work each morning involves a car journey, a walk and wait, a train journey, and a walk. Her car journey time, \(U\) minutes, from home to the village car park has a mean of 13 and a standard deviation of 3 . Her time, \(V\) minutes, to walk from the village car park to the village railway station and wait for a train to depart has a mean of 15 and a standard deviation of 6 . Her train journey time, \(W\) minutes, from the village railway station to the city centre railway station has a mean of 24 and a standard deviation of 4 . Her time, \(X\) minutes, to walk from the city centre railway station to her office has a mean of 9 and a standard deviation of 2 . The values of the product moment correlation coefficient for the above 4 variables are $$\rho _ { U V } = - 0.6 \quad \text { and } \quad \rho _ { U W } = \rho _ { U X } = \rho _ { V W } = \rho _ { V X } = \rho _ { W X } = 0$$
  1. Determine values for the mean and the variance of:
    1. \(M = U + V\);
    2. \(D = W - 2 U\);
    3. \(T = M + W + X\), given that \(\rho _ { M W } = \rho _ { M X } = 0\).
  2. Assuming that the variables \(M , D\) and \(T\) are normally distributed, determine the probability that, on a particular morning:
    1. Alyssa's journey time from leaving home to leaving the village railway station is exactly 30 minutes;
    2. Alyssa's train journey time is more than twice her car journey time;
    3. Alyssa's total journey time is between 50 minutes and 70 minutes.
AQA S3 2012 June Q7
7
  1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
    1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
    2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find, in terms of \(n\) and \(p\), an expression for \(\operatorname { Var } ( X )\).
  2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
    1. Use the first inequality to show that $$m \leqslant ( n + 1 ) p$$
    2. Given that the second inequality results in $$m \geqslant ( n + 1 ) p - 1$$ deduce that the distribution \(\mathrm { B } ( 10,0.65 )\) has one mode, and find the two values for the mode of the distribution \(B ( 35,0.5 )\).
  3. The random variable \(Y\) has a binomial distribution with parameters 4000 and 0.00095 . Use a distributional approximation to estimate \(\mathrm { P } ( Y \leqslant k )\), where \(k\) denotes the mode of \(Y\).
    (3 marks)
AQA S3 2013 June Q1
1 The number of telephone calls per hour to an out-of-hours doctors' service may be modelled by a Poisson distribution. The total number of telephone calls received during a random sample of 12 weekday night shifts, all of the same duration, was 392.
  1. Calculate an approximate \(98 \%\) confidence interval for the mean number of calls received per weekday night shift.
  2. The mean number of calls received during weekend shifts of 48 hours' total duration is 136.8 . Comment on a claim that the mean number of calls per hour during weekend shifts is greater than that during weekday night shifts, which are each of \(\mathbf { 1 4 }\) hours' duration.
    (3 marks)
AQA S3 2013 June Q2
2 On a rail route between two stations, A and \(\mathrm { B } , 90 \%\) of trains leave A on time and \(10 \%\) of trains leave A late. Of those trains that leave A on time, \(15 \%\) arrive at B early, \(75 \%\) arrive on time and \(10 \%\) arrive late. Of those trains that leave A late, \(35 \%\) arrive at B on time and \(65 \%\) arrive late.
  1. Represent this information by a fully-labelled tree diagram.
  2. Hence, or otherwise, calculate the probability that a train:
    1. arrives at B early or on time;
    2. left A on time, given that it arrived at B on time;
    3. left A late, given that it was not late in arriving at B .
  3. Two trains arrive late at B. Assuming that their journey times are independent, calculate the probability that exactly one train left A on time.
AQA S3 2013 June Q3
3 A builders' merchant's depot has two machines, X and Y , each of which can be used for filling bags with sand or gravel. The weight, in kilograms, delivered by machine X may be modelled by a normal distribution with mean \(\mu _ { \mathrm { X } }\) and standard deviation 25 . The weight, in kilograms, delivered by machine Y may be modelled by a normal distribution with mean \(\mu _ { \mathrm { Y } }\) and standard deviation 30 . Fred, the depot's yardman, records the weights, in kilograms, of a random sample of 10 bags of sand delivered by machine X as
\(\begin{array} { l l l l l l l l l l } 1055 & 1045 & 1000 & 985 & 1040 & 1025 & 1005 & 1030 & 1015 & 1060 \end{array}\)
He also records the weights, in kilograms, of a random sample of 8 bags of gravel delivered by machine Y as $$\begin{array} { l l l l l l l l } 1085 & 1055 & 1055 & 1000 & 1035 & 1050 & 1005 & 1075 \end{array}$$
  1. Construct a \(95 \%\) confidence interval for \(\mu _ { \mathrm { Y } } - \mu _ { \mathrm { X } }\), giving the limits to the nearest 5 kg .
  2. Dot, the depot's manager, commented that Fred's data collection may have been biased. Justify her comment and explain how the possible bias could have been eliminated.
    (2 marks)
AQA S3 2013 June Q4
4 An analysis of a sample of 250 patients visiting a medical centre showed that 38 per cent were aged over 65 years. An analysis of a sample of 100 patients visiting a dental practice showed that 21 per cent were aged over 65 years. Assume that each of these two samples has been randomly selected.
Investigate, at the \(5 \%\) level of significance, the hypothesis that the percentage of patients visiting the medical centre, who are aged over 65 years, exceeds that of patients visiting the dental practice, who are aged over 65 years, by more than 10 per cent.
AQA S3 2013 June Q5
5 The schedule for an organisation's afternoon meeting is as follows.
Session A (Speaker 1) 2.00 pm to 3.15 pm
Session B (Discussion) 3.15 pm to 3.45 pm
Session C (Speaker 2) \(\quad 3.45 \mathrm { pm }\) to 5.00 pm
Records show that:
the duration, \(X\), of Session A has mean 68 minutes and standard deviation 10 minutes;
the duration, \(Y\), of Session B has mean 25 minutes and standard deviation 5 minutes;
the duration, \(Z\), of Session C has mean 73 minutes and standard deviation 15 minutes;
and that: $$\rho _ { X Z } = 0 \quad \rho _ { X Y } = - 0.8 \quad \rho _ { Y Z } = 0$$
  1. Determine the means and the variances of:
    1. \(L = X + Z\);
    2. \(M = X + Y\).
  2. Assuming that \(L\) and \(M\) are each normally distributed, determine the probability that:
    1. the total time for the two speaker sessions is less than \(2 \frac { 1 } { 2 }\) hours;
    2. Session C is late in starting.
AQA S3 2013 June Q6
6 The demand for a WWSatNav at a superstore may be modelled by a Poisson distribution with a mean of 2.5 per day. The superstore is open 6 days each week, from Monday morning to Saturday evening.
    1. Determine the probability that the demand for WWSatNavs during a particular week is at most 18 .
    2. The superstore receives a delivery of WWSatNavs on each Sunday evening. The manager, Meena, requires that the probability of WWSatNavs being out of stock during a week should be at most \(5 \%\). Determine the minimum number of WWSatNavs that Meena requires to be in stock after a delivery.
    1. Use a distributional approximation to estimate the probability that the demand for WWSatNavs during a period of \(\mathbf { 2 }\) weeks is more than 35.
    2. Changes to the superstore's delivery schedule result in it receiving a delivery of WWSatNavs on alternate Sunday evenings. Meena now requires that the probability of WWSatNavs being out of stock during the 2 weeks following a delivery should be at most \(5 \%\). Use a distributional approximation to determine the minimum number of WWSatNavs that Meena now requires to be in stock after a delivery.
      (3 marks)
AQA S3 2013 June Q7
7 It is claimed that the proportion, \(P\), of people who prefer cooked fresh garden peas to cooked frozen garden peas is greater than 0.50 .
  1. In an attempt to investigate this claim, a sample of 50 people were each given an unlabelled portion of cooked fresh garden peas and an unlabelled portion of cooked frozen garden peas to taste. After tasting each portion, the people were each asked to state which of the two portions they preferred. Of the 50 people sampled, 29 preferred the cooked fresh garden peas. Assuming that the 50 people may be considered to constitute a random sample, use a binomial distribution and the \(10 \%\) level of significance to investigate the claim.
    (6 marks)
  2. It was then decided to repeat the tasting in part (a) but to involve a sample of 500 , rather than 50, people. Of the 500 people sampled, 271 preferred the cooked fresh garden peas.
    1. Assuming that the 500 people may be considered to constitute a random sample, use an approximation to the distribution of the sample proportion, \(\widehat { P }\), and the \(10 \%\) level of significance to again investigate the claim.
    2. The critical value of \(\widehat { P }\) for the test in part (b)(i) is 0.529 , correct to three significant figures. It is also given that, in fact, 55 per cent of people prefer cooked fresh garden peas. Estimate the power for a test of the claim that \(P > 0.50\) based on a random sample of 500 people and using the \(10 \%\) level of significance.
      (5 marks)
AQA S3 2014 June Q1
2 marks
1 A hotel's management is concerned about the quality of the free pens that it provides in its meeting rooms. The hotel's assistant manager tests a random sample of 200 such pens and finds that 23 of them fail to write immediately.
  1. Calculate an approximate \(\mathbf { 9 6 \% }\) confidence interval for the proportion of pens that fail to write immediately.
  2. The supplier of the pens to the hotel claims that at most 2 in 50 pens fail to write immediately. Comment, with numerical justification, on the supplier's claim.
    [0pt] [2 marks] QUESTION
    PART Answer space for question 1
AQA S3 2014 June Q2
6 marks
2 Each household within a district council's area has two types of wheelie-bin: a black one for general refuse and a green one for garden refuse. Each type of bin is emptied by the council fortnightly. The weight, in kilograms, of refuse emptied from a black bin can be modelled by the random variable \(B \sim \mathrm {~N} \left( \mu _ { B } , 0.5625 \right)\). The weight, in kilograms, of refuse emptied from a green bin can be modelled by the random variable \(G \sim \mathrm {~N} \left( \mu _ { G } , 0.9025 \right)\). The mean weight of refuse emptied from a random sample of 20 black bins was 21.35 kg . The mean weight of refuse emptied from an independent random sample of 15 green bins was 21.90 kg . Test, at the \(5 \%\) level of significance, the hypothesis that \(\mu _ { B } = \mu _ { G }\).
[0pt] [6 marks]
AQA S3 2014 June Q3
12 marks
3 An investigation was carried out into the type of vehicle being driven when its driver was caught speeding. The investigation was restricted to drivers who were caught speeding when driving vehicles with at least 4 wheels. An analysis of the results showed that \(65 \%\) were driving cars ( C ), \(20 \%\) were driving vans (V) and 15\% were driving lorries (L). Of those driving cars, \(30 \%\) were caught by fixed speed cameras (F), 55\% were caught by mobile speed cameras (M) and 15\% were caught by average speed cameras (A). Of those driving vans, \(35 \%\) were caught by fixed speed cameras (F), \(45 \%\) were caught by mobile speed cameras (M) and 20\% were caught by average speed cameras (A). Of those driving lorries, \(10 \%\) were caught by fixed speed cameras \(( \mathrm { F } )\), \(65 \%\) were caught by mobile speed cameras (M) and \(25 \%\) were caught by average speed cameras (A).
  1. Represent this information by a tree diagram on which are shown labels and percentages or probabilities.
  2. Hence, or otherwise, calculate the probability that a driver, selected at random from those caught speeding:
    1. was driving either a car or a lorry and was caught by a mobile speed camera;
    2. was driving a lorry, given that the driver was caught by an average speed camera;
    3. was not caught by a fixed speed camera, given that the driver was not driving a car.
      [0pt] [8 marks]
  3. Three drivers were selected at random from those caught speeding by fixed speed cameras. Calculate the probability that they were driving three different types of vehicle.
    [0pt] [4 marks]
AQA S3 2014 June Q4
8 marks
4 A sample of 50 male Eastern Grey kangaroos had a mean weight of 42.6 kg and a standard deviation of 6.2 kg . A sample of 50 male Western Grey kangaroos had a mean weight of 39.7 kg and a standard deviation of 5.3 kg .
  1. Construct a 98\% confidence interval for the difference between the mean weight of male Eastern Grey kangaroos and that of male Western Grey kangaroos.
    [0pt] [5 marks]
    1. What assumption about the selection of each of the two samples was it necessary to make in order that the confidence interval constructed in part (a) was valid?
      [0pt] [1 mark]
    2. Why was it not necessary to assume anything about the distributions of the weights of male kangaroos in order that the confidence interval constructed in part (a) was valid?
      [0pt] [2 marks]
AQA S3 2014 June Q5
4 marks
5 The numbers of daily morning operations, \(X\), and daily afternoon operations, \(Y\), in an operating theatre of a small private hospital can be modelled by the following bivariate probability distribution.
\multirow{2}{*}{}Number of morning operations ( \(\boldsymbol { X }\) )
23456\(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)
\multirow{3}{*}{Number of afternoon operations ( \(\boldsymbol { Y }\) )}30.000.050.200.200.050.50
40.000.150.100.050.000.30
50.050.050.100.000.000.20
\(\mathrm { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.050.250.400.250.051.00
    1. State why \(\mathrm { E } ( X ) = 4\) and show that \(\operatorname { Var } ( X ) = 0.9\).
    2. Given that $$\mathrm { E } ( Y ) = 3.7 , \operatorname { Var } ( Y ) = 0.61 \text { and } \mathrm { E } ( X Y ) = 14.4$$ calculate values for \(\operatorname { Cov } ( X , Y )\) and \(\rho _ { X Y }\).
  2. Calculate values for the mean and the variance of:
    1. \(T = X + Y\);
    2. \(\quad D = X - Y\).
      [0pt] [4 marks]
AQA S3 2014 June Q6
5 marks
6 Population \(A\) has a normal distribution with unknown mean \(\mu _ { A }\) and a variance of 18.8.
Population \(B\) has a normal distribution with unknown mean \(\mu _ { B }\) but with the same variance as Population \(A\). The random variables \(\bar { X } _ { A }\) and \(\bar { X } _ { B }\) denote the means of independent samples, each of size \(n\), from populations \(A\) and \(B\) respectively.
  1. Find an expression, in terms of \(n\), for \(\operatorname { Var } \left( \bar { X } _ { A } - \bar { X } _ { B } \right)\).
  2. Given that the width of a \(99 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\) is to be at most 5 , calculate the minimum value for \(n\).
    [0pt] [5 marks]
AQA S3 2014 June Q7
4 marks
7
  1. The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
    1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
    2. Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
  2. The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
    1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
    2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
      (A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
      (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
      (C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance.
      [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-22_2490_1728_219_141}
      \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-23_2490_1719_217_150}
      \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-24_2489_1728_221_141}
AQA S3 2015 June Q1
6 marks
1 A demographer measured the length of the right foot, \(x\) millimetres, and the length of the right hand, \(y\) millimetres, of each of a sample of 12 males aged between 19 years and 25 years. The results are given in the table.