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AQA S3 2009 June Q2
13 marks Moderate -0.3
2 A hotel chain has hotels in three types of location: city, coastal and country. The percentages of the chain's reservations for each of these locations are 30,55 and 15 respectively. Each of the chain's hotels offers three types of reservation: Bed \& Breakfast, Half Board and Full Board. The percentages of these types of reservation for each of the three types of location are shown in the table.
\multirow{2}{*}{}Type of location
CityCoastalCountry
\multirow{3}{*}{Type of reservation}Bed \Breakfast801030
Half Board156550
Full Board52520
For example, 80 per cent of reservations for hotels in city locations are for Bed \& Breakfast.
  1. For a reservation selected at random:
    1. show that the probability that it is for Bed \& Breakfast is 0.34 ;
    2. calculate the probability that it is for Half Board in a hotel in a coastal location;
    3. calculate the probability that it is for a hotel in a coastal location, given that it is for Half Board.
  2. A random sample of 3 reservations for Half Board is selected. Calculate the probability that these 3 reservations are for hotels in different types of location.
AQA S3 2009 June Q3
6 marks Standard +0.8
3 The proportion, \(p\), of an island's population with blood type \(\mathrm { A } \mathrm { Rh } ^ { + }\)is believed to be approximately 0.35 . A medical organisation, requiring a more accurate estimate, specifies that a \(98 \%\) confidence interval for \(p\) should have a width of at most 0.1 . Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the organisation's requirement.
AQA S3 2009 June Q4
8 marks Standard +0.8
4 Holly, a horticultural researcher, believes that the mean height of stems on Tahiti daffodils exceeds that on Jetfire daffodils by more than 15 cm . She measures the heights, \(x\) centimetres, of stems on a random sample of 65 Tahiti daffodils and finds that their mean, \(\bar { x }\), is 40.7 and that their standard deviation, \(s _ { x }\), is 3.4 . She also measures the heights, \(y\) centimetres, of stems on a random sample of 75 Jetfire daffodils and finds that their mean, \(\bar { y }\), is 24.4 and that their standard deviation, \(s _ { y }\), is 2.8 . Investigate, at the \(1 \%\) level of significance, Holly's belief.
AQA S3 2009 June Q5
10 marks Moderate -0.3
5 The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
  1. Given that $$\mathrm { E } ( X ) = n p \quad \text { and } \quad \mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$$ find an expression for \(\operatorname { Var } ( X )\).
  2. Given that \(X\) has a mean of 36 and a standard deviation of 4.8:
    1. find values for \(n\) and \(p\);
    2. use a distributional approximation to estimate \(\mathrm { P } ( 30 < X < 40 )\).
AQA S3 2009 June Q6
13 marks Moderate -0.3
6 The table shows the probability distribution for the number of weekday (Monday to Friday) morning newspapers, \(X\), purchased by the Reed household per week.
\(\boldsymbol { x }\)012345
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.160.150.250.250.150.04
  1. Find values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. The number of weekday (Monday to Friday) evening newspapers, \(Y\), purchased by the same household per week is such that $$\mathrm { E } ( Y ) = 2.0 , \quad \operatorname { Var } ( Y ) = 1.5 \quad \text { and } \quad \operatorname { Cov } ( X , Y ) = - 0.43$$ Find values for the mean and variance of:
    1. \(S = X + Y\);
    2. \(\quad D = X - Y\).
  3. The total cost per week, \(L\), of the Reed household's weekday morning and evening newspapers may be assumed to be normally distributed with a mean of \(\pounds 2.31\) and a standard deviation of \(\pounds 0.89\). The total cost per week, \(M\), of the household's weekend (Saturday and Sunday) newspapers may be assumed to be independent of \(L\) and normally distributed with a mean of \(\pounds 2.04\) and a standard deviation of \(\pounds 0.43\). Determine the probability that the total cost per week of the Reed household's newspapers is more than \(\pounds 5\).
AQA S3 2009 June Q7
17 marks Standard +0.8
7 The daily number of customers visiting a small arts and crafts shop may be modelled by a Poisson distribution with a mean of 24 .
  1. Using a distributional approximation, estimate the probability that there was a total of at most 150 customers visiting the shop during a given 6-day period.
  2. The shop offers a picture framing service. The daily number of requests, \(Y\), for this service may be assumed to have a Poisson distribution. Prior to the shop advertising this service in the local free newspaper, the mean value of \(Y\) was 2. Following the advertisement, the shop received a total of 17 requests for the service during a period of 5 days.
    1. Using a Poisson distribution, carry out a test, at the \(10 \%\) level of significance, to investigate the claim that the advertisement increased the mean daily number of requests for the shop's picture framing service.
    2. Determine the critical value of \(Y\) for your test in part (b)(i).
    3. Hence, assuming that the advertisement increased the mean value of \(Y\) to 3, determine the power of your test in part (b)(i).
AQA S3 2010 June Q2
8 marks Standard +0.3
2 Rodney and Derrick, two independent fruit and vegetable market stallholders, sell punnets of locally-grown raspberries from their stalls during June and July. The following information, based on independent random samples, was collected as part of an investigation by Trading Standards Officers.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Weight of raspberries in a punnet (grams)
\cline { 3 - 5 } \multicolumn{2}{c|}{}Sample sizeSample meanSample standard deviation, \(\boldsymbol { s }\)
\multirow{2}{*}{Stallholder}Rodney502255
\cline { 2 - 5 }Derrick752198
  1. Construct a \(99 \%\) confidence interval for the difference between the mean weight of raspberries in a punnet sold by Rodney and the mean weight of raspberries in a punnet sold by Derrick.
  2. What can be concluded from your confidence interval?
  3. In addition to weight, state one other factor that may influence whether customers buy raspberries from Rodney or from Derrick.
    \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-05_2484_1709_223_153}
AQA S3 2010 June Q3
7 marks Standard +0.8
3
The weekly number of hits, \(S\), on Sam's website may be modelled by a Poisson distribution with parameter \(\lambda _ { S }\). The weekly number of hits, \(T\), on Tina's website may be modelled by a Poisson distribution with parameter \(\lambda _ { T }\).
During a period of 40 weeks, the number of hits on Sam's website was 940.
During a period of 60 weeks, the number of hits on Tina's website was 1560.
Assuming that \(S\) and \(T\) are independent random variables, investigate, at the \(2 \%\) level of significance, Tina's claim that the mean weekly number of hits on her website is greater than that on Sam's website.
(7 marks)

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AQA S3 2010 June Q4
13 marks Standard +0.3
4 It is proposed to introduce, for all males at age 60, screening tests, A and B, for a certain disease. Test B is administered only when the result of Test A is inconclusive. It is known that 10\% of 60-year-old men suffer from the disease. For those 60 -year-old men suffering from the disease:
  • Test A is known to give a positive result, indicating a presence of the disease, in \(90 \%\) of cases, a negative result in \(2 \%\) of cases and a requirement for the administration of Test B in \(8 \%\) of cases;
  • Test B is known to give a positive result in \(98 \%\) of cases and a negative result in 2\% of cases.
For those 60 -year-old men not suffering from the disease:
  • Test A is known to give a positive result in \(1 \%\) of cases, a negative result in \(80 \%\) of cases and a requirement for the administration of Test B in 19\% of cases;
  • Test B is known to give a positive result in \(1 \%\) of cases and a negative result in \(99 \%\) of cases.
AQA S3 2010 June Q5
10 marks Standard +0.3
5 In the manufacture of desk drawer fronts, a machine cuts sheets of veneered chipboard into rectangular pieces of width \(W\) millimetres and height \(H\) millimetres. The 4 edges of each of these pieces are then covered with matching veneered tape. The distributions of \(W\) and \(H\) are such that $$\mathrm { E } ( W ) = 350 \quad \operatorname { Var } ( W ) = 5 \quad \mathrm { E } ( H ) = 210 \quad \operatorname { Var } ( H ) = 4 \quad \rho _ { W H } = 0.75$$
  1. Calculate the mean and the variance of the length of tape, \(T = 2 W + 2 H\), needed for the edges of a drawer front.
  2. A desk has 4 such drawers whose sizes may be assumed to be independent. Given that \(T\) may be assumed to be normally distributed, determine the probability that the total length of tape needed for the edges of the desk's 4 drawer fronts does not exceed 4.5 metres.
    \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-13_2484_1709_223_153}
AQA S3 2010 June Q6
18 marks Standard +0.8
6
  1. A district council claimed that more than 80 per cent of the complaints that it received about the delivery of its services were answered to the satisfaction of complainants before reaching formal status. An analysis of a random sample of 175 complaints revealed that 28 reached formal status.
    1. Construct an approximate \(95 \%\) confidence interval for the proportion of complaints that reach formal status.
    2. Hence comment on the council's claim.
  2. The district council also claimed that less than 40 per cent of all formal complaints were due to a failing in the delivery of its services. An analysis of the 50 formal complaints received during 2007/08 showed that 16 were due to a failing in the delivery of its services.
    1. Using an exact test, investigate the council's claim at the \(10 \%\) level of significance. The 50 formal complaints received during 2007/08 may be assumed to be a random sample.
    2. Determine the critical value for your test in part (b)(i).
    3. In fact, only 25 per cent of all formal complaints were due to a failing in the delivery of the council's services. Determine the probability of a Type II error for a test of the council's claim at the \(10 \%\) level of significance and based on the analysis of a random sample of 50 formal complaints.
      (4 marks)
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AQA S3 2010 June Q7
15 marks Challenging +1.2
7 The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
    1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
    2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), find, in terms of \(\lambda\), 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. Show that \(\lambda - 1 \leqslant m \leqslant \lambda\).
    2. Given that \(\lambda = 4.9\), determine \(\mathrm { P } ( X = m )\).
  3. The random variable \(Y\) has a Poisson distribution with mode \(d\) and standard deviation 15.5. Use a distributional approximation to estimate \(\mathrm { P } ( Y \geqslant d )\).
    \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-19_2484_1709_223_153}
AQA S3 2011 June Q1
7 marks Moderate -0.3
1 A consumer report claimed that more than 25 per cent of visitors to a theme park were dissatisfied with the catering facilities provided. In a survey, 375 visitors who had used the catering facilities were interviewed independently, and 108 of them stated that they were dissatisfied with the catering facilities provided.
  1. Test, at the \(2 \%\) level of significance, the consumer report's claim.
  2. State an assumption about the 375 visitors that was necessary in order for the hypothesis test in part (a) to be valid.
AQA S3 2011 June Q2
7 marks Standard +0.3
2 The number of emergency calls received by a fire station may be modelled by a Poisson distribution. During a given period of 13 weeks, the station received a total of 108 emergency calls.
  1. Construct an approximate \(98 \%\) confidence interval for the average weekly number of emergency calls received by the station.
  2. Hence comment on the station officer's claim that the station receives an average of one emergency call per day.
    (2 marks)
AQA S3 2011 June Q3
13 marks Standard +0.3
3 An IT help desk has three telephone stations: Alpha, Beta and Gamma. Each of these stations deals only with telephone enquiries. The probability that an enquiry is received at Alpha is 0.60 .
The probability that an enquiry is received at Beta is 0.25 .
The probability that an enquiry is received at Gamma is 0.15 . Each enquiry is resolved at the station that receives the enquiry. The percentages of enquiries resolved within various times at each station are shown in the table.
Time
\(\boldsymbol { \leq } \mathbf { 1 }\) hour\(\leq \mathbf { 2 4 }\) hours\(\leq 72\) hours
StationAlpha5580100
Beta6085100
Gamma4075100
For example:
80 per cent of enquiries received at Alpha are resolved within 24 hours;
25 per cent of enquiries received at Alpha take between 1 hour and 24 hours to resolve.
  1. Find the probability that an enquiry, selected at random, is:
    1. resolved at Gamma;
    2. resolved at Alpha within 1 hour;
    3. resolved within 24 hours;
    4. received at Beta, given that it is resolved within 24 hours.
  2. A random sample of 3 enquiries was selected. Given that all 3 enquiries were resolved within 24 hours, calculate the probability that they were all received at:
    1. Beta;
    2. the same station.
AQA S3 2011 June Q4
5 marks Standard +0.8
4
The waiting time at a hospital's A\&E department may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\frac { \mu } { 2 }\).
The department's manager wishes a \(95 \%\) confidence interval for \(\mu\) to be constructed such that it has a width of at most \(0.2 \mu\).
Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the manager's wish.
(5 marks)
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AQA S3 2011 June Q5
8 marks Standard +0.3
5
An examination of 160 e-mails received by Gopal showed that 72 had attachments. An examination of 250 e-mails received by Haley showed that 102 had attachments.
Stating two necessary assumptions about the selection of e-mails, construct an approximate \(99 \%\) confidence interval for the difference between the proportion of e-mails received by Gopal that have attachments and the proportion of e-mails received by Haley that have attachments.
(8 marks)
AQA S3 2011 June Q6
13 marks Challenging +1.2
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.
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AQA S3 2011 June Q7
9 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 \(\operatorname { Var } ( X ) = \mathrm { E } ( X )\).
  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.
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      \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-20_2486_1714_221_153}
AQA S3 2011 June Q8
13 marks Challenging +1.2
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
6 marks Moderate -0.8
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
7 marks Moderate -0.3
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
14 marks Moderate -0.5
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
6 marks Standard +0.8
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
10 marks Standard +0.3
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.