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Edexcel S2 Q7
18 marks Standard +0.3
7. In a competition at a funfair, participants have to stay on a log being rotated in a pool of water for as long as possible. The length of time, in tens of seconds, that the competitors stay on the log is modelled by the random variable \(T\) with the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k ( t - 3 ) ^ { 2 } , & 0 \leq t \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 1 } { 9 }\).
  2. Sketch f \(( t )\) for all values of \(t\).
  3. Show that the mean time that competitors stay on the \(\log\) is 7.5 seconds. When the competition is next run the organisers decide to make it easier at first by spinning the log more slowly and then increasing the speed of rotation. The length of time, in tens of seconds, that the competitors now stay on the log is modelled by the random variable \(S\) with the following probability density function: $$f ( s ) = \begin{cases} \frac { 1 } { 12 } \left( 8 - s ^ { 3 } \right) , & 0 \leq s \leq 2 \\ 0 , & \text { otherwise } \end{cases}$$
  4. Find the change in the mean time that competitors stay on the log.
Edexcel S2 Q1
7 marks Easy -1.2
  1. The random variable \(X\) follows a Poisson distribution with a mean of 1.4 Find \(\mathrm { P } ( X \leq 3 )\).
  2. The random variable \(Y\) follows a binomial distribution such that \(Y \sim \mathrm {~B} ( 20,0.6 )\). Find \(\mathrm { P } ( Y \leq 12 )\).
    (4 marks)
Edexcel S2 Q2
9 marks Moderate -0.8
2. A driving instructor keeps records of all the learners she has taught. In order to analyse her success rate she wishes to take a random sample of 120 of these learners.
  1. Suggest a suitable sampling frame and identify the sampling units. She believes that only 1 in 20 of the people she teaches fail to pass their test in their first two attempts. She decides to use her sample to test whether or not the proportion is different from this.
  2. Using a suitable approximation and stating clearly the hypotheses she should use, find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
  3. State the significance level of this test.
Edexcel S2 Q3
9 marks Standard +0.3
3. In an old computer game a white square representing a ball appears at random at the top of the playing area, which is 24 cm wide, and moves down the screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen when it appears. The distribution of \(X\) is rectangular over the interval [4,28].
  1. Find the mean and variance of \(X\).
  2. Find \(\mathrm { P } ( | X - 16 | < 3 )\). During a single game, a player receives 12 "balls".
  3. Find the probability that the ball appears within 3 cm of the middle of the top edge of the playing area more than four times in a single game.
    (3 marks)
Edexcel S2 Q4
14 marks Moderate -0.8
4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.
  1. State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case. Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
  2. less than two visitors in a 10 -minute interval,
  3. at least ten visitors in a 15-minute interval.
  4. Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.
    (5 marks)
Edexcel S2 Q5
17 marks Standard +0.3
5. Four coins are flipped together and the random variable \(H\) represents the number of heads obtained. Assuming that the coins are fair,
  1. suggest with reasons a suitable distribution for modelling \(H\) and give the value of any parameters needed,
  2. show that the probability of obtaining more heads than tails is \(\frac { 5 } { 16 }\). The four coins are flipped 5 times and more heads are obtained than tails 4 times.
  3. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the probability of getting more heads than tails being more than \(\frac { 5 } { 16 }\). Given that the four coins are all biased such that the chance of each one showing a head is 50\% more than the chance of it showing a tail,
  4. find the probability of obtaining more heads than tails when the four coins are flipped together.
Edexcel S2 Q6
19 marks Standard +0.3
6. The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} \frac { 1 } { 16 } x , & 2 \leq x \leq 6 \\ 0 , & \text { otherwise } \end{cases}$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Find \(\mathrm { E } ( X )\).
  3. Show that \(\operatorname { Var } ( X ) = \frac { 11 } { 9 }\).
  4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\).
  5. Show that the interquartile range of \(X\) is \(2 ( \sqrt { } 7 - \sqrt { 3 } )\). END
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.
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.
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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.
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