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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 S2 2016 June Q1
13 marks Moderate -0.3
1 The water in a pond contains three different species of a spherical green algae:
Volvox globator, at an average rate of 4.5 spheres per \(1 \mathrm {~cm} ^ { 3 }\);
Volvox aureus, at an average rate of 2.3 spheres per \(1 \mathrm {~cm} ^ { 3 }\);
Volvox tertius, at an average rate of 1.2 spheres per \(1 \mathrm {~cm} ^ { 3 }\).
Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond.
Find the probability that:
  1. a \(1 \mathrm {~cm} ^ { 3 }\) sample contains no more than 5 Volvox globator spheres;
  2. a \(1 \mathrm {~cm} ^ { 3 }\) sample contains at least 2 Volvox aureus spheres;
  3. a \(5 \mathrm {~cm} ^ { 3 }\) sample contains more than 8 but fewer than 12 Volvox tertius spheres;
  4. a \(0.1 \mathrm {~cm} ^ { 3 }\) sample contains a total of exactly 2 Volvox spheres;
  5. a \(1 \mathrm {~cm} ^ { 3 }\) sample contains at least 1 sphere of each of the three different species of algae.
    [0pt] [3 marks]
AQA S2 2016 June Q3
13 marks Moderate -0.8
3 Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.
\(\boldsymbol { x }\)0123456
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)00.190.260.200.130.070.15
  1. Find the probability that a member borrows more than 3 books.
  2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books.
  3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\).
  4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\).
  5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
    1. the mean amount that will be paid by a member;
    2. the standard deviation of the amount that will be paid by a member.
AQA S2 2016 June Q4
7 marks Easy -1.2
4 A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2 . The error, \(X ^ { \circ } \mathrm { C }\), resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \left\{ \begin{array} { l c } k & 0 \leqslant x \leqslant 0.1 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. State the value of \(k\).
  2. Find the probability that the error resulting from this rounding down is greater than \(0.03 ^ { \circ } \mathrm { C }\).
    1. State the value for \(\mathrm { E } ( X )\).
    2. Use integration to find the value for \(\mathrm { E } \left( X ^ { 2 } \right)\).
    3. Hence find the value for the standard deviation of \(X\).
      \includegraphics[max width=\textwidth, alt={}]{72aa9867-88c6-4b1b-97f7-bf4ba2da4031-12_1355_1707_1352_153}
AQA S2 2016 June Q5
13 marks Standard +0.3
5 A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.
Problems during first 3 monthsProblems during first year but after first 3 monthsNo problems during first yearTotal
Petrol engine1035170215
Diesel engine482335
Total1443193250
  1. Use a \(\chi ^ { 2 }\)-test to investigate, at the \(10 \%\) significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine.
  2. Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa.
    [0pt] [2 marks]
AQA S2 2016 June Q6
16 marks Standard +0.3
6 Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm . Gerald visited three islands, \(\mathrm { A } , \mathrm { B }\) and C , and measured the length, \(X\) centimetres, of each of a sample of \(n\) sand lizards on each island. The samples may be regarded as random. The data are shown in the table.
AQA S2 2016 June Q7
9 marks Standard +0.3
7 The continuous random variable \(X\) has a cumulative distribution function \(\mathrm { F } ( x )\), where $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 4 } ( x - 1 ) & 1 \leqslant x < 4 \\ \frac { 1 } { 16 } \left( 12 x - x ^ { 2 } - 20 \right) & 4 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{array} \right.$$
  1. Sketch the probability density function, \(\mathrm { f } ( x )\), on the grid below.
  2. Find the mean value of \(X\).
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.
    \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-05_2484_1709_223_153}
    \begin{center} \begin{tabular}{|l|l|} \hline & \begin{tabular}{l}