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AQA S1 2006 June Q1
8 marks Moderate -0.3
1 The table shows, for each of a random sample of 8 paperback fiction books, the number of pages, \(x\), and the recommended retail price, \(\pounds y\), to the nearest 10 p.
\(\boldsymbol { x }\)223276374433564612704766
\(\boldsymbol { y }\)6.504.005.508.004.505.008.005.50
    1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
    2. Interpret your value in the context of this question.
    3. Suggest one other variable, in addition to the number of pages, which may affect the recommended retail price of a paperback fiction book.
  2. The same 8 books were later included in a book sale. The value of the product moment correlation coefficient between the number of pages and the sale price was 0.959 , correct to three decimal places. What can be concluded from this value?
AQA S1 2006 June Q2
12 marks Moderate -0.8
2 The heights of sunflowers may be assumed to be normally distributed with a mean of 185 cm and a standard deviation of 10 cm .
  1. Determine the probability that the height of a randomly selected sunflower:
    1. is less than 200 cm ;
    2. is more than 175 cm ;
    3. is between 175 cm and 200 cm .
  2. Determine the probability that the mean height of a random sample of 4 sunflowers is more than 190 cm .
AQA S1 2006 June Q3
11 marks Moderate -0.8
3 A new car tyre is fitted to a wheel. The tyre is inflated to its recommended pressure of 265 kPa and the wheel left unused. At 3-month intervals thereafter, the tyre pressure is measured with the following results:
Time after fitting
\(( x\) months \()\)
03691215182124
Tyre pressure
\(( y\) kPa \()\)
265250240235225215210195180
    1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
    2. Interpret in context the value for the gradient of your line.
    3. Comment on the value for the intercept with the \(y\)-axis of your line.
  2. The tyre manufacturer states that, when one of these new tyres is fitted to the wheel of a car and then inflated to 265 kPa , a suitable regression equation is of the form $$y = 265 + b x$$ The manufacturer also states that, as the car is used, the tyre pressure will decrease at twice the rate of that found in part (a).
    1. Suggest a suitable value for \(b\).
    2. One of these new tyres is fitted to the wheel of a car and inflated to 265 kPa . The car is then used for 8 months, after which the tyre pressure is checked for the first time. Show that, accepting the manufacturer's statements, the tyre pressure can be expected to have fallen below its minimum safety value of 220 kPa .
      (2 marks)
AQA S1 2006 June Q4
7 marks Moderate -0.3
4 The weights of packets of sultanas may be assumed to be normally distributed with a standard deviation of 6 grams. The weights of a random sample of 10 packets were as follows: \(\begin{array} { l l l l l l l l l l } 498 & 496 & 499 & 511 & 503 & 505 & 510 & 509 & 513 & 508 \end{array}\)
    1. Construct a \(99 \%\) confidence interval for the mean weight of packets of sultanas, giving the limits to one decimal place.
    2. State why, in calculating your confidence interval, use of the Central Limit Theorem was not necessary.
    3. On each packet it states 'Contents 500 grams'. Comment on this statement using both the given sample and your confidence interval.
  2. Given that the mean weight of all packets of sultanas is 500 grams, state the probability that a 99\% confidence interval for the mean, calculated from a random sample of packets, will not contain 500 grams.
AQA S1 2006 June Q5
17 marks Standard +0.3
5 Kirk and Les regularly play each other at darts.
  1. The probability that Kirk wins any game is 0.3 , and the outcome of each game is independent of the outcome of every other game. Find the probability that, in a match of 15 games, Kirk wins:
    1. exactly 5 games;
    2. fewer than half of the games;
    3. more than 2 but fewer than 7 games.
  2. Kirk attends darts coaching sessions for three months. He then claims that he has a probability of 0.4 of winning any game, and that the outcome of each game is independent of the outcome of every other game.
    1. Assuming this claim to be true, calculate the mean and standard deviation for the number of games won by Kirk in a match of 15 games.
    2. To assess Kirk's claim, Les keeps a record of the number of games won by Kirk in a series of 10 matches, each of 15 games, with the following results: $$\begin{array} { l l l l l l l l l l } 8 & 5 & 6 & 3 & 9 & 12 & 4 & 2 & 6 & 5 \end{array}$$ Calculate the mean and standard deviation of these values.
    3. Hence comment on the validity of Kirk's claim.
AQA S1 2006 June Q6
Easy -1.3
6 A housing estate consists of 320 houses: 120 detached and 200 semi-detached. The numbers of children living in these houses are shown in the table.
\multirow{2}{*}{}Number of children
NoneOneTwoAt least threeTotal
Detached house24324123120
Semi-detached house40378835200
Total646912958320
A house on the estate is selected at random. \(D\) denotes the event 'the house is detached'. \(R\) denotes the event 'no children live in the house'. \(S\) denotes the event 'one child lives in the house'. \(T\) denotes the event 'two children live in the house'.
( \(D ^ { \prime }\) denotes the event 'not \(D\) '.)
  1. Find:
    1. \(\mathrm { P } ( D )\);
    2. \(\quad \mathrm { P } ( D \cap R )\);
    3. \(\quad \mathrm { P } ( D \cup T )\);
    4. \(\mathrm { P } ( D \mid R )\);
    5. \(\mathrm { P } \left( R \mid D ^ { \prime } \right)\).
    1. Name two of the events \(D , R , S\) and \(T\) that are mutually exclusive.
    2. Determine whether the events \(D\) and \(R\) are independent. Justify your answer.
  3. Define, in the context of this question, the event:
    1. \(D ^ { \prime } \cup T\);
    2. \(D \cap ( R \cup S )\).
AQA S1 2015 June Q1
6 marks Easy -1.2
1 The number of passengers getting off the 11.45 am train at a railway station on each of 35 days is summarised as follows.
AQA S1 2015 June Q2
10 marks Moderate -0.8
2 The length of aluminium baking foil on a roll may be modelled by a normal distribution with mean 91 metres and standard deviation 0.8 metres.
  1. Determine the probability that the length of foil on a particular roll is:
    1. less than 90 metres;
    2. not exactly 90 metres;
    3. between 91 metres and 92.5 metres.
  2. The length of cling film on a roll may also be modelled by a normal distribution but with mean 153 metres and standard deviation \(\sigma\) metres. It is required that \(1 \%\) of rolls of cling film should have a length less than 150 metres.
    Find the value of \(\sigma\) that is needed to satisfy this requirement.
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-04_1526_1714_1181_153}
AQA S1 2015 June Q3
11 marks Moderate -0.5
3 Fourteen candidates each sat two test papers, Paper 1 and Paper 2, on the same day. The marks, out of a total of 50, achieved by the students on each paper are shown in the table.
AQA S1 2015 June Q4
15 marks Moderate -0.8
4
  1. Chris shops at his local store on his way to and from work every Friday.
    The event that he buys a morning newspaper is denoted by \(M\), and the event that he buys an evening newspaper is denoted by \(E\). On any one Friday, Chris may buy neither, exactly one or both of these newspapers.
    1. Complete the table of probabilities, printed on the opposite page, where \(M ^ { \prime }\) and \(E ^ { \prime }\) denote the events 'not \(M\) ' and 'not \(E\) ' respectively.
    2. Hence, or otherwise, find the probability that, on any given Friday, Chris buys exactly one newspaper.
    3. Give a numerical justification for the following statement.
      'The events \(M\) and \(E\) are not mutually exclusive.'
  2. The event that Chris buys a morning newspaper on Saturday is denoted by \(S\), and the event that he buys a morning newspaper on the following day, Sunday, is denoted by \(T\). The event that he buys a morning newspaper on both Saturday and Sunday is denoted by \(S \cap T\). Each combination of the events \(S\) and \(T\) is independent of any combination of the events \(M\) and \(E\). However, the events \(S\) and \(T\) are not independent, with $$\mathrm { P } ( S ) = 0.85 , \quad \mathrm { P } ( T \mid S ) = 0.20 \quad \text { and } \quad \mathrm { P } \left( T \mid S ^ { \prime } \right) = 0.75$$ Find the probability that, on a particular Friday, Saturday and Sunday, Chris buys:
    1. all four newspapers;
    2. none of the four newspapers.
    1. State, as briefly as possible, in the context of the question, the event that is denoted by \(M \cap E ^ { \prime } \cap S \cap T ^ { \prime }\).
    2. Calculate the value of \(\mathrm { P } \left( M \cap E ^ { \prime } \cap S \cap T ^ { \prime } \right)\). \section*{Answer space for question 4}
      1. (i)
        \cline { 2 - 4 } \multicolumn{1}{c|}{}\(\boldsymbol { M }\)\(\boldsymbol { M } ^ { \prime }\)Total
        \(\boldsymbol { E }\)0.160.28
        \(\boldsymbol { E } ^ { \prime }\)
        Total0.601.00
        \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-11_2050_1707_687_153}
AQA S1 2015 June Q5
11 marks Moderate -0.8
5 The table shows the number of customers, \(x\), and the takings, \(\pounds y\), recorded to the nearest \(\pounds 10\), at a local butcher's shop on each of 10 randomly selected weekdays.
\(\boldsymbol { x }\)86606546719356817557
\(\boldsymbol { y }\)9407906205307701050690780860550
  1. The first 6 pairs of data values in this table are plotted on the scatter diagram shown on the opposite page. Plot the final 4 pairs of data values on the scatter diagram.
    1. Calculate the equation of the least squares regression line in the form \(y = a + b x\) and draw your line on the scatter diagram.
    2. Interpret your value for \(b\) in the context of the question.
    3. State why your value for \(a\) has no practical interpretation.
  3. Estimate, to the nearest \(\pounds 10\), the shop's takings when the number of customers is 50 .
    [0pt] [1 mark]
    \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-14_1255_1705_1448_155}
    Butcher's shop \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Answer space for question 5} \includegraphics[alt={},max width=\textwidth]{4c679380-894f-4d36-aec8-296b662058e2-15_2335_1760_372_100}
    \end{figure}
AQA S1 2015 June Q6
12 marks Moderate -0.8
6 Customers at a supermarket can pay at a checkout either by cash, debit card or credit card.
  1. The probability that a customer pays by cash is 0.22 . Calculate the probability that exactly 2 customers from a random sample of 24 customers pay by cash.
  2. The probability that a customer pays by debit card is 0.45 . Determine the probability that the number of customers who pay by debit card from a random sample of \(\mathbf { 4 0 }\) customers is:
    1. fewer than 20 ;
    2. more than 15 ;
    3. at least 12 but at most 24 .
  3. The random variable \(W\) denotes the number of customers who pay by credit card from a random sample of \(\mathbf { 2 0 0 }\) customers. Calculate values for the mean and the variance of \(W\).
    [0pt] [3 marks]
AQA S1 2015 June Q7
10 marks Moderate -0.3
7
  1. A greengrocer displays apples in trays. Each customer selects the apples he or she wishes to buy and puts them into a bag. Records show that the weight of such bags of apples may be modelled by a normal distribution with mean 1.16 kg and standard deviation 0.43 kg . Determine the probability that the mean weight of a random sample of 10 such bags of apples exceeds 1.25 kg .
  2. The greengrocer also displays pears in trays. Each customer selects the pears he or she wishes to buy and puts them into a bag. A random sample of 40 such bags of pears had a mean weight of 0.86 kg and a standard deviation of 0.65 kg .
    1. Construct a \(\mathbf { 9 6 \% }\) confidence interval for the mean weight of a bag of pears.
    2. Hence comment on a claim that customers wish to buy, on average, a greater weight of apples than of pears.
      [0pt] [2 marks]
AQA S1 2015 June Q1
4 marks Moderate -0.8
1
The table shows the annual gas consumption, \(x \mathrm { kWh }\), and the annual electricity consumption, \(y \mathrm { kWh }\), for a sample of 10 bungalows of similar size and occupancy.
\(\boldsymbol { x }\)21371185211522217312198542356120738221111789724523
\(\boldsymbol { y }\)2281232722212378278728563078264725662559
$$S _ { x x } = 76581640 \quad S _ { y y } = 694250 \quad S _ { x y } = 3629670$$
  1. Calculate the value of \(r _ { x y }\), the product moment correlation coefficient between \(x\) and \(y\).
  2. Interpret your value of \(r _ { x y }\) in the context of this question.
AQA S1 2015 June Q2
6 marks Easy -1.2
2 The table summarises the diameters, \(d\) millimetres, of a random sample of 60 new cricket balls to be used in junior cricket.
AQA S1 2015 June Q3
13 marks Moderate -0.8
3 A ferry sails once each day from port D to port A. The ferry departs from D on time or late but never early. However, the ferry can arrive at A early, on time or late. The probabilities for some combined events of departing from \(D\) and arriving at \(A\) are shown in the table below.
  1. Complete the table.
  2. Write down the probability that, on a particular day, the ferry:
    1. both departs and arrives on time;
    2. departs late.
  3. Find the probability that, on a particular day, the ferry:
    1. arrives late, given that it departed late;
    2. does not arrive late, given that it departed on time.
  4. On three particular days, the ferry departs from port D on time. Find the probability that, on these three days, the ferry arrives at port A early once, on time once and late once. Give your answer to three decimal places.
    [0pt] [4 marks]
    1. \begin{table}[h]
      \captionsetup{labelformat=empty} \caption{Answer space for question 3}
      \multirow{2}{*}{}Arrive at A
      EarlyOn timeLateTotal
      \multirow{2}{*}{Depart from D}On time0.160.560.08
      Late
      Total0.220.651.00
      \end{table}
AQA S1 2015 June Q4
15 marks Moderate -0.3
4 Stephan is a roofing contractor who is often required to replace loose ridge tiles on house roofs. In order to help him to quote more accurately the prices for such jobs in the future, he records, for each of 11 recently repaired roofs, the number of ridge tiles replaced, \(x _ { i }\), and the time taken, \(y _ { i }\) hours. His results are shown in the table.
Roof \(( \boldsymbol { i } )\)\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)\(\mathbf { 7 }\)\(\mathbf { 8 }\)\(\mathbf { 9 }\)\(\mathbf { 1 0 }\)\(\mathbf { 1 1 }\)
\(\boldsymbol { x } _ { \boldsymbol { i } }\)811141416202222252730
\(\boldsymbol { y } _ { \boldsymbol { i } }\)5.05.26.37.28.08.810.611.011.812.113.0
  1. The pairs of data values for roofs 1 to 7 are plotted on the scatter diagram shown on the opposite page. Plot the 4 pairs of data values for roofs 8 to 11 on the scatter diagram.
    1. Calculate the equation of the least squares regression line of \(y _ { i }\) on \(x _ { i }\), and draw your line on the scatter diagram.
    2. Interpret your values for the gradient and for the intercept of this regression line.
  3. Estimate the time that it would take Stephan to replace 15 loose ridge tiles on a house roof.
  4. Given that \(r _ { i }\) denotes the residual for the point representing roof \(i\) :
    1. calculate the value of \(r _ { 6 }\);
    2. state why the value of \(\sum _ { i = 1 } ^ { 11 } r _ { i }\) gives no useful information about the connection between the number of ridge tiles replaced and the time taken.
      [0pt] [1 mark]
      \section*{Answer space for question 4}
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-11_2385_1714_322_155}
AQA S1 2015 June Q5
12 marks Moderate -0.3
5
  1. Wooden lawn edging is supplied in 1.8 m length rolls. The actual length, \(X\) metres, of a roll may be modelled by a normal distribution with mean 1.81 and standard deviation 0.08 . Determine the probability that a randomly selected roll has length:
    1. less than 1.90 m ;
    2. greater than 1.85 m ;
    3. between 1.81 m and 1.85 m .
  2. Plastic lawn edging is supplied in 9 m length rolls. The actual length, \(Y\) metres, of a roll may be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). An analysis of a batch of rolls, selected at random, showed that $$\mathrm { P } ( Y < 9.25 ) = 0.88$$
    1. Use this probability to find the value of \(z\) such that $$9.25 - \mu = z \times \sigma$$ where \(z\) is a value of \(Z \sim \mathrm {~N} ( 0,1 )\).
    2. Given also that $$\mathrm { P } ( Y > 8.75 ) = 0.975$$ find values for \(\mu\) and \(\sigma\).
AQA S1 2015 June Q6
13 marks Moderate -0.8
6
  1. In a particular country, 35 per cent of the population is estimated to have at least one mobile phone. A sample of 40 people is selected from the population.
    Use the distribution \(\mathrm { B } ( 40,0.35 )\) to estimate the probability that the number of people in the sample that have at least one mobile phone is:
    1. at most 15 ;
    2. more than 10 ;
    3. more than 12 but fewer than 18 ;
    4. exactly equal to the mean of the distribution.
  2. In the same country, 70 per cent of households have a landline telephone connection. A sample of 50 households is selected from all households in the country.
    Stating a necessary condition regarding this selection, estimate the probability that fewer than 30 households have a landline telephone connection.
    [0pt] [4 marks]
AQA S1 2015 June Q7
12 marks Moderate -0.3
7
  1. The weight of a sack of mixed dog biscuits can be modelled by a normal distribution with a mean of 10.15 kg and a standard deviation of 0.3 kg . A pet shop purchases 12 such sacks that can be considered to be a random sample.
    Calculate the probability that the mean weight of the 12 sacks is less than 10 kg .
  2. The weight of dry cat food in a pouch can also be modelled by a normal distribution. The contents, \(x\) grams, of each of a random sample of 40 pouches were weighed. Subsequent analysis of these weights gave $$\bar { x } = 304.6 \quad \text { and } \quad s = 5.37$$
    1. Construct a \(99 \%\) confidence interval for the mean weight of dry cat food in a pouch. Give the limits to one decimal place.
    2. Comment, with justification, on each of the following two claims. Claim 1: The mean weight of dry cat food in a pouch is more than 300 grams.
      Claim 2: All pouches contain more than 300 grams of dry cat food.
      [0pt] [4 marks]
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-24_2288_1705_221_155}
AQA S2 2009 January Q1
11 marks Standard +0.3
1 Fortune High School gave its students a wider choice of subjects to study. The table shows the number of students, of each gender, who chose to study each of the additional subjects during the school year 2007/08.
\cline { 2 - 5 } \multicolumn{1}{c|}{}Bulgarian
Climate
Change
FinancePolish
Male7312540
Female2242219
Assuming that these data form a random sample, use a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether the choice of these subjects is independent of gender.
(11 marks)
AQA S2 2009 January Q2
9 marks Standard +0.3
2 A group of estate agents in a particular area claimed that, after the introduction of a new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area had not changed from 8 weeks.
  1. A random sample of 9 house purchases in the area revealed that their completion times, in weeks, were as follows. $$\begin{array} { l l l l l l l l l } 6 & 7 & 10 & 12 & 9 & 11 & 7 & 8 & 14 \end{array}$$ Assuming that completion times in the area are normally distributed with standard deviation 2.5 weeks, test, at the \(5 \%\) level of significance, the group's claim. (7 marks)
  2. It was subsequently discovered that, after the introduction of the new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area remained at 8 weeks. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out your hypothesis test in part (a). Give a reason for your answer.
    (2 marks)
AQA S2 2009 January Q3
14 marks Moderate -0.3
3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, \(X\), for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, \(Y\), for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.
  1. Calculate:
    1. \(\mathrm { P } ( X \leqslant 3 )\);
    2. \(\quad \mathrm { P } ( Y = 5 )\).
  2. The total daily demand for the device at Joe's two garages is denoted by \(T\).
    1. Write down the distribution of \(T\), stating any assumption that you make.
    2. Determine \(\mathrm { P } ( 6 < T < 12 )\).
    3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
    4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least \(99 \%\) of days.
AQA S2 2009 January Q4
6 marks Moderate -0.3
4 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - c \\ \frac { x + c } { 4 c } & - c \leqslant x \leqslant 3 c \\ 1 & x > 3 c \end{array} \right.$$ where \(c\) is a positive constant.
  1. Determine \(\mathrm { P } \left( - \frac { 3 c } { 4 } < X < \frac { 3 c } { 4 } \right)\).
  2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 c } & - c \leqslant x \leqslant 3 c \\ 0 & \text { otherwise } \end{array} \right.$$
  3. Hence, or otherwise, find expressions, in terms of \(c\), for:
    1. \(\mathrm { E } ( X )\);
    2. \(\operatorname { Var } ( X )\).
AQA S2 2009 January Q5
13 marks Standard +0.3
5 Jane, who supplies fruit to a jam manufacturer, knows that the weight of fruit in boxes that she sends to the manufacturer can be modelled by a normal distribution with unknown mean, \(\mu\) grams, and unknown standard deviation, \(\sigma\) grams. Jane selects a random sample of 16 boxes and, using the \(t\)-distribution, calculates correctly that a \(98 \%\) confidence interval for \(\mu\) is \(( 70.65,80.35 )\).
    1. Show that the sample mean is 75.5 grams.
    2. Find the width of the confidence interval.
    3. Calculate an estimate of the standard error of the mean.
    4. Hence, or otherwise, show that an unbiased estimate of \(\sigma ^ { 2 }\) is 55.6 , correct to three significant figures.
  2. Jane decides that the width of the \(98 \%\) confidence interval is too large. Construct a \(95 \%\) confidence interval for \(\mu\), based on her sample of 16 boxes.
  3. Jane is informed that the manufacturer would prefer the confidence interval to have a width of at most 5 grams.
    1. Write down a confidence interval for \(\mu\), again based on Jane's sample of 16 boxes, which has a width of 5 grams.
    2. Determine the percentage confidence level for your interval in part (c)(i).