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AQA S1 2010 January Q7
13 marks Standard +0.3
7 [Figure 1, printed on the insert, is provided for use in this question.]
Harold considers himself to be an expert in assessing the auction value of antiques. He regularly visits car boot sales to buy items that he then sells at his local auction rooms. Harold's father, Albert, who is not convinced of his son's expertise, collects the following data from a random sample of 12 items bought by Harold.
ItemPurchase price (£ \(\boldsymbol { x }\) )Auction price (£ y)
A2030
B3545
C1825
D5050
E4538
F5545
G4350
H8190
I9085
J30190
K5765
L11225
  1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
  2. Interpret your value in the context of this question.
    1. On Figure 1, complete the scatter diagram for these data.
    2. Comment on what this reveals.
  4. When items J and L are omitted from the data, it is found that $$S _ { x x } = 4854.4 \quad S _ { y y } = 4216.1 \quad S _ { x y } = 4268.8$$
    1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\) for the remaining 10 items.
    2. Hence revise as necessary your interpretation in part (b).
AQA S1 2005 June Q1
6 marks Easy -1.2
1 For each of a random sample of 10 customers, a store records the time, \(x\) minutes, spent shopping and the value, \(\pounds y\), to the nearest 10 p, of items purchased. The results are tabulated below.
Time (x)1345109172316216
Value (y)12.55.72.318.47.917.117.918.68.321.3
    1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
    2. Interpret your value in context.
  2. Write down the value of the product moment correlation coefficient if the time had been recorded in seconds and the value in pence to the nearest 10p.
AQA S1 2005 June Q2
15 marks Moderate -0.3
2 The weight, \(X\) grams, of a particular variety of orange is normally distributed with mean 205 and standard deviation 25.
  1. Determine the probability that the weight of an orange is:
    1. less than 250 grams;
    2. between 200 grams and 250 grams.
  2. A wholesaler decides to grade such oranges by weight. He decides that the smallest 30 per cent should be graded as small, the largest 20 per cent graded as large, and the remainder graded as medium. Determine, to one decimal place, the maximum weight of an orange graded as:
    1. small;
    2. medium.
  3. The weight, \(Y\) grams, of a second variety of orange is normally distributed with mean 175. Given that 90 per cent of these oranges weigh less than 200 grams, calculate the standard deviation of their weights.
    (4 marks)
AQA S1 2005 June Q3
11 marks Moderate -0.8
3 Fred and his daughter, Delia, support their town's rugby team. The probability that Fred watches a game is 0.8 . The probability that Delia watches a game is 0.9 when her father watches the game, and is 0.4 when her father does not watch the game.
  1. Calculate the probability that:
    1. both Fred and Delia watch a particular game;
    2. neither Fred nor Delia watch a particular game.
  2. Molly supports the same rugby team as Fred and Delia. The probability that Molly watches a game is 0.7 , and is independent of whether or not Fred or Delia watches the game. Calculate the probability that:
    1. all 3 supporters watch a particular game;
    2. exactly 2 of the 3 supporters watch a particular game.
AQA S1 2005 June Q4
12 marks Moderate -0.8
4 The time taken for a fax machine to scan an A4 sheet of paper is dependent, in part, on the number of lines of print on the sheet. The table below shows, for each of a random sample of 8 sheets of A4 paper, the number, \(x\), of lines of print and the scanning time, \(y\) seconds, taken by the fax machine.
Sheet\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)\(\mathbf { 7 }\)\(\mathbf { 8 }\)
\(\boldsymbol { x }\)1016232731353844
\(\boldsymbol { y }\)2.43.53.24.14.15.64.65.3
  1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
  2. The following table lists some of the residuals for the regression line.
    Sheet\(\mathbf { 1 }\)\(\mathbf { 2 }\)\(\mathbf { 3 }\)\(\mathbf { 4 }\)\(\mathbf { 5 }\)\(\mathbf { 6 }\)\(\mathbf { 7 }\)\(\mathbf { 8 }\)
    Residual- 0.1740.4180.085- 0.2540.906- 0.157
    1. Calculate the values of the residuals for sheets 3 and 7 .
    2. Hence explain what can be deduced about the regression line.
  3. The time, \(z\) seconds, to transmit an A4 page after scanning is given by: $$z = 0.80 + 0.05 x$$ Estimate the total time to scan and transmit an A4 page containing:
    1. 15 lines of print;
    2. 75 lines of print. In each case comment on the likely reliability of your estimate.
AQA S1 2005 June Q5
19 marks Moderate -0.3
5
  1. At a particular checkout in a supermarket, the probability that the barcode reader fails to read the barcode first time on any item is 0.07 , and is independent from item to item.
    1. Calculate the probability that, from a shopping trolley containing 17 items, the reader fails to read the barcode first time on exactly 2 of the items.
    2. Determine the probability that, from a shopping trolley containing 50 items, the reader fails to read the barcode first time on at most 5 of the items.
  2. At another checkout in the supermarket, the probability that a faulty barcode reader fails to read the barcode first time on any item is 0.55 , and is independent from item to item. Determine the probability that, from a shopping trolley containing 50 items, this reader fails to read the barcode first time on at least 30 of the items.
  3. At a third checkout in the supermarket, a record is kept of \(X\), the number of times per 50 items that the barcode reader fails to read a barcode first time. An analysis of the records gives a mean of 10 and a standard deviation of 6.8.
    1. Estimate \(p\), the probability that the barcode reader fails to read a barcode first time.
    2. Using your estimate of \(p\) and assuming that \(X\) can be modelled by a binomial distribution, estimate the standard deviation of \(X\).
    3. Hence comment on the assumption that \(X\) can be modelled by a binomial distribution.
AQA S1 2005 June Q6
12 marks Standard +0.3
6 On arrival at a business centre, all visitors are required to register at the reception desk. An analysis of the register, for a random sample of 100 days, results in the following information on the number, \(X\), of visitors per day.
Number of visitors per dayNumber of days
1-1013
11-2033
21-2517
26-3012
31-358
36-405
41-505
51-1007
Total100
  1. Calculate an estimate of:
    1. \(\mu\), the mean number of visitors per day;
    2. \(\sigma\), the standard deviation of the number of visitors per day.
  2. Give a reason, based upon the data provided, why \(X\) is unlikely to be normally distributed.
    1. Give a reason why \(\bar { X }\), the mean of a random sample of 100 observations on \(X\), may be assumed to be normally distributed.
    2. State, in terms of \(\mu\) and \(\sigma\), the mean and variance of \(\bar { X }\).
  4. Hence construct a \(99 \%\) confidence interval for \(\mu\).
  5. The receptionist claims that she registers on average more than 30 visitors per day, and frequently registers more than 50 visitors on any one day. Comment on each of these two claims.
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
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. \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]
  5. \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 \(\sigma\).