Questions S1 - A-Level Maths

Her journey time, $X$ minutes, by route A may be assumed to be normally distributed with a mean of 37 and a standard deviation of 8 . Determine:
1. $\mathrm { P } ( X < 45 )$;
2. $\mathrm { P } ( 30 < X < 45 )$.
Her journey time, $Y$ minutes, by route B may be assumed to be normally distributed with a mean of 40 and a standard deviation of $\sigma$. Given that $\mathrm { P } ( Y > 45 ) = 0.12$, calculate the value of $\sigma$.
If Monica leaves home at 8.15 am to walk to work hoping to arrive by 9.00 am , state, with a reason, which route she should take.
When Monica travels to work from home by car, her journey time, $W$ minutes, has a mean of 18 and a standard deviation of 12 . Estimate the probability that, for a random sample of 36 journeys to work from home by car, Monica's mean time is more than 20 minutes.
Indicate where, if anywhere, in this question you needed to make use of the Central Limit Theorem.

AQA S1 2007 January Q7

7 [Figure 1, printed on the insert, is provided for use in this question.]
Stan is a retired academic who supplements his pension by mowing lawns for customers who live nearby. As part of a review of his charges for this work, he measures the areas, $x \mathrm {~m} ^ { 2 }$, of a random sample of eight of his customers' lawns and notes the times, $y$ minutes, that it takes him to mow these lawns. His results are shown in the table.

Customer	$\mathbf { A }$	$\mathbf { B }$	$\mathbf { C }$	$\mathbf { D }$	$\mathbf { E }$	$\mathbf { F }$	$\mathbf { G }$	$\mathbf { H }$
$\boldsymbol { x }$	360	140	860	600	1180	540	260	480
$\boldsymbol { y }$	50	25	135	70	140	90	55	70

On Figure 1, plot a scatter diagram of these data.
Calculate the equation of the least squares regression line of $y$ on $x$. Draw your line on Figure 1.
Calculate the value of the residual for Customer H and indicate how your value is confirmed by your scatter diagram.
Given that Stan charges $\pounds 12$ per hour, estimate the charge for mowing a customer's lawn that has an area of $560 \mathrm {~m} ^ { 2 }$.

AQA S1 2010 January Q1

1 Draught excluder for doors and windows is sold in rolls of nominal length 10 metres.
The actual length, $X$ metres, of draught excluder on a roll may be modelled by a normal distribution with mean 10.2 and standard deviation 0.15 .

Determine:
1. $\mathrm { P } ( X < 10.5 )$;
2. $\mathrm { P } ( 10.0 < X < 10.5 )$.
A customer randomly selects six 10 -metre rolls of the draught excluder. Calculate the probability that all six rolls selected contain more than 10 metres of draught excluder.

AQA S1 2010 January Q2

2 Lizzie, the receptionist at a dental practice, was asked to keep a weekly record of the number of patients who failed to turn up for an appointment. Her records for the first 15 weeks were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 20 & 26 & 32 & a & 37 & 14 & 27 & 34 & 15 & 18 & b & 25 & 37 & 29 & 25 \end{array}$$ Unfortunately, Lizzie forgot to record the actual values for two of the 15 weeks, so she recorded them as $a$ and $b$. However, she did remember that $a < 10$ and that $b > 40$.

Calculate the median and the interquartile range of these 15 values.
Give a reason why, for these data:
1. the mode is not an appropriate measure of average;
2. the standard deviation cannot be used as a measure of spread.
Subsequent investigations revealed that the missing values were 8 and 43 . Calculate the mean and the standard deviation of the 15 values.

AQA S1 2010 January Q3

3 The table shows, for each of a random sample of 7 weeks, the number of customers, $x$, who purchased fuel from a filling station, together with the total volume, $y$ litres, of fuel purchased by these customers.

$\boldsymbol { x }$	230	184	165	147	241	174	210
$\boldsymbol { y }$	4551	3410	3252	3756	3787	4024	4254

Calculate the equation of the least squares regression line of $y$ on $x$.
Estimate the volume of fuel sold during a week in which 200 customers purchase fuel.
Comment on the likely reliability of your estimate in part (b), given that, for the regression line calculated in part (a), the values of the 7 residuals lie between approximately - 415 litres and + 430 litres.

AQA S1 2010 January Q4

4 Each school-day morning, three students, Rita, Said and Ting, travel independently from their homes to the same school by one of three methods: walk, cycle or bus. The table shows the probabilities of their independent daily choices.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Walk	Cycle	Bus
Rita	0.65	0.10	0.25
Said	0.40	0.45	0.15
Ting	0.25	0.55	0.20

Calculate the probability that, on any given school-day morning:
1. all 3 students walk to school;
2. only Rita travels by bus to school;
3. at least 2 of the 3 students cycle to school.
Ursula, a friend of Rita, never travels to school by bus. The probability that: Ursula walks to school when Rita walks to school is 0.9 ; Ursula cycles to school when Rita cycles to school is 0.7 . Calculate the probability that, on any given school-day morning, Rita and Ursula travel to school by:
1. the same method;
2. different methods.

AQA S1 2010 January Q5

5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows.
$\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}$

It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
1. Construct a $98 \%$ confidence interval for the mean weight, $\mu$ grams, of flour in a bag, giving the limits to four significant figures.
2. State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
3. If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for $\mu$ to be valid.
The statement ' 1 kg ' is printed on each bag. Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
Given that $\mu = 1000$, state the probability that a $98 \%$ confidence interval for $\mu$ will not contain 1000.
(l mark)

AQA S1 2010 January Q6

6 During the winter, the probability that Barry's cat, Sylvester, chooses to stay outside all night is 0.35 , and the cat's choice is independent from night to night.

Determine the probability that, during a period of 2 weeks ( 14 nights) in winter, Sylvester chooses to stay outside:
1. on at most 7 nights;
2. on at least 11 nights;
3. on more than 5 nights but fewer than 10 nights.
Calculate the probability that, during a period of $\mathbf { 3 }$ weeks in winter, Sylvester chooses to stay outside on exactly 4 nights.
Barry claims that, during the summer, the number of nights per week, $S$, on which Sylvester chooses to stay outside can be modelled by a binomial distribution with $n = 7$ and $p = \frac { 5 } { 7 }$.
1. Assuming that Barry's claim is correct, find the mean and the variance of $S$.
2. For a period of 13 weeks during the summer, the number of nights per week on which Sylvester chose to stay outside had a mean of 5 and a variance of 1.5 . Comment on Barry's claim.
  (2 marks)

AQA S1 2010 January Q7

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.

Item	Purchase price (£ $\boldsymbol { x }$ )	Auction price (£ y)
A	20	30
B	35	45
C	18	25
D	50	50
E	45	38
F	55	45
G	43	50
H	81	90
I	90	85
J	30	190
K	57	65
L	112	25

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
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.
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

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)	13	4	5	10	9	17	23	16	2	16
Value (y)	12.5	5.7	2.3	18.4	7.9	17.1	17.9	18.6	8.3	21.3

1. Calculate the value of the product moment correlation coefficient between $x$ and $y$.
2. Interpret your value in context.
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

2 The weight, $X$ grams, of a particular variety of orange is normally distributed with mean 205 and standard deviation 25.

Determine the probability that the weight of an orange is:
1. less than 250 grams;
2. between 200 grams and 250 grams.
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.
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

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.

Calculate the probability that:
1. both Fred and Delia watch a particular game;
2. neither Fred nor Delia watch a particular game.
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

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	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$	$\mathbf { 6 }$	$\mathbf { 7 }$	$\mathbf { 8 }$
$\boldsymbol { x }$	10	16	23	27	31	35	38	44
$\boldsymbol { y }$	2.4	3.5	3.2	4.1	4.1	5.6	4.6	5.3

Calculate the equation of the least squares regression line of $y$ on $x$.
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 - 0.174 0.418 0.085 - 0.254 0.906 - 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.
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

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.
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.
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

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 day	Number of days
1-10	13
11-20	33
21-25	17
26-30	12
31-35	8
36-40	5
41-50	5
51-100	7
Total	100

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.
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 $\bar { X }$.
Hence construct a $99 \%$ confidence interval for $\mu$.
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

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.

$\boldsymbol { x }$	223	276	374	433	564	612	704	766
$\boldsymbol { y }$	6.50	4.00	5.50	8.00	4.50	5.00	8.00	5.50

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.
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

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 .

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 .
Determine the probability that the mean height of a random sample of 4 sunflowers is more than 190 cm .

AQA S1 2006 June Q3

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:

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.
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

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.
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

5 Kirk and Les regularly play each other at darts.

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.
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

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
	None	One	Two	At least three	Total
Detached house	24	32	41	23	120
Semi-detached house	40	37	88	35	200
Total	64	69	129	58	320

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 $D$ '.)

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.
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

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

4 marks

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.

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.
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

2 marks

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

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.'
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
  $\boldsymbol { E }$ 0.16 0.28
  $\boldsymbol { E } ^ { \prime }$
  Total 0.60 1.00
  
  \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-11_2050_1707_687_153}

Customer	\(\mathbf { A }\)	\(\mathbf { B }\)	\(\mathbf { C }\)	\(\mathbf { D }\)	\(\mathbf { E }\)	\(\mathbf { F }\)	\(\mathbf { G }\)	\(\mathbf { H }\)
\(\boldsymbol { x }\)	360	140	860	600	1180	540	260	480
\(\boldsymbol { y }\)	50	25	135	70	140	90	55	70

Item	Purchase price (£ \(\boldsymbol { x }\) )	Auction price (£ y)
A	20	30
B	35	45
C	18	25
D	50	50
E	45	38
F	55	45
G	43	50
H	81	90
I	90	85
J	30	190
K	57	65
L	112	25

Sheet	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)	\(\mathbf { 7 }\)	\(\mathbf { 8 }\)
\(\boldsymbol { x }\)	10	16	23	27	31	35	38	44
\(\boldsymbol { y }\)	2.4	3.5	3.2	4.1	4.1	5.6	4.6	5.3

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(\boldsymbol { M }\)	\(\boldsymbol { M } ^ { \prime }\)	Total
\(\boldsymbol { E }\)	0.16		0.28
\(\boldsymbol { E } ^ { \prime }\)
Total		0.60	1.00

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