Questions - AQA S1 - A-Level Maths

Number of vehicles registered	0	1	2	$\geqslant 3$
Percentage of households	18	47	25	10

A random sample of 30 households within the city is selected. Use a binomial distribution with $n = 30$, together with relevant information from the table in each case, to find the probability that the sample contains:
1. exactly 3 households with no registered vehicles;
2. at most 5 households with three or more registered vehicles;
3. more than 10 households with at least two registered vehicles;
4. more than 5 households but fewer than 10 households with exactly two registered vehicles.
If a random sample of $\mathbf { 1 5 0 }$ households within the city were to be selected, estimate the mean and the variance for the number of households in the sample that would have either one or two registered vehicles.
[0pt] [2 marks]
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AQA S1 2014 June Q6

3 marks

6 A rubber seal is fitted to the bottom of a flood barrier. When no pressure is applied, the depth of the seal is 15 cm . When pressure is applied, a watertight seal is created between the flood barrier and the ground. The table shows the pressure, $x$ kilopascals ( kPa ), applied to the seal and the resultant depth, $y$ centimetres, of the seal.

$\boldsymbol { x }$	25	50	75	100	125	150	175	200	250	300
$\boldsymbol { y }$	14.7	13.4	12.8	11.9	11.0	10.3	9.7	9.0	7.5	6.7

1. State the value that you would expect for $a$ in the equation of the least squares regression line, $y = a + b x$.
2. Calculate the equation of the least squares regression line, $y = a + b x$.
3. Interpret, in context, your value for $b$.
Calculate an estimate of the depth of the seal when it is subjected to a pressure of 225 kPa .
1. Give a statistical reason as to why your equation is unlikely to give a realistic estimate of the depth of the seal if it were to be subjected to a pressure of 400 kPa .
2. Give a reason based on the context of this question as to why your equation will not give a realistic estimate of the depth of the seal if it were to be subjected to a pressure of 525 kPa .
  [0pt] [3 marks]
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-20_946_1709_1761_153}
  
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-21_2484_1707_221_153}
  
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-23_2484_1707_221_153}

AQA S1 2014 June Q7

7 The volume of water, $V$, used by a guest in an en suite shower room at a small guest house may be modelled by a random variable with mean $\mu$ litres and standard deviation 65 litres. A random sample of 80 guests using this shower room showed a mean usage of 118 litres of water.

1. Give a numerical justification as to why $V$ is unlikely to be normally distributed.
2. Explain why $\bar { V }$, the mean of a random sample of 80 observations of $V$, may be assumed to be approximately normally distributed.
1. Construct a $98 \%$ confidence interval for $\mu$.
2. Hence comment on a claim that $\mu$ is 140 .
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-24_1526_1709_1181_153}
  
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-25_2484_1707_221_153}
  \includegraphics[max width=\textwidth, alt={}, center]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-27_2490_1719_217_150}
  \includegraphics[max width=\textwidth, alt={}, center]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-28_2486_1728_221_141}

AQA S1 2016 June Q1

1 The table shows the heights, $x \mathrm {~cm}$, and the arm spans, $y \mathrm {~cm}$, of a random sample of 12 men aged between 21 years and 40 years.

$\boldsymbol { x }$	152	166	154	159	179	167	155	168	174	182	161	163
$\boldsymbol { y }$	143	154	151	153	168	160	146	163	170	175	155	158

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret, in context, your value calculated in part (a).

AQA S1 2016 June Q2

5 marks

2 A small chapel was open to visitors for 55 days during the summer of 2015. The table summarises the daily numbers of visitors.

Number of visitors	Number of days
20 or fewer	1
21	2
22	3
23	6
24	8
25	10
26	13
27	7
28	2
29	1
30 or more	2
Total	55

For these data:
1. state the modal value;
2. find values for the median and the interquartile range.
Name one measure of average and one measure of spread that cannot be calculated exactly from the data in the table.
[0pt] [2 marks]
Reference to the raw data revealed that the 3 unknown exact values in the table were 13,37 and 58. Making use of this additional information, together with the data in the table, calculate the value of each of the two measures that you named in part (b).
[0pt] [3 marks]

AQA S1 2016 June Q3

5 marks

3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.

\multirow{2}{*}{}		Charge band for course of treatment
		Band 0	Band 1	Band 2	Band 3	Total
\multirow{4}{*}{Age of patient (years)}	Under 19	32	43	5	0	80
	Between 19 and 40	17	62	22	3	104
	Between 41 and 65	28	82	35	31	176
	66 or over	13	53	68	6	140
	Total	90	240	130	40	500

Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
1. aged between 41 and 65;
2. aged 66 or over and charged at band 2;
3. aged between 19 and 40 and charged at most at band 1;
4. aged 41 or over, given that the patient was charged at band 2;
5. charged at least at band 2, given that the patient was not aged 66 or over.
Four patients at this dental surgery, not included in the above 500 patients, are selected at random. Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
[0pt] [5 marks]

AQA S1 2016 June Q4

2 marks

4 As part of her science project, a student found the mass, $y$ grams, of a particular compound that dissolved in 100 ml of water at each of 12 different set temperatures, $x ^ { \circ } \mathrm { C }$. The results are shown in the table.

$\boldsymbol { x }$	20	25	30	35	40	45	50	55	60	65	70	75
$\boldsymbol { y }$	242	262	269	290	298	310	326	355	359	375	390	412

Calculate the equation of the least squares regression line of $y$ on $x$.
Interpret, in context, your value for the gradient of this regression line.
Use your equation to estimate the mass of the compound which will dissolve in 100 ml of water at $68 ^ { \circ } \mathrm { C }$.
Given that the values of the 12 residuals for the regression line of $y$ on $x$ lie between - 7 and + 9 , comment, with justification, on the likely accuracy of your estimate in part (c).
[0pt] [2 marks]

AQA S1 2016 June Q5

7 marks

5 Still mineral water is supplied in 1.5-litre bottles. The actual volume, $X$ millilitres, in a bottle may be modelled by a normal distribution with mean $\mu = 1525$ and standard deviation $\sigma = 9.6$.

Determine the probability that the volume of water in a randomly selected bottle is:
1. less than 1540 ml ;
2. more than 1535 ml ;
3. between 1515 ml and 1540 ml ;
4. not 1500 ml .
The supplier requires that only 10 per cent of bottles should contain more than 1535 ml of water. Assuming that there has been no change in the value of $\sigma$, calculate the reduction in the value of $\mu$ in order to satisfy this requirement. Give your answer to one decimal place.
Sparkling spring water is supplied in packs of six 0.5 -litre bottles. The actual volume in a bottle may be modelled by a normal distribution with mean 508.5 ml and standard deviation 3.5 ml . Stating a necessary assumption, determine the probability that:
1. the volume of water in each of the 6 bottles from a randomly selected pack is more than 505 ml ;
2. the mean volume of water in the 6 bottles from a randomly selected pack is more than 505 ml .
  [0pt] [7 marks]

AQA S1 2016 June Q6

2 marks

6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.

Colour	Blue	Green	Red	Orange	Yellow	White
Proportion	0.25	0.25	0.18	0.12	0.15	0.05

A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with $n = 50$, together with relevant information from the table, to estimate the probability that this sample contains:
1. exactly 4 red loom bands;
2. at most 10 yellow loom bands;
3. at least 30 blue or green loom bands;
4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
The random variable $R$ denotes the number of red loom bands in a random sample of $\mathbf { 3 0 0 }$ loom bands selected from the box. Estimate values for the mean and the variance of $R$.
[0pt] [2 marks]

AQA S1 2016 June Q7

5 marks

7 Customers buying euros ( €) at a travel agency must pay for them in pounds ( $\pounds$ ). The amounts paid, $\pounds x$, by a sample of 40 customers were, in ascending order, as follows.

AQA S1 2005 January Q1

1 Each Monday, Azher has a stall at a town's outdoor market. The table below shows, for each of a random sample of 10 Mondays during 2003, the air temperature, $x ^ { \circ } \mathrm { C }$, at 9 am and Azher's takings, £y.

Monday	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$	$\mathbf { 6 }$	$\mathbf { 7 }$	$\mathbf { 8 }$	$\mathbf { 9 }$	$\mathbf { 1 0 }$
$\boldsymbol { x }$	2	6	9	18	1	3	7	12	13	4
$\boldsymbol { y }$	97	103	136	245	121	78	145	128	141	312

A scatter diagram of these data is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-2_901_1068_1078_447} Give two distinct comments, in context, on what this diagram reveals.
One of the Mondays is found to be Easter Monday, the busiest Monday market of the year. Identify which Monday this is most likely to be.
Removing the data for the Monday you identified in part (b), calculate the value of the product moment correlation coefficient for the remaining 9 pairs of values of $x$ and $y$.
Name one other variable that would have been likely to affect Azher's takings at this town's outdoor market.
(l mark)

AQA S1 2005 January Q2

2 The volume, in millilitres, of lemonade in mini-cans may be assumed to be normally distributed with a standard deviation of 3.5. The volumes, in millilitres, of lemonade in a random sample of 12 mini-cans were as follows.

155	148	156	149	147	156
157	156	150	154	148	154

Construct a $98 \%$ confidence interval for the mean volume of lemonade in a mini-can, giving the limits to one decimal place.
On each mini-can is printed " 150 ml ". Comment on this, using the given sample and your confidence interval in part (a).
State why, in part (a), use of the Central Limit Theorem was not necessary.

AQA S1 2005 January Q3

3 [Figure 1, printed on the insert, is provided for use in this question.]
A parcel delivery company has a depot on the outskirts of a town. Each weekday, a van leaves the depot to deliver parcels across a nearby area. The table below shows, for a random sample of 10 weekdays, the number, $x$, of parcels to be delivered and the total time, $y$ minutes, that the van is out of the depot.

$\boldsymbol { x }$	9	16	22	11	19	26	14	10	11	17
$\boldsymbol { y }$	79	127	172	109	152	214	131	80	94	148

On Figure 1, plot a scatter diagram of these data.
Calculate the equation of the least squares regression line of $y$ on $x$ and draw your line on Figure 1.
Use your regression equation to estimate the total time that the van is out of the depot when delivering:
1. 15 parcels;
2. 35 parcels. Comment on the likely reliability of each of your estimates.
The time that the van is out of the depot delivering parcels may be thought of as the time needed to travel to and from the area plus an amount of time proportional to the number of parcels to be delivered. Given that the regression line of $y$ on $x$ is of the form $y = a + b x$, give an interpretation, in context, for each of your values of $a$ and $b$.
(2 marks)

AQA S1 2005 January Q4

4 Chopped lettuce is sold in bags nominally containing 100 grams.
The weight, $X$ grams, of chopped lettuce, delivered by the machine filling the bags, may be assumed to be normally distributed with mean $\mu$ and standard deviation 4.

Assuming that $\mu = 106$, determine the probability that a randomly selected bag of chopped lettuce:
1. weighs less than 110 grams;
2. is underweight.
Determine the minimum value of $\mu$ so that at most 2 per cent of bags of chopped lettuce are underweight. Give your answer to one decimal place.
Boxes each contain 10 bags of chopped lettuce. The mean weight of a bag of chopped lettuce in a box is denoted by $\bar { X }$. Given that $\mu = 108.5$ :
1. write down values for the mean and variance of $\bar { X }$;
2. determine the probability that $\bar { X }$ exceeds 110 .

AQA S1 2005 January Q5

5 Each evening Aaron sets his alarm for 7 am. He believes that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.

Assuming that Aaron's belief is correct, determine the probability that, during a week (7 mornings), he wakes before his alarm rings:
1. on 2 or fewer mornings;
2. on more than 1 but fewer than 5 mornings.
Assuming that Aaron's belief is correct, calculate the probability that, during a 4 -week period, he wakes before his alarm rings on exactly 7 mornings.
Assuming that Aaron's belief is correct, calculate values for the mean and standard deviation of the number of mornings in a week when Aaron wakes before his alarm rings.
(2 marks)
During a 50-week period, Aaron records, each week, the number of mornings on which he wakes before his alarm rings. The results are as follows.
Number of mornings 0 1 2 3 4 5 6 7
Frequency 10 8 7 7 5 5 4 4
1. Calculate the mean and standard deviation of these data.
2. State, giving reasons, whether your answers to part (d)(i) support Aaron's belief that the probability that he wakes before his alarm rings each morning is 0.4 , and is independent from morning to morning.
  (3 marks)

AQA S1 2005 January Q6

6 The table below shows the numbers of males and females in each of three employment categories at a university on 31 July 2003.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Employment category
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Managerial	Academic	Support
Male	38	369	303
Female	26	275	643

An employee is selected at random. Determine the probability that the employee is:
1. female;
2. a female academic;
3. either female or academic or both;
4. female, given that the employee is academic.
Three employees are selected at random, without replacement. Determine the probability that:
1. all three employees are male;
2. exactly one employee is male.
The event "employee selected is academic" is denoted by $A$. The event "employee selected is female" is denoted by $F$. Describe in context, as simply as possible, the events denoted by:
1. $F \cap A$;
2. $F ^ { \prime } \cup A$.
  Surname Other Names
  Centre Number Candidate Number
  Candidate Signature
  General Certificate of Education
  January 2005
  Advanced Subsidiary Examination MS/SS1B AQA
  459:5EMLM
  : 11 P וPII " 1 : : ר
  ALLI.ub c \section*{STATISTICS} Unit Statistics 1B Insert for use in Question 3.
  Fill in the boxes at the top of this page.
  Fasten this insert securely to your answer book. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Scatter diagram for parcel deliveries by a van} \includegraphics[alt={},max width=\textwidth]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-8_2420_1664_349_175}
  \end{figure} Figure 1 (for Question 3)

AQA S1 2007 January Q1

1 The times, in seconds, taken by 20 people to solve a simple numerical puzzle were

17	19	22	26	28	31	34	36	38	39
41	42	43	47	50	51	53	55	57	58

Calculate the mean and the standard deviation of these times.
In fact, 23 people solved the puzzle. However, 3 of them failed to solve it within the allotted time of 60 seconds. Calculate the median and the interquartile range of the times taken by all 23 people.
(4 marks)
For the times taken by all 23 people, explain why:
1. the mode is not an appropriate numerical measure;
2. the range is not an appropriate numerical measure.

AQA S1 2007 January Q2

2 A hotel has 50 single rooms, 16 of which are on the ground floor. The hotel offers guests a choice of a full English breakfast, a continental breakfast or no breakfast. The probabilities of these choices being made are $0.45,0.25$ and 0.30 respectively. It may be assumed that the choice of breakfast is independent from guest to guest.

On a particular morning there are 16 guests, each occupying a single room on the ground floor. Calculate the probability that exactly 5 of these guests require a full English breakfast.
On a particular morning when there are 50 guests, each occupying a single room, determine the probability that:
1. at most 12 of these guests require a continental breakfast;
2. more than 10 but fewer than 20 of these guests require no breakfast.
When there are 40 guests, each occupying a single room, calculate the mean and the standard deviation for the number of these guests requiring breakfast.

AQA S1 2007 January Q3

3 Estimate, without undertaking any calculations, the value of the product moment correlation coefficient between the variables $x$ and $y$ in each of the three scatter diagrams.

\includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_631_659_516_301}
\includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_620_647_525_1119}
\includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_624_655_1279_303}
(5 marks)

AQA S1 2007 January Q4

Assuming that the 78 performances may be considered to be a random sample, construct a $90 \%$ confidence interval for the mean proceeds from the sales of programmes at an evening performance of this play.
Comment on the likely validity of the assumption in part (a) when constructing a confidence interval for the mean proceeds from the sales of programmes at an evening performance of:
1. this particular play;
2. any play.

AQA S1 2007 January Q5

5 Dafydd, Eli and Fabio are members of an amateur cycling club that holds a time trial each Sunday during the summer. The independent probabilities that Dafydd, Eli and Fabio take part in any one of these trials are $0.6,0.7$ and 0.8 respectively. Find the probability that, on a particular Sunday during the summer:

none of the three cyclists takes part;
Fabio is the only one of the three cyclists to take part;
exactly one of the three cyclists takes part;
either one or two of the three cyclists take part.

AQA S1 2007 January Q6

6 When Monica walks to work from home, she uses either route A or route B.

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.

Number of vehicles registered	0	1	2	\(\geqslant 3\)
Percentage of households	18	47	25	10

\(\boldsymbol { x }\)	25	50	75	100	125	150	175	200	250	300
\(\boldsymbol { y }\)	14.7	13.4	12.8	11.9	11.0	10.3	9.7	9.0	7.5	6.7

Monday	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)	\(\mathbf { 7 }\)	\(\mathbf { 8 }\)	\(\mathbf { 9 }\)	\(\mathbf { 1 0 }\)
\(\boldsymbol { x }\)	2	6	9	18	1	3	7	12	13	4
\(\boldsymbol { y }\)	97	103	136	245	121	78	145	128	141	312

Surname	Other Names
Centre Number		Candidate Number
Candidate Signature

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

Number of mornings	0	1	2	3	4	5	6	7
Frequency	10	8	7	7	5	5	4	4

Questions — AQA S1 (156 questions)

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