Questions - AQA - A-Level Maths

\multirow{2}{*}{}		Nationality
		English	Welsh	Scottish	Irish
\multirow[b]{2}{*}{Playing position}	Forward	14	5	2	6
	Back	8	7	2	6

A player was selected at random from the squad for a radio interview. Calculate the probability that the player was:
1. a Welsh back;
2. English;
3. not English;
4. Irish, given that the player was a back;
5. a forward, given that the player was not Scottish.
Four players were selected at random from the squad to visit a school. Calculate the probability that all four players were English.

AQA S1 2007 June Q3

5 marks Easy -1.2

A sample of 50 washed baking potatoes was selected at random from a large batch.
The weights of the 50 potatoes were found to have a mean of 234 grams and a standard deviation of 25.1 grams. Construct a $95 \%$ confidence interval for the mean weight of potatoes in the batch.
(4 marks)
The batch of potatoes is purchased by a market stallholder. He sells them to his customers by allowing them to choose any 5 potatoes for $\pounds 1$. Give a reason why such chosen potatoes are unlikely to represent a random sample from the batch.

AQA S1 2007 June Q4

12 marks Moderate -0.8

4 A library allows each member to have up to 15 books on loan at any one time. The table shows the numbers of books currently on loan to a random sample of 95 members of the library.

Number of books on loan	0	1	2	3	4	$5 - 9$	$10 - 14$	15
Number of members	4	13	24	17	15	11	5	6

For these data:
1. state values for the mode and range;
2. determine values for the median and interquartile range;
3. calculate estimates of the mean and standard deviation.
Making reference to your answers to part (a), give a reason for preferring:
1. the median and interquartile range to the mean and standard deviation for summarising the given data;
2. the mean and standard deviation to the mode and range for summarising the given data.
  (1 mark)

AQA S1 2007 June Q5

13 marks Moderate -0.8

5 Bob, a gardener, measures the time taken, $y$ minutes, for 60 grams of weedkiller pellets to dissolve in 10 litres of water at different set temperatures, $x ^ { \circ } \mathrm { C }$. His results are shown in the table.

$\boldsymbol { x }$	16	20	24	28	32	36	40	44	48	52	56
$\boldsymbol { y }$	4.7	4.3	3.8	3.5	3.0	2.7	2.4	2.0	1.8	1.6	1.1

State why the explanatory variable is temperature.
Calculate the equation of the least squares regression line $y = a + b x$.
1. Interpret, in the context of this question, your value for $b$.
2. Explain why no sensible practical interpretation can be given for your value of $a$.
1. Estimate the time taken to dissolve 60 grams of weedkiller pellets in 10 litres of water at $30 ^ { \circ } \mathrm { C }$.
2. Show why the equation cannot be used to make a valid estimate of the time taken to dissolve 60 grams of weedkiller pellets in 10 litres of water at $75 ^ { \circ } \mathrm { C }$. (2 marks)

AQA S1 2007 June Q6

13 marks Standard +0.3

6 Each weekday, Monday to Friday, Trina catches a train from her local station. She claims that the probability that the train arrives on time at the station is 0.4 , and that the train's arrival time is independent from day to day.

Assuming her claims to be true, determine the probability that the train arrives on time at the station:
1. on at most 3 days during a 2 -week period ( 10 days);
2. on more than 10 days but fewer than 20 days during an 8-week period.
1. Assuming Trina's claims to be true, determine the mean and standard deviation for the number of times during a week (5 days) that the train arrives on time at the station.
2. Each week, for a period of 13 weeks, Trina's travelling colleague, Suzie, records the number of times that the train arrives on time at the station. Suzie's results are
  2 2 4 1 2 3 3 2 2 0 3 2 0
  Calculate the mean and standard deviation of these values.
3. Hence comment on the likely validity of Trina's claims.

AQA S1 2007 June Q7

16 marks Moderate -0.3

Electra is employed by E \& G Ltd to install electricity meters in new houses on an estate. Her time, $X$ minutes, to install a meter may be assumed to be normally distributed with a mean of 48 and a standard deviation of 20 . Determine:
1. $\mathrm { P } ( X < 60 )$;
2. $\mathrm { P } ( 30 < X < 60 )$;
3. the time, $k$ minutes, such that $\mathrm { P } ( X < k ) = 0.9$.
Gazali is employed by E \& G Ltd to install gas meters in the same new houses. His time, $Y$ minutes, to install a meter has a mean of 37 and a standard deviation of 25 .
1. Explain why $Y$ is unlikely to be normally distributed.
2. State why $\bar { Y }$, the mean of a random sample of 35 gas meter installations, is likely to be approximately normally distributed.
3. Determine $\mathrm { P } ( \bar { Y } > 40 )$.

AQA S1 2008 June Q1

6 marks Moderate -0.8

1 The table shows the times taken, $y$ minutes, for a wood glue to dry at different air temperatures, $x ^ { \circ } \mathrm { C }$.

$\boldsymbol { x }$	10	12	15	18	20	22	25	28	30
$\boldsymbol { y }$	42.9	40.6	38.5	35.4	33.0	30.7	28.0	25.3	22.6

Calculate the equation of the least squares regression line $y = a + b x$.
Estimate the time taken for the glue to dry when the air temperature is $21 ^ { \circ } \mathrm { C }$.

AQA S1 2008 June Q2

9 marks Easy -1.8

2 A basket in a stationery store contains a total of 400 marker and highlighter pens. Of the marker pens, some are permanent and the rest are non-permanent. The colours and types of pen are shown in the table.

	Colour
Type	Black	Blue	Red	Green
Permanent marker	44	66	32	18
Non-permanent marker	36	53	21	10
Highlighter	0	41	37	42

A pen is selected at random from the basket. Calculate the probability that it is:

a blue pen;
a marker pen;
a blue pen or a marker pen;
a green pen, given that it is a highlighter pen;
a non-permanent marker pen, given that it is a red pen.

AQA S1 2008 June Q3

10 marks Easy -1.3

3 [Figure 1, printed on the insert, is provided for use in this question.]
The table shows, for each of a sample of 12 handmade decorative ceramic plaques, the length, $x$ millimetres, and the width, $y$ millimetres.

Plaque	$\boldsymbol { x }$	$\boldsymbol { y }$
A	232	109
B	235	112
C	236	114
D	234	118
E	230	117
F	230	113
G	246	121
H	240	125
I	244	128
J	241	122
K	246	126
L	245	123

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret your value in the context of this question.
On Figure 1, complete the scatter diagram for these data.
In fact, the 6 plaques $\mathrm { A } , \mathrm { B } , \ldots , \mathrm { F }$ are from a different source to the 6 plaques $\mathrm { G } , \mathrm { H } , \ldots , \mathrm { L }$. With reference to your scatter diagram, but without further calculations, estimate the value of the product moment correlation coefficient between $x$ and $y$ for each source of plaque.

AQA S1 2008 June Q4

6 marks Easy -1.2

4 The runs scored by a cricketer in 11 innings during the 2006 season were as follows. $$\begin{array} { l l l l l l l l l l l } 47 & 63 & 0 & 28 & 40 & 51 & a & 77 & 0 & 13 & 35 \end{array}$$ The exact value of $a$ was unknown but it was greater than 100 .

Calculate the median and the interquartile range of these 11 values.
Give a reason why, for these 11 values:
1. the mode is not an appropriate measure of average;
2. the range is not an appropriate measure of spread.

AQA S1 2008 June Q5

15 marks Standard +0.3

5 When a particular make of tennis ball is dropped from a vertical distance of 250 cm on to concrete, the height, $X$ centimetres, to which it first bounces may be assumed to be normally distributed with a mean of 140 and a standard deviation of 2.5.

Determine:
1. $\mathrm { P } ( X < 145 )$;
2. $\mathrm { P } ( 138 < X < 142 )$.
Determine, to one decimal place, the maximum height exceeded by $85 \%$ of first bounces.
Determine the probability that, for a random sample of 4 first bounces, the mean height is greater than 139 cm .

AQA S1 2008 June Q6

15 marks Moderate -0.8

6 For the adult population of the UK, 35 per cent of men and 29 per cent of women do not wear glasses or contact lenses.

Determine the probability that, in a random sample of 40 men:
1. at most 15 do not wear glasses or contact lenses;
2. more than 10 but fewer than 20 do not wear glasses or contact lenses.
Calculate the probability that, in a random sample of 10 women, exactly 3 do not wear glasses or contact lenses.
1. Calculate the mean and the variance for the number who do wear glasses or contact lenses in a random sample of 20 women.
2. The numbers wearing glasses or contact lenses in 10 groups, each of 20 women, had a mean of 16.5 and a variance of 2.50. Comment on the claim that these 10 groups were not random samples.

AQA S1 2008 June Q7

14 marks Moderate -0.3

7 Vernon, a service engineer, is expected to carry out a boiler service in one hour.
One hour is subtracted from each of his actual times, and the resulting differences, $x$ minutes, for a random sample of 100 boiler services are summarised in the table.

Difference	Frequency
$- 6 \leqslant x < - 4$	4
$- 4 \leqslant x < - 2$	9
$- 2 \leqslant x < 0$	13
$0 \leqslant x < 2$	27
$2 \leqslant x < 4$	21
$4 \leqslant x < 6$	15
$6 \leqslant x < 8$	7
$8 \leqslant x \leqslant 10$	4
Total	100

1. Calculate estimates of the mean and the standard deviation of these differences.
  (4 marks)
2. Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon's actual service times for this sample.
1. Construct an approximate $98 \%$ confidence interval for the mean time taken by Vernon to carry out a boiler service.
2. Give a reason why this confidence interval is approximate rather than exact.
Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour. Comment, with a justification, on each of these claims.

AQA S1 2009 June Q1

11 marks Easy -1.3

1 A large bookcase contains two types of book: hardback and paperback. The number of books of each type in each of four subject categories is shown in the table.

\multirow{2}{*}{}		Subject category
		Crime	Romance	Science fiction	Thriller	Total
\multirow{2}{*}{Type}	Hardback	8	16	18	18	60
	Paperback	16	40	14	30	100
	Total	24	56	32	48	160

A book is selected at random from the bookcase. Calculate the probability that the book is:
1. a paperback;
2. not science fiction;
3. science fiction or a hardback;
4. a thriller, given that it is a paperback.
Three books are selected at random, without replacement, from the bookcase. Calculate, to three decimal places, the probability that one is crime, one is romance and one is science fiction.

AQA S1 2009 June Q2

10 marks Moderate -0.8

2 Hermione, who is studying reptiles, measures the length, $x \mathrm {~cm}$, and the weight, $y$ grams, of a sample of 11 adult snakes of the same type. Her results are shown in the table.

AQA S1 2009 June Q3

10 marks Moderate -0.3

3 The weight, $X$ grams, of talcum powder in a tin may be modelled by a normal distribution with mean 253 and standard deviation $\sigma$.

Given that $\sigma = 5$, determine:
1. $\mathrm { P } ( X < 250 )$;
2. $\mathrm { P } ( 245 < X < 250 )$;
3. $\mathrm { P } ( X = 245 )$.
Assuming that the value of the mean remains unchanged, determine the value of $\sigma$ necessary to ensure that $98 \%$ of tins contain more than 245 grams of talcum powder.
(4 marks) \includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_38_118_440_159}
\includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_40_118_529_159}

AQA S1 2009 June Q4

8 marks Moderate -0.8

4 As part of an investigation, a chlorine block is immersed in a large tank of water held at a constant temperature. The block slowly dissolves, and its weight, $y$ grams, is noted $x$ days after immersion. The results are shown in the table.

$\boldsymbol { x }$ days	5	10	15	20	30	40	50	60	75
$\boldsymbol { y }$ grams	47	44	42	38	35	27	23	16	9

Calculate the equation of the least squares regression line of $y$ on $x$.
Hence estimate, to the nearest gram, the initial weight of the block.

A company which markets the chlorine blocks claims that a block will usually dissolve completely after about 13 weeks. Comment, with justification, on this claim.

PART PEFRENC

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.......... $\_\_\_\_$
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AQA S1 2009 June Q5

11 marks Moderate -0.3

5 A survey of all the households on an estate is undertaken to provide information on the number of children per household. The results, for the 99 households with children, are shown in the table.

Number of children $( \boldsymbol { x } )$	1	2	3	4	5	6	7
Number of households $( \boldsymbol { f } )$	14	35	25	13	9	2	1

For these 99 households, calculate values for:
1. the median and the interquartile range;
2. the mean and the standard deviation.
In fact, 163 households were surveyed, of which 64 contained no children.
1. For all 163 households, calculate the value for the mean number of children per household.
2. State whether the value for the standard deviation, when calculated for all 163 households, will be smaller than, the same as, or greater than that calculated in part (a)(ii).
3. It is claimed that, for all 163 households on the estate, the number of children per household may be modelled approximately by a normal distribution. Comment, with justification, on this claim. Your comment should refer to a fact other than the discrete nature of the data.
  
  \includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-11_2484_1709_223_153}

AQA S1 2009 June Q6

11 marks Moderate -0.8

The time taken, in minutes, by Domesat to install a domestic satellite system may be modelled by a normal distribution with unknown mean, $\mu$, and standard deviation 15 . The times taken, in minutes, for a random sample of 10 installations are as follows.
$\begin{array} { l l l l l l l l l l } 47 & 39 & 25 & 51 & 47 & 36 & 63 & 41 & 78 & 43 \end{array}$
Construct a $98 \%$ confidence interval for $\mu$.
The time taken, $Y$ minutes, by Teleair to erect a TV aerial and then connect it to a TV is known to have a mean of 108 and a standard deviation of 28. Estimate the probability that the mean of a random sample of 40 observations of $Y$ is more than 120 .
Indicate, with a reason, where, if at all, in this question you made use of the Central Limit Theorem.
(2 marks)
\includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-13_2484_1709_223_153}

AQA S1 2009 June Q7

14 marks Moderate -0.3

7 Mr Alott and Miss Fewer work in a postal sorting office.

The number of letters per batch, $R$, sorted incorrectly by Mr Alott when sorting batches of 50 letters may be modelled by the distribution $\mathrm { B } ( 50,0.15 )$. Determine:
1. $\mathrm { P } ( R < 10 )$;
2. $\mathrm { P } ( 5 \leqslant R \leqslant 10 )$.
It is assumed that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
1. Calculate the probability that, when sorting a batch of $\mathbf { 2 2 }$ letters, Miss Fewer sorts exactly 2 letters incorrectly.
2. Calculate the probability that, when sorting a batch of $\mathbf { 3 5 }$ letters, Miss Fewer sorts at least 1 letter incorrectly.
3. Calculate the mean and the variance for the number of letters sorted correctly by Miss Fewer when she sorts a batch of $\mathbf { 1 2 0 }$ letters.
4. Miss Fewer sorts a random sample of 20 batches, each containing 120 letters. The number of letters sorted correctly per batch has a mean of 112.8 and a variance of 56.86 . Comment on the assumptions that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
  \includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-15_2484_1709_223_153}

AQA S1 2010 June Q1

5 marks Moderate -0.5

1 The weight, $x \mathrm {~kg}$, and the engine power, $y \mathrm { bhp }$, of each car in a random sample of 10 hatchback cars are shown in the table.

$\boldsymbol { x }$	1196	1062	1335	1429	1012	1355	1145	1417	1275	1284
$\boldsymbol { y }$	123	88	150	158	69	120	94	143	107	128

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret your value in the context of the question.
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-03_2484_1709_223_153}

AQA S1 2010 June Q2

7 marks Easy -1.2

2 Before leaving for a tour of the UK during the summer of 2008, Eduardo was told that the UK price of a 1.5-litre bottle of spring water was about 50p. Whilst on his tour, Eduardo noted the prices, $x$ pence, which he paid for 1.5-litre bottles of spring water from 12 retail outlets. He then subtracted 50 p from each price and his resulting differences, in pence, were $$\begin{array} { l l l l l l l l l l l l } - 18 & - 11 & 1 & 15 & 7 & - 1 & 17 & - 16 & 18 & - 3 & 0 & 9 \end{array}$$

1. Calculate the mean and the standard deviation of these differences.
2. Hence calculate the mean and the standard deviation of the prices, $x$ pence, paid by Eduardo.
Based on an exchange rate of $€ 1.22$ to $\pounds 1$, calculate, in euros, the mean and the standard deviation of the prices paid by Eduardo.
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-05_2484_1709_223_153}

AQA S1 2010 June Q3

13 marks Standard +0.3

3 Each day, Margot completes the crossword in her local morning newspaper. Her completion times, $X$ minutes, can be modelled by a normal random variable with a mean of 65 and a standard deviation of 20 .

Determine:
1. $\mathrm { P } ( X < 90 )$;
2. $\mathrm { P } ( X > 60 )$.
Given that Margot's completion times are independent from day to day, determine the probability that, during a particular period of 6 days:
1. she completes one of the six crosswords in exactly 60 minutes;
2. she completes each crossword in less than 60 minutes;
3. her mean completion time is less than 60 minutes.
  
  \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-07_2484_1709_223_153}

AQA S1 2010 June Q4

14 marks Moderate -0.8

4 In a certain country, 15 per cent of the male population is left-handed.

Determine the probability that, in a random sample of 50 males from this country:
1. at most 10 are left-handed;
2. at least 5 are left-handed;
3. more than 6 but fewer than 12 are left-handed.
In the same country, 11 per cent of the female population is left-handed. Calculate the probability that, in a random sample of 35 females from this country, exactly 4 are left-handed.
A sample of 2000 people is selected at random from the population of the country. The proportion of males in the sample is 52 per cent. How many people in the sample would you expect to be left-handed?
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-09_2484_1709_223_153}

AQA S1 2010 June Q5

11 marks Easy -1.2

5 Hugh owns a small farm.

He has two sows, Josie and Rosie, which he feeds at a trough in their field at 8.00 am each day. The probability that Josie is waiting at the trough at 8.00 am on any given day is 0.90 . The probability that Rosie is waiting at the trough at 8.00 am on any given day is 0.70 when Josie is waiting at the trough, but is only 0.20 when Josie is not waiting at the trough. Calculate the probability that, at 8.00 am on a given day:
1. both sows are waiting at the trough;
2. neither sow is waiting at the trough;
3. at least one sow is waiting at the trough.
Hugh also has two cows, Daisy and Maisy. Each day at 4.00 pm , he collects them from the gate to their field and takes them to be milked. The probability, $\mathrm { P } ( D )$, that Daisy is waiting at the gate at 4.00 pm on any given day is 0.75 .
The probability, $\mathrm { P } ( M )$, that Maisy is waiting at the gate at 4.00 pm on any given day is 0.60 .
The probability that both Daisy and Maisy are waiting at the gate at 4.00 pm on any given day is 0.40 .
1. In the table of probabilities, $D ^ { \prime }$ and $M ^ { \prime }$ denote the events 'not $D$ ' and 'not $M$ ' respectively.

PART PEFRENC

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\(\_\_\_\_\)	\(\_\_\_\_\)


	\(\_\_\_\_\)


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	\includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-08_57_1681_2227_161}


	\(\_\_\_\_\)

Number of books on loan	0	1	2	3	4	\(5 - 9\)	\(10 - 14\)	15
Number of members	4	13	24	17	15	11	5	6

\(\boldsymbol { x }\)	16	20	24	28	32	36	40	44	48	52	56
\(\boldsymbol { y }\)	4.7	4.3	3.8	3.5	3.0	2.7	2.4	2.0	1.8	1.6	1.1

Difference	Frequency
\(- 6 \leqslant x < - 4\)	4
\(- 4 \leqslant x < - 2\)	9
\(- 2 \leqslant x < 0\)	13
\(0 \leqslant x < 2\)	27
\(2 \leqslant x < 4\)	21
\(4 \leqslant x < 6\)	15
\(6 \leqslant x < 8\)	7
\(8 \leqslant x \leqslant 10\)	4
Total	100

\(\boldsymbol { x }\) days	5	10	15	20	30	40	50	60	75
\(\boldsymbol { y }\) grams	47	44	42	38	35	27	23	16	9

Number of children \(( \boldsymbol { x } )\)	1	2	3	4	5	6	7
Number of households \(( \boldsymbol { f } )\)	14	35	25	13	9	2	1

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