Questions S1 - A-Level Maths

$\boldsymbol { x }$	5	10	15	20	25	30	35	40	45	50	55	60	65
$\boldsymbol { y }$	5.2	4.7	4.3	4.0	3.2	2.8	2.5	2.2	1.8	1.5	1.3	1.0	0.6

Explain why calorific value is the response variable.
Calculate the equation of the least squares regression line of $y$ on $x$, giving your answer in the form $y = a + b x$.
Interpret, in context, your values for $a$ and $b$.
Use your equation to estimate the wood's calorific value when it has a moisture content of 27 per cent.
Calculate the value of the residual for the point $( 35,2.5 )$.
Given that the values of the 13 residuals lie between - 0.28 and + 0.23 , comment on the likely accuracy of your estimate in part (d).
1. Give a general reason why your equation should not be used to estimate the wood's calorific value when it has a moisture content of 80 per cent.
2. Give a specific reason, based on the context of this question and with numerical support, why your equation cannot be used to estimate the wood's calorific value when it has a moisture content of 80 per cent.

AQA S1 2012 January Q6

6 Twins Alec and Eric are members of the same local cricket club and play for the club’s under 18 team. The probability that Alec is selected to play in any particular game is 0.85 .
The probability that Eric is selected to play in any particular game is 0.60 .
The probability that both Alec and Eric are selected to play in any particular game is 0.55 .

By using a table, or otherwise:
1. show that the probability that neither twin is selected for a particular game is 0.10 ;
2. find the probability that at least one of the twins is selected for a particular game;
3. find the probability that exactly one of the twins is selected for a particular game.
The probability that the twins' younger brother, Cedric, is selected for a particular game is:
0.30 given that both of the twins have been selected;
0.75 given that exactly one of the twins has been selected;
0.40 given that neither of the twins has been selected. Calculate the probability that, for a particular game:
1. all three brothers are selected;
2. at least two of the three brothers are selected.
  (6 marks)

AQA S1 2012 January Q7

7 A random sample of 50 full-time university employees was selected as part of a higher education salary survey. The annual salary in thousands of pounds, $x$, of each employee was recorded, with the following summarised results. $$\sum x = 2290.0 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 28225.50$$ Also recorded was the fact that 6 of the 50 salaries exceeded $\pounds 60000$.

1. Calculate values for $\bar { x }$ and $s$, where $s ^ { 2 }$ denotes the unbiased estimate of $\sigma ^ { 2 }$.
2. Hence show why the annual salary, $X$, of a full-time university employee is unlikely to be normally distributed. Give numerical support for your answer.
1. Indicate why the mean annual salary, $\bar { X }$, of a random sample of 50 full-time university employees may be assumed to be normally distributed.
2. Hence construct a $99 \%$ confidence interval for the mean annual salary of full-time university employees.
It is claimed that the annual salaries of full-time university employees have an average which exceeds $\pounds 55000$ and that more than $25 \%$ of such salaries exceed £60000. Comment on each of these two claims.

AQA S1 2013 January Q1

1 Bob, a church warden, decides to investigate the lifetime of a particular manufacturer's brand of beeswax candle. Each candle is 30 cm in length. From a box containing a large number of such candles, he selects one candle at random. He lights the candle and, after it has burned continuously for $x$ hours, he records its length, $y \mathrm {~cm}$, to the nearest centimetre. His results are shown in the table.

$\boldsymbol { x }$	5	10	15	20	25	30	35	40	45
$\boldsymbol { y }$	27	25	21	19	16	11	9	5	2

State the value that you would expect for $a$ in the equation of the least squares regression line, $y = a + b x$.
1. Calculate the equation of the least squares regression line, $y = a + b x$.
2. Interpret the value that you obtain for $b$.
3. It is claimed by the candle manufacturer that the total length of time that such candles are likely to burn for is more than 50 hours. Comment on this claim, giving a numerical justification for your answer.

AQA S1 2013 January Q2

2 The volume of Everwhite toothpaste in a pump-action dispenser may be modelled by a normal distribution with a mean of 106 ml and a standard deviation of 2.5 ml . Determine the probability that the volume of Everwhite in a randomly selected dispenser is:

less than 110 ml ;
more than 100 ml ;
between 104 ml and 108 ml ;
not exactly 106 ml .

AQA S1 2013 January Q3

3 Stopoff owns a chain of hotels. Guests are presented with the bills for their stays when they check out.

Assume that the number of bills that contain errors may be modelled by a binomial distribution with parameters $n$ and $p$, where $p = 0.30$. Determine the probability that, in a random sample of 40 bills:
1. at most 10 bills contain errors;
2. at least 15 bills contain errors;
3. exactly 12 bills contain errors.
Calculate the mean and the variance for each of the distributions $\mathrm { B } ( 16,0.20 )$ and $B ( 16,0.125 )$.
Stan, who is a travelling salesperson, always uses Stopoff hotels. He holds one of its diamond customer cards and so should qualify for special customer care. However, he regularly finds errors in his bills when he checks out. Each month, during a 12-month period, Stan stayed in Stopoff hotels on exactly 16 occasions. He recorded, each month, the number of occasions on which his bill contained errors. His recorded values were as follows. $$\begin{array} { l l l l l l l l l l l l } 2 & 1 & 4 & 3 & 1 & 3 & 0 & 3 & 1 & 0 & 5 & 1 \end{array}$$
1. Calculate the mean and the variance of these 12 values.
2. Hence state with reasons which, if either, of the distributions $\mathrm { B } ( 16,0.20 )$ and $B ( 16,0.125 )$ is likely to provide a satisfactory model for these 12 values.

AQA S1 2013 January Q4

4 Ashok is a work-experience student with an organisation that offers two separate professional examination papers, I and II. For each of a random sample of 12 students, A to L , he records the mark, $x$ per cent, achieved on Paper I, and the mark, $y$ per cent, achieved on Paper II.

\cline { 2 - 13 } \multicolumn{1}{c\|}{}	$\mathbf { A }$	$\mathbf { B }$	$\mathbf { C }$	$\mathbf { D }$	$\mathbf { E }$	$\mathbf { F }$	$\mathbf { G }$	$\mathbf { H }$	$\mathbf { I }$	$\mathbf { J }$	$\mathbf { K }$	$\mathbf { L }$
$\boldsymbol { x }$	34	46	53	62	67	72	60	54	70	71	82	85
$\boldsymbol { y }$	61	66	72	78	88	81	49	60	54	44	49	36

1. Calculate the value of the product moment correlation coefficient, $r$, between $x$ and $y$.
2. Interpret your value of $r$ in the context of this question.
1. Give two possible advantages of plotting data on a graph before calculating the value of a product moment correlation coefficient.
2. Complete the plotting of Ashok's data on the scatter diagram on page 5.
3. State what is now revealed by the scatter diagram.

Ashok subsequently discovers that students A to F have a more scientific background than students G to L. With reference to your scatter diagram, estimate the value of the product moment correlation coefficient for each of the two groups of students. You are not expected to calculate the two values.

\cline { 2 - 7 } \multicolumn{1}{c\|}{}	$\mathbf { G }$	$\mathbf { H }$	$\mathbf { I }$	$\mathbf { J }$	$\mathbf { K }$	$\mathbf { L }$
$\boldsymbol { x }$	60	54	70	71	82	85
$\boldsymbol { y }$	49	60	54	44	49	36

\section*{Examination Marks}

\includegraphics[max width=\textwidth, alt={}]{68830a6a-5479-4e5c-a845-a6536ab51cee-5_1616_1634_836_189}

AQA S1 2013 January Q5

5 Roger is an active retired lecturer. Each day after breakfast, he decides whether the weather for that day is going to be fine ( $F$ ), dull ( $D$ ) or wet ( $W$ ). He then decides on only one of four activities for the day: cycling ( $C$ ), gardening ( $G$ ), shopping ( $S$ ) or relaxing $( R )$. His decisions from day to day may be assumed to be independent. The table shows Roger's probabilities for each combination of weather and activity.

\multirow{2}{*}{}		Weather
		Fine ( $F$ )	Dull ( $D$ )	Wet ( $\boldsymbol { W }$ )
\multirow{4}{*}{Activity}	Cycling ( $\boldsymbol { C }$ )	0.30	0.10	0
	Gardening ( $\boldsymbol { G }$ )	0.25	0.05	0
	Shopping ( $\boldsymbol { S }$ )	0	0.10	0.05
	Relaxing ( $\boldsymbol { R }$ )	0	0.05	0.10

Find the probability that, on a particular day, Roger decided:
1. that it was going to be fine and that he would go cycling;
2. on either gardening or shopping;
3. to go cycling, given that he had decided that it was going to be fine;
4. not to relax, given that he had decided that it was going to be dull;
5. that it was going to be fine, given that he did not go cycling.
Calculate the probability that, on a particular Saturday and Sunday, Roger decided that it was going to be fine and decided on the same activity for both days.

AQA S1 2013 January Q6

The length of one-metre galvanised-steel straps used in house building may be modelled by a normal distribution with a mean of 1005 mm and a standard deviation of 15 mm . The straps are supplied to house builders in packs of 12, and the straps in a pack may be assumed to be a random sample. Determine the probability that the mean length of straps in a pack is less than one metre.
Tania, a purchasing officer for a nationwide house builder, measures the thickness, $x$ millimetres, of each of a random sample of 24 galvanised-steel straps supplied by a manufacturer. She then calculates correctly that the value of $\bar { x }$ is 4.65 mm .
1. Assuming that the thickness, $X \mathrm {~mm}$, of such a strap may be modelled by the distribution $\mathrm { N } \left( \mu , 0.15 ^ { 2 } \right)$, construct a $99 \%$ confidence interval for $\mu$.
2. Hence comment on the manufacturer's specification that the mean thickness of such straps is greater than 4.5 mm .

AQA S1 2013 January Q7

7 A machine, which cuts bread dough for loaves, can be adjusted to cut dough to any specified set weight. For any set weight, $\mu$ grams, the actual weights of cut dough are known to be approximately normally distributed with a mean of $\mu$ grams and a fixed standard deviation of $\sigma$ grams. It is also known that the machine cuts dough to within 10 grams of any set weight.

Estimate, with justification, a value for $\sigma$.
The machine is set to cut dough to a weight of 415 grams. As a training exercise, Sunita, the quality control manager, asked Dev, a recently employed trainee, to record the weight of each of a random sample of 15 such pieces of dough selected from the machine's output. She then asked him to calculate the mean and the standard deviation of his 15 recorded weights. Dev subsequently reported to Sunita that, for his sample, the mean was 391 grams and the standard deviation was 95.5 grams. Advise Sunita on whether or not each of Dev's values is likely to be correct. Give numerical support for your answers.
Maria, an experienced quality control officer, recorded the weight, $y$ grams, of each of a random sample of 10 pieces of dough selected from the machine's output when it was set to cut dough to a weight of 820 grams. Her summarised results were as follows. $$\sum y = 8210.0 \quad \text { and } \quad \sum ( y - \bar { y } ) ^ { 2 } = 110.00$$ Explain, with numerical justifications, why both of these values are likely to be correct.

AQA S1 2007 June Q1

1 The table shows the length, in centimetres, and maximum diameter, in centimetres, of each of 10 honeydew melons selected at random from those on display at a market stall.

Length	24	25	19	28	27	21	35	23	32	26
Maximum diameter	18	14	16	11	13	14	12	16	15	14

Calculate the value of the product moment correlation coefficient.
Interpret your value in the context of this question.

AQA S1 2007 June Q2

2 The British and Irish Lions 2005 rugby squad contained 50 players. The nationalities and playing positions of these players are shown in the table.

\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

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

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

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

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

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

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

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

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

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

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

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

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

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.

\(\boldsymbol { x }\)	5	10	15	20	25	30	35	40	45	50	55	60	65
\(\boldsymbol { y }\)	5.2	4.7	4.3	4.0	3.2	2.8	2.5	2.2	1.8	1.5	1.3	1.0	0.6

\cline { 2 - 13 } \multicolumn{1}{c\|}{}	\(\mathbf { A }\)	\(\mathbf { B }\)	\(\mathbf { C }\)	\(\mathbf { D }\)	\(\mathbf { E }\)	\(\mathbf { F }\)	\(\mathbf { G }\)	\(\mathbf { H }\)	\(\mathbf { I }\)	\(\mathbf { J }\)	\(\mathbf { K }\)	\(\mathbf { L }\)
\(\boldsymbol { x }\)	34	46	53	62	67	72	60	54	70	71	82	85
\(\boldsymbol { y }\)	61	66	72	78	88	81	49	60	54	44	49	36

\multirow{2}{*}{}		Weather
		Fine ( \(F\) )	Dull ( \(D\) )	Wet ( \(\boldsymbol { W }\) )
\multirow{4}{*}{Activity}	Cycling ( \(\boldsymbol { C }\) )	0.30	0.10	0
	Gardening ( \(\boldsymbol { G }\) )	0.25	0.05	0
	Shopping ( \(\boldsymbol { S }\) )	0	0.10	0.05
	Relaxing ( \(\boldsymbol { R }\) )	0	0.05	0.10

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

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

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