Questions - AQA S2 - A-Level Maths

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AQA S2 2005 June Q1

7 marks Standard +0.3

1 The number of cars, $X$, passing along a road each minute can be modelled by a Poisson distribution with a mean of 2.6.

Calculate $\mathrm { P } ( X = 2 )$.
1. Write down the distribution of $Y$, the number of cars passing along this road in a 5-minute interval.
2. Hence calculate the probability that at least 15 cars pass along this road in each of four successive 5 -minute intervals.

AQA S2 2005 June Q2

10 marks Standard +0.3

2 Syd, a snooker player, believes that the outcome of any frame of snooker in which he plays may be influenced by the time of day that the frame takes place. The results of 100 randomly selected frames of snooker, played by Syd, are recorded below.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Afternoon	Evening	Total
Win	30	24	54
Lose	18	28	46
Total	48	52	100

Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to test Syd's belief.
(10 marks)

AQA S2 2005 June Q3

8 marks Moderate -0.3

3 The heights, in metres, of a random sample of 10 students attending Higrade School are recorded below. $\begin{array} { l l l l l l l l l } 1.76 & 1.59 & 1.54 & 1.62 & 1.49 & 1.52 & 1.56 & 1.47 & 1.75 \end{array} 1.50$ Assume that the heights of students attending Higrade School are normally distributed.

Calculate unbiased estimates for the mean and variance of the heights of students attending Higrade School.
(3 marks)
Construct a 90\% confidence interval for the mean height of students attending Higrade School.
(5 marks)

AQA S2 2005 June Q4

7 marks Moderate -0.8

4 The error, $X$ millimetres, made when the heights of prospective members of a new gym club are measured can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \begin{cases} k & - 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

State the value of $k$.
Write down the value of $\mathrm { E } ( X )$.
Calculate $\mathrm { P } ( X > 0 )$.
The height of a randomly selected prospective member is measured. Find the probability that the magnitude of the error made exceeds 3.5 millimetres.

AQA S2 2005 June Q5

10 marks Moderate -0.5

5 The discrete random variable $R$ has the following probability distribution.

$\boldsymbol { r }$	1	2	4
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	$\frac { 1 } { 4 }$	$\frac { 1 } { 2 }$	$\frac { 1 } { 4 }$

Calculate exact values for $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$.
1. By tabulating the probability distribution for $X = \frac { 1 } { R ^ { 2 } }$, show that $\mathrm { E } ( X ) = \frac { 25 } { 64 }$.
2. Hence find the value of the mean of the area of a rectangle which has sides of length $\frac { 8 } { R }$ and $\left( R + \frac { 8 } { R } \right)$.
  (3 marks)

AQA S2 2005 June Q6

10 marks Standard +0.3

6 The contents, in millilitres, of cartons of milk produced at Kream Dairies, can be modelled by a normal distribution with mean 568 and variance $\sigma ^ { 2 }$. After receiving several complaints from their customers who thought that the average content of the cartons had been reduced, the production manager of Kream Dairies decided to investigate. A random sample of 8 cartons of milk was taken, revealing the following contents, in millilitres. $$\begin{array} { l l l l l l l l } 560 & 568 & 561 & 562 & 564 & 567 & 565 & 563 \end{array}$$ Investigate, at the $1 \%$ level of significance, whether the average content of cartons of milk is less than 568 millilitres.
(10 marks)

AQA S2 2005 June Q7

14 marks Standard +0.3

7 The time, $T$ hours, that the supporters of Bracken Football Club have to queue in order to obtain their Cup Final tickets has the following probability density function. $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 5 } & 0 \leqslant t < 3 \\ \frac { 1 } { 45 } t ( 6 - t ) & 3 \leqslant t \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

Sketch the graph of f.
Write down the value of $\mathrm { P } ( T = 3 )$.
Find the probability that a randomly selected supporter has to queue for at least 3 hours in order to obtain tickets.
Show that the median queuing time is 2.5 hours.
Calculate P (median $< T <$ mean).

AQA S2 2005 June Q8

9 marks Moderate -0.3

8 The mean age of people attending a large concert is claimed to be 35 years. A random sample of 100 people attending the concert was taken and their mean age was found to be 37.9 years.

Given that the standard deviation of the ages of the people attending the concert is 12 years, test, at the $1 \%$ level of significance, the claim that the mean age is 35 years.
(7 marks)
Explain, in the context of this question, the meaning of a Type II error.
(2 marks)

AQA S2 2006 June Q1

7 marks Standard +0.3

1 The number of A-grades, $X$, achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .

Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
The number of A-grades, $Y$, achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
1. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
2. What assumption did you make in answering part (b)(i)?

AQA S2 2006 June Q2

7 marks Moderate -0.3

2 The weights of lions kept in captivity at Wildcat Safari Park are normally distributed.
The weights, in kilograms, of a random sample of five lions were recorded as $$\begin{array} { l l l l l } 46 & 48 & 57 & 49 & 54 \end{array}$$

Construct a 95\% confidence interval for the mean weight of lions kept in captivity at Wildcat Safari Park.
State the probability that this confidence interval does not contain the mean weight of lions kept in captivity at Wildcat Safari Park.

AQA S2 2006 June Q3

8 marks Easy -1.2

3 Morecrest football team always scores at least one goal but never scores more than four goals in each game. The number of goals, $R$, scored in each game by the team can be modelled by the following probability distribution.

$\boldsymbol { r }$	1	2	3	4
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	$\frac { 7 } { 16 }$	$\frac { 5 } { 16 }$	$\frac { 3 } { 16 }$	$\frac { 1 } { 16 }$

Calculate exact values for the mean and variance of $R$.
Next season the team will play 32 games. They expect to win $90 \%$ of the games in which they score at least three goals, half of the games in which they score exactly two goals and $20 \%$ of the games in which they score exactly one goal. Find, for next season:
1. the number of games in which they expect to score at least three goals;
2. the number of games that they expect to win.

AQA S2 2006 June Q4

13 marks Moderate -0.3

4 It is claimed that the area within which a school is situated affects the age profile of the staff employed at that school. In order to investigate this claim, the age profiles of staff employed at two schools with similar academic achievements are compared. Academia High School, situated in a rural community, employs 120 staff whilst Best Manor Grammar School, situated in an inner-city community, employs 80 staff. The percentage of staff within each age group, for each school, is given in the table.

$\mathbf { 2 2 - } \mathbf { 3 4 }$

17.5

40.0

$\mathbf { 3 5 - } \mathbf { 3 9 }$

60.0

45.0

$\mathbf { 4 0 - } \mathbf { 5 9 }$

22.5

15.0

1. Form the data into a contingency table suitable for analysis using a $\chi ^ { 2 }$ distribution.
  (2 marks)
2. Use a $\chi ^ { 2 }$ test, at the $1 \%$ level of significance, to determine whether there is an association between the age profile of the staff employed and the area within which the school is situated.
Interpret your result in part (a)(ii) as it relates to the 22-34 age group.

AQA S2 2006 June Q5

10 marks Moderate -0.3

The continuous random variable $X$ follows a rectangular distribution with probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$
1. Write down $\mathrm { E } ( X )$.
2. Prove, using integration, that $$\operatorname { Var } ( X ) = \frac { 1 } { 12 } b ^ { 2 }$$
At an athletics meeting, the error, in seconds, made in recording the time taken to complete the 10000 metres race may be modelled by the random variable $T$, having the probability density function $$f ( t ) = \left\{ \begin{array} { c c } 5 & - 0.1 \leqslant t \leqslant 0.1 \\ 0 & \text { otherwise } \end{array} \right.$$ Calculate $\mathrm { P } ( | T | > 0.02 )$.

AQA S2 2006 June Q6

17 marks Standard +0.3

6 The lifetime, $X$ hours, of Everwhite camera batteries is normally distributed. The manufacturer claims that the mean lifetime of these batteries is 100 hours.

The members of a photography club suspect that the batteries do not last as long as is claimed by the manufacturer. In order to investigate their suspicion, the members test a random sample of five of these batteries and find the lifetimes, in hours, to be as follows: $$\begin{array} { l l l l l } 85 & 92 & 100 & 95 & 99 \end{array}$$ Test the members' suspicion at the $5 \%$ level of significance.
The manufacturer, believing that the mean lifetime of these batteries has not changed from 100 hours, decides to determine the lifetime, $x$ hours, of each of a random sample of 80 Everwhite camera batteries. The manufacturer obtains the following results, where $\bar { x }$ denotes the sample mean: $$\sum x = 8080 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 6399$$ Test the manufacturer's belief at the $5 \%$ level of significance.

AQA S2 2006 June Q7

15 marks Standard +0.3

7 The continuous random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } ( 2 x + 1 ) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 15 } ( 4 - x ) ^ { 2 } & 1 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

Sketch the graph of f.
1. Show that the cumulative distribution function, $\mathrm { F } ( x )$, for $0 \leqslant x \leqslant 1$ is $$\mathrm { F } ( x ) = \frac { 1 } { 5 } x ( x + 1 )$$
2. Hence write down the value of $\mathrm { P } ( X \leqslant 1 )$.
3. Find the value of $x$ for which $\mathrm { P } ( X \geqslant x ) = \frac { 17 } { 20 }$.
4. Find the lower quartile of the distribution.

AQA S2 2008 June Q1

9 marks Standard +0.3

1 It is thought that the incidence of asthma in children is associated with the volume of traffic in the area where they live. Two surveys of children were conducted: one in an area where the volume of traffic was heavy and the other in an area where the volume of traffic was light. For each area, the table shows the number of children in the survey who had asthma and the number who did not have asthma.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Asthma	No asthma	Total
Heavy traffic	52	58	110
Light traffic	28	62	90
Total	80	120	200

Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to determine whether the incidence of asthma in children is associated with the volume of traffic in the area where they live.
Comment on the number of children in the survey who had asthma, given that they lived in an area where the volume of traffic was heavy.

AQA S2 2008 June Q2

10 marks Standard +0.3

The number of telephone calls, $X$, received per hour for Dr Able may be modelled by a Poisson distribution with mean 6 . Determine $\mathrm { P } ( X = 8 )$.
The number of telephone calls, $Y$, received per hour for Dr Bracken may be modelled by a Poisson distribution with mean $\lambda$ and standard deviation 3 .
1. Write down the value of $\lambda$.
2. Determine $\mathrm { P } ( Y > \lambda )$.
1. Assuming that $X$ and $Y$ are independent Poisson variables, write down the distribution of the total number of telephone calls received per hour for Dr Able and Dr Bracken.
2. Determine the probability that a total of at most 20 telephone calls will be received during any one-hour period.
3. The total number of telephone calls received during each of 6 one-hour periods is to be recorded. Calculate the probability that a total of at least 21 telephone calls will be received during exactly 4 of these one-hour periods.

AQA S2 2008 June Q3

6 marks Moderate -0.3

3 Alan's company produces packets of crisps. The standard deviation of the weight of a packet of crisps is known to be 2.5 grams. Alan believes that, due to the extra demand on the production line at a busy time of the year, the mean weight of packets of crisps is not equal to the target weight of 34.5 grams. In an experiment set up to investigate Alan's belief, the weights of a random sample of 50 packets of crisps were recorded. The mean weight of this sample is 35.1 grams. Investigate Alan's belief at the $5 \%$ level of significance.

AQA S2 2008 June Q4

12 marks Standard +0.3

4 The delay, in hours, of certain flights from Australia may be modelled by the continuous random variable $T$, with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 2 } { 15 } t & 0 \leqslant t \leqslant 3 \\ 1 - \frac { 1 } { 5 } t & 3 \leqslant t \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$

Sketch the graph of f.
Calculate:
1. $\mathrm { P } ( T \leqslant 2 )$;
2. $\mathrm { P } ( 2 < T < 4 )$.
Determine $\mathrm { E } ( T )$.

AQA S2 2008 June Q5

8 marks Moderate -0.3

5 The weight of fat in a digestive biscuit is known to be normally distributed.
Pat conducted an experiment in which she measured the weight of fat, $x$ grams, in each of a random sample of 10 digestive biscuits, with the following results: $$\sum x = 31.9 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1.849$$

1. Construct a $99 \%$ confidence interval for the mean weight of fat in digestive biscuits.
2. Comment on a claim that the mean weight of fat in digestive biscuits is 3.5 grams.
If 200 such $99 \%$ confidence intervals were constructed, how many would you expect not to contain the population mean?

AQA S2 2008 June Q6

8 marks Standard +0.3

6 The management of the Wellfit gym claims that the mean cholesterol level of those members who have held membership of the gym for more than one year is 3.8 . A local doctor believes that the management's claim is too low and investigates by measuring the cholesterol levels of a random sample of 7 such members of the Wellfit gym, with the following results: $$\begin{array} { l l l l l l l } 4.2 & 4.3 & 3.9 & 3.8 & 3.6 & 4.8 & 4.1 \end{array}$$ Is there evidence, at the $5 \%$ level of significance, to justify the doctor's belief that the mean cholesterol level is greater than the management's claim? State any assumption that you make.

AQA S2 2008 June Q7

9 marks Easy -1.2

The number of text messages, $N$, sent by Peter each month on his mobile phone never exceeds 40. When $0 \leqslant N \leqslant 10$, he is charged for 5 messages.
When $10 < N \leqslant 20$, he is charged for 15 messages.
When $20 < N \leqslant 30$, he is charged for 25 messages.
When $30 < N \leqslant 40$, he is charged for 35 messages.
The number of text messages, $Y$, that Peter is charged for each month has the following probability distribution:
$\boldsymbol { y }$ 5 15 25 35
$\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )$ 0.1 0.2 0.3 0.4
1. Calculate the mean and the standard deviation of $Y$.
2. The Goodtime phone company makes a total charge for text messages, $C$ pence, each month given by $$C = 10 Y + 5$$ Calculate $\mathrm { E } ( C )$.
The number of text messages, $X$, sent by Joanne each month on her mobile phone is such that $$\mathrm { E } ( X ) = 8.35 \quad \text { and } \quad \mathrm { E } \left( X ^ { 2 } \right) = 75.25$$ The Newtime phone company makes a total charge for text messages, $T$ pence, each month given by $$T = 0.4 X + 250$$ Calculate $\operatorname { Var } ( T )$.

AQA S2 2008 June Q8

13 marks Moderate -0.3

8 The continuous random variable $X$ has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - 1 \\ \frac { x + 1 } { k + 1 } & - 1 \leqslant x \leqslant k \\ 1 & x > k \end{array} \right.$$ where $k$ is a positive constant.

Find, in terms of $k$, an expression for $\mathrm { P } ( X < 0 )$.
Determine an expression, in terms of $k$, for the lower quartile, $q _ { 1 }$.
Show that the probability density function of $X$ is defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { k + 1 } & - 1 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$
Given that $k = 11$ :
1. sketch the graph of f;
2. determine $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$;
3. show that $\mathrm { P } \left( q _ { 1 } < X < \mathrm { E } ( X ) \right) = 0.25$.

AQA S2 2011 June Q1

13 marks Moderate -0.3

1 The number of cars passing a speed camera on a main road between 9.30 am and 11.30 am may be modelled by a Poisson distribution with a mean rate of 2.6 per minute.

1. Write down the distribution of $X$, the number of cars passing the speed camera during a 5-minute interval between 9.30 am and 11.30 am .
2. Determine $\mathrm { P } ( X = 20 )$.
3. Determine $\mathrm { P } ( 6 \leqslant X \leqslant 18 )$.
Give two reasons why a Poisson distribution with mean 2.6 may not be a suitable model for the number of cars passing the speed camera during a 1 -minute interval between 8.00 am and 9.30 am on weekdays.
When $n$ cars pass the speed camera, the number of cars, $Y$, that exceed 60 mph may be modelled by the distribution $\mathrm { B } ( n , 0.2 )$. Given that $n = 20$, determine $\mathrm { P } ( Y \geqslant 5 )$.
Stating a necessary assumption, calculate the probability that, during a given 5-minute interval between 9.30 am and 11.30 am , exactly 20 cars pass the speed camera of which at least 5 are exceeding 60 mph .

AQA S2 2011 June Q2

11 marks Moderate -0.3

The continuous random variable $X$ has a rectangular distribution defined by the probability density function $$f ( x ) = \begin{cases} 0.01 \pi & u \leqslant x \leqslant 11 u \\ 0 & \text { otherwise } \end{cases}$$ where $u$ is a constant.
1. Show that $u = \frac { 10 } { \pi }$.
2. Using the formulae for the mean and the variance of a rectangular distribution, find, in terms of $\pi$, values for $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
3. Calculate exact values for the mean and the variance of the circumferences of circles having diameters of length $\left( X + \frac { 10 } { \pi } \right)$.

A machine produces circular discs which have an area of $Y \mathrm {~cm} ^ { 2 }$. The distribution of $Y$ has mean $\mu$ and variance 25 . A random sample of 100 such discs is selected. The mean area of the discs in this sample is calculated to be $40.5 \mathrm {~cm} ^ { 2 }$. Calculate a 95\% confidence interval for $\mu$. Emily believed that the performances of 16-year-old students in their GCSEs are associated with the schools that they attend. To investigate her belief, Emily collected data on the GCSE results for 2010 from four schools in her area. The table shows Emily's collected data, denoted by $O _ { i }$, together with the corresponding expected frequencies, $E _ { i }$, necessary for a $\chi ^ { 2 }$ test.

\multirow{2}{*}{}	$\boldsymbol { \geq } \mathbf { 5 }$ GCSEs		$\mathbf { 1 } \boldsymbol { \leqslant }$ GCSEs < $\mathbf { 5 }$		No GCSEs
	$O _ { i }$	$E _ { i }$	$O _ { i }$	$E _ { i }$	$O _ { i }$	$E _ { i }$
Jolliffe College for the Arts	187	193.15	93	90.62	30	26.23
Volpe Science Academy	175	184.43	97	86.52	24	25.05
Radok Music School	183	183.81	78	86.23	34	24.96
Bailey Language School	265	248.61	112	116.63	22	33.76

Emily used these values to correctly conduct a $\chi ^ { 2 }$ test at the $1 \%$ level of significance.

\(\boldsymbol { r }\)	1	2	4
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)

\(\boldsymbol { r }\)	1	2	3	4
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	\(\frac { 7 } { 16 }\)	\(\frac { 5 } { 16 }\)	\(\frac { 3 } { 16 }\)	\(\frac { 1 } { 16 }\)

\(\boldsymbol { y }\)	5	15	25	35
\(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	0.1	0.2	0.3	0.4

\multirow{2}{*}{}	\(\boldsymbol { \geq } \mathbf { 5 }\) GCSEs		\(\mathbf { 1 } \boldsymbol { \leqslant }\) GCSEs < \(\mathbf { 5 }\)		No GCSEs
	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)	\(O _ { i }\)	\(E _ { i }\)
Jolliffe College for the Arts	187	193.15	93	90.62	30	26.23
Volpe Science Academy	175	184.43	97	86.52	24	25.05
Radok Music School	183	183.81	78	86.23	34	24.96
Bailey Language School	265	248.61	112	116.63	22	33.76

Questions — AQA S2 (141 questions)

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AQA S2 2005 June Q1

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