Questions S2 - A-Level Maths

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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.

AQA S2 2011 June Q4

14 marks Standard +0.3

4 A discrete random variable $X$ has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 3 x } { 40 } & x = 1,2,3,4 \\ \frac { x } { 20 } & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$

Calculate $\mathrm { E } ( X )$.
Show that:
1. $\quad \mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 20 }$;
  (2 marks)
2. $\operatorname { Var } \left( \frac { 1 } { X } \right) = \frac { 7 } { 160 }$.
The discrete random variable $Y$ is such that $Y = \frac { 40 } { X }$. Calculate:
1. $\mathrm { P } ( Y < 20 )$;
2. $\mathrm { P } ( X < 4 \mid Y < 20 )$.

AQA S2 2011 June Q5

13 marks Standard +0.3

The lifetime of a new 16-watt energy-saving light bulb may be modelled by a normal random variable with standard deviation 640 hours. A random sample of 25 bulbs, taken by the manufacturer from this distribution, has a mean lifetime of 19700 hours. Carry out a hypothesis test, at the $1 \%$ level of significance, to determine whether the mean lifetime has changed from 20000 hours.
The lifetime of a new 11-watt energy-saving light bulb may be modelled by a normal random variable with mean $\mu$ hours and standard deviation $\sigma$ hours. The manufacturer claims that the mean lifetime of these energy-saving bulbs is 10000 hours. Christine, from a consumer organisation, believes that this is an overestimate. To investigate her belief, she carries out a hypothesis test at the $5 \%$ level of significance based on the null hypothesis $\mathrm { H } _ { 0 } : \mu = 10000$.
1. State the alternative hypothesis that should be used by Christine in this test.
2. From the lifetimes of a random sample of 16 bulbs, Christine finds that $s = 500$ hours. Determine the range of values for the sample mean which would lead Christine not to reject her null hypothesis.
3. It was later revealed that $\mu = 10000$. State which type of error, if any, was made by Christine if she concluded that her null hypothesis should not be rejected.
  (l mark)

AQA S2 2011 June Q6

14 marks Standard +0.3

6 The continuous random variable $X$ has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 4 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

The cumulative distribution function of $X$ is denoted by $\mathrm { F } ( x )$. Show that, for $0 \leqslant x \leqslant 1$, $$\mathrm { F } ( x ) = \frac { 1 } { 8 } x \left( x ^ { 2 } + 3 \right)$$
Hence, or otherwise, verify that the median value of $X$ is 1 .
Show that the upper quartile, $q$, satisfies the equation $q ^ { 2 } - 5 q + 5 = 0$ and hence that $q = \frac { 1 } { 2 } ( 5 - \sqrt { 5 } )$.
Calculate the exact value of $\mathrm { P } ( q < X < 1.5 )$.

AQA S2 2012 June Q1

8 marks Moderate -0.3

1 At the start of the 2012 season, the ages of the members of the Warwickshire Acorns Cricket Club could be modelled by a normal random variable, $X$ years, with mean $\mu$ and standard deviation $\sigma$. The ages, $x$ years, of a random sample of 15 such members are summarised below. $$\sum x = 546 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1407.6$$

Construct a $98 \%$ confidence interval for $\mu$, giving the limits to one decimal place.
(6 marks)
At the start of the 2005 season, the mean age of the members was 40.0 years. Use your confidence interval constructed in part (a) to indicate, with a reason, whether the mean age had changed.

AQA S2 2012 June Q2

8 marks Moderate -0.3

2 The times taken to complete a round of golf at Slowpace Golf Club may be modelled by a random variable with mean $\mu$ hours and standard deviation 1.1 hours. Julian claims that, on average, the time taken to complete a round of golf at Slowpace Golf Club is greater than 4 hours. The times of 40 randomly selected completed rounds of golf at Slowpace Golf Club result in a mean of 4.2 hours.

Investigate Julian's claim at the $5 \%$ level of significance.
If the actual mean time taken to complete a round of golf at Slowpace Golf Club is 4.5 hours, determine whether a Type I error, a Type II error or neither was made in the test conducted in part (a). Give a reason for your answer.

AQA S2 2012 June Q3

7 marks Moderate -0.8

3 The continuous random variable $X$ has a cumulative distribution function defined by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x < - 5 \\ \frac { x + 5 } { 20 } & - 5 \leqslant x \leqslant 15 \\ 1 & x > 15 \end{array} \right.$$

Show that, for $- 5 \leqslant x \leqslant 15$, the probability density function, $\mathrm { f } ( x )$, of $X$ is given by $\mathrm { f } ( x ) = \frac { 1 } { 20 }$.
(1 mark)
Find:
1. $\mathrm { P } ( X \geqslant 7 )$;
2. $\mathrm { P } ( X \neq 7 )$;
3. $\mathrm { E } ( X )$;
4. $\mathrm { E } \left( 3 X ^ { 2 } \right)$.

AQA S2 2012 June Q4

13 marks Moderate -0.3

4 A house has a total of five bedrooms, at least one of which is always rented.
The probability distribution for $R$, the number of bedrooms that are rented at any given time, is given by $$\mathrm { P } ( R = r ) = \begin{cases} 0.5 & r = 1 \\ 0.4 ( 0.6 ) ^ { r - 1 } & r = 2,3,4 \\ 0.0296 & r = 5 \end{cases}$$

Complete the table below.
Find the probability that fewer than 3 bedrooms are not rented at any given time.
1. Find the value of $\mathrm { E } ( R )$.
2. Show that $\mathrm { E } \left( R ^ { 2 } \right) = 4.8784$ and hence find the value of $\operatorname { Var } ( R )$.
Bedrooms are rented on a monthly basis. The monthly income, $\pounds M$, from renting bedrooms in the house may be modelled by $$M = 1250 R - 282$$ Find the mean and the standard deviation of $M$.
$\boldsymbol { r }$ 1 2 3 4 5
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$ 0.5 0.0296

AQA S2 2012 June Q5

13 marks Standard +0.3

The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable $X$ having a Poisson distribution with mean 8.5. Determine the probability that, in any particular year, there are:
1. at least 9 minor accidents;
2. more than 5 but fewer than 10 minor accidents.
The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable $Y$ having a Poisson distribution with mean 1.5. Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable $T$ having the probability distribution $$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l } \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots \\ 0 & \text { otherwise } \end{array} \right.$$ Assuming that the number of minor accidents is independent of the number of major accidents:
1. state the value of $\lambda$;
2. determine $\mathrm { P } ( T > 16 )$;
3. calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.

AQA S2 2012 June Q6

11 marks Standard +0.3

6 Fiona, a lecturer in a school of engineering, believes that there is an association between the class of degree obtained by her students and the grades that they had achieved in A-level Mathematics. In order to investigate her belief, she collected the relevant data on the performances of a random sample of 200 recent graduates who had achieved grades A or B in A-level Mathematics. These data are tabulated below.

\multirow{2}{*}{}		Class of degree
		1	2(i)	2(ii)	3	Total
\multirow{2}{*}{A-level grade}	A	20	36	22	2	80
	B	9	55	48	8	120
	Total	29	91	70	10	200

Conduct a $\chi ^ { 2 }$ test, at the $1 \%$ level of significance, to determine whether Fiona's belief is justified.
Make two comments on the degree performance of those students in this sample who achieved a grade B in A-level Mathematics.

AQA S2 2012 June Q7

15 marks Standard +0.3

7 A continuous random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } ( 4 - x ) & 1 \leqslant x \leqslant 3 \\ \frac { 1 } { 6 } & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

Draw the graph of f on the grid on page 6 .
Prove that the mean of $X$ is $2 \frac { 5 } { 9 }$.
Calculate the exact value of:
1. $\mathrm { P } ( X > 2.5 )$;
2. $\mathrm { P } ( 1.5 < X < 4.5 )$;
3. $\mathrm { P } ( X > 2.5$ and $1.5 < X < 4.5 )$;
4. $\mathrm { P } ( X > 2.5 \mid 1.5 < X < 4.5 )$. \includegraphics[max width=\textwidth, alt={}, center]{bc21c177-6cd8-4c79-8782-d17f0238ce17-6_1340_1363_317_383}

\(\boldsymbol { r }\)	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

\(\boldsymbol { r }\)	1	2	3	4	5
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.5				0.0296

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AQA S2 2012 June Q4

AQA S2 2012 June Q5

AQA S2 2012 June Q6

AQA S2 2012 June Q7