Questions - AQA S2 - A-Level Maths

Determine the probability that, during a given 8 -hour period, the number of telephone calls received that request an urgent visit by an engineer is:
1. at most 5 ;
2. exactly 7 ;
3. at least 5 but fewer than 10 .
Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer.
The IT company has 4 engineers available for urgent visits and it may be assumed that each of these engineers takes exactly 1 hour for each such visit. At 10 am on a particular day, all 4 engineers are available for urgent visits.
1. State the maximum possible number of telephone calls received between 10 am and 11 am that request an urgent visit and for which an engineer is immediately available.
  (1 mark)
2. Calculate the probability that at 11 am an engineer will not be immediately available to make an urgent visit.
Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer.
(1 mark)
\includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-11_2484_1709_223_153}

\includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-12_2484_1712_223_153}

\includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-13_2484_1709_223_153}

AQA S2 2010 June Q6

The number of strokes, $R$, taken by the members of Duffers Golf Club to complete the first hole may be modelled by the following discrete probability distribution.
$\boldsymbol { r }$ $\leqslant 2$ 3 4 5 6 7 8 $\geqslant 9$
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$ 0 0.1 0.2 0.3 0.25 0.1 0.05 0
1. Determine the probability that a member, selected at random, takes at least 5 strokes to complete the first hole.
2. Calculate $\mathrm { E } ( R )$.
3. Show that $\operatorname { Var } ( R ) = 1.66$.
The number of strokes, $S$, taken by the members of Duffers Golf Club to complete the second hole may be modelled by the following discrete probability distribution.
$\boldsymbol { s }$ $\leqslant 2$ 3 4 5 6 7 8 $\geqslant 9$
$\mathbf { P } ( \boldsymbol { S } = \boldsymbol { s } )$ 0 0.15 0.4 0.3 0.1 0.03 0.02 0
Assuming that $R$ and $S$ are independent:
1. show that $\mathrm { P } ( R + S \leqslant 8 ) = 0.24$;
2. calculate the probability that, when 5 members are selected at random, at least 4 of them complete the first two holes in fewer than 9 strokes;
3. calculate $\mathrm { P } ( R = 4 \mid R + S \leqslant 8 )$.
  \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-15_2484_1709_223_153}
  
  \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-16_2484_1712_223_153}
  
  \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-17_2484_1709_223_153}

AQA S2 2010 June Q7

7 The random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x \leqslant 1
\frac { 1 } { 18 } ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

State values for the median and the lower quartile of $X$.
Show that, for $1 \leqslant x \leqslant 4$, the cumulative distribution function, $\mathrm { F } ( x )$, of $X$ is given by $$\mathrm { F } ( x ) = 1 + \frac { 1 } { 54 } ( x - 4 ) ^ { 3 }$$ (You may assume that $\int ( x - 4 ) ^ { 2 } \mathrm {~d} x = \frac { 1 } { 3 } ( x - 4 ) ^ { 3 } + c$.)
Determine $\mathrm { P } ( 2 \leqslant X \leqslant 3 )$.
1. Show that $q$, the upper quartile of $X$, satisfies the equation $( q - 4 ) ^ { 3 } = - 13.5$.
2. Hence evaluate $q$ to three decimal places.
  \includegraphics[max width=\textwidth, alt={}]{c31c5c67-834e-42ce-b4af-555890c393d5-19_2484_1709_223_153}

AQA S2 2011 June Q1

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

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

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

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

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

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

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

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

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

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

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

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}

AQA S2 2013 June Q1

1 Gemma, a biologist, studies guillemots, which are a species of seabird. She has found that the weight of an adult guillemot may be modelled by a normal distribution with mean $\mu$ grams. During 2012, she measured the weight, $x$ grams, of each of a random sample of 9 adult guillemots and obtained the following results. $$\sum x = 8532 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 38538$$

Construct a 98\% confidence interval for $\mu$ based on these data.
The corresponding confidence interval for $\mu$ obtained by Gemma based on a random sample of 9 adult guillemots measured during 2011 was $( 927,1063 )$, correct to the nearest gram.
1. Find the mean weight of guillemots in this sample.
2. Studies of some other species of seabird have suggested that their mean weights were less in 2012 than in 2011. Comment on whether Gemma's two confidence intervals provide evidence that the mean weight of guillemots was less in 2012 than in 2011.
  (2 marks)

AQA S2 2013 June Q2

2 A town council wanted residents to apply for grants that were available for home insulation. In a trial, a random sample of 200 residents was encouraged, either in a letter or by a phone call, to apply for the grants. The outcomes are shown in the table.

	Applied for grant	Did not apply for grant	Total
Letter	30	130	160
Phone call	14	26	40
Total	44	156	200

The council believed that a phone call was more effective than a letter in encouraging people to apply for a grant. Use a $\chi ^ { 2 }$-test to investigate this belief at the $5 \%$ significance level.
After the trial, all the residents in the town were encouraged, either in a letter or by a phone call, to apply for the grants. It was found that there was no association between the method of encouragement and the outcome. State, with a reason, whether a Type I error, a Type II error or neither occurred in carrying out the test in part (a).
(2 marks)

AQA S2 2013 June Q3

3 Mehreen lives a 2-minute walk away from a tram stop. Trams run every 10 minutes into the city centre, taking 20 minutes to get there. Every morning, Mehreen leaves her house, walks to the tram stop and catches the first tram that arrives. When she arrives at the city centre, she then has a 5-minute walk to her office. The total time, $T$ minutes, for Mehreen's journey from house to office may be modelled by a rectangular distribution with probability density function $$\mathrm { f } ( t ) = \begin{cases} 0.1 & a \leqslant t \leqslant b
0 & \text { otherwise } \end{cases}$$

1. Explain why $a = 27$.
2. State the value of $b$.
Find the values of $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.
Find the probability that the time for Mehreen's journey is within 5 minutes of half an hour.

AQA S2 2013 June Q4

4 Gamma-ray bursts (GRBs) are pulses of gamma rays lasting a few seconds, which are produced by explosions in distant galaxies. They are detected by satellites in orbit around Earth. One particular satellite detects GRBs at a constant average rate of 3.5 per week (7 days). You may assume that the detection of GRBs by this satellite may be modelled by a Poisson distribution.

Find the probability that the satellite detects:
1. exactly 4 GRBs during one particular week;
2. at least 2 GRBs on one particular day;
3. more than 10 GRBs but fewer than 20 GRBs during the 28 days of February 2013.
Give one reason, apart from the constant average rate, why it is likely that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
(1 mark)

AQA S2 2013 June Q5

5 In a computer game, players try to collect five treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable $X$. The probability distribution of $X$ is defined by $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 2 } & x = 1,2,3,4
k & x = 5
0 & \text { otherwise } \end{array} \right.$$

1. Show that $k = \frac { 1 } { 20 }$.
2. Calculate the value of $\mathrm { E } ( X )$.
3. Show that $\operatorname { Var } ( X ) = 1.5275$.
4. Find the probability that Isaac collects more than 2 treasures.
The number of points that Isaac scores for collecting treasures is $Y$ where $$Y = 100 X - 50$$ Calculate the mean and the standard deviation of $Y$.

AQA S2 2013 June Q6

6 A supermarket buys pears from a local supplier. The supermarket requires the mean weight of the pears to be at least 175 grams. William, the fresh-produce manager at the supermarket, suspects that the latest batch of pears delivered does not meet this requirement.

William weighs a random sample of 6 pears, obtaining the following weights, in grams. $$\begin{array} { l l l l l l } 160.6 & 155.4 & 181.3 & 176.2 & 162.3 & 172.8 \end{array}$$ Previous batches of pears have had weights that could be modelled by a normal distribution with standard deviation 9.4 grams. Assuming that this still applies, show that a hypothesis test at the $5 \%$ level of significance supports William's suspicion.
(7 marks)
William then weighs a random sample of 20 pears. The mean of this sample is 169.4 grams and $s = 11.2$ grams, where $s ^ { 2 }$ is an unbiased estimate of the population variance. Assuming that the population from which this sample is taken has a normal distribution but with unknown standard deviation, test William's suspicion at the $\mathbf { 1 \% }$ level of significance.
Give a reason why the probability of a Type I error occurring was smaller when conducting the test in part (b) than when conducting the test in part (a).

AQA S2 2013 June Q7

7 A continuous random variable $X$ has the probability density function defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & 0 \leqslant x \leqslant 1
\frac { 1 } { 3 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

Sketch the graph of f on the axes below.
1. Find the cumulative distribution function, F , for $0 \leqslant x \leqslant 1$.
2. Hence, or otherwise, find the value of the lower quartile of $X$.
1. Show that the cumulative distribution function for $1 \leqslant x \leqslant 2$ is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 3 } \left( 5 x - x ^ { 2 } - 3 \right)$$
2. Hence, or otherwise, find the value of the upper quartile of $X$.
  \includegraphics[max width=\textwidth, alt={}, center]{03c1e107-3377-4b0d-9daf-7f70233c18b5-5_554_1050_1217_424}

AQA S2 2014 June Q1

7 marks

1 Vanya collected five samples of air and measured the carbon dioxide content of each sample, in parts per million by volume (ppmv). The results were as follows. $$\begin{array} { l l l l l } 387 & 375 & 382 & 379 & 381 \end{array}$$

Assuming that these data form a random sample from a normal distribution with mean $\mu$ ppmv, construct a $90 \%$ confidence interval for $\mu$.
[0pt] [6 marks]
Vanya repeated her sampling procedure on each of 30 days and, for each day's results, a $90 \%$ confidence interval for $\mu$ was constructed. On how many of these 30 days would you expect $\mu$ to lie outside that day's confidence interval?
[0pt] [1 mark]

AQA S2 2014 June Q2

1 marks

2 A large multinational company recruits employees from all four countries in the UK. For a sample of 250 recruits, the percentages of males and females from each of the countries are shown in Table 1. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1}

\cline { 2 - 5 } \multicolumn{1}{c|}{}

Add the frequencies to the contingency table, Table 2, below.
Carry out a $\chi ^ { 2 }$-test at the $10 \%$ significance level to investigate whether there is an association between country and gender of recruits.
By comparing observed and expected values, make one comment about the distribution of female recruits.
[0pt] [1 mark] \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 2}
England Scotland Wales Northern Ireland Total
Male 145
Female 105
Total 250
\end{table}

\(\boldsymbol { r }\)	\(\leqslant 2\)	3	4	5	6	7	8	\(\geqslant 9\)
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0	0.1	0.2	0.3	0.25	0.1	0.05	0

\(\boldsymbol { s }\)	\(\leqslant 2\)	3	4	5	6	7	8	\(\geqslant 9\)
\(\mathbf { P } ( \boldsymbol { S } = \boldsymbol { s } )\)	0	0.15	0.4	0.3	0.1	0.03	0.02	0

\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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