Questions - AQA S2 - A-Level Maths

Age	Pass	Fail
$\mathbf { 1 7 } - \mathbf { 1 8 }$	28	20
$\mathbf { 1 9 } - \mathbf { 3 0 }$	2	14
$\mathbf { 3 1 } - \mathbf { 3 9 }$	12	33
$\mathbf { 4 0 } - \mathbf { 6 0 }$	6	5

Use a $\chi ^ { 2 }$ test at the $1 \%$ level of significance to investigate Julie's belief.
Interpret your result in part (a) as it relates to the 17-18 age group.

AQA S2 2010 January Q5

In a remote African village, it is known that 70 per cent of the villagers have a particular blood disorder. A medical research student selects 25 of the villagers at random. Using a binomial distribution, calculate the probability that more than 15 of these 25 villagers have this blood disorder.
1. In towns and cities in Asia, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 2.6 per 100000 people. A town in Asia with a population of 100000 is selected. Determine the probability that at most 5 people have this blood disorder.
2. In towns and cities in South America, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 49 per million people. A town in South America with a population of 100000 is selected. Calculate the probability that exactly 10 people have this blood disorder.
3. The random variable $T$ denotes the total number of people in the two selected towns who have this blood disorder. Write down the distribution of $T$ and hence determine $\mathrm { P } ( T > 16 )$.

AQA S2 2010 January Q6

Ali has a bag of 10 balls, of which 5 are red and 5 are blue. He asks Ben to select 5 of these balls from the bag at random. The probability distribution of $X$, the number of red balls that Ben selects, is given in Table 1. \begin{table}[h]

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

$\boldsymbol { x }$	0	1	2	3	4	5
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	$\frac { 1 } { 252 }$	$\frac { 25 } { 252 }$	$\frac { 100 } { 252 }$	$a$	$\frac { 25 } { 252 }$	$\frac { 1 } { 252 }$

\end{table}

State the value of $a$.
Hence write down the value of $\mathrm { E } ( X )$.
Determine the standard deviation of $X$.

Ali decides to play a game with Joanne using the same 10 balls. Joanne is asked to select 2 balls from the bag at random. Ali agrees to pay Joanne 90 p if the two balls that she selects are the same colour, but nothing if they are different colours. Joanne pays 50 p to play the game. The probability distribution of $Y$, the number of red balls that Joanne selects, is given in Table 2. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 2}
$\boldsymbol { y }$ 0 1 2
$\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )$ $\frac { 2 } { 9 }$ $\frac { 5 } { 9 }$ $\frac { 2 } { 9 }$
\end{table}
1. Determine whether Joanne can expect to make a profit or a loss from playing the game once.
2. Hence calculate the expected size of this profit or loss after Joanne has played the game 100 times.
  (3 marks)

AQA S2 2010 January Q7

7 Jim , a mathematics teacher, knows that the marks, $X$, achieved by his students can be modelled by a normal distribution with unknown mean $\mu$ and unknown variance $\sigma ^ { 2 }$. Jim selects 12 students at random and from their marks he calculates that $\bar { x } = 64.8$ and $s ^ { 2 } = 93.0$.

1. An estimate for the standard error of the sample mean is $d$. Show that $d ^ { 2 } = 7.75$.
2. Construct an $80 \%$ confidence interval for $\mu$.
1. Write down a confidence interval for $\mu$, based on Jim's sample of 12 students, which has a width of 10 marks.
2. Determine the percentage confidence level for the interval found in part (b)(i).

AQA S2 2010 January Q8

8 The continuous random variable $X$ has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1
( x - 2 ) ^ { 2 } & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

Sketch the graph of f.
Calculate $\mathrm { P } ( X \leqslant 1 )$.
Show that $\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 5 }$.
1. Given that $\mathrm { E } ( X ) = \frac { 19 } { 24 }$ and that $\operatorname { Var } ( X ) = \frac { 499 } { k }$, find the numerical value of $k$.
2. Find $\mathrm { E } \left( 5 X ^ { 2 } + 24 X - 3 \right)$.
3. Find $\operatorname { Var } ( 12 X - 5 )$.

AQA S2 2011 January Q1

1 A factory produces bottles of brown sauce and bottles of tomato sauce.

The content, $Y$ grams, of a bottle of brown sauce is normally distributed with mean $\mu _ { Y }$ and variance 4. A quality control inspection found that the mean content, $\bar { y }$ grams, of a random sample of 16 bottles of brown sauce was 450 . Construct a $95 \%$ confidence interval for $\mu _ { Y }$.
The content, $X$ grams, of a bottle of tomato sauce is normally distributed with mean $\mu _ { X }$ and variance $\sigma ^ { 2 }$. A quality control inspection found that the content, $x$ grams, of a random sample of 9 bottles of tomato sauce was summarised by $$\sum x = 4950 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 334$$
1. Construct a 90\% confidence interval for $\mu _ { X }$.
2. Holly, the supervisor at the factory, claims that the mean content of a bottle of tomato sauce is 545 grams. Comment, with a justification, on Holly's claim. State the level of significance on which your conclusion is based.
  (3 marks)

AQA S2 2011 January Q2

2 It is claimed that the way in which students voted at a particular general election was independent of their gender. In order to investigate this claim, 480 male and 540 female students who voted at this general election were surveyed. These students may be regarded as a random sample. The percentages of males and females who voted for the different parties are recorded in the table.

	Conservative	Labour	Liberal Democrat	Other parties
Male	32.5	30	25	12.5
Female	40	25	20	15

Complete the contingency table below.
Hence determine, at the $1 \%$ level of significance, whether the way in which students voted at this general election was independent of their gender.
Conservative Labour Liberal Democrat Other parties Total
Male 480
Female 540
Total 1020

AQA S2 2011 January Q3

3 Lucy is the captain of her school's cricket team.
The number of catches, $X$, taken by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.6 . The number of run-outs, $Y$, effected by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.15 .

Find:
1. $\mathrm { P } ( X \leqslant 1 )$;
2. $\mathrm { P } ( X \leqslant 1$ and $Y \geqslant 1 )$.
State the assumption that you made in answering part (a)(ii).
During a particular season, Lucy plays in 16 cricket matches.
1. Calculate the probability that the number of catches taken by Lucy during this season is exactly 10 .
2. Determine the probability that the total number of catches taken and run-outs effected by Lucy during this season is at least 15 .

AQA S2 2011 January Q4

A red biased tetrahedral die is rolled. The number, $X$, on the face on which it lands has the probability distribution given by
$\boldsymbol { x }$ 1 2 3 4
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$ 0.2 0.1 0.4 0.3
1. Calculate $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
2. The red die is now rolled three times. The random variable $S$ is the sum of the three numbers obtained. Find $\mathrm { E } ( S )$ and $\operatorname { Var } ( S )$.
A blue biased tetrahedral die is rolled. The number, $Y$, on the face on which it lands has the probability distribution given by $$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3
\frac { 7 } { 10 } & y = 4 \end{cases}$$ The random variable $T$ is the value obtained when the number on the face on which it lands is multiplied by 3 . Calculate $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.
Calculate:
1. $\mathrm { P } ( X > 1 )$;
2. $\mathrm { P } ( X + T \leqslant 9$ and $X > 1 )$;
3. $\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )$.

AQA S2 2011 January Q5

5 In 2001, the mean height of students at the end of their final year at Bright Hope Secondary School was 165 centimetres. In 2010, David and James selected a random sample of 100 students who were at the end of their final year at this school. They recorded these students' heights, $x$ centimetres, and found that $\bar { x } = 167.1$ and $s ^ { 2 } = 101.2$. To investigate the claim that the mean height had increased since 2001, David and James each correctly conducted a hypothesis test. They used the same null hypothesis and the same alternative hypothesis. However, David used a $5 \%$ level of significance whilst James used a $1 \%$ level of significance.

1. Write down the null and alternative hypotheses that both David and James used.
  (l mark)
2. Determine the outcome of each of the two hypothesis tests, giving each conclusion in context.
3. State why both David and James made use of the Central Limit Theorem in their hypothesis tests.
It was later found that, in 2010, the mean height of students at the end of their final year at Bright Hope Secondary School was actually 165 centimetres. Giving a reason for your answer in each case, determine whether a Type I error or a Type II error or neither was made in the hypothesis test conducted by:
1. David;
2. James.

AQA S2 2011 January Q6

6 The continuous random variable $X$ has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} \frac { 3 } { 8 } x ^ { 2 } & 0 \leqslant x \leqslant \frac { 1 } { 2 }
\frac { 3 } { 32 } & \frac { 1 } { 2 } \leqslant x \leqslant 11
0 & \text { otherwise } \end{cases}$$

Sketch the graph of f.
Show that:
1. $\quad \mathrm { P } \left( X \geqslant 8 \frac { 1 } { 3 } \right) = \frac { 1 } { 4 }$;
2. $\quad \mathrm { P } ( X \geqslant 3 ) = \frac { 3 } { 4 }$.
Hence write down the exact value of:
1. the interquartile range of $X$;
2. the median, $m$, of $X$.
Find the exact value of $\mathrm { P } ( X < m \mid X \geqslant 3 )$.

AQA S2 2012 January Q1

1 Josephine accurately measures the widths of A4 sheets of paper and then rounds the widths to the nearest 0.1 cm . The rounding error, $X$ centimetres, follows a rectangular distribution. A randomly selected A4 sheet of paper is measured to be 21.1 cm in width.

Write down the limits between which the true width of this A4 sheet of paper lies.
(1 mark)
Write down the value of $\mathrm { E } ( X )$ and determine the exact value of the standard deviation of $X$.
Calculate $\mathrm { P } ( - 0.01 \leqslant X \leqslant 0.03 )$.

AQA S2 2012 January Q2

A particular bowling club has a large number of members. Their ages may be modelled by a normal random variable, $X$, with standard deviation 7.5 years. On 30 June 2010, Ted, the club secretary, concerned about the ageing membership, selected a random sample of 16 members and calculated their mean age to be 65.0 years.
1. Carry out a hypothesis test, at the $5 \%$ level of significance, to determine whether the mean age of the club's members has changed from its value of 61.4 years on 30 June 2000.
2. Comment on the likely number of members who were under the age of 25 on 30 June 2010, giving a numerical reason for your answer.
During 2011, in an attempt to encourage greater participation in the sport, the club ran a recruitment drive. After the recruitment drive, the ages of members of the bowling club may be modelled by a normal random variable, $Y$ years, with mean $\mu$ and standard deviation $\sigma$. The ages, $y$ years, of a random sample of 12 such members are summarised below. $$\sum y = 702 \quad \text { and } \quad \sum ( y - \bar { y } ) ^ { 2 } = 88.25$$
1. Construct a $90 \%$ confidence interval for $\mu$, giving the limits to one decimal place.
2. Use your confidence interval to state, with a reason, whether the recruitment drive lowered the average age of the club's members.

AQA S2 2012 January Q3

Table 1 contains the observed frequencies, $a , b , c$ and $d$, relating to the two attributes, $X$ and $Y$, required to perform a $\chi ^ { 2 }$ test. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 1}
\cline { 2 - 4 } \multicolumn{1}{c|}{} $\boldsymbol { Y }$ Not $\boldsymbol { Y }$ Total
$\boldsymbol { X }$ $a$ $b$ $m$
Not $\boldsymbol { X }$ $c$ $d$ $n$
Total $p$ $q$ $N$
\end{table}
1. Write down, in terms of $m , n , p , q$ and $N$, expressions for the 4 expected frequencies corresponding to $a , b , c$ and $d$.
2. Hence prove that the sum of the expected frequencies is $N$.
Andy, a tennis player, wishes to investigate the possible effect of wind conditions on the results of his matches. The results of his matches for the 2011 season are represented in Table 2. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 2}
\cline { 2 - 4 } \multicolumn{1}{c|}{} Windy Not windy Total
Won 15 18 33
Lost 12 5 17
Total 27 23 50
\end{table} Conduct a $\chi ^ { 2 }$ test, at the $10 \%$ level of significance, to investigate whether there is an association between Andy's results and wind conditions.
(8 marks)

AQA S2 2012 January Q4

A discrete random variable $X$ has a probability function defined by $$\mathrm { P } ( X = x ) = \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } \quad \text { for } x = 0,1,2,3,4 , \ldots \ldots$$
1. State the name of the distribution of $X$.
2. Write down, in terms of $\lambda$, expressions for $\mathrm { E } ( 3 X - 1 )$ and $\operatorname { Var } ( 3 X - 1 )$.
3. Write down an expression for $\mathrm { P } ( X = x + 1 )$, and hence show that $$\mathrm { P } ( X = x + 1 ) = \frac { \lambda } { x + 1 } \mathrm { P } ( X = x )$$
The number of cars and the number of coaches passing a certain road junction may be modelled by independent Poisson distributions.
1. On a winter morning, an average of 500 cars per hour and an average of 10 coaches per hour pass this junction. Determine the probability that a total of at least 10 such vehicles pass this junction during a particular 1 -minute interval on a winter morning.
2. On a summer morning, an average of 836 cars per hour and an average of 22 coaches per hour pass this junction. Calculate the probability that a total of at most 3 such vehicles pass this junction during a particular 1 -minute interval on a summer morning. Give your answer to two significant figures.
  (3 marks)

AQA S2 2012 January Q5

Joshua plays a game in which he repeatedly tosses an unbiased coin. His game concludes when he obtains either a head or 5 tails in succession. The random variable $N$ denotes the number of tosses of his coin required to conclude a game. By completing Table 3 below, calculate $\mathrm { E } ( N )$.
Joshua's sister, Ruth, plays a separate game in which she repeatedly tosses a coin that is biased in such a way that the probability of a head in a single toss of her coin is $\frac { 1 } { 4 }$. Her game also concludes when she obtains either a head or 5 tails in succession. The random variable $M$ denotes the number of tosses of her coin required to conclude her game. Complete Table 4 below.
1. Joshua and Ruth play their games simultaneously. Calculate the probability that Joshua and Ruth will conclude their games in an equal number of tosses of their coins.
2. Joshua and Ruth play their games simultaneously on 3 occasions. Calculate the probability that, on at least 2 of these occasions, their games will be concluded in an equal number of tosses of their coins. Give your answer to three decimal places.
  (4 marks) \begin{table}[h]
  \captionsetup{labelformat=empty} \caption{Table 3}
  $\boldsymbol { n }$ 1 2 3 4 5
  $\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )$ $\frac { 1 } { 8 }$ $\frac { 1 } { 16 }$
  \end{table} \begin{table}[h]
  \captionsetup{labelformat=empty} \caption{Table 4}
  $\boldsymbol { m }$ 1 2 3 4 5
  $\mathbf { P } ( \boldsymbol { M } = \boldsymbol { m } )$ $\frac { 1 } { 4 }$ $\frac { 3 } { 16 }$
  \end{table}

AQA S2 2012 January Q6

6 The random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( x + 7 ) & 1 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$

Sketch the graph of f.
Find the exact value of $\mathrm { E } ( X )$.
Prove that the distribution function F , for $1 \leqslant x \leqslant 5$, is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 80 } ( x + 15 ) ( x - 1 )$$
Hence, or otherwise:
1. find $\mathrm { P } ( 2.5 \leqslant X \leqslant 4.5 )$;
2. show that the median, $m$, of $X$ satisfies the equation $m ^ { 2 } + 14 m - 55 = 0$.
Calculate the value of the median of $X$, giving your answer to three decimal places.

AQA S2 2013 January Q1

1 Dimitra is an athlete who competes in 400 m races. The times, in seconds, for her first six races of the 2012 season were $$\begin{array} { l l l l l l } 54.86 & 53.09 & 53.75 & 52.88 & 51.97 & 51.81 \end{array}$$

Assuming that these data form a random sample from a normal distribution, construct a $95 \%$ confidence interval for the mean time of Dimitra's races in the 2012 season, giving the limits to two decimal places.
For the 2011 season, Dimitra's mean time for her races was 53.41 seconds. After her first six races of the 2012 season, her coach claimed that the data showed that she would be more successful in races during the 2012 season than during the 2011 season. Make two comments about the coach's claim.

AQA S2 2013 January Q2

2 A large estate agency would like all the properties that it handles to be sold within three months. A manager wants to know whether the type of property affects the time taken to sell it. The data for a random sample of properties sold are tabulated below.

\multirow{2}{*}{}	Type of property
	Flat	Terraced	Semidetached	Detached	Total
Sold within three months	4	34	28	18	84
Sold in more than three months	9	18	8	6	41
Total	13	52	36	24	125

Conduct a $\chi ^ { 2 }$-test, at the $10 \%$ level of significance, to determine whether there is an association between the type of property and the time taken to sell it. Explain why it is necessary to combine two columns before carrying out this test.
The manager plans to spend extra money on advertising for one type of property in an attempt to increase the number sold within three months. Explain why the manager might choose:
1. terraced properties;
2. flats.
  (2 marks)

AQA S2 2013 January Q3

3 A large office block is busy during the five weekdays, Monday to Friday, and less busy during the two weekend days, Saturday and Sunday. The block is illuminated by fluorescent light tubes which frequently fail and must be replaced with new tubes by John, the caretaker. The number of fluorescent tubes that fail on a particular weekday can be modelled by a Poisson distribution with mean 1.5. The number of fluorescent tubes that fail on a particular weekend day can be modelled by a Poisson distribution with mean 0.5 .

Find the probability that:
1. on one particular Monday, exactly 3 fluorescent light tubes fail;
2. during the two days of a weekend, more than 1 fluorescent light tube fails;
3. during a complete seven-day week, fewer than 10 fluorescent light tubes fail.
John keeps a supply of new fluorescent light tubes. More new tubes are delivered every Monday morning to replace those that he has used during the previous week. John wants the probability that he runs out of new tubes before the next Monday morning to be less than 1 per cent. Find the minimum number of new tubes that he should have available on a Monday morning.
Give a reason why a Poisson distribution with mean 0.375 is unlikely to provide a satisfactory model for the number of fluorescent light tubes that fail between 1 am and 7 am on a weekday.

AQA S2 2013 January Q4

4 A continuous random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 3
9 k & 3 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

Sketch the graph of f.
Show that the value of $k$ is $\frac { 1 } { 18 }$.
1. Write down the median value of $X$.
2. Calculate the value of the lower quartile of $X$.

AQA S2 2013 January Q5

5 Aiden takes his car to a garage for its MOT test. The probability that his car will need to have $X$ tyres replaced is shown in the table.

$\boldsymbol { x }$	0	1	2	3	4
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.1	0.35	0.25	0.2	0.1

Show that the mean of $X$ is 1.85 and calculate the variance of $X$.
The charge for the MOT test is $\pounds c$ and the cost of each new tyre is $\pounds n$. The total amount that Aiden must pay the garage is $\pounds T$.
1. Express $T$ in terms of $c , n$ and $X$.
2. Hence, using your results from part (a), find expressions for $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.

AQA S2 2013 January Q6

6 The time, in weeks, that a patient must wait to be given an appointment in Holmsoon Hospital may be modelled by a random variable $T$ having the cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0
\frac { t ^ { 3 } } { 216 } & 0 \leqslant t \leqslant 6
1 & t > 6 \end{cases}$$

Find, to the nearest day, the time within which 90 per cent of patients will have been given an appointment.
Find the probability density function of $T$ for all values of $t$.
Calculate the mean and the variance of $T$.
Calculate the probability that the time that a patient must wait to be given an appointment is more than one standard deviation above the mean.

AQA S2 2013 January Q7

7 A factory produces 3-litre bottles of mineral water. The volume of water in a bottle has previously had a mean value of 3020 millilitres. Following a stoppage for maintenance, the volume of water, $x$ millilitres, in each of a random sample of 100 bottles is measured and the following data obtained, where $y = x - 3000$. $$\sum y = 1847.0 \quad \sum ( y - \bar { y } ) ^ { 2 } = 6336.00$$

Carry out a hypothesis test, at the $5 \%$ significance level, to investigate whether the mean volume of water in a bottle has changed.
(8 marks)
Subsequent measurements establish that the mean volume of water in a bottle produced by the factory after the stoppage is 3020 millilitres. State whether a Type I error, a Type II error or no error was made when carrying out the test in part (a).
(l mark)

\(\boldsymbol { x }\)	0	1	2	3	4	5
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	\(\frac { 1 } { 252 }\)	\(\frac { 25 } { 252 }\)	\(\frac { 100 } { 252 }\)	\(a\)	\(\frac { 25 } { 252 }\)	\(\frac { 1 } { 252 }\)

\(\boldsymbol { y }\)	0	1	2
\(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	\(\frac { 2 } { 9 }\)	\(\frac { 5 } { 9 }\)	\(\frac { 2 } { 9 }\)

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(\boldsymbol { Y }\)	Not \(\boldsymbol { Y }\)	Total
\(\boldsymbol { X }\)	\(a\)	\(b\)	\(m\)
Not \(\boldsymbol { X }\)	\(c\)	\(d\)	\(n\)
Total	\(p\)	\(q\)	\(N\)

\(\boldsymbol { n }\)	1	2	3	4	5
\(\mathbf { P } ( \boldsymbol { N } = \boldsymbol { n } )\)			\(\frac { 1 } { 8 }\)		\(\frac { 1 } { 16 }\)

\(\boldsymbol { m }\)	1	2	3	4	5
\(\mathbf { P } ( \boldsymbol { M } = \boldsymbol { m } )\)	\(\frac { 1 } { 4 }\)	\(\frac { 3 } { 16 }\)

Age	Pass	Fail
\(\mathbf { 1 7 } - \mathbf { 1 8 }\)	28	20
\(\mathbf { 1 9 } - \mathbf { 3 0 }\)	2	14
\(\mathbf { 3 1 } - \mathbf { 3 9 }\)	12	33
\(\mathbf { 4 0 } - \mathbf { 6 0 }\)	6	5

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Windy	Not windy	Total
Won	15	18	33
Lost	12	5	17
Total	27	23	50

Questions — AQA S2 (139 questions)

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