Questions - AQA S2 - A-Level Maths

$\boldsymbol { x }$	2 or fewer	3	4	5	6	7	8 or more
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0	0.1	0.2	$a$	0.3	$b$	0

Two pods are picked randomly from the box. Find the probability that the number of peas in each pod is at most 4.
It is given that $\mathrm { E } ( X ) = 5.1$.
1. Determine the values of $a$ and $b$.
2. Hence show that $\operatorname { Var } ( X ) = 1.29$.
3. Some children play a game with the pods, randomly picking a pod and scoring points depending on the number of peas in the pod. For each pod picked, the number of points scored, $N$, is found by doubling the number of peas in the pod and then subtracting 5. Find the mean and the standard deviation of $N$.
  [0pt] [3 marks]

AQA S2 2014 June Q4

4 A continuous random variable $X$ has a probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b
0 & \text { otherwise } \end{cases}$$ where $b > a > 0$.

1. Prove that $k = b - a$.
2. Write down the value of $\mathrm { E } ( X )$.
3. Show, by integration, that $\mathrm { E } \left( X ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)$.
4. Hence derive a simplified formula for $\operatorname { Var } ( X )$.
Given that $a = 4$ and $\operatorname { Var } ( X ) = 3$, find the numerical value of $\mathrm { E } ( X )$.
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AQA S2 2014 June Q5

3 marks

5 Peter, a geologist, is studying pebbles on a beach. He uses a frame, called a quadrat, to enclose an area of the beach. He then counts the number of quartz pebbles, $X$, within the quadrat. He repeats this procedure 40 times, obtaining the following summarised data. $$\sum x = 128 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 126.4$$ Peter believes that the distribution of $X$ can be modelled by a Poisson distribution with $\lambda = 3.2$.

Use the summarised data to support Peter's belief.
Using Peter's model, calculate the probability that:
1. a single placing of the quadrat contains more than 5 quartz pebbles;
2. a single placing of the quadrat contains at least 3 quartz pebbles but fewer than 8 quartz pebbles;
3. when the quadrat is placed twice, at least one placing contains no quartz pebbles.
Peter also models the number of flint pebbles enclosed by the quadrat by a Poisson distribution with mean 5 . He assumes that the number of flint pebbles enclosed by the quadrat is independent of the number of quartz pebbles enclosed by the quadrat. Using Peter's models, calculate the probability that a single placing of the quadrat contains a total of either 9 or 10 pebbles which are quartz or flint.
[0pt] [3 marks]

AQA S2 2014 June Q6

15 marks

6 South Riding Alarms (SRA) maintains household burglar-alarm systems. The company aims to carry out an annual service of a system in a mean time of 20 minutes.
Technicians who carry out an annual service must record the times at which they start and finish the service.

Gary is employed as a technician by SRA and his manager, Rajul, calculates the times taken for 8 annual services carried out by Gary. The results, in minutes, are as follows. $$\begin{array} { l l l l l l l l } 24 & 25 & 29 & 16 & 18 & 27 & 19 & 23 \end{array}$$ Assume that these times may be regarded as a random sample from a normal distribution. Carry out a hypothesis test, at the $10 \%$ significance level, to examine whether the mean time for an annual service carried out by Gary is 20 minutes.
[0pt] [8 marks]
Rajul suspects that Gary may be taking longer than 20 minutes on average to carry out an annual service. Rajul therefore calculates the times taken for 100 annual services carried out by Gary. Assume that these times may also be regarded as a random sample from a normal distribution but with a standard deviation of 4.6 minutes. Find the highest value of the sample mean which would not support Rajul's suspicion at the $5 \%$ significance level. Give your answer to two decimal places.
[0pt] [4 marks]
$7 \quad$ A continuous random variable $X$ has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 4 } { 5 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 20 } ( x - 3 ) ( 3 x - 11 ) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
Find $\mathrm { P } ( X < 1 )$.
1. Show that, for $1 \leqslant x \leqslant 3$, the cumulative distribution function, $\mathrm { F } ( x )$, is given by $$\mathrm { F } ( x ) = \frac { 1 } { 20 } \left( x ^ { 3 } - 10 x ^ { 2 } + 33 x - 16 \right)$$
2. Hence verify that the median value of $X$ lies between 1.13 and 1.14 .
  [0pt] [3 marks] QUESTION
  PART Answer space for question 7
  REFERENCE REFERENCE
  \includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-20_2290_1707_221_153}

AQA S2 2015 June Q1

4 marks

1 In a survey of the tideline along a beach, plastic bottles were found at a constant average rate of 280 per kilometre, and drinks cans were found at a constant average rate of 220 per kilometre. It may be assumed that these objects were distributed randomly and independently. Calculate the probability that:

a 10 m length of tideline along this beach contains no more than 5 plastic bottles;
a 20 m length of tideline along this beach contains exactly 2 drinks cans;
a 30 m length of tideline along this beach contains a total of at least 12 but fewer than 18 of these two types of object.
[0pt] [4 marks]

AQA S2 2015 June Q2

3 marks

2 The continuous random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b
0 & \text { otherwise } \end{cases}$$

Write down, in terms of $a$ and $b$, the value of $k$.
1. Given that $\mathrm { E } ( X ) = 1$ and $\operatorname { Var } ( X ) = 3$, find the values of $a$ and $b$.
2. Four independent values of $X$ are taken. Find the probability that exactly one of these four values is negative.
  [0pt] [3 marks]

AQA S2 2015 June Q3

2 marks

3 A machine fills bags with frozen peas. Measurements taken over several weeks have shown that the standard deviation of the weights of the filled bags of peas has been 2.2 grams. Following maintenance on the machine, a quality control inspector selected 8 bags of peas. The weights, in grams, of the bags were $$\begin{array} { l l l l l l l l } 910.4 & 908.7 & 907.2 & 913.2 & 905.6 & 911.1 & 909.5 & 907.9 \end{array}$$ It may be assumed that the bags constitute a random sample from a normal distribution.

Giving the limits to four significant figures, calculate a 95\% confidence interval for the mean weight of a bag of frozen peas filled by the machine following the maintenance:
1. assuming that the standard deviation of the weights of the bags of peas is known to be 2.2 grams;
2. assuming that the standard deviation of the weights of the bags of peas may no longer be 2.2 grams.
The weight printed on the bags of peas is 907 grams. One of the inspector's concerns is that bags should not be underweight. Make two comments about this concern with regard to the data and your calculated confidence intervals.
[0pt] [2 marks]

AQA S2 2015 June Q4

2 marks

4 Wellgrove village has a main road running through it that has a 40 mph speed limit. The villagers were concerned that many vehicles travelled too fast through the village, and so they set up a device for measuring the speed of vehicles on this main road. This device indicated that the mean speed of vehicles travelling through Wellgrove was 44.1 mph . In an attempt to reduce the mean speed of vehicles travelling through Wellgrove, life-size photographs of a police officer were erected next to the road on the approaches to the village. The speed, $X \mathrm { mph }$, of a sample of 100 vehicles was then measured and the following data obtained. $$\sum x = 4327.0 \quad \sum ( x - \bar { x } ) ^ { 2 } = 925.71$$

State an assumption that must be made about the sample in order to carry out a hypothesis test to investigate whether the desired reduction in mean speed had occurred.
Given that the assumption that you stated in part (a) is valid, carry out such a test, using the $1 \%$ level of significance.
Explain, in the context of this question, the meaning of:
1. a Type I error;
2. a Type II error.
  [0pt] [2 marks]

AQA S2 2015 June Q5

1 marks

5 In a particular town, a survey was conducted on a sample of 200 residents aged 41 years to 50 years. The survey questioned these residents to discover the age at which they had left full-time education and the greatest rate of income tax that they were paying at the time of the survey. The summarised data obtained from the survey are shown in the table.

\multirow{2}{*}{Greatest rate of income tax paid}	Age when leaving education (years)			\multirow[b]{2}{*}{Total}
	16 or less	17 or 18	19 or more
Zero	32	3	4	39
Basic	102	12	17	131
Higher	17	5	8	30
Total	151	20	29	200

Use a $\chi ^ { 2 }$-test, at the $5 \%$ level of significance, to investigate whether there is an association between age when leaving education and greatest rate of income tax paid.
It is believed that residents of this town who had left education at a later age were more likely to be paying the higher rate of income tax. Comment on this belief.
[0pt] [1 mark]

AQA S2 2015 June Q6

6 The continuous random variable $X$ has the cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 2 } x - \frac { 1 } { 16 } x ^ { 2 } & 0 \leqslant x \leqslant 4
1 & x > 4 \end{cases}$$

Find the probability that $X$ lies between 0.4 and 0.8 .
Show that the probability density function, $\mathrm { f } ( x )$, of $X$ is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 8 } x & 0 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
1. Find the value of $\mathrm { E } ( X )$.
2. Show that $\operatorname { Var } ( X ) = \frac { 8 } { 9 }$.
The continuous random variable $Y$ is defined by $$Y = 3 X - 2$$ Find the values of $\mathrm { E } ( Y )$ and $\operatorname { Var } ( Y )$.

AQA S2 2015 June Q7

5 marks

7 Each week, a newsagent stocks 5 copies of the magazine Statistics Weekly. A regular customer always buys one copy. The demand for additional copies may be modelled by a Poisson distribution with mean 2. The number of copies sold in a week, $X$, has the probability distribution shown in the table, where probabilities are stated correct to three decimal places.

$\boldsymbol { x }$	1	2	3	4	5
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.135	0.271	0.271	$a$	$b$

Show that, correct to three decimal places, the values of $a$ and $b$ are 0.180 and 0.143 respectively.
Find the values of $\mathrm { E } ( X )$ and $\mathrm { E } \left( X ^ { 2 } \right)$, showing the calculations needed to obtain these values, and hence calculate the standard deviation of $X$.
The newsagent makes a profit of $\pounds 1$ on each copy of Statistics Weekly that is sold and loses 50 p on each copy that is not sold. Find the mean weekly profit for the newsagent from sales of this magazine.
Assuming that the weekly demand remains the same, show that the mean weekly profit from sales of Statistics Weekly will be greater if the newsagent stocks only 4 copies.
[0pt] [5 marks]
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AQA S2 2016 June Q1

7 marks

1 The water in a pond contains three different species of a spherical green algae:
Volvox globator, at an average rate of 4.5 spheres per $1 \mathrm {~cm} ^ { 3 }$;
Volvox aureus, at an average rate of 2.3 spheres per $1 \mathrm {~cm} ^ { 3 }$;
Volvox tertius, at an average rate of 1.2 spheres per $1 \mathrm {~cm} ^ { 3 }$.
Individual Volvox spheres may be considered to occur randomly and independently of all other Volvox spheres. Random samples of water are collected from this pond.
Find the probability that:

a $1 \mathrm {~cm} ^ { 3 }$ sample contains no more than 5 Volvox globator spheres;
a $1 \mathrm {~cm} ^ { 3 }$ sample contains at least 2 Volvox aureus spheres;
a $5 \mathrm {~cm} ^ { 3 }$ sample contains more than 8 but fewer than 12 Volvox tertius spheres;
a $0.1 \mathrm {~cm} ^ { 3 }$ sample contains a total of exactly 2 Volvox spheres;
a $1 \mathrm {~cm} ^ { 3 }$ sample contains at least 1 sphere of each of the three different species of algae.
[0pt] [3 marks]

AQA S2 2016 June Q3

3 Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, $X$, follows the distribution shown in the table.

$\boldsymbol { x }$	0	1	2	3	4	5	6
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0	0.19	0.26	0.20	0.13	0.07	0.15

Find the probability that a member borrows more than 3 books.
Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books.
Show that the mean of $X$ is 3.08, and calculate the variance of $X$.
One of the library staff notices that the values of the mean and the variance of $X$ are similar and suggests that a Poisson distribution could be used to model $X$. Without further calculations, give two reasons why a Poisson distribution would not be suitable to model $X$.
The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
1. the mean amount that will be paid by a member;
2. the standard deviation of the amount that will be paid by a member.

AQA S2 2016 June Q4

4 A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2 . The error, $X ^ { \circ } \mathrm { C }$, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \left\{ \begin{array} { l c } k & 0 \leqslant x \leqslant 0.1
0 & \text { otherwise } \end{array} \right.$$

State the value of $k$.
Find the probability that the error resulting from this rounding down is greater than $0.03 ^ { \circ } \mathrm { C }$.
1. State the value for $\mathrm { E } ( X )$.
2. Use integration to find the value for $\mathrm { E } \left( X ^ { 2 } \right)$.
3. Hence find the value for the standard deviation of $X$.
  \includegraphics[max width=\textwidth, alt={}]{72aa9867-88c6-4b1b-97f7-bf4ba2da4031-12_1355_1707_1352_153}

AQA S2 2016 June Q5

2 marks

5 A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.

	Problems during first 3 months	Problems during first year but after first 3 months	No problems during first year	Total
Petrol engine	10	35	170	215
Diesel engine	4	8	23	35
Total	14	43	193	250

Use a $\chi ^ { 2 }$-test to investigate, at the $10 \%$ significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine.
Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems. Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa.
[0pt] [2 marks]

AQA S2 2016 June Q6

2 marks

6 Gerald is a scientist who studies sand lizards. He believes that sand lizards on islands are, on average, shorter than those on the mainland. The population of sand lizards on the mainland has a mean length of 18.2 cm and a standard deviation of 1.8 cm . Gerald visited three islands, $\mathrm { A } , \mathrm { B }$ and C , and measured the length, $X$ centimetres, of each of a sample of $n$ sand lizards on each island. The samples may be regarded as random. The data are shown in the table.

AQA S2 2016 June Q7

7 The continuous random variable $X$ has a cumulative distribution function $\mathrm { F } ( x )$, where $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1
\frac { 1 } { 4 } ( x - 1 ) & 1 \leqslant x < 4
\frac { 1 } { 16 } \left( 12 x - x ^ { 2 } - 20 \right) & 4 \leqslant x \leqslant 6
1 & x > 6 \end{array} \right.$$

Sketch the probability density function, $\mathrm { f } ( x )$, on the grid below.
Find the mean value of $X$.

AQA S2 2011 June Q3

State the null hypothesis that Emily used.
Find the value of the test statistic, $X ^ { 2 }$, giving your answer to one decimal place.
State, in context, the conclusion that Emily should reach based on the results of her $\chi ^ { 2 }$ test.
Make one comment on the GCSE performances of 16-year-old students attending Bailey Language School.
Emily's friend, Joanna, used the same data to correctly conduct a $\chi ^ { 2 }$ test using the $10 \%$ level of significance. State, with justification, the conclusion that Joanna should reach.

AQA S2 2009 January Q1

1 Fortune High School gave its students a wider choice of subjects to study. The table shows the number of students, of each gender, who chose to study each of the additional subjects during the school year 2007/08.

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

Assuming that these data form a random sample, use a $\chi ^ { 2 }$ test, at the $10 \%$ level of significance, to test whether the choice of these subjects is independent of gender.
(11 marks)

AQA S2 2009 January Q2

2 A group of estate agents in a particular area claimed that, after the introduction of a new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area had not changed from 8 weeks.

A random sample of 9 house purchases in the area revealed that their completion times, in weeks, were as follows. $$\begin{array} { l l l l l l l l l } 6 & 7 & 10 & 12 & 9 & 11 & 7 & 8 & 14 \end{array}$$ Assuming that completion times in the area are normally distributed with standard deviation 2.5 weeks, test, at the $5 \%$ level of significance, the group's claim. (7 marks)
It was subsequently discovered that, after the introduction of the new search procedure at the Land Registry, the mean completion time for the purchase of a house in the area remained at 8 weeks. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out your hypothesis test in part (a). Give a reason for your answer.
(2 marks)

AQA S2 2009 January Q3

3 Joe owns two garages, Acefit and Bestjob, each specialising in the fitting of the latest satellite navigation device. The daily demand, $X$, for the device at Acefit garage may be modelled by a Poisson distribution with mean 3.6. The daily demand, $Y$, for the device at Bestjob garage may be modelled by a Poisson distribution with mean 4.4.

Calculate:
1. $\mathrm { P } ( X \leqslant 3 )$;
2. $\quad \mathrm { P } ( Y = 5 )$.
The total daily demand for the device at Joe's two garages is denoted by $T$.
1. Write down the distribution of $T$, stating any assumption that you make.
2. Determine $\mathrm { P } ( 6 < T < 12 )$.
3. Calculate the probability that the total demand for the device will exceed 14 on each of two consecutive days. Give your answer to one significant figure.
4. Determine the minimum number of devices that Joe should have in stock if he is to meet his total demand on at least $99 \%$ of days.

AQA S2 2009 January Q4

4 The continuous random variable $X$ has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - c
\frac { x + c } { 4 c } & - c \leqslant x \leqslant 3 c
1 & x > 3 c \end{array} \right.$$ where $c$ is a positive constant.

Determine $\mathrm { P } \left( - \frac { 3 c } { 4 } < X < \frac { 3 c } { 4 } \right)$.
Show that the probability density function, $\mathrm { f } ( x )$, of $X$ is $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 c } & - c \leqslant x \leqslant 3 c
0 & \text { otherwise } \end{array} \right.$$
Hence, or otherwise, find expressions, in terms of $c$, for:
1. $\mathrm { E } ( X )$;
2. $\operatorname { Var } ( X )$.

AQA S2 2009 January Q5

5 Jane, who supplies fruit to a jam manufacturer, knows that the weight of fruit in boxes that she sends to the manufacturer can be modelled by a normal distribution with unknown mean, $\mu$ grams, and unknown standard deviation, $\sigma$ grams. Jane selects a random sample of 16 boxes and, using the $t$-distribution, calculates correctly that a $98 \%$ confidence interval for $\mu$ is $( 70.65,80.35 )$.

1. Show that the sample mean is 75.5 grams.
2. Find the width of the confidence interval.
3. Calculate an estimate of the standard error of the mean.
4. Hence, or otherwise, show that an unbiased estimate of $\sigma ^ { 2 }$ is 55.6 , correct to three significant figures.
Jane decides that the width of the $98 \%$ confidence interval is too large. Construct a $95 \%$ confidence interval for $\mu$, based on her sample of 16 boxes.
Jane is informed that the manufacturer would prefer the confidence interval to have a width of at most 5 grams.
1. Write down a confidence interval for $\mu$, again based on Jane's sample of 16 boxes, which has a width of 5 grams.
2. Determine the percentage confidence level for your interval in part (c)(i).

AQA S2 2009 January Q6

6 A small supermarket has a total of four checkouts, at least one of which is always staffed. The probability distribution for $R$, the number of checkouts that are staffed at any given time, is $$\mathrm { P } ( R = r ) = \left\{ \begin{array} { c l } \frac { 2 } { 3 } \left( \frac { 1 } { 3 } \right) ^ { r - 1 } & r = 1,2,3
k & r = 4 \end{array} \right.$$

Show that $k = \frac { 1 } { 27 }$.
Find the probability that, at any given time, there will be at least 3 checkouts that are staffed.
It is suggested that the total number of customers, $C$, that can be served at the checkouts per hour may be modelled by $$C = 27 R + 5$$ Find:
1. $\mathrm { E } ( C )$;
2. the standard deviation of $C$.

AQA S2 2009 January Q7

7 The continuous random variable $X$ has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 16 } x ^ { 3 } & 0 \leqslant x \leqslant 2
\frac { 1 } { 6 } ( 5 - x ) & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{array} \right.$$

Sketch the graph of f.
Prove that the cumulative distribution function of $X$ for $2 \leqslant x \leqslant 5$ can be written in the form $$\mathrm { F } ( x ) = 1 - \frac { 1 } { 12 } ( 5 - x ) ^ { 2 }$$
Hence, or otherwise, determine $\mathrm { P } ( X \geqslant 3 \mid X \leqslant 4 )$.

\(\boldsymbol { x }\)	1	2	3	4	5
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.135	0.271	0.271	\(a\)	\(b\)

Questions — AQA S2 (139 questions)

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