Questions - AQA S2 - A-Level Maths

1. Calculate $\mathrm { P } ( X = 2 )$.
2. Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.
1. Write down the distribution of $Y$, the number of patients contracting this superbug in a given 6-month period.
2. Find the probability that at least 12 patients will contract this superbug during a given 6-month period.
State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.

AQA S2 2006 January Q2

2 Year 12 students at Newstatus School choose to participate in one of four sports during the Spring term. The students' choices are summarised in the table.

	Squash	Badminton	Archery	Hockey	Total
Male	5	16	30	19	70
Female	4	20	33	53	110
Total	9	36	63	72	180

Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to determine whether the choice of sport is independent of gender.
Interpret your result in part (a) as it relates to students choosing hockey.

AQA S2 2006 January Q3

3 The time, $T$ minutes, that parents have to wait before seeing a mathematics teacher at a school parents' evening can be modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$. At a recent parents' evening, a random sample of 9 parents was asked to record the times that they waited before seeing a mathematics teacher. The times, in minutes, are $$\begin{array} { l l l l l l l l l } 5 & 12 & 10 & 8 & 7 & 6 & 9 & 7 & 8 \end{array}$$

Construct a $90 \%$ confidence interval for $\mu$.
Comment on the headteacher's claim that the mean time that parents have to wait before seeing a mathematics teacher is 5 minutes.

AQA S2 2006 January Q4

A random variable $X$ has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} k & a < x < b
0 & \text { otherwise } \end{cases}$$
1. Show that $k = \frac { 1 } { b - a }$.
2. Prove, using integration, that $\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )$.
The error, $X$ grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: $$f ( x ) = \begin{cases} k & - 2 < x < 4
0 & \text { otherwise } \end{cases}$$
1. Write down the value of the mean, $\mu$, of $X$.
2. Evaluate the standard deviation, $\sigma$, of $X$.
3. Hence find $\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)$.

AQA S2 2006 January Q5

5 The Globe Express agency organises trips to the theatre. The cost, $\pounds X$, of these trips can be modelled by the following probability distribution:

$\boldsymbol { x }$	40	45	55	74
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.30	0.24	0.36	0.10

Calculate the mean and standard deviation of $X$.
For special celebrity charity performances, Globe Express increases the cost of the trips to $\pounds Y$, where $$Y = 10 X + 250$$ Determine the mean and standard deviation of $Y$.

AQA S2 2006 January Q6

6 In previous years, the marks obtained in a French test by students attending Topnotch College have been modelled satisfactorily by a normal distribution with a mean of 65 and a standard deviation of 9 . Teachers in the French department at Topnotch College suspect that this year their students are, on average, underachieving. In order to investigate this suspicion, the teachers selected a random sample of 35 students to take the French test and found that their mean score was 61.5.

Investigate, at the $5 \%$ level of significance, the teachers' suspicion.
Explain, in the context of this question, the meaning of a Type I error.

AQA S2 2006 January Q7

7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, $T$ hours, is modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1
0 & \text { otherwise } \end{cases}$$

Show that $\mathrm { E } ( T ) = \frac { 8 } { 15 }$.
1. Find the cumulative distribution function, $\mathrm { F } ( t )$, for $0 \leqslant t \leqslant 1$.
2. Hence, or otherwise, for a commuter selected at random, find $$\mathrm { P } ( \text { mean } < T < \text { median } )$$

AQA S2 2006 January Q8

8 Bottles of sherry nominally contain 1000 millilitres. After the introduction of a new method of filling the bottles, there is a suspicion that the mean volume of sherry in a bottle has changed. In order to investigate this suspicion, a random sample of 12 bottles of sherry is taken and the volume of sherry in each bottle is measured. The volumes, in millilitres, of sherry in these bottles are found to be

996	1006	1009	999	1007	1003
998	1010	997	996	1008	1007

Assuming that the volume of sherry in a bottle is normally distributed, investigate, at the $5 \%$ level of significance, whether the mean volume of sherry in a bottle differs from 1000 millilitres.

AQA S2 2007 January Q1

1 Alan's journey time, in minutes, to travel home from work each day is known to be normally distributed with mean $\mu$. Alan records his journey time, in minutes, on a random sample of 8 days as being $$\begin{array} { l l l l l l l l } 36 & 38 & 39 & 40 & 50 & 35 & 36 & 42 \end{array}$$ Construct a $95 \%$ confidence interval for $\mu$.

AQA S2 2007 January Q2

2 The number of computers, $A$, bought during one day from the Amplebuy computer store can be modelled by a Poisson distribution with a mean of 3.5. The number of computers, $B$, bought during one day from the Bestbuy computer store can be modelled by a Poisson distribution with a mean of 5.0 .

1. Calculate $\mathrm { P } ( A = 4 )$.
2. Determine $\mathrm { P } ( B \leqslant 6 )$.
3. Find the probability that a total of fewer than 10 computers is bought from these two stores on one particular day.
Calculate the probability that a total of fewer than 10 computers is bought from these two stores on at least 4 out of 5 consecutive days.
The numbers of computers bought from the Choicebuy computer store over a 10-day period are recorded as $$\begin{array} { l l l l l l l l l l } 8 & 12 & 6 & 6 & 9 & 15 & 10 & 8 & 6 & 12 \end{array}$$
1. Calculate the mean and variance of these data.
2. State, giving a reason based on your results in part (c)(i), whether or not a Poisson distribution provides a suitable model for these data.

AQA S2 2007 January Q3

3 The handicap committee of a golf club has indicated that the mean score achieved by the club's members in the past was 85.9 . A group of members believes that recent changes to the golf course have led to a change in the mean score achieved by the club's members and decides to investigate this belief. A random sample of the scores, $x$, of 100 club members was taken and is summarised by $$\sum x = 8350 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 15321$$ where $\bar { x }$ denotes the sample mean.
Test, at the $5 \%$ level of significance, the group's belief that the mean score of 85.9 has changed.

AQA S2 2007 January Q4

4 The number of fish, $X$, caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution:

$\boldsymbol { x }$	1	2	3	4	5	6	$\geqslant 7$
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.01	0.05	0.14	0.30	$k$	0.12	0

Find the value of $k$.
Find:
1. $\mathrm { E } ( X )$;
2. $\operatorname { Var } ( X )$.
When Pearl sells her fish, she earns a profit, in pounds, given by $$Y = 5 X + 2$$ Find:
1. $\mathrm { E } ( Y )$;
2. the standard deviation of $Y$.

AQA S2 2007 January Q5

5 Jasmine's French teacher states that a homework assignment should take, on average, 30 minutes to complete. Jasmine believes that he is understating the mean time that the assignment takes to complete and so decides to investigate. She records the times, in minutes, that it takes for a random sample of 10 students to complete the French assignment, with the following results: $$\begin{array} { l l l l l l l l l l } 29 & 33 & 36 & 42 & 30 & 28 & 31 & 34 & 37 & 35 \end{array}$$

Test, at the $1 \%$ level of significance, Jasmine's belief that her French teacher has understated the mean time that it should take to complete the homework assignment.
State an assumption that you must make in order for the test used in part (a) to be valid.

AQA S2 2007 January Q6

6 The waiting time, $T$ minutes, before being served at a local newsagents can be modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 8 } t ^ { 2 } & 0 \leqslant t < 1
\frac { 1 } { 16 } ( t + 5 ) & 1 \leqslant t \leqslant 3
0 & \text { otherwise } \end{cases}$$

Sketch the graph of f.
For a customer selected at random, calculate $\mathrm { P } ( T \geqslant 1 )$.
1. Show that the cumulative distribution function for $1 \leqslant t \leqslant 3$ is given by $$\mathrm { F } ( t ) = \frac { 1 } { 32 } \left( t ^ { 2 } + 10 t - 7 \right)$$
2. Hence find the median waiting time.

AQA S2 2007 January Q7

7 A statistics unit is required to determine whether or not there is an association between students' performances in mathematics at Key Stage 3 and at GCE. A survey of the results of 500 students showed the following information:

\multirow{2}{*}{}		GCE Grade				\multirow[b]{2}{*}{Total}
		A	B	C	Below C
\multirow{3}{*}{Key Stage 3 Level}	8	60	55	47	43	205
	7	55	32	31	26	144
	6	40	38	35	38	151
	Total	155	125	113	107	500

Use a $\chi ^ { 2 }$ test at the $10 \%$ level of significance to determine whether there is an association between students' performances in mathematics at Key Stage 3 and at GCE.
Comment on the number of students who gained a grade A at GCE having gained a level 7 at Key Stage 3.

AQA S2 2007 January Q8

8 The continuous random variable $X$ has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant - 4
\frac { x + 4 } { 9 } & - 4 \leqslant x \leqslant 5
1 & x \geqslant 5 \end{array} \right.$$

Determine the probability density function, $\mathrm { f } ( x )$, of $X$.
Sketch the graph of f .
Determine $\mathrm { P } ( X > 2 )$.
Evaluate the mean and variance of $X$.

AQA S2 2008 January Q1

1 David claims that customers have to queue at a supermarket checkout for more than 5 minutes, on average. The queuing times, $x$ minutes, of 40 randomly selected customers result in $\bar { x } = 5.5$ and $s ^ { 2 } = 1.31$. Investigate, at the $1 \%$ level of significance, David's claim.

AQA S2 2008 January Q2

2 A new information technology centre is advertising places on its one-week residential computer courses.

The number of places, $X$, booked each week on the publishing course may be modelled by a Poisson distribution with a mean of 9.0.
1. State the standard deviation of $X$.
2. Calculate $\mathrm { P } ( 6 < X < 12 )$.
The number of places booked each week on the web design course may be modelled by a Poisson distribution with a mean of 2.5.
1. Write down the distribution for $T$, the total number of places booked each week on the publishing and web design courses.
2. Hence calculate the probability that, during a given week, a total of fewer than 2 places are booked.
The number of places booked on the database course during each of a random sample of 10 weeks is as follows: $$\begin{array} { l l l l l l l l l l } 14 & 15 & 8 & 16 & 18 & 4 & 10 & 12 & 15 & 8 \end{array}$$ By calculating appropriate numerical measures, state, with a reason, whether or not the Poisson distribution $\mathrm { Po } ( 12.0 )$ could provide a suitable model for the number of places booked each week on the database course.

AQA S2 2008 January Q3

The continuous random variable $T$ follows a rectangular distribution with probability density function given by $$\mathrm { f } ( t ) = \left\{ \begin{array} { l c } k & - a \leqslant t \leqslant b
0 & \text { otherwise } \end{array} \right.$$
1. Express $k$ in terms of $a$ and $b$.
2. Prove, using integration, that $\mathrm { E } ( T ) = \frac { 1 } { 2 } ( b - a )$.
The error, in minutes, made by a commuter when estimating the journey time by train into London may be modelled by the random variable $T$ with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 10 } & - 4 \leqslant t \leqslant 6
0 & \text { otherwise } \end{array} \right.$$
1. Write down the value of $\mathrm { E } ( T )$.
2. Calculate $\mathrm { P } ( T < - 3$ or $T > 3 )$.

AQA S2 2008 January Q4

4 A speed camera was used to measure the speed, $V$ mph, of John's serves during a tennis singles championship. For 10 randomly selected serves, $$\sum v = 1179 \quad \text { and } \quad \sum ( v - \bar { v } ) ^ { 2 } = 1014.9$$ where $\bar { v }$ is the sample mean.

Construct a $99 \%$ confidence interval for the mean speed of John's serves at this tennis championship, stating any assumption that you make.
(7 marks)
Hence comment on John's claim that, at this championship, he consistently served at speeds in excess of 130 mph .
(1 mark)

AQA S2 2008 January Q5

5 A discrete random variable $X$ has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { x } { 20 } & x = 1,2,3,4,5
\frac { x } { 24 } & x = 6
0 & \text { otherwise } \end{array} \right.$$

Calculate $\mathrm { P } ( X \geqslant 5 )$.
1. Show that $\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 24 }$.
2. Hence, or otherwise, show that $\operatorname { Var } \left( \frac { 1 } { X } \right) = 0.036$, correct to three decimal places.
Calculate the mean and the variance of $A$, the area of rectangles having sides of length $X + 3$ and $\frac { 1 } { X }$.

AQA S2 2008 January Q6

6 A survey is carried out in an attempt to determine whether the salary achieved by the age of 30 is associated with having had a university education. The results of this survey are given in the table.

	Salary < £30000	Salary $\boldsymbol { \geq }$ £30000	Total
University education	52	78	130
No university education	63	57	120
Total	115	135	250

Use a $\chi ^ { 2 }$ test, at the $10 \%$ level of significance, to determine whether the salary achieved by the age of 30 is associated with having had a university education.
What do you understand by a Type I error in this context?

AQA S2 2008 January Q7

7 The waiting time, $X$ minutes, for fans to gain entrance to see an event may be modelled by a continuous random variable having the distribution function defined by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 2 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 54 } \left( x ^ { 3 } - 12 x ^ { 2 } + 48 x - 10 \right) & 1 \leqslant x \leqslant 4
1 & x > 4 \end{cases}$$

1. Sketch the graph of F.
2. Explain why the value of $q _ { 1 }$, the lower quartile of $X$, is $\frac { 1 } { 2 }$.
3. Show that the upper quartile, $q _ { 3 }$, satisfies $1.6 < q _ { 3 } < 1.7$.
The probability density function of $X$ is defined by $$\mathrm { f } ( x ) = \begin{cases} \alpha & 0 \leqslant x \leqslant 1
\beta ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
1. Show that the exact values of $\alpha$ and $\beta$ are $\frac { 1 } { 2 }$ and $\frac { 1 } { 18 }$ respectively.
2. Hence calculate $\mathrm { E } ( X )$.

AQA S2 2010 January Q1

1 Roger claims that, on average, his journey time from home to work each day is greater than 45 minutes. The times, $x$ minutes, of 30 randomly selected journeys result in $\bar { x } = 45.8$ and $s ^ { 2 } = 4.8$.
Investigate Roger's claim at the $1 \%$ level of significance.

AQA S2 2010 January Q2

2 The error, in minutes, made by Paul in estimating the time that he takes to complete a college assignment may be modelled by the random variable $T$ with probability density function $$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 30 } & - 5 \leqslant t \leqslant 25
0 & \text { otherwise } \end{array} \right.$$

Find:
1. $\mathrm { E } ( T )$;
  (1 mark)
2. $\quad \operatorname { Var } ( T )$.
Calculate the probability that Paul will make an error of magnitude at least 2 minutes when estimating the time that he takes to complete a given assignment.

\(\boldsymbol { x }\)	40	45	55	74
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.30	0.24	0.36	0.10

\(\boldsymbol { x }\)	1	2	3	4	5	6	\(\geqslant 7\)
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.01	0.05	0.14	0.30	\(k\)	0.12	0

	Salary < £30000	Salary \(\boldsymbol { \geq }\) £30000	Total
University education	52	78	130
No university education	63	57	120
Total	115	135	250

Questions — AQA S2 (139 questions)

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