Questions S2 - A-Level Maths

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.

AQA S2 2010 January Q3

3 Lorraine bought a new golf club. She then practised with this club by using it to hit golf balls on a golf range. After several such practice sessions, she believed that there had been no change from 190 metres in the mean distance that she had achieved when using her old club. To investigate this belief, she measured, at her next practice session, the distance, $x$ metres, of each of a random sample of 10 shots with her new club. Her results gave $$\sum x = 1840 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1240$$ Investigate Lorraine's belief at the $2 \%$ level of significance, stating any assumption that you make.
(7 marks)

AQA S2 2010 January Q4

4 Julie, a driving instructor, believes that the first-time performances of her students in their driving tests are associated with their ages. Julie's records of her students' first-time performances in their driving tests are shown in the table.

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 $\mathrm { P } ( X < m \mid X \geqslant 3 )$.

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

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

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

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