Questions S2 - A-Level Maths

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)

AQA S2 2005 June Q1

1 The number of cars, $X$, passing along a road each minute can be modelled by a Poisson distribution with a mean of 2.6.

Calculate $\mathrm { P } ( X = 2 )$.
1. Write down the distribution of $Y$, the number of cars passing along this road in a 5-minute interval.
2. Hence calculate the probability that at least 15 cars pass along this road in each of four successive 5 -minute intervals.

AQA S2 2005 June Q2

2 Syd, a snooker player, believes that the outcome of any frame of snooker in which he plays may be influenced by the time of day that the frame takes place. The results of 100 randomly selected frames of snooker, played by Syd, are recorded below.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Afternoon	Evening	Total
Win	30	24	54
Lose	18	28	46
Total	48	52	100

Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to test Syd's belief.
(10 marks)

AQA S2 2005 June Q3

3 The heights, in metres, of a random sample of 10 students attending Higrade School are recorded below.
$\begin{array} { l l l l l l l l l } 1.76 & 1.59 & 1.54 & 1.62 & 1.49 & 1.52 & 1.56 & 1.47 & 1.75 \end{array} 1.50$ Assume that the heights of students attending Higrade School are normally distributed.

Calculate unbiased estimates for the mean and variance of the heights of students attending Higrade School.
(3 marks)
Construct a 90\% confidence interval for the mean height of students attending Higrade School.
(5 marks)

AQA S2 2005 June Q4

4 The error, $X$ millimetres, made when the heights of prospective members of a new gym club are measured can be modelled by a rectangular distribution with the following probability density function. $$f ( x ) = \begin{cases} k & - 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$

State the value of $k$.
Write down the value of $\mathrm { E } ( X )$.
Calculate $\mathrm { P } ( X > 0 )$.
The height of a randomly selected prospective member is measured. Find the probability that the magnitude of the error made exceeds 3.5 millimetres.

AQA S2 2005 June Q5

5 The discrete random variable $R$ has the following probability distribution.

$\boldsymbol { r }$	1	2	4
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	$\frac { 1 } { 4 }$	$\frac { 1 } { 2 }$	$\frac { 1 } { 4 }$

Calculate exact values for $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$.
1. By tabulating the probability distribution for $X = \frac { 1 } { R ^ { 2 } }$, show that $\mathrm { E } ( X ) = \frac { 25 } { 64 }$.
2. Hence find the value of the mean of the area of a rectangle which has sides of length $\frac { 8 } { R }$ and $\left( R + \frac { 8 } { R } \right)$.
  (3 marks)

AQA S2 2005 June Q6

6 The contents, in millilitres, of cartons of milk produced at Kream Dairies, can be modelled by a normal distribution with mean 568 and variance $\sigma ^ { 2 }$. After receiving several complaints from their customers who thought that the average content of the cartons had been reduced, the production manager of Kream Dairies decided to investigate. A random sample of 8 cartons of milk was taken, revealing the following contents, in millilitres. $$\begin{array} { l l l l l l l l } 560 & 568 & 561 & 562 & 564 & 567 & 565 & 563 \end{array}$$ Investigate, at the $1 \%$ level of significance, whether the average content of cartons of milk is less than 568 millilitres.
(10 marks)

AQA S2 2005 June Q7

7 The time, $T$ hours, that the supporters of Bracken Football Club have to queue in order to obtain their Cup Final tickets has the following probability density function. $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 5 } & 0 \leqslant t < 3
\frac { 1 } { 45 } t ( 6 - t ) & 3 \leqslant t \leqslant 6
0 & \text { otherwise } \end{cases}$$

Sketch the graph of f.
Write down the value of $\mathrm { P } ( T = 3 )$.
Find the probability that a randomly selected supporter has to queue for at least 3 hours in order to obtain tickets.
Show that the median queuing time is 2.5 hours.
Calculate P (median $< T <$ mean).

AQA S2 2005 June Q8

8 The mean age of people attending a large concert is claimed to be 35 years. A random sample of 100 people attending the concert was taken and their mean age was found to be 37.9 years.

Given that the standard deviation of the ages of the people attending the concert is 12 years, test, at the $1 \%$ level of significance, the claim that the mean age is 35 years.
(7 marks)
Explain, in the context of this question, the meaning of a Type II error.
(2 marks)

AQA S2 2006 June Q1

1 The number of A-grades, $X$, achieved in total by students at Lowkey School in their Mathematics examinations each year can be modelled by a Poisson distribution with a mean of 3 .

Determine the probability that, during a 5 -year period, students at Lowkey School achieve a total of more than 18 A -grades in their Mathematics examinations. (3 marks)
The number of A-grades, $Y$, achieved in total by students at Lowkey School in their English examinations each year can be modelled by a Poisson distribution with a mean of 7 .
1. Determine the probability that, during a year, students at Lowkey School achieve a total of fewer than 15 A -grades in their Mathematics and English examinations.
2. What assumption did you make in answering part (b)(i)?

AQA S2 2006 June Q2

2 The weights of lions kept in captivity at Wildcat Safari Park are normally distributed.
The weights, in kilograms, of a random sample of five lions were recorded as $$\begin{array} { l l l l l } 46 & 48 & 57 & 49 & 54 \end{array}$$

Construct a 95\% confidence interval for the mean weight of lions kept in captivity at Wildcat Safari Park.
State the probability that this confidence interval does not contain the mean weight of lions kept in captivity at Wildcat Safari Park.

AQA S2 2006 June Q3

3 Morecrest football team always scores at least one goal but never scores more than four goals in each game. The number of goals, $R$, scored in each game by the team can be modelled by the following probability distribution.

$\boldsymbol { r }$	1	2	3	4
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	$\frac { 7 } { 16 }$	$\frac { 5 } { 16 }$	$\frac { 3 } { 16 }$	$\frac { 1 } { 16 }$

Calculate exact values for the mean and variance of $R$.
Next season the team will play 32 games. They expect to win $90 \%$ of the games in which they score at least three goals, half of the games in which they score exactly two goals and $20 \%$ of the games in which they score exactly one goal. Find, for next season:
1. the number of games in which they expect to score at least three goals;
2. the number of games that they expect to win.

AQA S2 2006 June Q4

4 It is claimed that the area within which a school is situated affects the age profile of the staff employed at that school. In order to investigate this claim, the age profiles of staff employed at two schools with similar academic achievements are compared. Academia High School, situated in a rural community, employs 120 staff whilst Best Manor Grammar School, situated in an inner-city community, employs 80 staff. The percentage of staff within each age group, for each school, is given in the table.

$\mathbf { 2 2 - } \mathbf { 3 4 }$

17.5

40.0

$\mathbf { 3 5 - } \mathbf { 3 9 }$

60.0

45.0

$\mathbf { 4 0 - } \mathbf { 5 9 }$

22.5

15.0

1. Form the data into a contingency table suitable for analysis using a $\chi ^ { 2 }$ distribution.
  (2 marks)
2. Use a $\chi ^ { 2 }$ test, at the $1 \%$ level of significance, to determine whether there is an association between the age profile of the staff employed and the area within which the school is situated.
Interpret your result in part (a)(ii) as it relates to the 22-34 age group.

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

\(\boldsymbol { r }\)	1	2	4
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)

\(\boldsymbol { r }\)	1	2	3	4
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	\(\frac { 7 } { 16 }\)	\(\frac { 5 } { 16 }\)	\(\frac { 3 } { 16 }\)	\(\frac { 1 } { 16 }\)

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

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

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