Questions - AQA S3 - A-Level Maths

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
(2 marks)
Assuming that these data come from a bivariate normal distribution, investigate, at the $1 \%$ level of significance, the claim that for junior athletes there is a positive correlation between $x$ and $y$.
Interpret your conclusion in the context of this question.

AQA S3 2008 June Q2

2 A survey of a random sample of 200 passengers on UK internal flights revealed that 132 of them were on business trips.

Construct an approximate $98 \%$ confidence interval for the proportion of passengers on UK internal flights that are on business trips.
Hence comment on the claim that more than 60 per cent of passengers on UK internal flights are on business trips.

AQA S3 2008 June Q3

3 Pitted black olives in brine are sold in jars labelled " 340 grams net weight". Two machines, A and B, independently fill these jars with olives before the brine is added. The weight, $X$ grams, of olives delivered by machine A may be modelled by a normal distribution with mean $\mu _ { X }$ and standard deviation 4.5. The weight, $Y$ grams, of olives delivered by machine B may be modelled by a normal distribution with mean $\mu _ { Y }$ and standard deviation 5.7. The mean weight of olives from a random sample of 10 jars filled by machine A is found to be 157 grams, whereas that from a random sample of 15 jars filled by machine $B$ is found to be 162 grams. Test, at the $1 \%$ level of significance, the hypothesis that $\mu _ { X } = \mu _ { Y }$.
(6 marks)

AQA S3 2008 June Q4

4 A manufacturer produces three models of washing machine: basic, standard and deluxe. An analysis of warranty records shows that $25 \%$ of faults are on basic machines, $60 \%$ are on standard machines and 15\% are on deluxe machines. For basic machines, 30\% of faults reported during the warranty period are electrical, $50 \%$ are mechanical and $20 \%$ are water-related. For standard machines, 40\% of faults reported during the warranty period are electrical, $45 \%$ are mechanical and 15\% are water-related. For deluxe machines, $55 \%$ of faults reported during the warranty period are electrical, $35 \%$ are mechanical and $10 \%$ are water-related.

Draw a tree diagram to represent the above information.
Hence, or otherwise, determine the probability that a fault reported during the warranty period:
1. is electrical;
2. is on a deluxe machine, given that it is electrical.
A random sample of 10 electrical faults reported during the warranty period is selected. Calculate the probability that exactly 4 of them are on deluxe machines.

AQA S3 2008 June Q5

5 The daily number of emergency calls received from district A may be modelled by a Poisson distribution with a mean of $\lambda _ { \mathrm { A } }$. The daily number of emergency calls received from district B may be modelled by a Poisson distribution with a mean of $\lambda _ { \mathrm { B } }$. During a period of 184 days, the number of emergency calls received from district A was 3312, whilst the number received from district B was 2760.

Construct an approximate $95 \%$ confidence interval for $\lambda _ { \mathrm { A } } - \lambda _ { \mathrm { B } }$.
State one assumption that is necessary in order to construct the confidence interval in part (a).

AQA S3 2008 June Q6

6 An aircraft, based at airport A, flies regularly to and from airport B.
The aircraft's flying time, $X$ minutes, from A to B has a mean of 128 and a variance of 50 .
The aircraft's flying time, $Y$ minutes, on the return flight from B to A is such that $$\mathrm { E } ( Y ) = 112 , \quad \operatorname { Var } ( Y ) = 50 \quad \text { and } \quad \rho _ { X Y } = - 0.4$$

Given that $F = X + Y$ :
1. find the mean of $F$;
2. show that the variance of $F$ is 60 .
At airport B , the stopover time, $S$ minutes, is independent of $F$ and has a mean of 75 and a variance of 36 . Find values for the mean and the variance of:
1. $T = F + S$;
2. $M = F - 3 S$.
Hence, assuming that $T$ and $M$ are normally distributed, determine the probability that, on a particular round trip of the aircraft from A to B and back to A :
1. the time from leaving A to returning to A exceeds 300 minutes;
2. the stopover time is greater than one third of the total flying time.

AQA S3 2008 June Q7

The random variable $X$ has a Poisson distribution with $\mathrm { E } ( X ) = \lambda$.
1. Prove, from first principles, that $\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }$.
2. Hence deduce that $\operatorname { Var } ( X ) = \lambda$.
The independent Poisson random variables $X _ { 1 }$ and $X _ { 2 }$ are such that $\mathrm { E } \left( X _ { 1 } \right) = 5$ and $\mathrm { E } \left( X _ { 2 } \right) = 2$. The random variables $D$ and $F$ are defined by $$D = X _ { 1 } - X _ { 2 } \quad \text { and } \quad F = 2 X _ { 1 } + 10$$
1. Determine the mean and the variance of $D$.
2. Determine the mean and the variance of $F$.
3. For each of the variables $D$ and $F$, give a reason why the distribution is not Poisson.
The daily number of black printer cartridges sold by a shop may be modelled by a Poisson distribution with a mean of 5 . Independently, the daily number of colour printer cartridges sold by the same shop may be modelled by a Poisson distribution with a mean of 2. Use a distributional approximation to estimate the probability that the total number of black and colour printer cartridges sold by the shop during a 4 -week period ( 24 days) exceeds 175.

AQA S3 2009 June Q1

1 An analysis of a random sample of 150 urban dwellings for sale showed that 102 are semi-detached. An analysis of an independent random sample of 80 rural dwellings for sale showed that 36 are semi-detached.

Construct an approximate $99 \%$ confidence interval for the difference between the proportion of urban dwellings for sale that are semi-detached and the proportion of rural dwellings for sale that are semi-detached.
Hence comment on the claim that there is no difference between these two proportions.

AQA S3 2009 June Q2

2 A hotel chain has hotels in three types of location: city, coastal and country. The percentages of the chain's reservations for each of these locations are 30,55 and 15 respectively. Each of the chain's hotels offers three types of reservation: Bed \& Breakfast, Half Board and Full Board. The percentages of these types of reservation for each of the three types of location are shown in the table.

\multirow{2}{*}{}		Type of location
		City	Coastal	Country
\multirow{3}{*}{Type of reservation}	Bed \	Breakfast	80	10	30
	Half Board	15	65	50
	Full Board	5	25	20

For example, 80 per cent of reservations for hotels in city locations are for Bed \& Breakfast.

For a reservation selected at random:
1. show that the probability that it is for Bed \& Breakfast is 0.34 ;
2. calculate the probability that it is for Half Board in a hotel in a coastal location;
3. calculate the probability that it is for a hotel in a coastal location, given that it is for Half Board.
A random sample of 3 reservations for Half Board is selected. Calculate the probability that these 3 reservations are for hotels in different types of location.

AQA S3 2009 June Q3

3 The proportion, $p$, of an island's population with blood type $\mathrm { A } \mathrm { Rh } ^ { + }$is believed to be approximately 0.35 . A medical organisation, requiring a more accurate estimate, specifies that a $98 \%$ confidence interval for $p$ should have a width of at most 0.1 . Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the organisation's requirement.

AQA S3 2009 June Q4

4 Holly, a horticultural researcher, believes that the mean height of stems on Tahiti daffodils exceeds that on Jetfire daffodils by more than 15 cm . She measures the heights, $x$ centimetres, of stems on a random sample of 65 Tahiti daffodils and finds that their mean, $\bar { x }$, is 40.7 and that their standard deviation, $s _ { x }$, is 3.4 . She also measures the heights, $y$ centimetres, of stems on a random sample of 75 Jetfire daffodils and finds that their mean, $\bar { y }$, is 24.4 and that their standard deviation, $s _ { y }$, is 2.8 . Investigate, at the $1 \%$ level of significance, Holly's belief.

AQA S3 2009 June Q5

5 The random variable $X$ has a binomial distribution with parameters $n$ and $p$.

Given that $$\mathrm { E } ( X ) = n p \quad \text { and } \quad \mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$$ find an expression for $\operatorname { Var } ( X )$.
Given that $X$ has a mean of 36 and a standard deviation of 4.8:
1. find values for $n$ and $p$;
2. use a distributional approximation to estimate $\mathrm { P } ( 30 < X < 40 )$.

AQA S3 2009 June Q6

6 The table shows the probability distribution for the number of weekday (Monday to Friday) morning newspapers, $X$, purchased by the Reed household per week.

$\boldsymbol { x }$	0	1	2	3	4	5
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.16	0.15	0.25	0.25	0.15	0.04

Find values for $\mathrm { E } ( X )$ and $\operatorname { Var } ( X )$.
The number of weekday (Monday to Friday) evening newspapers, $Y$, purchased by the same household per week is such that $$\mathrm { E } ( Y ) = 2.0 , \quad \operatorname { Var } ( Y ) = 1.5 \quad \text { and } \quad \operatorname { Cov } ( X , Y ) = - 0.43$$ Find values for the mean and variance of:
1. $S = X + Y$;
2. $\quad D = X - Y$.
The total cost per week, $L$, of the Reed household's weekday morning and evening newspapers may be assumed to be normally distributed with a mean of $\pounds 2.31$ and a standard deviation of $\pounds 0.89$. The total cost per week, $M$, of the household's weekend (Saturday and Sunday) newspapers may be assumed to be independent of $L$ and normally distributed with a mean of $\pounds 2.04$ and a standard deviation of $\pounds 0.43$. Determine the probability that the total cost per week of the Reed household's newspapers is more than $\pounds 5$.

AQA S3 2009 June Q7

7 The daily number of customers visiting a small arts and crafts shop may be modelled by a Poisson distribution with a mean of 24 .

Using a distributional approximation, estimate the probability that there was a total of at most 150 customers visiting the shop during a given 6-day period.
The shop offers a picture framing service. The daily number of requests, $Y$, for this service may be assumed to have a Poisson distribution. Prior to the shop advertising this service in the local free newspaper, the mean value of $Y$ was 2. Following the advertisement, the shop received a total of 17 requests for the service during a period of 5 days.
1. Using a Poisson distribution, carry out a test, at the $10 \%$ level of significance, to investigate the claim that the advertisement increased the mean daily number of requests for the shop's picture framing service.
2. Determine the critical value of $Y$ for your test in part (b)(i).
3. Hence, assuming that the advertisement increased the mean value of $Y$ to 3, determine the power of your test in part (b)(i).

AQA S3 2010 June Q2

2 Rodney and Derrick, two independent fruit and vegetable market stallholders, sell punnets of locally-grown raspberries from their stalls during June and July. The following information, based on independent random samples, was collected as part of an investigation by Trading Standards Officers.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Weight of raspberries in a punnet (grams)
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Sample size	Sample mean	Sample standard deviation, $\boldsymbol { s }$
\multirow{2}{*}{Stallholder}	Rodney	50	225	5
\cline { 2 - 5 }	Derrick	75	219	8

Construct a $99 \%$ confidence interval for the difference between the mean weight of raspberries in a punnet sold by Rodney and the mean weight of raspberries in a punnet sold by Derrick.
What can be concluded from your confidence interval?
In addition to weight, state one other factor that may influence whether customers buy raspberries from Rodney or from Derrick.
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AQA S3 2010 June Q3

3
The weekly number of hits, $S$, on Sam's website may be modelled by a Poisson distribution with parameter $\lambda _ { S }$. The weekly number of hits, $T$, on Tina's website may be modelled by a Poisson distribution with parameter $\lambda _ { T }$.
During a period of 40 weeks, the number of hits on Sam's website was 940.
During a period of 60 weeks, the number of hits on Tina's website was 1560.
Assuming that $S$ and $T$ are independent random variables, investigate, at the $2 \%$ level of significance, Tina's claim that the mean weekly number of hits on her website is greater than that on Sam's website.
(7 marks)
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AQA S3 2010 June Q4

4 It is proposed to introduce, for all males at age 60, screening tests, A and B, for a certain disease. Test B is administered only when the result of Test A is inconclusive. It is known that 10\% of 60-year-old men suffer from the disease. For those 60 -year-old men suffering from the disease:

Test A is known to give a positive result, indicating a presence of the disease, in $90 \%$ of cases, a negative result in $2 \%$ of cases and a requirement for the administration of Test B in $8 \%$ of cases;
Test B is known to give a positive result in $98 \%$ of cases and a negative result in 2\% of cases.

For those 60 -year-old men not suffering from the disease:

Test A is known to give a positive result in $1 \%$ of cases, a negative result in $80 \%$ of cases and a requirement for the administration of Test B in 19\% of cases;
Test B is known to give a positive result in $1 \%$ of cases and a negative result in $99 \%$ of cases.
1. Draw a tree diagram to represent the above information.
2. 1. Hence, or otherwise, determine the probability that:
    (A) a 60-year-old man, suffering from the disease, tests negative;
    (B) a 60-year-old man, not suffering from the disease, tests positive.
  2. A random sample of ten thousand 60-year-old men is given the screening tests. Calculate, to the nearest 10, the number who you would expect to be given an incorrect diagnosis.
3. Determine the probability that:
  1. a 60-year-old man suffers from the disease given that the tests provide a positive result;
  2. a 60-year-old man does not suffer from the disease given that the tests provide a negative result.

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AQA S3 2010 June Q5

5 In the manufacture of desk drawer fronts, a machine cuts sheets of veneered chipboard into rectangular pieces of width $W$ millimetres and height $H$ millimetres. The 4 edges of each of these pieces are then covered with matching veneered tape. The distributions of $W$ and $H$ are such that $$\mathrm { E } ( W ) = 350 \quad \operatorname { Var } ( W ) = 5 \quad \mathrm { E } ( H ) = 210 \quad \operatorname { Var } ( H ) = 4 \quad \rho _ { W H } = 0.75$$

Calculate the mean and the variance of the length of tape, $T = 2 W + 2 H$, needed for the edges of a drawer front.
A desk has 4 such drawers whose sizes may be assumed to be independent. Given that $T$ may be assumed to be normally distributed, determine the probability that the total length of tape needed for the edges of the desk's 4 drawer fronts does not exceed 4.5 metres.
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AQA S3 2010 June Q6

A district council claimed that more than 80 per cent of the complaints that it received about the delivery of its services were answered to the satisfaction of complainants before reaching formal status. An analysis of a random sample of 175 complaints revealed that 28 reached formal status.
1. Construct an approximate $95 \%$ confidence interval for the proportion of complaints that reach formal status.
2. Hence comment on the council's claim.
The district council also claimed that less than 40 per cent of all formal complaints were due to a failing in the delivery of its services. An analysis of the 50 formal complaints received during 2007/08 showed that 16 were due to a failing in the delivery of its services.
1. Using an exact test, investigate the council's claim at the $10 \%$ level of significance. The 50 formal complaints received during 2007/08 may be assumed to be a random sample.
2. Determine the critical value for your test in part (b)(i).
3. In fact, only 25 per cent of all formal complaints were due to a failing in the delivery of the council's services. Determine the probability of a Type II error for a test of the council's claim at the $10 \%$ level of significance and based on the analysis of a random sample of 50 formal complaints.
  (4 marks)
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AQA S3 2010 June Q7

7 The random variable $X$ has a Poisson distribution with parameter $\lambda$.

1. Prove, from first principles, that $\mathrm { E } ( X ) = \lambda$.
2. Hence, given that $\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }$, find, in terms of $\lambda$, an expression for $\operatorname { Var } ( X )$.
The mode, $m$, of $X$ is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
1. Show that $\lambda - 1 \leqslant m \leqslant \lambda$.
2. Given that $\lambda = 4.9$, determine $\mathrm { P } ( X = m )$.
The random variable $Y$ has a Poisson distribution with mode $d$ and standard deviation 15.5. Use a distributional approximation to estimate $\mathrm { P } ( Y \geqslant d )$.
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AQA S3 2011 June Q1

1 A consumer report claimed that more than 25 per cent of visitors to a theme park were dissatisfied with the catering facilities provided. In a survey, 375 visitors who had used the catering facilities were interviewed independently, and 108 of them stated that they were dissatisfied with the catering facilities provided.

Test, at the $2 \%$ level of significance, the consumer report's claim.
State an assumption about the 375 visitors that was necessary in order for the hypothesis test in part (a) to be valid.

AQA S3 2011 June Q2

2 The number of emergency calls received by a fire station may be modelled by a Poisson distribution. During a given period of 13 weeks, the station received a total of 108 emergency calls.

Construct an approximate $98 \%$ confidence interval for the average weekly number of emergency calls received by the station.
Hence comment on the station officer's claim that the station receives an average of one emergency call per day.
(2 marks)

AQA S3 2011 June Q3

3 An IT help desk has three telephone stations: Alpha, Beta and Gamma. Each of these stations deals only with telephone enquiries. The probability that an enquiry is received at Alpha is 0.60 .
The probability that an enquiry is received at Beta is 0.25 .
The probability that an enquiry is received at Gamma is 0.15 . Each enquiry is resolved at the station that receives the enquiry. The percentages of enquiries resolved within various times at each station are shown in the table.

\multirow{2}{*}{}		Time
		$\boldsymbol { \leqslant } \mathbf { 1 }$ hour	$\leqslant \mathbf { 2 4 }$ hours	$\leqslant 72$ hours
\multirow{3}{*}{Station}	Alpha	55	80	100
	Beta	60	85	100
	Gamma	40	75	100

For example:
80 per cent of enquiries received at Alpha are resolved within 24 hours;
25 per cent of enquiries received at Alpha take between 1 hour and 24 hours to resolve.

Find the probability that an enquiry, selected at random, is:
1. resolved at Gamma;
2. resolved at Alpha within 1 hour;
3. resolved within 24 hours;
4. received at Beta, given that it is resolved within 24 hours.
A random sample of 3 enquiries was selected. Given that all 3 enquiries were resolved within 24 hours, calculate the probability that they were all received at:
1. Beta;
2. the same station.
  \begin{center} \begin{tabular}{|l|l|l|} \hline & & \begin{tabular}{l}

AQA S3 2011 June Q4

4
The waiting time at a hospital's A\&E department may be modelled by a normal distribution with mean $\mu$ and standard deviation $\frac { \mu } { 2 }$.
The department's manager wishes a $95 \%$ confidence interval for $\mu$ to be constructed such that it has a width of at most $0.2 \mu$.
Calculate, to the nearest 10, an estimate of the minimum sample size necessary in order to achieve the manager's wish.
(5 marks)
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AQA S3 2011 June Q5

5
An examination of 160 e-mails received by Gopal showed that 72 had attachments. An examination of 250 e-mails received by Haley showed that 102 had attachments.
Stating two necessary assumptions about the selection of e-mails, construct an approximate $99 \%$ confidence interval for the difference between the proportion of e-mails received by Gopal that have attachments and the proportion of e-mails received by Haley that have attachments.
(8 marks)
\end{tabular}
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\(\boldsymbol { x }\)	0	1	2	3	4	5
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.16	0.15	0.25	0.25	0.15	0.04

\multirow{2}{*}{}		Time
		\(\boldsymbol { \leqslant } \mathbf { 1 }\) hour	\(\leqslant \mathbf { 2 4 }\) hours	\(\leqslant 72\) hours
\multirow{3}{*}{Station}	Alpha	55	80	100
	Beta	60	85	100
	Gamma	40	75	100

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