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AQA S3 2010 June Q3

7 marks Standard +0.8

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

13 marks Standard +0.3

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.

\includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-09_2484_1709_223_153}

\includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-10_2484_1712_223_153}

\includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-11_2484_1709_223_153}

AQA S3 2010 June Q5

10 marks Standard +0.3

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

18 marks Standard +0.8

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

15 marks Challenging +1.2

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 $\mathrm { P } ( Y \geqslant d )$.
\includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-19_2484_1709_223_153}

AQA S3 2011 June Q1

7 marks Moderate -0.3

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

7 marks Standard +0.3

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

13 marks Standard +0.3

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}

1. Show that $\operatorname { Cov } \left( X _ { 1 } , X _ { 2 } \right) = - 4600$.
2. Given that $X _ { 1 } - X _ { 2 }$ may be assumed to be normally distributed, determine the probability that the difference between the weights of a selected pair of dressed pheasants exceeds 250 grams.
The weight of a box is independent of the total weight of a pair of dressed pheasants, and is normally distributed with a mean of 500 grams and a standard deviation of 40 grams. Determine the probability that a box containing a pair of dressed pheasants weighs less than 2750 grams.
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AQA S3 2011 June Q7

9 marks Standard +0.8

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 $\operatorname { Var } ( X ) = \mathrm { E } ( X )$.
The random variable $Y$ has a Poisson distribution with $\mathrm { E } ( Y ) = 2.5$. Given that $Z = 4 Y + 30$ :
1. show that $\operatorname { Var } ( Z ) = \mathrm { E } ( Z )$;
2. give a reason why the distribution of $Z$ is not Poisson.
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AQA S3 2011 June Q8

13 marks Challenging +1.2

8 The tensile strength of rope is measured in kilograms. The standard deviation of the tensile strength of a particular design of 10 mm diameter rope is known to be 285 kilograms. A retail organisation, which buys such rope from two manufacturers, A and B , wishes to compare their ropes for mean tensile strength. The mean tensile strength, $\bar { x }$, of a random sample of 80 lengths from manufacturer A was 3770 kilograms. The mean tensile strength, $\bar { y }$, of a random sample of 120 lengths from manufacturer B was 3695 kilograms.

1. Test, at the $5 \%$ level of significance, the hypothesis that there is no difference between the mean tensile strength of rope from manufacturer A and that of rope from manufacturer B.
2. Why was it not necessary to know the distributions of tensile strength in order for your test in part (a)(i) to be valid?
1. Deduce that, for your test in part (a)(i), the critical values of $( \bar { x } - \bar { y } )$ are $\pm 80.63$, correct to two decimal places.
2. In fact, the mean tensile strength of rope from manufacturer A exceeds that of rope from manufacturer B by 125 kilograms. Determine the probability of a Type II error for a test of the hypothesis in part (a)(i) at the $5 \%$ level of significance, based upon a random sample of 80 lengths from manufacturer A and a random sample of 120 lengths from manufacturer B. (4 marks)

AQA S3 2012 June Q1

6 marks Moderate -0.8

1 A wildlife expert measured the neck lengths, $x$ metres, and the tail lengths, $y$ metres, of a sample of 12 mature male giraffes as part of a study into their physical characteristics. The results are shown in the table.

AQA S3 2012 June Q2

7 marks Moderate -0.3

2 As part of a comparison of two varieties of cucumber, Fanfare and Marketmore, random samples of harvested cucumbers of each variety were selected and their lengths measured, in centimetres. The results are summarised in the table.

\multirow{2}{*}{}		\multirow[b]{2}{*}{Sample size}	Length (cm)
			Sample mean	Sample standard deviation
\multirow{2}{*}{Cucumber variety}	Fanfare	50	22.0	1.31
	Marketmore	75	21.6	0.702

Test, at the $1 \%$ level of significance, the hypothesis that there is no difference between the mean length of harvested Fanfare cucumbers and that of harvested Marketmore cucumbers.
In addition to length, name one other characteristic of cucumbers that could be used for comparative purposes.

AQA S3 2012 June Q3

14 marks Moderate -0.5

3 A hotel has three types of room: double, twin and suite. The percentage of rooms in the hotel of each type is 40,45 and 15 respectively. Each room in the hotel may be occupied by $0,1,2$, or 3 or more people. The proportional occupancy of each type of room is shown in the table.

AQA S3 2012 June Q4

6 marks Standard +0.8

4 The manager of a medical centre suspects that patients using repeat prescriptions were requesting, on average, more items during 2011 than during 2010. The mean number of items on a repeat prescription during 2010 was 2.6.
An analysis of a random sample of 250 repeat prescriptions during 2011 showed a total of 688 items requested. The number of items requested on a repeat prescription may be modelled by a Poisson distribution. Use a distributional approximation to investigate, at the $5 \%$ level of significance, the manager's suspicion.

AQA S3 2012 June Q5

10 marks Standard +0.3

5 A random sample of 125 people was selected from a council's electoral roll. Of these, 68 were in favour of a proposed local building plan.

Construct an approximate 98\% confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal.
Calculate, to the nearest 5, an estimate of the minimum sample size necessary in order that an approximate $98 \%$ confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal has a width of at most 10 per cent.

AQA S3 2012 June Q6

17 marks Standard +0.8

6 Alyssa lives in the country but works in a city centre.
Her journey to work each morning involves a car journey, a walk and wait, a train journey, and a walk. Her car journey time, $U$ minutes, from home to the village car park has a mean of 13 and a standard deviation of 3 . Her time, $V$ minutes, to walk from the village car park to the village railway station and wait for a train to depart has a mean of 15 and a standard deviation of 6 . Her train journey time, $W$ minutes, from the village railway station to the city centre railway station has a mean of 24 and a standard deviation of 4 . Her time, $X$ minutes, to walk from the city centre railway station to her office has a mean of 9 and a standard deviation of 2 . The values of the product moment correlation coefficient for the above 4 variables are $$\rho _ { U V } = - 0.6 \quad \text { and } \quad \rho _ { U W } = \rho _ { U X } = \rho _ { V W } = \rho _ { V X } = \rho _ { W X } = 0$$

Determine values for the mean and the variance of:
1. $M = U + V$;
2. $D = W - 2 U$;
3. $T = M + W + X$, given that $\rho _ { M W } = \rho _ { M X } = 0$.
Assuming that the variables $M , D$ and $T$ are normally distributed, determine the probability that, on a particular morning:
1. Alyssa's journey time from leaving home to leaving the village railway station is exactly 30 minutes;
2. Alyssa's train journey time is more than twice her car journey time;
3. Alyssa's total journey time is between 50 minutes and 70 minutes.

AQA S3 2012 June Q7

15 marks Challenging +1.2

The random variable $X$ has a binomial distribution with parameters $n$ and $p$.
1. Prove, from first principles, that $\mathrm { E } ( X ) = n p$.
2. Hence, given that $\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$, find, in terms of $n$ and $p$, 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. Use the first inequality to show that $$m \leqslant ( n + 1 ) p$$
2. Given that the second inequality results in $$m \geqslant ( n + 1 ) p - 1$$ deduce that the distribution $\mathrm { B } ( 10,0.65 )$ has one mode, and find the two values for the mode of the distribution $B ( 35,0.5 )$.
The random variable $Y$ has a binomial distribution with parameters 4000 and 0.00095 . Use a distributional approximation to estimate $\mathrm { P } ( Y \leqslant k )$, where $k$ denotes the mode of $Y$.
(3 marks)

AQA S3 2013 June Q1

8 marks Standard +0.3

1 The number of telephone calls per hour to an out-of-hours doctors' service may be modelled by a Poisson distribution. The total number of telephone calls received during a random sample of 12 weekday night shifts, all of the same duration, was 392.

Calculate an approximate $98 \%$ confidence interval for the mean number of calls received per weekday night shift.
The mean number of calls received during weekend shifts of 48 hours' total duration is 136.8 . Comment on a claim that the mean number of calls per hour during weekend shifts is greater than that during weekday night shifts, which are each of $\mathbf { 1 4 }$ hours' duration.
(3 marks)

AQA S3 2013 June Q2

14 marks Moderate -0.5

2 On a rail route between two stations, A and $\mathrm { B } , 90 \%$ of trains leave A on time and $10 \%$ of trains leave A late. Of those trains that leave A on time, $15 \%$ arrive at B early, $75 \%$ arrive on time and $10 \%$ arrive late. Of those trains that leave A late, $35 \%$ arrive at B on time and $65 \%$ arrive late.

Represent this information by a fully-labelled tree diagram.
Hence, or otherwise, calculate the probability that a train:
1. arrives at B early or on time;
2. left A on time, given that it arrived at B on time;
3. left A late, given that it was not late in arriving at B .
Two trains arrive late at B. Assuming that their journey times are independent, calculate the probability that exactly one train left A on time.

AQA S3 2013 June Q3

9 marks Standard +0.3

3 A builders' merchant's depot has two machines, X and Y , each of which can be used for filling bags with sand or gravel. The weight, in kilograms, delivered by machine X may be modelled by a normal distribution with mean $\mu _ { \mathrm { X } }$ and standard deviation 25 . The weight, in kilograms, delivered by machine Y may be modelled by a normal distribution with mean $\mu _ { \mathrm { Y } }$ and standard deviation 30 . Fred, the depot's yardman, records the weights, in kilograms, of a random sample of 10 bags of sand delivered by machine X as
$\begin{array} { l l l l l l l l l l } 1055 & 1045 & 1000 & 985 & 1040 & 1025 & 1005 & 1030 & 1015 & 1060 \end{array}$
He also records the weights, in kilograms, of a random sample of 8 bags of gravel delivered by machine Y as $$\begin{array} { l l l l l l l l } 1085 & 1055 & 1055 & 1000 & 1035 & 1050 & 1005 & 1075 \end{array}$$

Construct a $95 \%$ confidence interval for $\mu _ { \mathrm { Y } } - \mu _ { \mathrm { X } }$, giving the limits to the nearest 5 kg .
Dot, the depot's manager, commented that Fred's data collection may have been biased. Justify her comment and explain how the possible bias could have been eliminated.
(2 marks)

AQA S3 2013 June Q4

8 marks Standard +0.8

4 An analysis of a sample of 250 patients visiting a medical centre showed that 38 per cent were aged over 65 years. An analysis of a sample of 100 patients visiting a dental practice showed that 21 per cent were aged over 65 years. Assume that each of these two samples has been randomly selected.
Investigate, at the $5 \%$ level of significance, the hypothesis that the percentage of patients visiting the medical centre, who are aged over 65 years, exceeds that of patients visiting the dental practice, who are aged over 65 years, by more than 10 per cent.

AQA S3 2013 June Q5

10 marks Standard +0.3

5 The schedule for an organisation's afternoon meeting is as follows.
Session A (Speaker 1) 2.00 pm to 3.15 pm
Session B (Discussion) 3.15 pm to 3.45 pm
Session C (Speaker 2) $\quad 3.45 \mathrm { pm }$ to 5.00 pm
Records show that:
the duration, $X$, of Session A has mean 68 minutes and standard deviation 10 minutes;
the duration, $Y$, of Session B has mean 25 minutes and standard deviation 5 minutes;
the duration, $Z$, of Session C has mean 73 minutes and standard deviation 15 minutes;
and that: $$\rho _ { X Z } = 0 \quad \rho _ { X Y } = - 0.8 \quad \rho _ { Y Z } = 0$$

Determine the means and the variances of:
1. $L = X + Z$;
2. $M = X + Y$.
Assuming that $L$ and $M$ are each normally distributed, determine the probability that:
1. the total time for the two speaker sessions is less than $2 \frac { 1 } { 2 }$ hours;
2. Session C is late in starting.

\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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