Questions S3 - A-Level Maths

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Edexcel S3 2017 June Q5

10 marks Moderate -0.3

5. Paul takes the company bus to work. According to the bus timetable he should arrive at work at 0831. Paul believes the bus is not reliable and often arrives late. Paul decides to test the arrival time of the bus and carries out a survey. He records the values of the random variable $$X = \text { number of minutes after } 0831 \text { when the bus arrives. }$$ His results are summarised below. $$n = 15 \quad \sum x = 60 \quad \sum x ^ { 2 } = 1946$$

Calculate unbiased estimates of the mean, $\mu$, and the variance of $X$. Using the mean of Paul's sample and given $X \sim \mathrm {~N} \left( \mu , 10 ^ { 2 } \right)$
1. calculate a 95\% confidence interval for the mean arrival time at work for this company bus.
2. State an assumption you made about the values in the sample obtained by Paul.
Comment on Paul's belief. Justify your answer.

Edexcel S3 2017 June Q6

9 marks Standard +0.3

6. An engineer has developed a new battery. She claims that the new battery will last more than 8 hours longer, on average, than the old battery. To test the claim, the engineer randomly selects a sample of 50 new batteries and 40 old batteries. She records how long each battery lasts, $x$ hours for the new batteries and $y$ hours for the old batteries. The results are summarised in the table below.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	$n$	Sample mean	$s ^ { 2 }$
New battery	50	$\bar { x } = 83$	7
Old battery	40	$\bar { y } = 74$	6

Test, at the $5 \%$ level of significance, whether or not there is evidence to support the engineer's claim. State your hypotheses and show your working clearly.
Explain the relevance of the Central Limit Theorem to the test in part (a).

Edexcel S3 2017 June Q7

16 marks Challenging +1.2

7. Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.

Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.
Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar. A random sample of 5 medium bags of sugar is taken.
Find the value of $d$ so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than $d$ grams.

Edexcel S3 2018 June Q1

13 marks Standard +0.3

Phil measures the concentration of a radioactive element, $c$, and the amount of dissolved solids, $a$, of 8 random samples of groundwater. His results are shown in the table below.

Sample	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
$c$	625	700	650	645	720	600	825	665
$a$	1.28	1.30	1.00	1.20	1.55	1.15	1.40	1.45

Given that $$\mathrm { S } _ { c c } = 34787.5 \quad \mathrm {~S} _ { a a } = 0.2172875 \quad \mathrm {~S} _ { c a } = 47.7625$$

calculate, to 3 decimal places, the product moment correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids for these groundwater samples.
Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the concentration of this radioactive element and the amount of dissolved solids in groundwater. Use a $5 \%$ significance level. State your hypotheses clearly.
Calculate, to 3 decimal places, Spearman's rank correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids.
Use your value of Spearman's rank correlation coefficient to test for evidence of a positive correlation between the concentration of the radioactive element and the amount of dissolved solids. Use a $5 \%$ significance level. State your hypotheses clearly.
Using your conclusions in part (b) and part (d), comment on the possible relationship between these variables.

Edexcel S3 2018 June Q2

13 marks Standard +0.3

Merchandise is sold at concerts. The manager of a concert claims that the mean value of merchandise sold to premium ticket holders is more than $\pounds 6$ greater than the mean value of merchandise sold to standard ticket holders.
1. Given that all the tickets for the next concert have been sold, describe how a stratified sample should be taken at the concert.
The mean value of merchandise sold to a random sample of 60 standard ticket holders at the concert is $\pounds 15$ with a standard deviation of $\pounds 10$. The mean value of merchandise sold to a random sample of 55 premium ticket holders at the concert is $\pounds 23$ with a standard deviation of $\pounds 8$.
Test the manager's claim at the $5 \%$ level of significance. State your hypotheses clearly.
For the test in part (b), state whether or not it is necessary to assume that values of merchandise sold have normal distributions. Give a reason for your answer.
REA

Edexcel S3 2018 June Q3

10 marks Standard +0.3

A random sample of repair times, in hours, was taken for an electronic component. The 4 observed times are shown below.
1.3
1.7
1.4
1.8
1. Calculate unbiased estimates of the mean and the variance of the population of repair times for this electronic component.
The population standard deviation of the repair times for this electronic component is known to be 0.5 hours. An estimate of the population mean is required to be within 0.1 hours of its true value with a probability of at least 0.99
Find the minimum sample size required.

Edexcel S3 2018 June Q4

9 marks Standard +0.3

The waiting times, in minutes, of patients at a doctor's surgery follows a normal distribution with unknown mean $\mu$ and known standard deviation $\sigma$

A random sample of 120 patients was taken.

Find, in the form $k \sigma$, the width of a $99 \%$ confidence interval for $\mu$ based on this sample. Give the value of $k$ to 2 decimal places. A further random sample of 100 patients from the surgery gave a $90 \%$ confidence interval for $\mu$ of $( 5.14,6.25 )$
Use this confidence interval to determine whether or not it provides evidence that $\mu = 6$ State the hypotheses being tested here and write down the significance level being used. You do not need to carry out any further calculations.
Find the value of $\sigma$

Edexcel S3 2018 June Q5

12 marks Challenging +1.2

5. The weights, in kg , of cars may be assumed to follow the normal distribution $\mathrm { N } \left( 1000,250 ^ { 2 } \right)$. The weights, in kg , of lorries may be assumed to follow the normal distribution $\mathrm { N } \left( 2800,650 ^ { 2 } \right)$. A lorry and a car are chosen at random.

Find the probability that the lorry weighs more than 3 times the weight of the car. A ferry carries vehicles across a river. The ferry is designed to carry a maximum weight of 20000 kg .
One morning, 8 cars and 3 lorries drive on to the ferry. Find the probability that their total weight will exceed the recommended maximum weight of 20000 kg .
State a necessary assumption needed for the calculation in part (b).

Edexcel S3 2018 June Q6

18 marks Standard +0.3

David carries out an experiment with 4 identical dice, each with faces numbered 1 to 6 . He rolls the 4 dice and counts the number of dice showing an even number on the uppermost face. He repeats this 150 times. The results are summarised in the table below.

No. of dice showing an even number	0	1	2	3	4
Frequency	12	45	36	39	18

David defines the random variable $C$ as the number of dice showing an even number on the uppermost face when the four dice are thrown. David claims that $C \sim \mathrm {~B} ( 4,0.5 )$

Stating your hypotheses clearly and using a $1 \%$ level of significance, test David's claim. Show your working clearly. John claims that $C \sim \mathrm {~B} ( 4 , p )$
Calculate an estimate of the value of $p$ from the summary of the results of David's experiment. Show your working clearly. John decides to test his claim. He calculates expected frequencies using the results of David's experiment and obtains the following table.
No. of dice showing an even number 0 1 2 3 4
Expected frequency 8.65 36.00 $d$ 39.00 $e$
Calculate, to 2 decimal places, the value of $d$ and the value of $e$
State suitable hypotheses to test John's claim. John obtained a test statistic of 16.9 and carries out a test at the $1 \%$ level of significance.
State what conclusion John should make about his claim.
END

AQA S3 2008 June Q1

7 marks Moderate -0.3

1 The best performances of a random sample of 20 junior athletes in the long jump, $x$ metres, and in the high jump, $y$ metres, were recorded. The following statistics were calculated from the results. $$S _ { x x } = 7.0036 \quad S _ { y y } = 0.8464 \quad S _ { x y } = 1.3781$$

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

8 marks Moderate -0.3

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

6 marks Standard +0.3

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

10 marks Moderate -0.8

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

7 marks Standard +0.8

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

18 marks Standard +0.3

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

19 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 $\operatorname { Var } ( X ) = \lambda$.
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

8 marks Standard +0.3

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

13 marks Moderate -0.3

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.

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

13 marks Moderate -0.3

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

17 marks Standard +0.8

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

8 marks Standard +0.3

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

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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\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(n\)	Sample mean	\(s ^ { 2 }\)
New battery	50	\(\bar { x } = 83\)	7
Old battery	40	\(\bar { y } = 74\)	6

Sample	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
\(c\)	625	700	650	645	720	600	825	665
\(a\)	1.28	1.30	1.00	1.20	1.55	1.15	1.40	1.45

No. of dice showing an even number	0	1	2	3	4
Expected frequency	8.65	36.00	\(d\)	39.00	\(e\)

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

Questions S3 (621 questions)

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Edexcel S3 2017 June Q5

Edexcel S3 2017 June Q6

Edexcel S3 2017 June Q7

Edexcel S3 2018 June Q1

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Edexcel S3 2018 June Q3

Edexcel S3 2018 June Q4

Edexcel S3 2018 June Q5

Edexcel S3 2018 June Q6

AQA S3 2008 June Q1

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