Questions S3 - A-Level Maths

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AQA S3 2013 June Q7

15 marks Standard +0.3

7 It is claimed that the proportion, $P$, of people who prefer cooked fresh garden peas to cooked frozen garden peas is greater than 0.50 .

In an attempt to investigate this claim, a sample of 50 people were each given an unlabelled portion of cooked fresh garden peas and an unlabelled portion of cooked frozen garden peas to taste. After tasting each portion, the people were each asked to state which of the two portions they preferred. Of the 50 people sampled, 29 preferred the cooked fresh garden peas. Assuming that the 50 people may be considered to constitute a random sample, use a binomial distribution and the $10 \%$ level of significance to investigate the claim.
(6 marks)
It was then decided to repeat the tasting in part (a) but to involve a sample of 500 , rather than 50, people. Of the 500 people sampled, 271 preferred the cooked fresh garden peas.
1. Assuming that the 500 people may be considered to constitute a random sample, use an approximation to the distribution of the sample proportion, $\widehat { P }$, and the $10 \%$ level of significance to again investigate the claim.
2. The critical value of $\widehat { P }$ for the test in part (b)(i) is 0.529 , correct to three significant figures. It is also given that, in fact, 55 per cent of people prefer cooked fresh garden peas. Estimate the power for a test of the claim that $P > 0.50$ based on a random sample of 500 people and using the $10 \%$ level of significance.
  (5 marks)

AQA S3 2014 June Q1

2 marks Moderate -0.8

1 A hotel's management is concerned about the quality of the free pens that it provides in its meeting rooms. The hotel's assistant manager tests a random sample of 200 such pens and finds that 23 of them fail to write immediately.

Calculate an approximate $\mathbf { 9 6 \% }$ confidence interval for the proportion of pens that fail to write immediately.
The supplier of the pens to the hotel claims that at most 2 in 50 pens fail to write immediately. Comment, with numerical justification, on the supplier's claim.
[0pt] [2 marks] QUESTION
PART Answer space for question 1

AQA S3 2014 June Q2

6 marks Standard +0.3

2 Each household within a district council's area has two types of wheelie-bin: a black one for general refuse and a green one for garden refuse. Each type of bin is emptied by the council fortnightly. The weight, in kilograms, of refuse emptied from a black bin can be modelled by the random variable $B \sim \mathrm {~N} \left( \mu _ { B } , 0.5625 \right)$. The weight, in kilograms, of refuse emptied from a green bin can be modelled by the random variable $G \sim \mathrm {~N} \left( \mu _ { G } , 0.9025 \right)$. The mean weight of refuse emptied from a random sample of 20 black bins was 21.35 kg . The mean weight of refuse emptied from an independent random sample of 15 green bins was 21.90 kg . Test, at the $5 \%$ level of significance, the hypothesis that $\mu _ { B } = \mu _ { G }$.
[0pt] [6 marks]

AQA S3 2014 June Q3

12 marks Moderate -0.3

3 An investigation was carried out into the type of vehicle being driven when its driver was caught speeding. The investigation was restricted to drivers who were caught speeding when driving vehicles with at least 4 wheels. An analysis of the results showed that $65 \%$ were driving cars ( C ), $20 \%$ were driving vans (V) and 15\% were driving lorries (L). Of those driving cars, $30 \%$ were caught by fixed speed cameras (F), 55\% were caught by mobile speed cameras (M) and 15\% were caught by average speed cameras (A). Of those driving vans, $35 \%$ were caught by fixed speed cameras (F), $45 \%$ were caught by mobile speed cameras (M) and 20\% were caught by average speed cameras (A). Of those driving lorries, $10 \%$ were caught by fixed speed cameras $( \mathrm { F } )$, $65 \%$ were caught by mobile speed cameras (M) and $25 \%$ were caught by average speed cameras (A).

Represent this information by a tree diagram on which are shown labels and percentages or probabilities.
Hence, or otherwise, calculate the probability that a driver, selected at random from those caught speeding:
1. was driving either a car or a lorry and was caught by a mobile speed camera;
2. was driving a lorry, given that the driver was caught by an average speed camera;
3. was not caught by a fixed speed camera, given that the driver was not driving a car.
  [0pt] [8 marks]
Three drivers were selected at random from those caught speeding by fixed speed cameras. Calculate the probability that they were driving three different types of vehicle.
[0pt] [4 marks]

AQA S3 2014 June Q4

8 marks Moderate -0.3

4 A sample of 50 male Eastern Grey kangaroos had a mean weight of 42.6 kg and a standard deviation of 6.2 kg . A sample of 50 male Western Grey kangaroos had a mean weight of 39.7 kg and a standard deviation of 5.3 kg .

Construct a 98\% confidence interval for the difference between the mean weight of male Eastern Grey kangaroos and that of male Western Grey kangaroos.
[0pt] [5 marks]
1. What assumption about the selection of each of the two samples was it necessary to make in order that the confidence interval constructed in part (a) was valid?
  [0pt] [1 mark]
2. Why was it not necessary to assume anything about the distributions of the weights of male kangaroos in order that the confidence interval constructed in part (a) was valid?
  [0pt] [2 marks]

AQA S3 2014 June Q5

4 marks Moderate -0.3

5 The numbers of daily morning operations, $X$, and daily afternoon operations, $Y$, in an operating theatre of a small private hospital can be modelled by the following bivariate probability distribution.

\multirow{2}{*}{}		Number of morning operations ( $\boldsymbol { X }$ )
		2	3	4	5	6	$\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )$
\multirow{3}{*}{Number of afternoon operations ( $\boldsymbol { Y }$ )}	3	0.00	0.05	0.20	0.20	0.05	0.50
	4	0.00	0.15	0.10	0.05	0.00	0.30
	5	0.05	0.05	0.10	0.00	0.00	0.20
	$\mathrm { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0.05	0.25	0.40	0.25	0.05	1.00

1. State why $\mathrm { E } ( X ) = 4$ and show that $\operatorname { Var } ( X ) = 0.9$.
2. Given that $$\mathrm { E } ( Y ) = 3.7 , \operatorname { Var } ( Y ) = 0.61 \text { and } \mathrm { E } ( X Y ) = 14.4$$ calculate values for $\operatorname { Cov } ( X , Y )$ and $\rho _ { X Y }$.
Calculate values for the mean and the variance of:
1. $T = X + Y$;
2. $\quad D = X - Y$.
  [0pt] [4 marks]

AQA S3 2014 June Q6

5 marks Standard +0.3

6 Population $A$ has a normal distribution with unknown mean $\mu _ { A }$ and a variance of 18.8.
Population $B$ has a normal distribution with unknown mean $\mu _ { B }$ but with the same variance as Population $A$. The random variables $\bar { X } _ { A }$ and $\bar { X } _ { B }$ denote the means of independent samples, each of size $n$, from populations $A$ and $B$ respectively.

Find an expression, in terms of $n$, for $\operatorname { Var } \left( \bar { X } _ { A } - \bar { X } _ { B } \right)$.
Given that the width of a $99 \%$ confidence interval for $\mu _ { A } - \mu _ { B }$ is to be at most 5 , calculate the minimum value for $n$.
[0pt] [5 marks]

AQA S3 2014 June Q7

4 marks Challenging +1.2

The random variable $X$ has a Poisson distribution with parameter $\lambda$.
1. Prove, from first principles, that $\mathrm { E } ( X ) = \lambda$.
2. Given that $\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }$, deduce that $\operatorname { Var } ( X ) = \lambda$.
The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter $\lambda$.
1. An analysis of one ball of string, selected at random, showed 15 faults. Using an exact test, investigate the claim that $\lambda > 10$. Use the $5 \%$ level of significance.
2. A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
  (A) Using an approximate test, re-investigate the claim that $\lambda > 10$. Use the $5 \%$ level of significance.
  (B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
  (C) Given that, in fact, $\lambda = 12$, estimate the probability of a Type II error for a test of the claim that $\lambda > 10$ based upon a random sample of 20 balls of string and using the $5 \%$ level of significance.
  [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-22_2490_1728_219_141} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-23_2490_1719_217_150} \includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-24_2489_1728_221_141}

AQA S3 2015 June Q1

6 marks Moderate -0.8

1 A demographer measured the length of the right foot, $x$ millimetres, and the length of the right hand, $y$ millimetres, of each of a sample of 12 males aged between 19 years and 25 years. The results are given in the table.

AQA S3 2015 June Q2

8 marks Standard +0.3

2 Emilia runs an online perfume business from home. She believes that she receives more orders on Mondays than on Fridays. She checked this during a period of 26 weeks and found that she received a total of 507 orders on the Mondays and a total of 416 orders on the Fridays. The daily numbers of orders that Emilia receives may be modelled by independent Poisson distributions with means $\lambda _ { \mathrm { M } }$ for Mondays and $\lambda _ { \mathrm { F } }$ for Fridays.

Construct an approximate $99 \%$ confidence interval for $\lambda _ { \mathrm { M } } - \lambda _ { \mathrm { F } }$.
Hence comment on Emilia's belief.

AQA S3 2015 June Q3

12 marks Moderate -0.3

3 A particular brand of spread is produced in three varieties: standard, light and very light. During a marketing campaign, the producer advertises that some cartons of spread contain coupons worth $\pounds 1 , \pounds 2$ or $\pounds 4$. For each variety of spread, the proportion of cartons containing coupons of each value is shown in the table.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Variety
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Standard	Light	Very light
No coupon	0.70	0.65	0.55
£1 coupon	0.20	0.25	0.30
£2 coupon	0.08	0.06	0.10
£4 coupon	0.02	0.04	0.05

For example, the probability that a carton of standard spread contains a coupon worth $\pounds 2$ is 0.08 . In a large batch of cartons, 55 per cent contain standard spread, 30 per cent contain light spread and 15 per cent contain very light spread.

A carton of spread is selected at random from the batch. Find the probability that the carton:
1. contains standard spread and a coupon worth $\pounds 1$;
2. does not contain a coupon;
3. contains light spread, given that it does not contain a coupon;
4. contains very light spread, given that it contains a coupon.
A random sample of 3 cartons is selected from the batch. Given that all of these 3 cartons contain a coupon, find the probability that they each contain a different variety of spread.
[0pt] [4 marks]

AQA S3 2015 June Q4

17 marks Moderate -0.3

A large survey in the USA establishes that 60 per cent of its residents own a smartphone. A survey of 250 UK residents reveals that 164 of them own a smartphone.
Assuming that these 250 UK residents may be regarded as a random sample, investigate the claim that the percentage of UK residents owning a smartphone is the same as that in the USA. Use the 5\% level of significance.
A random sample of 40 residents in a market town reveals that 5 of them own a 4 G mobile phone. Use an exact test to investigate, at the $5 \%$ level of significance, the belief that fewer than 25 per cent of the town's residents own a 4 G mobile phone.
A marketing company needs to estimate the proportion of residents in a large city who own a 4 G mobile phone. It wishes to estimate this proportion to within 0.05 with a confidence of 98\%. Given that the proportion is known to be at most 30 per cent, estimate the sample size necessary in order to meet the company's need.
[0pt] [5 marks]

AQA S3 2015 June Q5

16 marks Standard +0.3

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. Given that $\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$, find an expression for $\operatorname { Var } ( X )$.
1. The random variable $Y$ has a binomial distribution with $\mathrm { E } ( Y ) = 3$ and $\operatorname { Var } ( Y ) = 2.985$. Find values for $n$ and $p$.
2. The random variable $U$ has $\mathrm { E } ( U ) = 5$ and $\operatorname { Var } ( U ) = 6.25$. Show that $U$ does not have a binomial distribution.
The random variable $V$ has the distribution $\operatorname { Po } ( 5 )$ and $W = 2 V + 10$. Show that $\mathrm { E } ( W ) = \operatorname { Var } ( W )$ but that $W$ does not have a Poisson distribution.
The probability that, in a particular country, a person has blood group AB negative is 0.2 per cent. A sample of 5000 people is selected. Given that the sample may be assumed to be random, use a distributional approximation to estimate the probability that at least 6 people but at most 12 people have blood group AB negative.
[0pt] [3 marks]

AQA S3 2015 June Q6

16 marks Challenging +1.2

The independent random variables $S$ and $L$ have means $\mu _ { S }$ and $\mu _ { L }$ respectively, and a common variance of $\sigma ^ { 2 }$. The variable $\bar { S }$ denotes the mean of a random sample of $n$ observations on $S$ and the variable $\bar { L }$ denotes the mean of a random sample of $n$ observations on $L$. Find a simplified expression, in terms of $\sigma ^ { 2 }$, for the variance of $\bar { L } - 2 \bar { S }$.
A machine fills both small bottles and large bottles with shower gel. It is known that the volume of shower gel delivered by the machine is normally distributed with a standard deviation of 8 ml .
1. A random sample of 25 small bottles filled by the machine contained a mean volume of $\bar { s } = 258 \mathrm { ml }$ of shower gel. An independent random sample of 25 large bottles filled by the machine contained a mean volume of $\bar { l } = 522 \mathrm { ml }$ of shower gel. Investigate, at the $10 \%$ level of significance, the hypothesis that the mean volume of shower gel in a large bottle is more than twice that in a small bottle.
  [0pt] [7 marks]
2. Deduce that, for the test of the hypothesis in part (b)(i), the critical value of $\bar { L } - 2 \bar { S }$ is 4.585 , correct to three decimal places.
  [0pt] [2 marks]
3. In fact, the mean volume of shower gel in a large bottle exceeds twice that in a small bottle by 10 ml . Determine the probability of a Type II error for a test of the hypothesis in part (b)(i) at the 10\% level of significance, based upon random samples of 25 small bottles and 25 large bottles.
  [0pt] [4 marks]

Edexcel S3 Q1

6 marks Moderate -0.5

A museum is open to the public for six hours a day from Monday to Friday every week. The number of visitors, $V$, to the museum on ten randomly chosen days were as follows:

$$\begin{array} { l l l l l l l l l l } 182 & 172 & 113 & 99 & 168 & 183 & 135 & 129 & 150 & 108 \end{array}$$

Calculate an unbiased estimate of the mean of $V$. Assuming that $V$ is normally distributed with a variance of 130 ,
find a 95\% confidence interval for the mean of $V$.

Edexcel S3 Q2

7 marks Easy -1.3

2. (a) Explain what is meant by a simple random sample.
(b) Explain briefly how you could use a table of random numbers to select a simple random sample of size 12 from a list of the 70 junior members of a tennis club.
(c) Give an example of a situation in which you might choose to take a stratified sample and explain why.

Edexcel S3 Q3

11 marks Standard +0.3

3. The time that a school pupil spends on French homework each week is normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes. The time that this pupil spends on English homework each week is normally distributed with a mean of 1 hour 30 minutes and a standard deviation of 18 minutes. Find the probability that in a randomly chosen week

the pupil spends more than 2 hours in total doing French and English homework,
the pupil spends more than twice as long doing English homework as he spends doing French homework.
(6 marks)

Edexcel S3 Q4

11 marks Standard +0.3

4. A group of 40 males and 40 females were asked which of three "Reality TV" shows they liked most - Watched, Stranded or One-2-Win. The results were as follows:

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Watched	Stranded	One-2-Win
Males	21	6	13
Females	15	10	15

Stating your hypotheses clearly, test at the $10 \%$ level whether or not there is a significant difference in the preferences of males and females.

Edexcel S3 Q5

12 marks Standard +0.3

5. A marathon runner believes that she is more likely to win a medal at her national championships the higher the temperature is on the day of the race. She records the temperature at the start of each of eight races against fields of a similar standard and her finishing position in each race. Her results are shown in the table below.

Temperature $\left( { } ^ { \circ } \mathrm { C } \right)$	16	9	11	5	7	21	12	15
Finishing position	2	15	5	19	10	4	6	11

Calculate Spearman's rank correlation coefficient for these data.
Using a 5\% level of significance and stating your hypotheses clearly, interpret your result. Another runner suggests that she should use her time in each race instead of her finishing position and calculate the product moment correlation coefficient for the data.
Comment on this suggestion.

Edexcel S3 Q6

12 marks Standard +0.3

6. The weight of a particular electrical component is normally distributed with a mean of 46.7 grams and a variance of 1.8 grams $^ { 2 }$. The component is sold in boxes of 12 .

State the distribution of the mean weight of the components in one box.
Find the probability that the mean weight of the components in a randomly chosen box is more than 47 grams.
(3 marks)
After a break in production the component manufacturer wishes to find out if the mean weight of the components has changed. A random sample of 30 components is found to have a mean weight of 46.5 grams.
Assuming that the variance of the weight of the components is unchanged, test at the $5 \%$ level of significance if there has been any change in the mean weight of the components.
(7 marks)

Edexcel S3 Q7

16 marks Standard +0.3

7. A student collects data on whether competitors in local tennis tournaments are right, or left-handed. The table below shows the number of left-handed players who reached the last 16 for fifty tournaments.

No. of Left-handed Players	0	1	2	3	4	$\geq 5$
No. of Tournaments	4	12	18	11	5	0

The student believes that a binomial distribution with $n = 16$ and $p = 0.1$ could be a suitable model for these data.

Stating your hypotheses clearly test the student's model at the $5 \%$ level of significance.
(13 marks)
To improve the model the student decides to estimate $p$ using the data in the table. Using this value of $p$ to calculate expected frequencies the student had 5 classes after combining and calculated that $\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.127$
Test at the $5 \%$ level of significance whether or not the binomial distribution is a suitable model for the number of left-handed players who reach the last 16 in local tennis tournaments. \section*{END}

Edexcel S3 Q1

4 marks Easy -1.2

A Veterinary Surgeon wishes to survey a stratified sample of size 100 from those people who have pets registered at her surgery. The list below shows the strata to be used and the number in each group.

people who own just dogs - 165 ,
people who own just cats - 140 ,
people who own just small mammals - 105,
others, including those who own more than one type of pet - 90 .
1. Find how many members of each group should be included in the sample.
2. Give two advantages of using stratified sampling.

Edexcel S3 Q2

9 marks Standard +0.3

A psychologist is investigating the numbers people choose when asked to pick a number at random in a given interval. He finds that when asked to pick a number between 0 and 100 people are less likely to pick certain numbers, such as multiples of ten. He believes, however that if people are asked to pick an odd number between 0 and 100 they are equally likely to pick a number ending in any of the digits $1,3,5,7$ or 9 .

To test this theory he asks 80 people to pick an odd number between 0 and 100 and records the last digit of the numbers chosen. His results are shown in the table below.

Last Digit	1	3	5	7	9
Frequency	16	20	14	17	13

Stating your hypotheses clearly and using a 10\% level of significance test the psychologist's theory.
(9 marks)

Edexcel S3 Q3

10 marks Standard +0.3

3. A clothes manufacturer wishes to find out if adult females have become taller on average since twenty years ago when their mean height was 5 ft 6 inches. Studies over time have shown that the standard deviation of the height of adult females has been fairly constant at 2.3 inches. The manager wishes to test if the mean height is now more than 5 ft 6 inches and takes a sample of 150 adult females.

Stating your hypotheses clearly, find the critical region for the mean height of the sample for a test at the $5 \%$ level of significance. The total height of the females in the sample is 832 ft .
Carry out the test making your conclusion clear.

Edexcel S3 Q4

12 marks Standard +0.3

4. For a project a student collects data on engine size and sales over a period of time for the models of cars made by one particular manufacturer. Her results are shown in the table below.

Calculate Spearman's rank correlation coefficient for these data.
Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is any evidence of correlation.
Explain why it is more appropriate to use Spearman's rank correlation coefficient for this test than the product moment correlation coefficient.
(2 marks)

\multirow{2}{*}{}		Number of morning operations ( \(\boldsymbol { X }\) )
		2	3	4	5	6	\(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)
\multirow{3}{*}{Number of afternoon operations ( \(\boldsymbol { Y }\) )}	3	0.00	0.05	0.20	0.20	0.05	0.50
	4	0.00	0.15	0.10	0.05	0.00	0.30
	5	0.05	0.05	0.10	0.00	0.00	0.20
	\(\mathrm { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.05	0.25	0.40	0.25	0.05	1.00

Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\)	16	9	11	5	7	21	12	15
Finishing position	2	15	5	19	10	4	6	11

No. of Left-handed Players	0	1	2	3	4	\(\geq 5\)
No. of Tournaments	4	12	18	11	5	0

Questions S3 (621 questions)

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AQA S3 2013 June Q7

AQA S3 2014 June Q1

AQA S3 2014 June Q2

AQA S3 2014 June Q3

AQA S3 2014 June Q4

AQA S3 2014 June Q5

AQA S3 2014 June Q6

AQA S3 2014 June Q7

AQA S3 2015 June Q1

AQA S3 2015 June Q2

AQA S3 2015 June Q3

AQA S3 2015 June Q4

AQA S3 2015 June Q5

AQA S3 2015 June Q6

Edexcel S3 Q1

Edexcel S3 Q2

Edexcel S3 Q3

Edexcel S3 Q4

Edexcel S3 Q5

Edexcel S3 Q6

Edexcel S3 Q7

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Edexcel S3 Q4