Questions S3 - A-Level Maths

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Edexcel S3 Q5

12 marks Standard +0.3

5. A child is playing with a set of red and blue wooden cubes. The side length of the red cubes is normally distributed with a mean of 14.5 cm and a variance of $16.0 \mathrm {~cm} ^ { 2 }$. The side length of the blue cubes is normally distributed with a mean of 12.2 cm and a variance of $9.0 \mathrm {~cm} ^ { 2 }$.

Find the probability that a randomly chosen red cube will have a side length of more than 3 cm greater than a randomly chosen blue cube. The child makes two towers, one from 4 red cubes and one from 5 blue cubes. Assuming that the cubes for each colour of tower were chosen at random,
find the probability that the red tower is taller than the blue tower.
Explain why the assumption that the cubes for each tower were chosen at random is unlikely to be realistic.

Edexcel S3 Q6

14 marks Standard +0.3

6. A market researcher recorded the number of adverts for vehicles in each of three categories on ITV, Channel 4 and Channel 5 over a period of time. The results are shown in the table below.

	ITV	Channel 4	Channel 5
Family Saloon	69	35	28
Sports Car	20	28	18
Off-road Vehicle	12	22	8

Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence of the proportion of adverts for each type of vehicle being dependent on the channel.
Suggest a reason for your result in part (a).

Edexcel S3 Q7

14 marks Standard +0.3

7. (a) Briefly state the central limit theorem. A student throws ten dice and records the number of sixes showing. The dice are fair, numbered 1 to 6 on the faces.
(b) Write down the distribution of the number of sixes obtained when the ten dice are thrown.
(c) Find the mean and variance of this distribution. The student throws the ten dice 100 times, recording the number of sixes showing each time.
(d) Find the probability that the mean number of sixes obtained is more than 1.8

Edexcel S3 Q1

5 marks Easy -1.8

A personnel manager has details on all company employees and wishes to consult a sample of them on a possible change to the company's hours of business. She decides to take a stratified sample based on different age groups.
1. Give one advantage of using stratified sampling in this situation.
The manager needs to select a sample of size 10 , without replacement, from a list of 65 employees aged 16 to 25 . She numbers these employees from 01 to 65 in alphabetical order and uses the table of random numbers given in the formula book. She starts with the top of the sixth two-digit column and works down. The first two numbers she writes down are 30 and 47.
Find the other eight numbers in the sample.
Suggest another factor that might be useful to consider in deciding on the strata.
(1 mark)

Edexcel S3 Q2

6 marks Standard +0.3

2. A Geography teacher is interested in the link between mathematical ability and the ability to visualise three-dimensional situations. He gives a group of 15 students a test and records each student's score, $m$, on the mathematics questions and each student's score, $v$, on the visiospatial questions. He calculates the following summary statistics: $$S _ { m m } = 3747.73 , \quad S _ { v v } = 2791.33 , \quad S _ { m v } = 2564.33$$

Calculate the product moment correlation coefficient for these data.
Stating your hypotheses clearly and using a $5 \%$ level of significance test the theory that students who are good at Mathematics tend to have better visio-spatial awareness.
(4 marks)

Edexcel S3 Q3

9 marks Standard +0.3

3. A random variable $X$ is distributed normally with a standard deviation of 6.8 Sixty observations of $X$ are made and found to have a mean of 31.4

Find a 90\% confidence interval for the mean of $X$.
How many observations of $X$ would be needed in order to obtain a $90 \%$ confidence interval for the mean of $X$ with a width of less than 1.5
(5 marks)

Edexcel S3 Q4

12 marks Standard +0.3

4. A paranormal investigator invites couples who believe they have a telepathic connection to participate in a trial. With each couple one person looks at a card with one of five shapes on it and the other person says which of the shapes they think it is. This is repeated six times and the number of correct answers recorded. The results from 120 couples are given below.

Number Correct	0	1	2	3	4	5	6
Number of Couples	26	56	28	8	2	0	0

The investigator wishes to see if this data fits a binomial distribution with parameters $n = 6$ and $p = \frac { 1 } { 5 }$ and calculates to 2 decimal places the expected frequencies given below.

Number Correct	0	1	2	3	4	5	6
Expected Frequency				9.83	1.84	0.18	0.01

Find the other expected frequencies.
Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not the distribution is an appropriate model.
Comment on your findings.

Edexcel S3 Q5

13 marks Standard +0.3

5. A Policy Unit wished to find out whether attitudes to the European Union varied with age. It conducted a survey asking 200 individuals to which of three age groups they belonged and whether they regarded themselves as generally pro-Europe or Eurosceptic. The results are shown in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Pro-Europe	Eurosceptic
$18 - 34$ years	43	21
$35 - 54$ years	30	36
55 years or over	27	43

Stating your hypotheses clearly, test at the $5 \%$ level of significance whether attitudes to Europe are associated with age.
(11 marks)
The survey also asked people if they voted at the last election. When the above test was repeated using only the results from those who had voted a value of 4.872 was calculated for $\sum \frac { ( O - E ) ^ { 2 } } { E }$. No classes were combined.
Find if this value leads to a different result.

Edexcel S3 Q6

14 marks Challenging +1.2

6. Four swimmers, $A , B , C$ and $D$, are to be used in a $4 \times 100$ metres freestyle relay. The time for each swimmer to complete a leg follows a normal distribution. The mean and standard deviation, in seconds, of the time for each swimmer to complete a leg and the order in which they are to swim are shown in the table below.

	mean	standard deviation
$1 ^ { \text {st } }$ leg $- A$	63.1	1.2
$2 ^ { \text {nd } }$ leg $- B$	65.7	1.5
$3 ^ { \text {rd } } \operatorname { leg } - C$	65.4	1.8
$4 ^ { \text {th } }$ leg - $D$	62.5	0.9

Find the probability that the total time for first two legs is less than the total time for the last two.
(6 marks)
The total time for another team to complete this relay is normally distributed with a mean of 259.0 seconds and a standard deviation of 3.4 seconds. The two teams are to compete over four races.
Find the probability that the first team wins all four races, assuming that the team's performances are not affected by previous results.
(8 marks)

Edexcel S3 Q7

16 marks Standard +0.3

7. A telephone company believes that, for young people, the average length of a telephone call on a land line is longer than on a mobile, due to the difference in price. The company collected data on the time, $t$ minutes, of 500 calls made by young people on mobiles and the data is summarised by $$\Sigma t = 7335 , \quad \Sigma t ^ { 2 } = 172040 .$$

Calculate unbiased estimates of the mean and variance of $t$. For 200 calls made on land lines by the same young people, unbiased estimates of the mean and variance of the call length were 15.9 minutes and 108.5 minutes ${ } ^ { 2 }$ respectively.
Stating your hypotheses clearly, test at the $5 \%$ level whether or not there is evidence that longer calls are made on land lines than on mobiles.
(9 marks)
Explain the importance of the central limit theorem in carrying out the test in part (b).

AQA S3 2006 June Q1

8 marks Moderate -0.3

1 A council claims that 80 per cent of households are generally satisfied with the services it provides. A random sample of 250 households shows that 209 are generally satisfied with the council's provision of services.

Construct an approximate $95 \%$ confidence interval for the proportion of households that are generally satisfied with the council's provision of services.
Hence comment on the council's claim.

AQA S3 2006 June Q2

7 marks Standard +0.3

2 The table below shows the heart rates, $x$ beats per minute, and the systolic blood pressures, $y$ milligrams of mercury, of a random sample of 10 patients undergoing kidney dialysis.

Patient	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$	$\mathbf { 6 }$	$\mathbf { 7 }$	$\mathbf { 8 }$	$\mathbf { 9 }$	$\mathbf { 1 0 }$
$\boldsymbol { x }$	83	86	88	92	94	98	101	111	115	121
$\boldsymbol { y }$	157	172	161	154	171	169	179	180	192	182

Calculate the value of the product moment correlation coefficient for these data.
Assuming that these data come from a bivariate normal distribution, investigate, at the $1 \%$ level of significance, the claim that, for patients undergoing kidney dialysis, there is a positive correlation between heart rate and systolic blood pressure.

AQA S3 2006 June Q3

11 marks Moderate -0.3

3 Each enquiry received by a business support unit is dealt with by Ewan, Fay or Gaby. The probabilities of them dealing with an enquiry are $0.2,0.3$ and 0.5 respectively. Of enquiries dealt with by Ewan, 60\% are answered immediately, 25\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Fay, 75\% are answered immediately, 15\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Gaby, 90\% are answered immediately and the remainder are answered at a later date.

Determine the probability that an enquiry:
1. is dealt with by Gaby and answered immediately;
2. is answered immediately;
3. is dealt with by Gaby, given that it is answered immediately.
Determine the probability that an enquiry is dealt with by Ewan, given that it is answered later the same day.

AQA S3 2006 June Q4

6 marks Moderate -0.3

4 The table below shows the probability distribution for the number of students, $R$, attending classes for a particular mathematics module.

$\boldsymbol { r }$	6	7	8
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	0.1	0.6	0.3

Find values for $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$.
The number of students, $S$, attending classes for a different mathematics module is such that $$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$ Find values for the mean and variance of:
1. $T = R + S$;
2. $\quad D = S - R$.

AQA S3 2006 June Q5

12 marks Standard +0.3

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a $98 \%$ confidence interval for the average number of letters per week received at work by Rosa.
2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.

AQA S3 2006 June Q6

8 marks Challenging +1.2

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

Prove that $\mathrm { E } ( X ) = \lambda$.
By first proving that $\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }$, or otherwise, prove that $\operatorname { Var } ( X ) = \lambda$.

AQA S3 2006 June Q7

19 marks Challenging +1.2

7 A shop sells cooked chickens in two sizes: medium and large.
The weights, $X$ grams, of medium chickens may be assumed to be normally distributed with mean $\mu _ { X }$ and standard deviation 45. The weights, $Y$ grams, of large chickens may be assumed to be normally distributed with mean $\mu _ { Y }$ and standard deviation 65. A random sample of 20 medium chickens had a mean weight, $\bar { x }$ grams, of 936 .
A random sample of 10 large chickens had the following weights in grams: $$\begin{array} { l l l l l l l l l l } 1165 & 1202 & 1077 & 1144 & 1195 & 1275 & 1136 & 1215 & 1233 & 1288 \end{array}$$

Calculate the mean weight, $\bar { y }$ grams, of this sample of large chickens.
Hence investigate, at the $1 \%$ level of significance, the claim that the mean weight of large chickens exceeds that of medium chickens by more than 200 grams.
1. Deduce that, for your test in part (b), the critical value of $( \bar { y } - \bar { x } )$ is 253.24, correct to two decimal places.
2. Hence determine the power of your test in part (b), given that $\mu _ { Y } - \mu _ { X } = 275$.
3. Interpret, in the context of this question, the value that you obtained in part (c)(ii).
  (3 marks)

AQA S3 2007 June Q1

8 marks Moderate -0.3

1 As part of an investigation into the starting salaries of graduates in a European country, the following information was collected.

\multirow{2}{*}{}		Starting salary (€)
	Sample size	Sample mean	Sample standard deviation
Science graduates	175	19268	7321
Arts graduates	225	17896	8205

Stating a necessary assumption about the samples, construct a $98 \%$ confidence interval for the difference between the mean starting salary of science graduates and that of arts graduates.
What can be concluded from your confidence interval?

AQA S3 2007 June Q2

11 marks Moderate -0.8

2 A hill-top monument can be visited by one of three routes: road, funicular railway or cable car. The percentages of visitors using these routes are 25, 35 and 40 respectively. The age distribution, in percentages, of visitors using each route is shown in the table. For example, 15 per cent of visitors using the road were under 18 .

\multirow{2}{*}{}		Percentage of visitors using
		Road	Funicular railway	Cable car
\multirow{3}{*}{Age (years)}	Under 18	15	25	10
	18 to 64	80	60	55
	Over 64	5	15	35

Calculate the probability that a randomly selected visitor:

who used the road is aged 18 or over;
is aged between 18 and 64;
used the funicular railway and is aged over 64;
used the funicular railway, given that the visitor is aged over 64.

AQA S3 2007 June Q3

11 marks Standard +0.8

3 Kutz and Styler are two unisex hair salons. An analysis of a random sample of 150 customers at Kutz shows that 28 per cent are male. An analysis of an independent random sample of 250 customers at Styler shows that 34 per cent are male.

Test, at the $5 \%$ level of significance, the hypothesis that there is no difference between the proportion of male customers at Kutz and that at Styler.
State, with a reason, the probability of making a Type I error in the test in part (a) if, in fact, the actual difference between the two proportions is 0.05 .

The random variable $X$ has a binomial distribution with parameters $n$ and $p$.
1. Prove that $\mathrm { E } ( X ) = n p$.
2. Given that $\mathrm { E } \left( X ^ { 2 } \right) - \mathrm { E } ( X ) = n ( n - 1 ) p ^ { 2 }$, show that $\operatorname { Var } ( X ) = n p ( 1 - p )$.
3. Given that $X$ is found to have a mean of 3 and a variance of 2.97, find values for $n$ and $p$.
4. Hence use a distributional approximation to estimate $\mathrm { P } ( X > 2 )$.
Dressher is a nationwide chain of stores selling women's clothes. It claims that the probability that a customer who buys clothes from its stores uses a Dressher store card is 0.45 . Assuming this claim to be correct, use a distributional approximation to estimate the probability that, in a random sample of 500 customers who buy clothes from Dressher stores, at least half of them use a Dressher store card.

AQA S3 2007 June Q7

12 marks Standard +0.8

7 In a town, the total number, $R$, of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 . A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.

Using the $10 \%$ level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of $R$.
Determine, for your test in part (a), the critical region for $R$.
Assuming that the offer for sale of houses on the new housing development has reduced the mean value of $R$ to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.
(4 marks)

OCR MEI S3 Q2

20 marks Standard +0.3

2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable $X$ with mean 180 and standard deviation 12, independently for all questions.

One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
She models the time (in minutes) to check a question by the Normally distributed random variable $Y$ with mean 50 and standard deviation 6, independently for all questions and independently of $X$. Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable $\frac { 1 } { 4 } X$, where $X$ is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant $k$ representing "thinking time" plus a random variable $T$ representing the time required to write the question itself, independently for all questions.
Taking $k$ as 45 and $T$ as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with $k = 30$ and $T$ replaced by $\frac { 3 } { 5 } T$. Find the probability as given by this model that a question can be prepared in less than $1 \frac { 3 } { 4 }$ hours.

	mean	standard deviation
\(1 ^ { \text {st } }\) leg \(- A\)	63.1	1.2
\(2 ^ { \text {nd } }\) leg \(- B\)	65.7	1.5
\(3 ^ { \text {rd } } \operatorname { leg } - C\)	65.4	1.8
\(4 ^ { \text {th } }\) leg - \(D\)	62.5	0.9

Patient	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)	\(\mathbf { 7 }\)	\(\mathbf { 8 }\)	\(\mathbf { 9 }\)	\(\mathbf { 1 0 }\)
\(\boldsymbol { x }\)	83	86	88	92	94	98	101	111	115	121
\(\boldsymbol { y }\)	157	172	161	154	171	169	179	180	192	182

\(\boldsymbol { r }\)	6	7	8
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.1	0.6	0.3

Questions S3 (621 questions)

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Edexcel S3 Q5

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OCR MEI S3 Q2