Questions S3 - A-Level Maths

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

AQA S3 2015 June Q3

4 marks

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

5 marks

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

3 marks

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

13 marks

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]

AQA S3 2016 June Q1

2 marks

1 In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote. Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.

Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'.
Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}]{f536a1ad-333a-47ec-a076-ec8497c1d8fc-02_1380_1707_1327_153}

AQA S3 2016 June Q2

11 marks

2 A plane flies regularly between airports D and T with an intermediate stop at airport M . The time of the plane's departure from, or arrival at, each airport is classified as either early, on time, or late. On $90 \%$ of flights, the plane departs from D on time, and on $10 \%$ of flights, it departs from D late. Of those flights that depart from D on time, $65 \%$ then depart from M on time and $35 \%$ depart from M late. Of those flights that depart from D late, $15 \%$ then depart from M on time and $85 \%$ depart from M late. Any flight that departs from M on time has probability 0.25 of arriving at T early, probability 0.60 of arriving at T on time and probability 0.15 of arriving at T late. Any flight that departs from M late has probability 0.10 of arriving at T early, probability 0.20 of arriving at T on time and probability 0.70 of arriving at T late.

Represent this information by a tree diagram on which labels and percentages or probabilities are shown.
Hence, or otherwise, calculate the probability that the plane:
1. arrives at T on time;
2. arrives at T on time, given that it departed from D on time;
3. does not arrive at T late, given that it departed from D on time;
4. does not arrive at T late, given that it departed from M on time.
Three independent flights of the plane depart from $D$ on time. Calculate the probability that two flights arrive at T on time and that one flight arrives at T early.
[0pt] [4 marks]

AQA S3 2016 June Q4

4 Ben is a fencing contractor who is often required to repair a garden fence by replacing a broken post between fence panels, as illustrated.
\includegraphics[max width=\textwidth, alt={}, center]{f536a1ad-333a-47ec-a076-ec8497c1d8fc-10_364_789_388_694} The tasks involved are as follows.
$U :$ detach the two fence panels from the broken post
$V$ : remove the broken post
$W$ : insert a new post
$X$ : attach the two fence panels to the new post
The mean and the standard deviation of the time, in minutes, for each of these tasks are shown in the table.

The random variables $U , V , W$ and $X$ are pairwise independent, except for $V$ and $W$ for which $\rho _ { V W } = 0.25$.

Determine values for the mean and the variance of:
1. $R = U + X$;
2. $F = V + W$;
3. $T = R + F$;
4. $D = W - V$.
Assuming that each of $R , F , T$ and $D$ is approximately normally distributed, determine the probability that:
1. the total time taken by Ben to repair a garden fence is less than 3 hours;
2. the time taken by Ben to insert a new post is at least 1 hour more than the time taken by him to remove the broken post.

AQA S3 2016 June Q5

6 marks

The random variable $X$, which has distribution $\mathrm { N } \left( \mu _ { X } , \sigma ^ { 2 } \right)$, is independent of the random variable $Y$, which has distribution $\mathrm { N } \left( \mu _ { Y } , \sigma ^ { 2 } \right)$. In order to test $\mathrm { H } _ { 0 } : \mu _ { X } = 1.5 \mu _ { Y }$, samples of size $n$ are taken on each of $X$ and $Y$ and the random variable $\bar { D }$ is defined as $$\bar { D } = \bar { X } - 1.5 \bar { Y }$$ State the distribution of $\bar { D }$ assuming that $\mathrm { H } _ { 0 }$ is true.
A machine that fills bags with rice delivers weights that are normally distributed with a standard deviation of 4.5 grams. The machine fills two sizes of bags: large and extra-large.
The mean weight of rice in a random sample of 50 large bags is 1509 grams.
The mean weight of rice in an independent random sample of 50 extra-large bags is 2261 grams. Test, at the $5 \%$ level of significance, the claim that, on average, the rice in an extra-large bag is $1 \frac { 1 } { 2 }$ times as heavy as that in a large bag.
[0pt] [6 marks]

AQA S3 2016 June Q6

6 marks

The discrete random variable $X$ has probability distribution given by $$\mathrm { P } ( X = x ) = \begin{cases} \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } & x = 0,1,2 , \ldots
0 & \text { otherwise } \end{cases}$$ Show that $\mathrm { E } ( X ) = \lambda$ and that $\operatorname { Var } ( X ) = \lambda$.
In light-weight chain, faults occur randomly and independently, and at a constant average rate of 0.075 per metre.
1. Calculate the probability that there are no faults in a 10 -metre length of this chain.
2. Use a distributional approximation to estimate the probability that, in a 500 -metre reel of light-weight chain, there are:
  (A) fewer than 30 faults;
  (B) at least 35 faults but at most 45 faults.
As part of an investigation into the quality of a new design of medium-weight chain, a sample of fifty 10 -metre lengths was selected. Subsequent analysis revealed a total of 49 faults.
Assuming that faults occur randomly and independently, and at a constant average rate, construct an approximate $98 \%$ confidence interval for the average number of faults per metre.
[0pt] [6 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

Edexcel S3 Q1

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

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

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

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

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

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

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

(a) Explain briefly the method of quota sampling.
(b) Give one disadvantage of quota sampling compared with stratified sampling.
(c) Describe a situation in which you would choose to use quota sampling rather than stratified sampling and explain why.
(2 marks)
Commentators on a game of cricket say that a certain batsman is "playing shots all round the ground". A sports statistician wishes to analyse this claim and records the direction of shots played by the batsman during the course of his innings. She divides the $360 ^ { \circ }$ around the batsman into six sectors, measuring the angle of each shot clockwise from the line between the wickets, and obtains the following results:

Sector	$0 ^ { \circ } -$	$45 ^ { \circ } -$	$90 ^ { \circ } -$	$180 ^ { \circ } -$	$270 ^ { \circ } -$	$315 ^ { \circ } - 360 ^ { \circ }$
No. of Shots	18	19	15	20	9	15

Stating your hypotheses clearly and using a $5 \%$ level of significance test whether or not these data can be modelled by a continuous uniform distribution.
(9 marks)

Edexcel S3 Q3

3. A film-buff is interested in how long it takes for the credits to roll at the end of a movie. She takes a random sample of 20 movies from those that she has bought on DVD and finds that the credits on these films last for a total of 46 minutes and 15 seconds

Assuming that the time for the credits to roll follows a Normal distribution with a standard deviation of 23 seconds, use her data to calculate a $90 \%$ confidence interval for the mean time taken for the credits to roll.
(5 marks)
Find the minimum number of movies she would need to have included in her sample for her confidence interval to have a width of less than 10 seconds.
(5 marks)
Explain why her sample might not be representative of all movies.

Edexcel S3 Q4

4. A hospital administrator is assessing staffing needs for its Accident and Emergency Department at different times of day. The administrator already has data on the number of admissions at different times of day but needs to know if the proportion of the cases that are serious remains constant. Staff are asked to assess whether each person arriving at Accident and Emergency has a "minor" or "serious" problem and the results for three different time periods are shown below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Minor	Serious
8 a.m. - 6 p.m.	45	11
6 p.m. - 2 a.m.	49	22
2 a.m. - 8 a.m.	14	7

Stating your hypotheses clearly, test at the $5 \%$ level of significance whether or not there is evidence of the proportion of serious injuries being different at different times of day.
(11 marks)

Edexcel S3 Q5

5. In a competition, a wine-enthusiast has to rank ten bottles of wine, $A$ to $J$, in order starting with the one he thinks is the most expensive. The table below shows his rankings and the actual order according to price.

Rank	1	2	3	4	5	6	7	8	9	10
Enthusiast	D	$C$	J	$A$	$G$	$F$	$B$	E	I	H
Price	A	$C$	D	$H$	$J$	B	$F$	I	$G$	$E$

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 evidence of positive correlation.
Explain briefly how you would have been able to carry out the test if bottles $B$ and $F$ had the same price.

Edexcel S3 Q6

6. A researcher collects data on the height of boys aged between nine and nine and-a-half years and their diet. The data on the height, $V$ cm, of the 80 boys who had always eaten a vegetarian diet is summarised by $$\Sigma V = 10367 , \quad \Sigma V ^ { 2 } = 1350314 .$$

Calculate unbiased estimates of the mean and variance of $V$. The researcher calculates unbiased estimates of the mean and variance of the height of boys whose diet has included meat from a sample of size 280, giving values of 130.5 cm and $96.24 \mathrm {~cm} ^ { 2 }$ respectively.
Stating your hypotheses clearly, test at the $1 \%$ level whether or not there is a significant difference in the heights of boys of this age according to whether or not they have a vegetarian diet.
(8 marks)

Edexcel S3 Q7

7. An examiner believes that once she has marked the first 20 papers the time it takes her to mark one paper for a particular exam follows a Normal distribution. Having already marked more than 20 papers for each of the $P 1$, M1 and S1 modules set one summer, the mean and standard deviation, in seconds, of the time it takes her to mark a paper for each module are as shown in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Mean	Standard Deviation
P1	252	17
M1	314	42
S1	284	29

Find the probability that the difference in the time it takes her to mark two randomly chosen $P 1$ papers is less than 5 seconds.
(6 marks)
Find the probability that it takes her less than 10 hours to mark $45 M 1$ and $80 S 1$ papers. \section*{END}

Edexcel S3 Q1

A researcher wishes to take a sample of size 9 , without replacement, from a list of 72 people involved in the trial of a new computer keyboard. She numbers the people from 01 to 72 and uses the table of random numbers given in the formula book. She starts with the left-hand side of the sixth row of the table and works across the row. The first two numbers she writes down are 56 and 32 .
1. Find the other six numbers in the sample.
2. Give one advantage and one disadvantage of using random numbers when taking a sample.
  (2 marks)
3. The length of time that registered customers spend on each visit to a supermarket's website is normally distributed with a mean of 28.5 minutes and a standard deviation of 7.2 minutes.
Eight visitors to the site are selected at random and the length of time, $T$ minutes, that each stays is recorded.
Write down the distribution of $\bar { T }$, the mean time spent at the site by these eight visitors.
(2 marks)
Find $\mathrm { P } ( 25 < \bar { T } < 30 )$.
(4 marks)

Edexcel S3 Q3

3. The discrete random variable $X$ has the probability distribution given below.

$x$	2	4	7	$k$
$\mathrm { P } ( X = x )$	0.05	0.15	0.3	0.5

Find the mean of $X$ in terms of $k$.
Find the bias in using ( $2 \bar { X } - 5$ ) as an estimator of $k$. Fifty observations of $X$ were made giving a sample mean of 8.34 correct to 3 significant figures.
Calculate an unbiased estimate of $k$.
(2 marks)

Rank	1	2	3	4	5	6	7	8	9	10
Enthusiast	D	\(C\)	J	\(A\)	\(G\)	\(F\)	\(B\)	E	I	H
Price	A	\(C\)	D	\(H\)	\(J\)	B	\(F\)	I	\(G\)	\(E\)

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

Sector	\(0 ^ { \circ } -\)	\(45 ^ { \circ } -\)	\(90 ^ { \circ } -\)	\(180 ^ { \circ } -\)	\(270 ^ { \circ } -\)	\(315 ^ { \circ } - 360 ^ { \circ }\)
No. of Shots	18	19	15	20	9	15

\(x\)	2	4	7	\(k\)
\(\mathrm { P } ( X = x )\)	0.05	0.15	0.3	0.5

Questions S3 (597 questions)

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