Questions - WJEC - A-Level Maths

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WJEC Unit 1 Specimen Q4

Identify the statement which is false. Find a counter example to show that this statement is in fact false.
Identify the statement which is true. Give a proof to show that this statement is in fact true. \item Figure 1 shows a sketch of the graph of $y = f ( x )$. The graph has a minimum point at $( - 3 , - 4 )$ and intersects the $x$-axis at the points $( - 8,0 )$ and $( 2,0 )$. \end{enumerate} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-3_540_992_422_518} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
Sketch the graph of $y = f ( x + 3 )$, indicating the coordinates of the stationary point and the coordinates of the points of intersection of the graph with the $x$-axis.
Figure 2 shows a sketch of the graph having one of the following equations with an appropriate value of either $p , q$ or $r$.
$y = f ( p x )$, where $p$ is a constant
$y = f ( x ) + q$, where $q$ is a constant
$y = r f ( x )$, where $r$ is a constant \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-3_513_1072_1683_587} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Write down the equation of the graph sketched in Figure 2, together with the value of the corresponding constant.

WJEC Further Unit 1 2018 June Q2

Find the values of $\alpha , \beta$, and $\gamma$.
Find the cubic equation with roots $3 \alpha , 3 \beta , 3 \gamma$. Give your answer in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d = 0$, where $a , b , c , d$ are constants to be determined.

WJEC Further Unit 3 2018 June Q2

Calculate the distance $A P$ when $P$ is instantaneously at rest for the first time, giving your answer correct to 2 decimal places.
Estimate the distance $A P$ when $P$ is instantaneously at rest for the second time and clearly state one assumption that you have made in making your estimate. \item The position vector $\mathbf { x }$ metres at time $t$ seconds of an object of mass 3 kg may be modelled by \end{enumerate} $$\mathbf { x } = 3 \sin t \mathbf { i } - 4 \cos 2 t \mathbf { j } + 5 \sin t \mathbf { k }$$
Find an expression for the velocity vector $\mathbf { v } \mathrm { ms } ^ { - 1 }$ at time $t$ seconds and determine the least value of $t$ when the object is instantaneously at rest.
Write down the momentum vector at time $t$ seconds.
Find, in vector form, an expression for the force acting on the object at time $t$ seconds.

WJEC Further Unit 3 Specimen Q2

Show that the speed of $P$ when it first begins to move in a circle is $\sqrt { 3 g }$.
In the subsequent motion, when the string first makes an angle of $45 ^ { \circ }$ with the downwards vertical,
1. calculate the speed $v$ of $P$,
2. determine the tension in the string. \item At time $t = 0 \mathrm {~s}$, the position vector of an object $A$ is $\mathbf { i } \mathrm { m }$ and the position vector of another object $B$ is $3 \mathbf { i } \mathrm {~m}$. The constant velocity vector of $A$ is $2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } \mathrm {~ms} ^ { - 1 }$ and the constant velocity vector of $B$ is $\mathbf { i } + 3 \mathbf { j } - 5 \mathbf { k } \mathrm {~ms} ^ { - 1 }$. Determine the value of $t$ when $A$ and $B$ are closest together and find the least distance between $A$ and $B$. \item Relative to a fixed origin $O$, the position vector $\mathbf { r } \mathrm { m }$ at time $t \mathrm {~s}$ of a particle $P$, of mass 0.4 kg , is given by \end{enumerate} $$\mathbf { r } = \mathrm { e } ^ { 2 t } \mathbf { i } + \sin ( 2 t ) \mathbf { j } + \cos ( 2 t ) \mathbf { k }$$
Show that the velocity vector $\mathbf { v }$ and the position vector $\mathbf { r }$ are never perpendicular to each other.
Given that the speed of $P$ at time $t$ is $v _ { \mathrm { ms } } ^ { - 1 }$, show that $$v ^ { 2 } = 4 \mathrm { e } ^ { 4 t } + 4$$
Find the kinetic energy of $P$ at time $t$.
Calculate the work done by the force acting on $P$ in the interval $0 < t < 1$.
Determine an expression for the rate at which the force acting on $P$ is working at time $t$.

WJEC Unit 2 Specimen Q1

The events $\mathrm { A } , B$ are such that $P ( A ) = 0.2 , P ( B ) = 0.3$. Determine the value of $P ( A \cup B )$ when
1. $A , B$ are mutually exclusive,
2. $A , B$ are independent,
3. $\quad A \subset B$.
4. Dewi, a candidate in an election, believes that $45 \%$ of the electorate intend to vote for him. His agent, however, believes that the support for him is less than this. Given that $p$ denotes the proportion of the electorate intending to vote for Dewi,
5. state hypotheses to be used to resolve this difference of opinion.
They decide to question a random sample of 60 electors. They define the critical region to be $X \leq 20$, where $X$ denotes the number in the sample intending to vote for Dewi.
1. Determine the significance level of this critical region.
2. If in fact $p$ is actually 0.35 , calculate the probability of a Type II error.
3. Explain in context the meaning of a Type II error.
4. Explain briefly why this test is unsatisfactory. How could it be improved while keeping approximately the same significance level?

WJEC Unit 2 Specimen Q3

3. Cars arrive at random at a toll bridge at a mean rate of 15 per hour.

Explain briefly why the Poisson distribution could be used to model the number of cars arriving in a particular time interval.
Phil stands at the bridge for 20 minutes. Determine the probability that he sees exactly 6 cars arrive.
Using the statistical tables provided, find the time interval (in minutes) for which the probability of more than 10 cars arriving is approximately 0.3 .

WJEC Unit 2 Specimen Q4

4. A researcher wishes to investigate the relationship between the amount of carbohydrate and the number of calories in different fruits. He compiles a list of 90 different fruits, e.g. apricots, kiwi fruits, raspberries. As he does not have enough time to collect data for each of the 90 different fruits, he decides to select a simple random sample of 14 different fruits from the list. For each fruit selected, he then uses a dieting website to find the number of calories (kcal) and the amount of carbohydrate (g) in a typical adult portion (e.g. a whole apple, a bunch of 10 grapes, half a cup of strawberries). He enters these data into a spreadsheet for analysis.

Explain how the random number function on a calculator could be used to select this sample of 14 different fruits.
The scatter graph represents 'Number of calories' against 'Carbohydrate' for the sample of 14 different fruits.
1. Describe the correlation between 'Number of calories' and 'Carbohydrate'.
2. Interpret the correlation between 'Number of calories' and 'Carbohydrate' in this context.
  \includegraphics[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-3_810_1154_1315_539}
The equation of the regression line for this dataset is: $$\text { 'Number of calories' } = 12.4 + 2.9 \times \text { 'Carbohydrate' }$$
1. Interpret the gradient of the regression line in this context.
2. Explain why it is reasonable for the regression line to have a non-zero intercept in this context.

WJEC Unit 2 Specimen Q5

5. Gareth has a keen interest in pop music. He recently read the following claim in a music magazine. \section*{In the pop industry most songs on the radio are not longer than three minutes.}

He decided to investigate this claim by recording the lengths of the top 50 singles in the UK Official Singles Chart for the week beginning 17 June 2016. (A 'single' in this context is one digital audio track.) Comment on the suitability of this sample to investigate the magazine's claim.
Gareth recorded the data in the table below.
Length of singles for top 50 UK Official Chart singles, 17 June 2016
2.5-(3.0) 3.0-(3.5) 3.5-(4.0) 4.0-(4.5) 4.5-(5.0) 5.0-(5.5) 5.5-(6.0) 6.0-(6.5) 6.5-(7.0) 7.0-(7.5)
3 17 22 7 0 0 0 0 0 1
He used these data to produce a graph of the distributions of the lengths of singles
\includegraphics[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-4_860_1435_1343_379} State two corrections that Gareth needs to make to the histogram so that it accurately represents the data in the table.
Gareth also produced a box plot of the lengths of singles. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Length of single for top 50 UK Official Singles Chart 17 June 2016} \includegraphics[alt={},max width=\textwidth]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-5_504_812_406_644}
\end{figure} He sees that there is one obvious outlier.
1. What will happen to the mean if the outlier is removed?
2. What will happen to the standard deviation if the outlier is removed?

Gareth decided to remove the outlier. He then produced a table of summary statistics.

Use the appropriate statistics from the table to show, by calculation, that the maximum value for the length of a single is not an outlier.

		Summary statistics Length of single for top 50 UK Official Singles Chart (minutes)
\multirow{2}{*}{Length of single}	N	Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
	49	3.57	0.393	2.77	3.26	3.60	3.89	4.38

State, with a reason, whether these statistics support the magazine's claim.

Gareth also calculated summary statistics for the lengths of 30 singles selected at random from his personal collection.

		Summary statistics Length of single for Gareth's random sample of 30 singles (minutes)
\multirow{2}{*}{Length of single}	N	Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
	30	3.13	0.364	2.58	2.73	2.92	3.22	3.95

Compare and contrast the distribution of lengths of singles in Gareth's personal collection with the distribution in the top 50 UK Official Singles Chart.

WJEC Unit 2 Specimen Q6

A small object, of mass 0.02 kg , is dropped from rest from the top of a building which is 160 m high.
1. Calculate the speed of the object as it hits the ground.
2. Determine the time taken for the object to reach the ground.
3. State one assumption you have made in your solution.
4. The diagram below shows two particles $A$ and $B$, of mass 2 kg and 5 kg respectively, which are connected by a light inextensible string passing over a fixed smooth pulley. Initially, $B$ is held at rest with the string just taut. It is then released.
  \includegraphics[max width=\textwidth, alt={}, center]{dfe44f43-5e4d-4b8b-a581-f7889abc5cda-6_515_238_1023_868}
5. Calculate the magnitude of the acceleration of $A$ and the tension in the string.
6. What assumption does the word 'light' in the description of the string enable you to make in your solution?
7. A particle $P$, of mass 3 kg , moves along the horizontal $x$-axis under the action of a resultant force $F \mathrm {~N}$. Its velocity $v \mathrm {~ms} ^ { - 1 }$ at time $t$ seconds is given by
$$v = 12 t - 3 t ^ { 2 }$$
Given that the particle is at the origin $O$ when $t = 1$, find an expression for the displacement of the particle from $O$ at time $t \mathrm {~s}$.
Find an expression for the acceleration of the particle at time $t \mathrm {~s}$.

WJEC Unit 2 Specimen Q9

9. A truck of mass 180 kg runs on smooth horizontal rails. A light inextensible rope is attached to the front of the truck. The rope runs parallel to the rails until it passes over a light smooth pulley. The rest of the rope hangs down a vertical shaft. When the truck is required to move, a load of $M \mathrm {~kg}$ is attached to the end of the rope in the shaft and the brakes are then released.

Find the tension in the rope when the truck and the load move with an acceleration of magnitude $0.8 \mathrm {~ms} ^ { - 2 }$ and calculate the corresponding value of $M$.
In addition to the assumptions given in the question, write down one further assumption that you have made in your solution to this problem and explain how that assumption affects both of your answers.

WJEC Unit 2 Specimen Q10

10. Two forces $\mathbf { F }$ and $\mathbf { G }$ acting on an object are such that $$\begin{aligned} & \mathbf { F } = \mathbf { i } - 8 \mathbf { j }
& \mathbf { G } = 3 \mathbf { i } + 11 \mathbf { j } \end{aligned}$$ The object has a mass of 3 kg . Calculate the magnitude and direction of the acceleration of the object.

WJEC Unit 3 2019 June Q1

$\mathbf { 1 }$ & $\mathbf { 0 }$
\hline \end{tabular} \end{center} a) Differentiate each of the following functions with respect to $x$. i) $x ^ { 5 } \ln x$
ii) $\frac { \mathrm { e } ^ { 3 x } } { x ^ { 3 } - 1 }$
iii) $( \tan x + 7 x ) ^ { \frac { 1 } { 2 } }$
b) A function is defined implicitly by $$3 y + 4 x y ^ { 2 } - 5 x ^ { 3 } = 8$$ Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
□
1
The function $f ( x )$ is defined by $$f ( x ) = \frac { \sqrt { x ^ { 2 } - 1 } } { x }$$ with domain $x \geqslant 1$.
a) Find an expression for $f ^ { - 1 } ( x )$. State the domain for $f ^ { - 1 }$ and sketch both $f ( x )$ and $f ^ { - 1 } ( x )$ on the same diagram.
b) Explain why the function $f f ( x )$ cannot be formed.
□
1
A chord $A B$ subtends an angle $\theta$ radians at the centre of a circle. The chord divides the circle into two segments whose areas are in the ratio $1 : 2$.
\includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-5_572_576_1197_749}
a) Show that $\sin \theta = \theta - \frac { 2 \pi } { 3 }$.
b) i) Show that $\theta$ lies between $2 \cdot 6$ and $2 \cdot 7$.
ii) Starting with $\theta _ { 0 } = 2 \cdot 6$, use the Newton-Raphson Method to find the value of $\theta$ correct to three decimal places.

WJEC Unit 4 Specimen Q1

It is known that $4 \%$ of a population suffer from a certain disease. When a diagnostic test is applied to a person with the disease, it gives a positive response with probability 0.98 . When the test is applied to a person who does not have the disease, it gives a positive response with probability 0.01 .
1. Using a tree diagram, or otherwise, show that the probability of a person who does not have the disease giving a negative response is 0.9504 .
The test is applied to a randomly selected member of the population.
Find the probability that a positive response is obtained.
Given that a positive response is obtained, find the probability that the person has the disease.

WJEC Unit 4 Specimen Q2

2. Mary and Jeff are archers and one morning they play the following game. They shoot an arrow at a target alternately, starting with Mary. The winner is the first to hit the target. You may assume that, with each shot, Mary has a probability 0.25 of hitting the target and Jeff has a probability $p$ of hitting the target. Successive shots are independent.

Determine the probability that Jeff wins the game
i) with his first shot,
ii) with his second shot.
Show that the probability that Jeff wins the game is $$\frac { 3 p } { 1 + 3 p }$$
Find the range of values of $p$ for which Mary is more likely to win the game than Jeff.

WJEC Unit 4 Specimen Q3

3. A string of length 60 cm is cut a random point.

Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than $100 \mathrm {~cm} ^ { 2 }$.

WJEC Unit 4 Specimen Q4

4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects $2 p$ coins that are less than 6.12 grams or greater than 8.12 grams.

The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Weight of UK two pence coins} \includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
\end{figure} Explain why the weight of 2 p coins can be modelled using a normal distribution.
Assume the distribution of the weight of $2 p$ coins is normally distributed. Calculate the proportion of $2 p$ coins that are rejected by this brand of coin counting machine.
A manager suspects that a large batch of $2 p$ coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.
Summary statistics
Mean
Standard
deviation
Minimum
Lower
quartile
Median
Upper
quartile
Maximum
6.89 0.296 6.45 6.63 6.88 7.08 7.48
i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.
ii) Assuming the population standard deviation is 0.357 grams, test at the $1 \%$ significance level whether the mean weight of the $2 p$ coins in this batch is less than 7.12 grams.

WJEC Unit 4 Specimen Q5

5. A hotel owner in Cardiff is interested in what factors hotel guests think are important when staying at a hotel. From a hotel booking website he collects the ratings for 'Cleanliness', 'Location', 'Comfort' and 'Value for money' for a random sample of 17 Cardiff hotels.
(Each rating is the average of all scores awarded by guests who have contributed reviews using a scale from 1 to 10 , where 10 is 'Excellent'.) The scatter graph shows the relationship between 'Value for money' and 'Cleanliness' for the sample of Cardiff hotels.
\includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-4_693_1033_749_516}

The product moment correlation coefficient for 'Value for money' and 'Cleanliness' for the sample of 17 Cardiff hotels is 0.895 . Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether this correlation is significant. State your conclusion in context.
The hotel owner also wishes to investigate whether 'Value for money' has a significant correlation with 'Cost per night'. He used a statistical analysis package which provided the following output which includes the Pearson correlation coefficient of interest and the corresponding $p$-value.
Value for money Cost per night
Value for money 1
Cost per night
0.047
$( 0.859 )$
1
Comment on the correlation between 'Value for money' and 'Cost per night'.

WJEC Unit 4 Specimen Q6

An object of mass 4 kg is moving on a horizontal plane under the action of a constant force $4 \mathbf { i } - 12 \mathbf { j } \mathrm {~N}$. At time $t = 0 \mathrm {~s}$, its position vector is $7 \mathbf { i } - 26 \mathbf { j }$ with respect to the origin $O$ and its velocity vector is $- \mathbf { i } + 4 \mathbf { j }$.
1. Determine the velocity vector of the object at time $t = 5 \mathrm {~s}$.
2. Calculate the distance of the object from the origin when $t = 2 \mathrm {~s}$.
3. The diagram below shows an object of weight 160 N at a point $C$, supported by two cables $A C$ and $B C$ inclined at angles of $23 ^ { \circ }$ and $40 ^ { \circ }$ to the horizontal respectively.
  \includegraphics[max width=\textwidth, alt={}, center]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-5_444_919_973_612}
4. Find the tension in $A C$ and the tension in $B C$.
5. State two modelling assumptions you have made in your solution.
6. The rate of change of a population of a colony of bacteria is proportional to the size of the population $P$, with constant of proportionality $k$. At time $t = 0$ (hours), the size of the population is 10 .
7. Find an expression, in terms of $k$, for $P$ at time $t$.
8. Given that the population doubles after 1 hour, find the time required for the population to reach 1 million.
9. A particle of mass 12 kg lies on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.8 . The particle is at rest. It is then subjected to a horizontal tractive force of magnitude 75 N .
  Determine the magnitude of the frictional force acting on the particle, giving a reason for your answer.
10. A body is projected at time $t = 0 \mathrm {~s}$ from a point $O$ with speed $V \mathrm {~ms} ^ { - 1 }$ in a direction inclined at an angle of $\theta$ to the horizontal.
11. Write down expressions for the horizontal and vertical components $x \mathrm {~m}$ and $y \mathrm {~m}$ of its displacement from $O$ at time $t \mathrm {~s}$.
12. Show that the range $R \mathrm {~m}$ on a horizontal plane through the point of projection is given by
$$R = \frac { V ^ { 2 } } { g } \sin 2 \theta$$
Given that the maximum range is 392 m , find, correct to one decimal place,
i) the speed of projection,
ii) the time of flight,
iii) the maximum height attained.

WJEC Further Unit 5 2022 June Q1

Rachel records the times taken, in minutes, to cycle into town from her house on a random sample of 10 days. Her results are shown below.

$$\begin{array} { l l l l l l l l l l } 15 \cdot 5 & 14 \cdot 9 & 16 \cdot 2 & 17 \cdot 3 & 14 \cdot 8 & 14 \cdot 2 & 16 \cdot 0 & 14 \cdot 2 & 15 \cdot 5 & 15 \cdot 1 \end{array}$$ Assuming that these data come from a normal distribution with mean $\mu$ and variance $0 \cdot 9$, calculate a $90 \%$ confidence interval for $\mu$.

WJEC Further Unit 5 2022 June Q2

2. Geraint is a beekeeper. The amounts of honey, $X \mathrm {~kg}$, that he collects annually, from each hive are modelled by the normal distribution $\mathrm { N } \left( 15,5 ^ { 2 } \right)$. At location $A$, Geraint has three hives and at location $B$ he has five hives. You may assume that the amounts of honey collected from the eight hives are independent of each other.

1. Find the probability that Geraint collects more than 14 kg of honey from the first hive at location $A$.
2. Find the probability that he collects more than 14 kg of honey from exactly two out of the three hives at location $A$.
Find the probability that the total amount of honey that Geraint collects from all eight hives is more than 160 kg .
Find the probability that Geraint collects at least twice as much honey from location B as from location A.

WJEC Further Unit 5 2022 June Q3

3. A statistics teacher wants to investigate whether students from the north of a county and students from the south of the same county feel similarly stressed about examinations. The teacher carries out a psychometric test on 10 randomly selected students to give a score between 0 (low stress) and 100 (high stress) to measure their stress levels before a set of examinations. The results are shown in the table below.

Student	Area	Stress Level
Heledd	North	67
Mair	North	55
Hywel	South	26
Gwyn	South	70
Liam	South	36
Marcin	South	57
Gosia	South	32
Kestutas	North	64
Erica	North	60
Tomos	North	22

State one reason why a Mann-Whitney test is appropriate.
Conduct a Mann-Whitney test at a significance level as close to $5 \%$ as possible. State your conclusion clearly.
How could this investigation be improved?

WJEC Further Unit 5 2022 June Q4

4. The Department of Health recommends that adults aged 18 to 65 should take part in at least 150 minutes of aerobic exercise per week. The results of a survey show that 940 out of 2000 randomly selected adults aged 18 to 65 in Wales take part in at least 150 minutes of aerobic exercise per week.

Calculate an approximate $95 \%$ confidence interval for the proportion of adults aged 18 to 65 in Wales who take part in at least 150 minutes of aerobic exercise per week.
Give two reasons why the interval is approximate.
Suppose that a $99 \%$ confidence interval is required, and that the width of the interval is to be no greater than $0 \cdot 04$. Estimate the minimum additional number of adults to be surveyed to satisfy this requirement.

WJEC Further Unit 5 2022 June Q5

5. A laboratory carrying out screening for a certain blood disorder claims that the average time taken for test results to be returned is 38 hours. A reporter for a national newspaper suspects that the results take longer, on average, to be returned than claimed by the laboratory. The reporter finds the time, $x$ hours, for 50 randomly selected results, in order to conduct a hypothesis test. The following summary statistics were obtained. $$\sum x = 2163 \quad \sum x ^ { 2 } = 98508$$

Calculate the $p$-value for the reporter's hypothesis test, and complete the test using a $5 \%$ level of significance. Hence write a headline for the reporter to use.
Explain the relevance or otherwise of the Central Limit Theorem to your answer in part (a).
Briefly explain why a random sample is preferable to taking a batch of 50 consecutive results.
On another occasion, the reporter took a different random sample of 10 results.
1. State, with a reason, what type of hypothesis test the reporter should use on this occasion.
2. State one assumption required to carry out this test.

WJEC Further Unit 5 2022 June Q6

6. A zoologist knows that the median body length of adults in a species of fire-bellied toads is 4.2 cm . The zoologist thinks he has discovered a new subspecies of fire-bellied toads. If there is sufficient evidence to suggest the median body length differs from 4.2 cm , he will continue his studies to confirm whether he has discovered a new subspecies. Otherwise, he will abandon his studies on fire-bellied toads. The lengths of 10 randomly selected adult toads from the group being investigated are given below. $$\begin{array} { l l l l l l l l l l } 5 \cdot 0 & 3 \cdot 2 & 4 \cdot 9 & 4 \cdot 0 & 3 \cdot 3 & 4 \cdot 2 & 6 \cdot 1 & 4 \cdot 3 & 4 \cdot 8 & 5 \cdot 9 \end{array}$$ Carry out a suitable Wilcoxon signed rank test at a significance level as close to $1 \%$ as possible and give your conclusion in context.

WJEC Further Unit 5 2022 June Q7

7.
\includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral $A B C D$, where $\widehat { B A D } = \alpha , \widehat { B C D } = \beta$ and $\alpha + \beta = 180 ^ { \circ }$. These angles are measured.
The random variables $X$ and $Y$ denote the measured values, in degrees, of $\widehat { B A D }$ and $\widehat { B C D }$ respectively. You are given that $X$ and $Y$ are independently normally distributed with standard deviation $\sigma$ and means $\alpha$ and $\beta$ respectively.

Calculate, correct to two decimal places, the probability that $X + Y$ will differ from $180 ^ { \circ }$ by less than $\sigma$.
Show that $T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )$ is an unbiased estimator for $\alpha$ and verify that it is a better estimator than $X$ for $\alpha$.
Now consider $T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)$.
1. Show that $T _ { 2 }$ is an unbiased estimator for $\alpha$ for all values of $\lambda$.
2. Find $\operatorname { Var } \left( T _ { 2 } \right)$ in terms of $\lambda$ and $\sigma$.
3. Hence determine the value of $\lambda$ which gives the best unbiased estimator for $\alpha$.

Length of singles for top 50 UK Official Chart singles, 17 June 2016
2.5-(3.0)	3.0-(3.5)	3.5-(4.0)	4.0-(4.5)	4.5-(5.0)	5.0-(5.5)	5.5-(6.0)	6.0-(6.5)	6.5-(7.0)	7.0-(7.5)
3	17	22	7	0	0	0	0	0	1

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