Questions - WJEC - A-Level Maths

Show that $f ^ { \prime \prime } ( x ) = - 2 \mathrm { e } ^ { x } \sin x$.
Determine the Maclaurin series for $f ( x )$ as far as the $x ^ { 4 }$ term.
Hence, by differentiating your series, determine the Maclaurin series for $\mathrm { e } ^ { x } \sin x$ as far as the $x ^ { 3 }$ term.
The equation $$10 \mathrm { e } ^ { x } \sin x - 11 x = 0$$ has a small positive root. Determine its approximate value, giving your answer correct to three decimal places.

WJEC Further Unit 5 2019 June Q1

A coffee shop produces biscuits to sell. The masses, in grams, of the biscuits follow a normal distribution with mean $\mu$. Eight biscuits are chosen at random and their masses, in grams, are recorded. The results are given below.
$\begin{array} { l l l l l l l l } 32 \cdot 1 & 29 \cdot 9 & 31 \cdot 0 & 31 \cdot 1 & 32 \cdot 5 & 30 \cdot 8 & 30 \cdot 7 & 31 \cdot 5 \end{array}$
1. Calculate a 95\% confidence interval for $\mu$ based on this sample.
2. Explain the relevance or otherwise of the Central Limit Theorem in your calculations.
3. The continuous random variable $X$ is uniformly distributed over the interval $( \theta - 1 , \theta + 5 )$, where $\theta$ is an unknown constant.
4. Find the mean and the variance of $X$.
5. Let $\bar { X }$ denote the mean of a random sample of 9 observations of $X$. Find, in terms of $\bar { X }$, an unbiased estimator for $\theta$ and determine its standard error.
6. The rules for the weight of a cricket ball state:
  "The ball, when new, shall weigh not less than $\mathbf { 1 5 5 . 9 ~ g }$, nor more than $\mathbf { 1 6 3 ~ g }$."
  A company produces cricket balls whose weights are normally distributed. It wants $99 \%$ of the balls it produces to be an acceptable weight.
7. What is the largest acceptable standard deviation?
The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.
Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball.

WJEC Further Unit 5 2019 June Q4

4. Rugby players sometimes use protein powder to aid muscle increase. The monthly weight gains of rugby players taking protein powder may be modelled by a normal distribution having a standard deviation of 40 g and a mean which may depend on the type of protein powder they consume. A rugby team coach gives the same amount of protein powder over a trial month to 22 randomly selected players. Protein powder $A$ was used by 12 players, randomly selected, and their mean weight gain was 900 g . Protein powder $B$ was used by the other 10 players and their mean weight gain was 870 g . Let $\mu _ { A }$ and $\mu _ { B }$ be the mean monthly weight gains, in grams, of the populations of rugby players who use protein powder $A$ and protein powder $B$ respectively.

Calculate a 98\% confidence interval for $\mu _ { A } - \mu _ { B }$.
In the given context, what can you conclude from your answer to part (a)? Give a reason for your answer.
Find the confidence level of the largest confidence interval that would lead the coach to favour protein powder $A$ over protein powder $B$.
State one non-statistical assumption you have made in order to reach these conclusions.

WJEC Further Unit 5 2019 June Q5

5. To qualify as a music examiner, a trainee must listen to a series of performances by 8 randomly chosen students. An experienced examiner and the trainee both award scores for each of the 8 performances. In order for the trainee to qualify, there must not be a significant difference between the average scores given by the experienced examiner and the trainee.

Explain why the Wilcoxon signed rank test is appropriate. The scores awarded are shown below.
Student A B C D E F G H
Experienced Examiner 108 109 92 95 145 148 134 120
Trainee 114 116 95 93 137 144 133 110
1. Carry out an appropriate Wilcoxon signed rank test on this dataset, using a $5 \%$ significance level.
2. What conclusion should be reached about the suitability of the trainee to qualify?

WJEC Further Unit 5 2019 June Q6

6. A manufacturer of batteries for electric cars claims that an hour of charge can power a certain model of car to travel for an average of 123 miles. An electric car company and a consumer, Hopcyn, both wish to test the validity of the manufacturer's claim.

Explain why Hopcyn may want to use a one-sided test and why the car company may want to use a two-sided test. To test the validity of this claim, Hopcyn collects data from a random sample of 90 drivers of this model of car to see how far they travelled, $X$ miles, on an hour of charge. He produced the following summary statistics. $$\sum x = 11007 \quad \sum x ^ { 2 } = 1361913$$
1. Assuming Hopcyn uses a one-sided test, state the hypotheses.
2. Test at the $5 \%$ significance level whether the manufacturer's claim is correct.

WJEC Further Unit 5 2019 June Q7

7. Nathan believes that shearers from Wales can shear more sheep, on average, in a given time than shearers from New Zealand. He takes a random sample of 8 shearers from Wales and 7 shearers from New Zealand. The numbers below indicate how many sheep were sheared in 45 minutes by the 15 shearers.

Wales:	60	53	42	38	37	36	31	28
New Zealand:	39	35	27	26	17	16	15

Use a Mann-Whitney U test at the $1 \%$ significance level to test whether Nathan is correct. You must state your hypotheses clearly and state the critical region.

WJEC Further Unit 5 2019 June Q8

8. The random variable $X$ has probability density function $$\begin{array} { l l } f ( x ) = 1 + \frac { 3 \lambda x } { 2 } & \text { for } - \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }
f ( x ) = 0 & \text { otherwise } \end{array}$$ where $\lambda$ is an unknown parameter such that $- 1 \leqslant \lambda \leqslant 1$.

1. Find $\mathrm { E } ( X )$ in terms of $\lambda$.
2. Show that $\operatorname { Var } ( X ) = \frac { 16 - 3 \lambda ^ { 2 } } { 192 }$.
Show that $\mathrm { P } ( X > 0 ) = \frac { 8 + 3 \lambda } { 16 }$. In order to estimate $\lambda , n$ independent observations of $X$ are made. The number of positive observations obtained is denoted by $Y$ and the sample mean is denoted by $\bar { X }$.
1. Identify the distribution of $Y$.
2. Show that $T _ { 1 }$ is an unbiased estimator for $\lambda$, where $$T _ { 1 } = \frac { 16 Y } { 3 n } - \frac { 8 } { 3 }$$
1. Show that $\operatorname { Var } \left( T _ { 1 } \right) = \frac { 64 - 9 \lambda ^ { 2 } } { 9 n }$.
2. Given that $T _ { 2 }$ is also an unbiased estimator for $\lambda$, where $$T _ { 2 } = 8 \bar { X }$$ find an expression for $\operatorname { Var } \left( T _ { 2 } \right)$ in terms of $\lambda$ and $n$.
3. Hence, giving a reason, determine which is the better estimator, $T _ { 1 }$ or $T _ { 2 }$. \section*{END OF PAPER}

WJEC Further Unit 5 2023 June Q1

The average time it takes for a new kettle to boil, when full of water, is 305 seconds. An old kettle will take longer, on average, to boil. Alun suspects that a particular kettle is an old kettle. He boils the full kettle on 9 occasions and the times taken, in seconds, are shown below.
305
295
310
310
315
307
300
311
306

You may assume the times taken to boil the full kettle are normally distributed.

Calculate unbiased estimates for the mean and variance of the times taken to boil the full kettle.
Test, at the $5 \%$ level of significance, whether there is evidence to suggest that this is an old kettle.
State a factor that Alun should control when carrying out this investigation.

WJEC Further Unit 5 2023 June Q2

2. The random variables $X$ and $Y$ are independent, with $X$ having mean $\mu$ and variance $\sigma ^ { 2 }$, and $Y$ having mean $\mu$ and variance $k \sigma ^ { 2 }$, where $k$ is a positive constant. Let $\bar { X }$ denote the mean of a random sample of 20 observations of $X$, and let $\bar { Y }$ denote the mean of a random sample of 25 observations of $Y$.

Given that $T _ { 1 } = \frac { 3 \bar { X } + 7 \bar { Y } } { 10 }$, show that $T _ { 1 }$ is an unbiased estimator for $\mu$.
Given that $T _ { 2 } = \frac { \bar { X } + a ^ { 2 } \bar { Y } } { 1 + a } , a > 0$, and $T _ { 2 }$ is an unbiased estimator for $\mu$, prove that $a = 1$.
Find and simplify expressions for the variances of $T _ { 1 }$ and $T _ { 2 }$.
Show that the value of $k$ for which $T _ { 1 }$ and $T _ { 2 }$ are equally good estimators is $\frac { 5 } { 6 }$.
Given that $T _ { 3 } = ( 1 - \lambda ) \bar { X } + \lambda \bar { Y }$, find an expression for $\lambda$, in terms of $k$, for which $T _ { 3 }$ has the smallest possible variance.

WJEC Further Unit 5 2023 June Q3

3. Athletes who compete in the 400 m event have resting heart rates (RHR), measured in beats per minute, which are normally distributed with known standard deviation $4 \cdot 7$. A random sample of 90 athletes who compete in the 400 m event is taken. Their resting heart rates are summarised by $$\sum x = 4014 \quad \text { and } \quad \sum x ^ { 2 } = 182257 .$$

Find a $99 \%$ confidence interval for the mean of the RHR of athletes who compete in the 400 m event. Give the limits of your interval correct to 1 decimal place.
Without doing any further calculation, explain how the width of a $95 \%$ confidence interval would compare to the width of your interval in part (a). Athletes who compete in the discus event have RHR which are normally distributed with known standard deviation $\sigma$. A random sample of 100 athletes who compete in the discus event is taken. A 95\% confidence interval for the mean of the RHR is calculated as [49•4, 52•6].
Determine the value of $\sigma$ that was used to calculate this confidence interval.
Referring to the confidence intervals, state, with a reason, what can be said about the RHR of athletes who compete in the 400 m event compared to the RHR of athletes who compete in the discus event.

WJEC Further Unit 5 2023 June Q4

4. Llŷr believes that he will have more social media followers by appearing on a certain Welsh television show. To investigate his belief, he collects data on 9 randomly selected contestants who have appeared on the show. Llŷr records the number of social media followers one week before and one week after the contestants appeared on the show. The data he collects are shown in the table below.

Contestant	A	B	C	D	E	F	G	H	1
Before	480	1008	0	344	351	781	876	741	457
After	841	998	751	344	954	542	820	1011	644

1. Carry out a Wilcoxon signed-rank test on this data set, at a significance level as close to 10\% as possible.
2. Suggest a possible course of action that Llŷr might take.
Give two reasons why the Wilcoxon signed-rank test is appropriate in this case.

WJEC Further Unit 5 2023 June Q5

5. The masses, $X$, in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.

Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
The mean mass, in kg , of a random sample of $n$ men drawn from this distribution is $\bar { X }$. Given that $\mathrm { P } ( \bar { X } > 80 )$ is approximately $0 \cdot 007$, find $n$. The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
State a modelling assumption you have made in calculating your answer for part (c).

WJEC Further Unit 5 2023 June Q6

6. A triathlon race organiser wishes to know whether competitors who are members of a triathlon club race more frequently than competitors who are not members of a triathlon club. Six competitors from a triathlon club and six competitors who are not members of a triathlon club are selected at random. The table below shows the number of triathlon races they each entered last year.

Use a Mann-Whitney U test at a significance level as close to $5 \%$ as possible to carry out the race organiser's investigation.
Briefly explain why a Wilcoxon signed-rank test is not appropriate in this case.

WJEC Further Unit 5 2023 June Q7

7. Branwen intends to buy a new bike, either a Cannotrek or a Bianchondale. If there is evidence that the difference in the mean times on the two bikes over a 10 km time trial is more than 1.25 minutes, she will buy the faster bike. Otherwise, she will base her decision on other factors. She negotiates a test period to try both bikes. The times, in minutes, taken by Branwen to complete a 10 km time trial on the Cannotrek may be modelled by a normal distribution with mean $\mu _ { C }$ and standard deviation $0 \cdot 75$. The times, in minutes, taken by Branwen to complete a 10 km time trial on the Bianchondale may be modelled by a normal distribution with mean $\mu _ { B }$ and standard deviation $0 \cdot 6$. During the test period, she completes 6 time trials with a mean time of 19.5 minutes on the Cannotrek, and 5 time trials with a mean time of 17.3 minutes on the Bianchondale. She calculates a $p \%$ confidence interval for $\mu _ { C } - \mu _ { B }$.

What would be the largest value of $p$ that would lead Branwen to base her purchasing decision on the time trials, without considering other factors?
State an assumption you have made in part (a).

WJEC Further Unit 5 2024 June Q1

During practice sessions, a basketball coach makes his players run several 'line drills'.
1. He records the times taken, in seconds, by one of his players to run the first 'line drill' on a random sample of 8 practice sessions. The results are shown below.
  $\begin{array} { l l l l l l l l } 29.4 & 31.1 & 28.9 & 30.0 & 29.9 & 30.4 & 29.7 & 30.2 \end{array}$
  Assuming that these data come from a normal distribution with mean $\mu$ and variance $0 \cdot 6$, calculate a $95 \%$ confidence interval for $\mu$.
2. State the two ways in which the method used to calculate the confidence interval in part (a) would change if the variance were unknown.
3. During a practice session, a player recorded a mean time of 35.6 seconds for 'line drills'.
  1. Give a reason why this player may not be the same as the player in part (a).
  2. Give a reason why this player could be the same as the player in part (a).
4. In country $A$, the median daily caffeine intake per student who drinks coffee is 120 mg . A university professor who oversees a foreign exchange programme believes that students visiting from country B drink more coffee and therefore have a greater daily caffeine intake from coffee.
On a randomly chosen day, the caffeine intake, in mg , from coffee consumption by each of 15 randomly selected students from country B is given below.
136 149 202 0 110 0 100 180
0 187 0 0 138 197 115
The professor suspects that the students with zero caffeine intake do not drink coffee, and decides to ignore those students and instead focus on the coffee-drinking students.
Conduct an appropriate Wilcoxon test at a significance level as close to $5 \%$ as possible. State your conclusion in context.
State one limitation of this investigation.

WJEC Further Unit 5 2024 June Q3

3. Tony runs a pie stand that sells two types of pie outside a football ground. He wants to estimate the probability that a customer will buy a steak pie rather than a vegetable pie. He conducts a survey by randomly selecting customers and recording their choice of pie. When he feels he has enough data, he notes that 55 customers bought steak pies and 25 bought vegetable pies.

Calculate an approximate $90 \%$ confidence interval for $p$, the probability that a randomly selected customer buys a steak pie.
Suppose that Tony carries out 50 such surveys and calculates $90 \%$ confidence intervals for each survey. Determine the expected number of these confidence intervals that would contain the true value of $p$.

WJEC Further Unit 5 2024 June Q4

4. The sports performance director at a university wishes to investigate whether there is a difference in the means of the specific gravities of blood of cyclists and runners. She models the distribution of specific gravity for cyclists as $\mathrm { N } \left( \mu _ { X } , 8 ^ { 2 } \right)$ and for runners as $\mathrm { N } \left( \mu _ { Y } , 10 ^ { 2 } \right)$.

State suitable hypotheses for this investigation.
The mean specific gravity of blood of a random sample of 40 cyclists from the university was 1063. The mean specific gravity of blood of a random sample of 40 runners from the same university was 1060.
Calculate and interpret the $p$-value for the data.
Suppose now that both samples were of size $n$, instead of 40. Find the least value of $n$ that would ensure that an observed difference of 3 in the mean specific gravities would be significant at the $1 \%$ level.

WJEC Further Unit 5 2024 June Q5

5. The probability density function of the continuous random variable $X$ is given by $$\begin{array} { l l } f ( x ) = \frac { 3 x ^ { 2 } } { \alpha ^ { 3 } } & \text { for } 0 \leqslant x \leqslant \alpha
f ( x ) = 0 & \text { otherwise. } \end{array}$$ $\bar { X }$ is the mean of a random sample of $n$ observations of $X$.

1. Show that $U = \frac { 4 \bar { X } } { 3 }$ is an unbiased estimator for $\alpha$.
2. If $\alpha$ is an integer, what is the smallest value of $n$ that gives a rational value for the standard error of $U$ ?
$\quad \bar { X } _ { 1 }$ and $\bar { X } _ { 2 }$ are the means of independent random samples of $X$, each of size $n$. The estimator $V = 4 \bar { X } _ { 1 } - \frac { 8 } { 3 } \bar { X } _ { 2 }$ is also an unbiased estimator for $\alpha$.
1. Show that $\frac { \operatorname { Var } ( U ) } { \operatorname { Var } ( V ) } = \frac { 1 } { 13 }$.
2. Hence state, with a reason, which of $U$ or $V$ is the better estimator.

WJEC Further Unit 5 2024 June Q6

6. Alana is a PhD student researching language acquisition. She gives one group of randomly selected participants, Group A, 4 minutes to memorise 40 words that are similar in meaning. She gives a different, randomly selected group of participants, Group B, 4 minutes to memorise 40 words that are different in meaning. Alana believes that the students in Group B will do better than the students in Group A. The following results are the number of words recalled on testing the students from the two groups.

Group A	32	8	24	16	10	20	22	18	23	21	26	14
Group B	30	29	11	25	38	36	28	12	17

Conduct a Mann-Whitney U test at a significance level as close as possible to $5 \%$ to test Alana's belief.

WJEC Further Unit 5 2024 June Q7

7. A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.

The mass $W$, in kg , of wheat stored in each individual container is normally distributed with mean $\mu$ and standard deviation $0 \cdot 6$. Given that, for containers of wheat, $10 \%$ store less than 19 kg , find the value of $\mu$.
The mass $X$, in kg , of corn stored in each individual container is normally distributed with mean $20 \cdot 1$ and standard deviation $1 \cdot 2$.
Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg .
The mass $Y$, in kg, of einkorn stored in each individual container is normally distributed with mean $22 \cdot 2$ and standard deviation $1 \cdot 5$. The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.
Calculate the probability that the farmer's wife will move
1. the einkorn,
2. the corn.
The mass $E$, in kg , of emmer stored in each individual container is normally distributed with mean $10 \cdot 5$ and standard deviation $\sigma$. The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208 .
1. Find the value of $\sigma$ that the farmer's son used.
2. Explain why the value of $\sigma$ that he used is unreasonable.
  Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Further Unit 5 Specimen Q1

Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, $X$ minutes, is modelled by the normal distribution $\mathrm { N } \left( 32,4 ^ { 2 } \right)$. You may assume that the times taken to complete the crossword on successive days are independent.
1. (i) Find the upper quartile of $X$ and explain its meaning in context.
  (ii) Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes.
2. Belle also does the crossword every day and the time that she takes to complete the crossword, $Y$ minutes, is modelled by the normal distribution $\mathrm { N } \left( 18,2 ^ { 2 } \right)$. Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword.
3. A factory manufactures a certain type of string. In order to ensure the quality of the product, a random sample of 10 pieces of string is taken every morning and the breaking strength of each piece, in Newtons, is measured. One morning, the results are as follows.
$$\begin{array} { l l l l l l l l l l } 68 \cdot 1 & 70 \cdot 4 & 68 \cdot 6 & 67 \cdot 7 & 71 \cdot 3 & 67 \cdot 6 & 68 \cdot 9 & 70 \cdot 2 & 68 \cdot 4 & 69 \cdot 8 \end{array}$$ You may assume that this is a random sample from a normal distribution with unknown mean $\mu$ and unknown variance $\sigma ^ { 2 }$.
Determine a 95\% confidence interval for $\mu$.
The factory manager is given these results and he asks 'Can I assume that the confidence interval that you have given me contains $\mu$ with probability 0.95 ?' Explain why the answer to this question is no and give a correct interpretation.

WJEC Further Unit 5 Specimen Q3

3. A motoring organisation wishes to determine whether or not the petrol consumption of two different car models $A$ and $B$ are the same. A trial is therefore carried out in which 6 cars of each model are given 10 litres of petrol and driven at a predetermined speed around a track until the petrol is used up. The distances travelled, in miles, are shown below

Model A:	86.3	84.2	85.8	83.1	84.7	85.3
Model B:	84.9	85.9	84.8	86.5	85.2	85.5

It is proposed to use a test with significance level $5 \%$ based on the Mann-Whitney statistic $U$.

State suitable hypotheses.
Find the critical region for the test.
Determine the value of $U$ for the above data and state your conclusion in context. You must justify your answer.

WJEC Further Unit 5 Specimen Q4

4. (a) In an opinion poll of 1800 people, 1242 said that they preferred red wine to white wine. Calculate a 95\% confidence interval for the proportion of people in the population who prefer red wine to white wine.
(b) In another opinion poll of 1000 people on the same subject, the following confidence interval was calculated.
[0pt] [0.672, 0.732]. Determine

the number of people in the sample who stated that they prefer red wine to white wine,
the confidence level of the confidence interval, giving your answer as a percentage correct to three significant figures.

WJEC Further Unit 5 Specimen Q5

5. A new species of animal has been found on an uninhabited island. A zoologist wishes to investigate whether or not there is a difference in the mean weights of males and females of the species. She traps some of the animals and weighs them with the following results.

Males (kg)	$5 \cdot 3,4 \cdot 6,5 \cdot 2,4 \cdot 5,4 \cdot 3,5 \cdot 5,5 \cdot 0,4 \cdot 8$
Females (kg)	$4 \cdot 9,5 \cdot 0,4 \cdot 1,4 \cdot 6,4 \cdot 3,5 \cdot 3,4 \cdot 2,4 \cdot 5,4 \cdot 8,4 \cdot 9$

You may assume that these are random samples from normal populations with a common standard deviation of 0.5 kg .

State suitable hypotheses for this investigation.
Determine the $p$-value of these results and state your conclusion in context.

WJEC Further Unit 5 Specimen Q6

6. A medical student is investigating two different methods, A and B , of measuring a patient's blood pressure. He believes that Method B gives, on average, a higher reading than Method A so he defines the following hypotheses.
$H _ { 0 }$ : There is on average no difference in the readings obtained using Methods A and B;
$H _ { 1 }$ : The reading obtained using Method B is on average higher than the reading obtained using Method A. He selects 10 patients at random and he measures their blood pressures using both methods. He obtains the following results.

Patient	A	B	C	D	E	F	G	H	I	J
Method A	121	133	119	142	151	139	161	148	151	125
Method B	126	131	127	152	145	151	157	155	160	126

Carry out an appropriate Wilcoxon signed rank test on this data set, using a 5\% significance level.
State what conclusion the medical student should reach, justifying your answer.

Student	A	B	C	D	E	F	G	H
Experienced Examiner	108	109	92	95	145	148	134	120
Trainee	114	116	95	93	137	144	133	110

Males (kg)	\(5 \cdot 3,4 \cdot 6,5 \cdot 2,4 \cdot 5,4 \cdot 3,5 \cdot 5,5 \cdot 0,4 \cdot 8\)
Females (kg)	\(4 \cdot 9,5 \cdot 0,4 \cdot 1,4 \cdot 6,4 \cdot 3,5 \cdot 3,4 \cdot 2,4 \cdot 5,4 \cdot 8,4 \cdot 9\)

136	149	202	0	110	0	100	180
0	187	0	0	138	197	115

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