Questions Further Unit 5 (36 questions)

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WJEC Further Unit 5 2024 June Q4
11 marks Standard +0.3
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 \(N\left(\mu_c, 8^2\right)\) and for runners as \(N\left(\mu_r, 10^2\right)\).
  1. State suitable hypotheses for this investigation. [1]
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.
  1. Calculate and interpret the \(p\)-value for the data. [6]
  2. 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. [4]
WJEC Further Unit 5 2024 June Q5
19 marks Challenging +1.8
The probability density function of the continuous random variable \(X\) is given by $$f(x) = \frac{3x^2}{\alpha^3} \quad \text{for } 0 \leq x \leq \alpha$$ $$f(x) = 0 \quad \text{otherwise.}$$ \(\overline{X}\) is the mean of a random sample of \(n\) observations of \(X\).
    1. Show that \(U = \frac{4\overline{X}}{3}\) is an unbiased estimator for \(\alpha\). [5]
    2. If \(\alpha\) is an integer, what is the smallest value of \(n\) that gives a rational value for the standard error of \(U\)? [9]
  2. \(\overline{X}_1\) and \(\overline{X}_2\) are the means of independent random samples of \(X\), each of size \(n\). The estimator \(V = 4\overline{X}_1 - \frac{8}{3}\overline{X}_2\) is also an unbiased estimator for \(\alpha\).
    1. Show that \(\frac{\text{Var}(U)}{\text{Var}(V)} = \frac{1}{13}\). [4]
    2. Hence state, with a reason, which of \(U\) or \(V\) is the better estimator. [1]
WJEC Further Unit 5 2024 June Q6
6 marks Standard +0.8
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 A32824161020221823212614
Group B302911253836281217
Conduct a Mann-Whitney U test at a significance level as close as possible to 5\% to test Alana's belief. [6]
WJEC Further Unit 5 2024 June Q7
19 marks Standard +0.8
A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.
  1. The mass \(W\), in kg, of wheat stored in each individual container is normally distributed with mean \(\mu\) and standard deviation 0.6. Given that, for containers of wheat, 10\% store less than 19 kg, find the value of \(\mu\). [3]
The mass \(X\), in kg, of corn stored in each individual container is normally distributed with mean 20.1 and standard deviation 1.2.
  1. Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg. [3]
The mass \(Y\), in kg, of einkorn stored in each individual container is normally distributed with mean 22.2 and standard deviation 1.5. 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.
  1. Calculate the probability that the farmer's wife will move
    1. the einkorn,
    2. the corn. [5]
  2. The mass \(E\), in kg, of emmer stored in each individual container is normally distributed with mean 10.5 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. [8]
WJEC Further Unit 5 Specimen Q1
13 marks Standard +0.8
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 \(\mathrm{N}(32, 4^2)\). You may assume that the times taken to complete the crossword on successive days are independent.
    1. Find the upper quartile of \(X\) and explain its meaning in context.
    2. Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes. [7]
  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 \(\mathrm{N}(18, 2^2)\). 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. [6]
WJEC Further Unit 5 Specimen Q2
11 marks Standard +0.3
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. $$68.1 \quad 70.4 \quad 68.6 \quad 67.7 \quad 71.3 \quad 67.6 \quad 68.9 \quad 70.2 \quad 68.4 \quad 69.8$$ You may assume that this is a random sample from a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma^2\).
  1. Determine a 95% confidence interval for \(\mu\). [9]
  2. 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. [2]
WJEC Further Unit 5 Specimen Q3
9 marks Challenging +1.2
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: \(86.3 \quad 84.2 \quad 85.8 \quad 83.1 \quad 84.7 \quad 85.3\) Model B: \(84.9 \quad 85.9 \quad 84.8 \quad 86.5 \quad 85.2 \quad 85.5\) It is proposed to use a test with significance level 5% based on the Mann-Whitney statistic \(U\).
  1. State suitable hypotheses. [2]
  2. Find the critical region for the test. [3]
  3. Determine the value of \(U\) for the above data and state your conclusion in context. You must justify your answer. [4]
WJEC Further Unit 5 Specimen Q4
12 marks Standard +0.3
  1. 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. [6]
  2. In another opinion poll of 1000 people on the same subject, the following confidence interval was calculated. \([0.672, 0.732]\). Determine
    1. the number of people in the sample who stated that they prefer red wine to white wine,
    2. the confidence level of the confidence interval, giving your answer as a percentage correct to three significant figures. [6]
WJEC Further Unit 5 Specimen Q5
10 marks Standard +0.3
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. \begin{align} \text{Males (kg)} &\quad 5.3, 4.6, 5.2, 4.5, 4.3, 5.5, 5.0, 4.8
\text{Females (kg)} &\quad 4.9, 5.0, 4.1, 4.6, 4.3, 5.3, 4.2, 4.5, 4.8, 4.9 \end{align} You may assume that these are random samples from normal populations with a common standard deviation of 0.5 kg.
  1. State suitable hypotheses for this investigation. [1]
  2. Determine the \(p\)-value of these results and state your conclusion in context. [9]
WJEC Further Unit 5 Specimen Q6
8 marks Standard +0.8
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.
PatientABCDEFGHIJ
Method A121133119142151139161148151125
Method B126131127152145151157155160126
  1. Carry out an appropriate Wilcoxon signed rank test on this data set, using a 5% significance level. [6]
  2. State what conclusion the medical student should reach, justifying your answer. [2]
WJEC Further Unit 5 Specimen Q7
17 marks Challenging +1.3
The discrete random variable \(X\) has the following probability distribution, where \(\theta\) is an unknown parameter belonging to the interval \(\left(0, \frac{1}{3}\right)\).
Value of \(X\)135
Probability\(\theta\)\(1 - 3\theta\)\(2\theta\)
  1. Obtain an expression for \(E(X)\) in terms of \(\theta\) and show that $$\text{Var}(X) = 4\theta(3 - \theta).$$ [4] In order to estimate the value of \(\theta\), a random sample of \(n\) observations on \(X\) was obtained and \(\bar{X}\) denotes the sample mean.
    1. Show that $$V = \frac{\bar{X} - 3}{2}$$ is an unbiased estimator for \(\theta\).
    2. Find an expression for the variance of \(V\). [4]
  3. Let \(Y\) denote the number of observations in the random sample that are equal to 1. Show that $$W = \frac{Y}{n}$$ is an unbiased estimator for \(\theta\) and find an expression for \(\text{Var}(W)\). [5]
  4. Determine which of \(V\) and \(W\) is the better estimator, explaining your method clearly. [4]