Questions — WJEC (504 questions)

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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]