Questions Further Unit 2 (40 questions)

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WJEC Further Unit 2 2018 June Q6
10 marks Moderate -0.3
A student, considering options for the future, collects data on education and salary. The table below shows the highest level of education attained and the salary bracket of a random sample of 664 people.
Fewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualificationTotal
Less than £20 0001832202810108
£20 000 to £60 000509511215550462
More than £60 0003222935594
Total7114916121865664
By conducting a chi-squared test for independence, the student investigates the relationship between the highest level of education attained and the salary earned.
  1. State the null and alternative hypotheses. [1]
  2. The table below shows the expected values. Calculate the value of \(k\). [2]
    Expected valuesFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £20 000\(k\)24·2326·1935·4610·57
    £20 000 to £60 00049·40103·67112·02151·6845·23
    More than £60 00010·0521·0922·7930·869·20
  3. The following computer output is obtained. Calculate the values of \(m\) and \(n\). [2]
    Chi Squared ContributionsFewer than 5 GCSE5 or more GCSE3 A LevelsUniversity degreePost graduate qualification
    Less than £20 0003·604530799\(m\)1·461651·56860·03098
    £20 000 to £60 0000·0072727350·725354E-060·072640·50396
    More than £60 0004·9466198630·038971·690810·55498\(n\)
    X-squared = 19·61301, df = 8, p-value = 0·0119
    1. Without carrying out any further calculations, explain how X-squared = 19·61301 (the \(\chi^2\) test statistic) was calculated. [2]
    2. Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c). [2]
  5. The student says that the highest levels of education lead to the highest paying jobs. Comment on the accuracy of the student's statement. [1]
WJEC Further Unit 2 2018 June Q7
7 marks Moderate -0.3
A university professor conducted some research into factors that affect job satisfaction. The four factors considered were Interesting work, Good wages, Job security and Appreciation of work done. The professor interviewed workers at 14 different companies and asked them to rate their companies on each of the factors. The workers' ratings were averaged to give each company a score out of 5 on each factor. Each company was also given a score out of 100 for Job satisfaction. The following graph shows the part of the research concerning Job Satisfaction versus Interesting work. \includegraphics{figure_2}
  1. Calculate the equation of the least squares regression line of Job satisfaction (\(y\)) on Interesting work (\(x\)), given the following summary statistics. [5] \(\sum x = 46 \cdot 2\), \quad \(\sum y = 898\), \quad \(S_{xx} = 3 \cdot 48\) \(S_{xy} = 49 \cdot 45\), \quad \(S_{yy} = 1437 \cdot 714\), \quad \(n = 14\)
  2. Give two reasons why it would be inappropriate for the professor to use this equation to calculate the score for Interesting work from a Job satisfaction score of 90. [2]
WJEC Further Unit 2 2023 June Q1
7 marks Moderate -0.8
The random variable \(X\) has mean 17 and variance 64. The independent random variable \(Y\) has mean 10 and variance 16. Find the value of
  1. E\((4Y - 2X + 1)\), [2]
  2. Var\((4Y - 5X + 3)\), [2]
  3. E\((X^2 Y)\). [3]
WJEC Further Unit 2 2023 June Q2
8 marks Moderate -0.3
For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \((19, 20)\).
  1. Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\). [6]
  2. State two reasons why your prediction in part (a) may not be reliable. [2]
WJEC Further Unit 2 2023 June Q3
11 marks Standard +0.3
It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.
  1. Find the probability that the lifetime of a randomly selected hair dryer is between 1·8 and 2·5 years. [4]
  2. Given that 20% of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\). [3]
  3. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers. [3]
  4. State one assumption that you have made in part (c). [1]
WJEC Further Unit 2 2023 June Q4
12 marks Standard +0.3
A continuous random variable \(X\) has cumulative distribution function \(F\) given by $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{1}{4}x & \text{for } 0 \leqslant x \leqslant 2, \\ \frac{1}{480}x^4 + \frac{7}{15} & \text{for } 2 < x \leqslant b, \\ 1 & \text{for } x > b. \end{cases}$$
  1. Show that \(b = 4\). [2]
  2. Find P\((X \leqslant 2 \cdot 5)\). [2]
  3. Write down the value of the lower quartile of \(X\). [1]
  4. Find the value of the upper quartile of \(X\). [3]
  5. Find, correct to three significant figures, the value of \(k\) that satisfies the equation P\((X > 3 \cdot 5) = \text{P}(X < k)\). [4]
WJEC Further Unit 2 2023 June Q5
12 marks Standard +0.3
  1. Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient. [2]
  2. A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors.
    TractorFarmer's rankPTO (horsepower)Price (£1000s)
    A177·580
    B687·945
    C553·047
    D441·053
    E2112·060
    F390·061
    Spearman's rank correlation coefficient between the farmer's ranks and the price is 0·9429.
    1. Test at the 5% significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm. [4]
    2. Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO. [4]
    3. How should the tractor salesman interpret the results in (i) and (ii)? [2]
WJEC Further Unit 2 2023 June Q6
20 marks Standard +0.8
A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that 10% of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution B\((20, 0 \cdot 1)\). The numbers of boats that were not taken for 110 randomly selected days are given below.
Number of boats not taken012345 or more
Frequency1035292583
  1. State suitable hypotheses to carry out a goodness of fit test. [1]
  2. Here is part of the table for a \(\chi^2\) goodness of fit test on the data.
    Number of boats not taken012345 or more
    Observed1035292583
    Expected\(f\)29·72\(g\)20·919·884·75
    1. Calculate the values of \(f\) and \(g\).
    2. By completing the test, give the conclusion the manager should reach. [10]
The cost of hiring a boat is £15. Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give £20 (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.
    1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
    2. Hence calculate the expected daily net income for the company following the manager's decision. [8]
  2. Is the manager justified in her decision? Give a reason for your answer. [1]
WJEC Further Unit 2 Specimen Q1
7 marks Challenging +1.8
The random variable \(X\) has mean14 and standard deviation 5. The independent random variable \(Y\) has mean 12 and standard deviation 3. The random variable \(W\) is given by \(W = XY\). Find the value of
  1. E(W), [1]
  2. Var(W). [6]
WJEC Further Unit 2 Specimen Q2
13 marks Standard +0.3
The queueing times, \(T\) minutes, of customers at a local Post Office are modelled by the probability density function $$f(t) = \frac{1}{2500}t(100-t^2) \quad \text{for } 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise.}$$
  1. Determine the mean queueing time. [3]
    1. Find the cumulative distribution function, \(F(t)\), of \(T\).
    2. Find the probability that a randomly chosen customer queues for more than 5 minutes.
    3. Find the median queueing time. [10]
WJEC Further Unit 2 Specimen Q3
9 marks Standard +0.3
A class of 8 students sit examinations in History and Geography. The marks obtained by these students are given below.
StudentABCDEFGH
History mark7359834957826760
Geography mark5551585944664967
  1. Calculate Spearman's rank correlation coefficient for this data set. [6]
  2. Hence determine whether or not, at the 5% significance level, there is evidence of a positive association between marks in History and marks in Geography. [2]
  3. Explain why it might not have been appropriate to use Pearson's product moment correlation coefficient to test association using this data set. [1]
WJEC Further Unit 2 Specimen Q4
9 marks Moderate -0.8
A year 12 student wishes to study at a Welsh university. For a randomly chosen year between 2000 and 2017 she collected data for seven universities in Wales from the Complete University Guide website. The data are for the variables: • 'Entry standards' – the average UCAS tariff score of new undergraduate students; • 'Student satisfaction' – a measure of student views of the teaching quality at the university taken from the National Student Survey (maximum 5); • 'Graduate prospects' – a measure of the employability of a university's first degree graduates (maximum 100); • 'Research quality' – a measure of the quality of the research undertaken in the university (maximum 4).
  1. Pearson's product-moment correlation coefficients, for each pairing of the four variables, are shown in the table below. Discuss the correlation between graduate prospects and the other three variables. [2]
    VariableEntry standardsStudent satisfactionGraduate prospectsResearch quality
    Entry standards1
    Student satisfaction-0.0301
    Graduate prospects0.7720.2361
    Research quality0.8660.0660.8271
  2. Calculate the equation of the least squares regression line to predict 'Entry standards'(y) from 'Research quality'(x), given the summary statistics: $$\sum x = 22.24, \sum y = 2522, S_{xx} = 1.0542, S_{xy} = 20193.5, S_{yy} = 122.72.$$ [5]
  3. The data for one of the Welsh universities are missing. This university has a research quality of 3.00. Use your equation to predict the entry standard for this university. [2]
WJEC Further Unit 2 Specimen Q5
10 marks Standard +0.3
The manager of a hockey team studies last season's results and puts forward the theory that the number of goals scored per match by her team can be modelled by a Poisson distribution with mean 2.0. The number of goals scored during the season are summarised below.
Goals scored01234 or more
Frequency61115108
  1. State suitable hypotheses to carry out a goodness of fit test. [1]
  2. Carry out a \(\chi^2\) goodness of fit test on this data set, using a 5% level of significance and draw a conclusion in context. [9]
WJEC Further Unit 2 Specimen Q6
10 marks Standard +0.3
Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5t\).
  1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m. [3]
  2. A customer arrives at exactly 11 a.m.
    1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P(T > t) = e^{-0.5t}.$$
    2. Hence find the probability density function, \(f(t)\), of \(T\).
    3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers. [7]
WJEC Further Unit 2 Specimen Q7
12 marks Moderate -0.5
The Pew Research Center's Internet Project offers scholars access to raw data sets from their research. One of the Pew Research Center's projects was on teenagers and technology. A random sample of American families was selected to complete a questionnaire. For each of their children, between and including the ages of 13 and 15, parents of these families were asked: Do you know your child's password for any of [his/her] social media accounts? Responses to this question were received from 493 families. The table below provides a summary of their responses.
Age (years)\multirow{2}{*}{Total}
\cline{2-4} Parent know password131415
Yes767567218
No66103106275
Total142178173493
  1. A test for significance is to be undertaken to see whether there is an association between whether a parent knows any of their child's social media passwords and the age of the child.
    1. Clearly state the null and alternative hypotheses.
    2. Obtain the expected value that is missing from the table below, indicating clearly how it is calculated from the data values given in the table above.
    Expected values:
    Age (years)
    \cline{2-4} Parent knows password131415
    Yes62.7978.7176.50
    No99.2996.50
    1. Obtain the two chi-squared contributions that are missing from the table below.
    Chi-squared contributions:
    Age (years)
    \cline{2-4} Parent knows password131415
    Yes0.1751.180
    No2.2030.935
    The following output was obtained from the statistical package that was used to undertake the analysis: Pearson chi-squared (2) = 7.409 \quad \(p\)-value = 0.0305
    1. Indicate how the degrees of freedom have been calculated for the chi-squared statistic.
    2. Interpret the output obtained from the statistical test in terms of the initial hypotheses. [10]
  2. Comment on the nature of the association observed, based on the contributions to the test statistic calculated in (a). [2]