Questions Further Unit 2 (35 questions)

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WJEC Further Unit 2 2024 June Q3
  1. A company makes bags. The table below shows the number of bags sold on a random sample of 50 days. A manager believes that the number of bags sold per day can be modelled by the Poisson distribution with mean \(2 \cdot 2\).
Number of
bags sold
012345 or more
Frequency71011967
  1. Carry out a chi-squared goodness of fit test, using a \(10 \%\) significance level.
  2. A chi-squared goodness of fit test for the Poisson distribution with mean \(2 \cdot 5\) is conducted. This uses the same number of degrees of freedom as part (a) and gives a test statistic of 1.53 . State, with a reason, which of these two Poisson models is a better fit for the data.
WJEC Further Unit 2 2024 June Q4
4. An author poses the following question: Does using cash for transactions affect people's financial behaviour?
She collects data on 'Cash transactions as a \% of all transactions' and 'Household debt as a \(\%\) of net disposable income' from a random sample of 25 countries. The table below shows the data she collected. There are missing values, \(p\) and \(q\), for Malta and Denmark respectively.
CountryCash transactions as a \% of all transactions \(\boldsymbol { x }\)Household debt as a \% of net disposable income \(\boldsymbol { y }\)CountryCash transactions as a \% of all transactions \(\boldsymbol { x }\)Household debt as a \% of net disposable income \(\boldsymbol { y }\)
Malta92\(p\)France68120
Mexico90-14Luxembourg64177
Greece88107Belgium63113
Spain87110Finland54137
Italy8687Estonia4882
Austria8591The Netherlands45247
Portugal81131UK42147
Slovenia8056Australia37214
Germany8095USA32109
Ireland79154Sweden20187
Slovakia7874South Korea14182
Lithuania7546Denmark\(q\)261
Latvia7143
The summary statistics and scatter diagram below are for the other 23 countries. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Household debt versus Cash transactions} \includegraphics[alt={},max width=\textwidth]{1538fa56-5b61-40ec-bb02-cf1ed9da5eb0-13_664_1296_511_379}
\end{figure} $$\begin{gathered} \sum x = 1467 \sum y = 2695 \sum x ^ { 2 } = 105073 \quad S _ { x x } = 11503 \cdot 91304 \quad S _ { y y } = 78669 \cdot 30435
\sum y ^ { 2 } = 394453 \sum x y = 152999 \quad S _ { x y } = - 18895 \cdot 13043 \end{gathered}$$
  1. Using the summary statistics for the 23 countries, calculate and interpret Pearson's product moment correlation coefficient.
  2. Calculate the equation of the least squares regression line of Household debt as a \% of net disposable income \(( y )\) on Cash transactions as a \% of all transactions ( \(x\) ). The regression line \(x\) on \(y\) is given below. $$x = - 0 \cdot 24 y + 91 \cdot 92$$
  3. By selecting the appropriate regression line in each case, estimate the values of \(p\) and \(q\) in the table.
  4. Comment on the reliability of your answers in part (c).
  5. Interpret the negative value of \(y\) for Mexico.
WJEC Further Unit 2 2024 June Q5
5. Lily is interested in the relationship between the way in which students learned Welsh and their attitude towards the Welsh language. Students were categorised as having learned Welsh in one of three ways:
  • from one Welsh-speaking parent/carer at home,
  • from two Welsh-speaking parents/carers at home,
  • at school only, for those with no Welsh-speaking parents/carers at home.
The students were asked to rate their attitude towards the Welsh language from 'Very negative' to 'Very positive'. The following data for a random sample of 253 students were collected as part of a project.
Learned Welsh
AttitudeFrom two parents/carersFrom one parent/carerAt school onlyTotal
Very negative2143046
Slightly negative4202145
Neutral1217837
Slightly positive21191151
Very positive25212874
Total649198253
Lily intends to carry out a chi-squared test for independence at the \(5 \%\) level. She produces the following tables which are incomplete.
Expected FrequenciesLearned Welsh
AttitudeFrom two parents/carersFrom one parent/carerAt school only
Very negative11.6416.5517.82
Slightly negative11.3816.1917.43
Neutral9.3613.3114.33
Slightly positive12.9018.3419.75
Very positiveF26.6228.66
Chi-Squared ContributionsLearned Welsh
AttitudeFrom two parents/carersFrom one parent/carerAt school only
Very negative7.980.398.33
Slightly negative\(4 \cdot 79\)0.900.73
Neutral\(0 \cdot 74\)1.02G
Slightly positive5.080.023.88
Very positive2.111.190.02
Total20.703.52H
  1. Calculate the values of \(F , G\) and \(H\).
  2. Carry out Lily's chi-squared test for independence at the \(5 \%\) level.
  3. By referring to the figures in the tables on pages 16 and 17, give two comments on the relationship between the way students learned Welsh and their attitude towards the Welsh language.
WJEC Further Unit 2 2024 June Q6
6. Penelope makes 8 cakes per week. Each cake costs \(\pounds 20\) to make and sells for \(\pounds 60\). She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is \(0 \cdot 3\). She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is \(\pounds 206\). Construct a probability distribution for the weekly profit.
Additional page, if required. number Write the question number(s) in the left-hand margin. 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 2 Specimen Q1
  1. 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 \(W = X Y\). Find the value of
    1. \(\quad \mathrm { E } ( W )\),
    2. \(\quad \operatorname { Var } ( W )\).
    3. The queueing times, \(T\) minutes, of customers at a local Post Office are modelled by the probability density function
    $$\begin{array} { l l } f ( t ) = \frac { 1 } { 2500 } t \left( 100 - t ^ { 2 } \right) & \text { for } 0 \leq t \leq 10
    f ( t ) = 0 & \text { otherwise. } \end{array}$$
  2. Determine the mean queueing time.
    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.
WJEC Further Unit 2 Specimen Q3
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.
  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.
  3. Explain why it might not have been appropriate to use Pearson's product moment correlation coefficient to test association using this data set.
WJEC Further Unit 2 Specimen Q4
4. 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.
VariableEntry standardsStudent satisfactionGraduate prospectsResearch quality
Entry standards1
Student satisfaction-0.0301
Graduate prospects0.7720.2361
Research quality0.8660.0660.8271
  • Calculate the equation of the least squares regression line to predict 'Entry standards'( \(y )\) from 'Research quality'( \(x\) ), given the summary statistics: $$\sum x = 22.24 , \sum y = 2522 , S _ { x x } = 1.0542 , S _ { y y } = 20193.5 , S _ { x y } = 122.72 .$$
  • 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.
    •
    WJEC Further Unit 2 Specimen Q5
    5. 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.
    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.
    WJEC Further Unit 2 Specimen Q6
    6. Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5 t\).
    1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m.
    2. A customer arrives at exactly 11 a.m.
      1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P ( T > t ) = \mathrm { e } ^ { - 0.5 t }$$
      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.
    WJEC Further Unit 2 Specimen Q7
    7. 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)Total
    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)
        Parent knows
        password
        		\(\mathbf { 1 3 }\)\(\mathbf { 1 4 }\)\(\mathbf { 1 5 }\)
        Yes62.7978.7176.50
        No99.2996.50
      3. Obtain the two chi-squared contributions that are missing from the table below. Chi-squared contributions:
        Age (years)
        Parent knows
        password
        		\(\mathbf { 1 3 }\)\(\mathbf { 1 4 }\)\(\mathbf { 1 5 }\)
        Yes0.1751.180
        No2.2030.935
        The following output was obtained from the statistical package that was used to undertake the analysis: $$\text { Pearson chi-squared } ( 2 ) = 7.409 \quad p \text {-value } = 0.0305$$
      4. Indicate how the degrees of freedom have been calculated for the chi-squared statistic.
      5. Interpret the output obtained from the statistical test in terms of the initial hypotheses.
    2. Comment on the nature of the association observed, based on the contributions to the test statistic calculated in (a).