Question 4 - A-Level Maths

WJEC Further Unit 2 2024 June — Question 4 12 marks

Exam Board	WJEC
Module	Further Unit 2 (Further Unit 2)
Year	2024
Session	June
Marks	12
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear regression
Type	Find unknown values from regression
Difficulty	Standard +0.8 This is a multi-step regression problem requiring students to work backwards from summary statistics to find missing values, involving understanding of how adding data points affects sums and regression calculations. While conceptually straightforward for Further Maths students familiar with regression formulas, it requires careful algebraic manipulation and understanding of how summary statistics change when data is added. More challenging than routine regression questions but not requiring novel insight.
Spec	5.08a Pearson correlation: calculate pmcc 5.09a Dependent/independent variables 5.09b Least squares regression: concepts 5.09c Calculate regression line

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.

Country	Cash transactions as a \% of all transactions $\boldsymbol { x }$	Household debt as a \% of net disposable income $\boldsymbol { y }$	Country	Cash transactions as a \% of all transactions $\boldsymbol { x }$	Household debt as a \% of net disposable income $\boldsymbol { y }$
Malta	92	$p$	France	68	120
Mexico	90	-14	Luxembourg	64	177
Greece	88	107	Belgium	63	113
Spain	87	110	Finland	54	137
Italy	86	87	Estonia	48	82
Austria	85	91	The Netherlands	45	247
Portugal	81	131	UK	42	147
Slovenia	80	56	Australia	37	214
Germany	80	95	USA	32	109
Ireland	79	154	Sweden	20	187
Slovakia	78	74	South Korea	14	182
Lithuania	75	46	Denmark	$q$	261
Latvia	71	43

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} $$\begin{gathered} \sum x = 1467 \sum y = 2695 \sum x ^ { 2 } = 105073 \quad S _ { x x } = 11503 \cdot 91304 \quad S _ { y y } = 78669 \cdot 30435 \\ \sum y ^ { 2 } = 394453 \sum x y = 152999 \quad S _ { x y } = - 18895 \cdot 13043 \end{gathered}$$

Using the summary statistics for the 23 countries, calculate and interpret Pearson's product moment correlation coefficient.
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. $$x = - 0 \cdot 24 y + 91 \cdot 92$$
By selecting the appropriate regression line in each case, estimate the values of $p$ and $q$ in the table.
Comment on the reliability of your answers in part (c).
Interpret the negative value of $y$ for Mexico.

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Question 4:

Answer	Marks
4	12

Question 4:
4 | 12

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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.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Country & Cash transactions as a \% of all transactions $\boldsymbol { x }$ & Household debt as a \% of net disposable income $\boldsymbol { y }$ & Country & Cash transactions as a \% of all transactions $\boldsymbol { x }$ & Household debt as a \% of net disposable income $\boldsymbol { y }$ \\
\hline
Malta & 92 & $p$ & France & 68 & 120 \\
\hline
Mexico & 90 & -14 & Luxembourg & 64 & 177 \\
\hline
Greece & 88 & 107 & Belgium & 63 & 113 \\
\hline
Spain & 87 & 110 & Finland & 54 & 137 \\
\hline
Italy & 86 & 87 & Estonia & 48 & 82 \\
\hline
Austria & 85 & 91 & The Netherlands & 45 & 247 \\
\hline
Portugal & 81 & 131 & UK & 42 & 147 \\
\hline
Slovenia & 80 & 56 & Australia & 37 & 214 \\
\hline
Germany & 80 & 95 & USA & 32 & 109 \\
\hline
Ireland & 79 & 154 & Sweden & 20 & 187 \\
\hline
Slovakia & 78 & 74 & South Korea & 14 & 182 \\
\hline
Lithuania & 75 & 46 & Denmark & $q$ & 261 \\
\hline
Latvia & 71 & 43 & \multicolumn{3}{|c|}{} \\
\hline
\end{tabular}
\end{center}

The summary statistics and scatter diagram below are for the other 23 countries.

\begin{figure}[h]
\begin{center}
\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{center}
\end{figure}

$$\begin{gathered}
\sum x = 1467 \sum y = 2695 \sum x ^ { 2 } = 105073 \quad S _ { x x } = 11503 \cdot 91304 \quad S _ { y y } = 78669 \cdot 30435 \\
\sum y ^ { 2 } = 394453 \sum x y = 152999 \quad S _ { x y } = - 18895 \cdot 13043
\end{gathered}$$
\begin{enumerate}[label=(\alph*)]
\item Using the summary statistics for the 23 countries, calculate and interpret Pearson's product moment correlation coefficient.
\item 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.

$$x = - 0 \cdot 24 y + 91 \cdot 92$$
\item By selecting the appropriate regression line in each case, estimate the values of $p$ and $q$ in the table.
\item Comment on the reliability of your answers in part (c).
\item Interpret the negative value of $y$ for Mexico.
\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 2 2024 Q4 [12]}}

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