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.