In a science investigation into energy conservation in the home, a student is collecting data on the time taken for an electric kettle to boil as the volume of water in the kettle is varied. The student's data are shown in the table below, where $v$ litres is the volume of water in the kettle and $t$ seconds is the time taken for the kettle to boil (starting with the water at room temperature in each case). Also shown are summary statistics and a scatter diagram on which the regression line of $t$ on $v$ is drawn.

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$v$ & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\
\hline
$t$ & 44 & 78 & 114 & 156 & 172 \\
\hline
\end{tabular}
\end{center}

$n = 5$, $\Sigma v = 3.0$, $\Sigma t = 564$, $\Sigma v^2 = 2.20$, $\Sigma vt = 405.2$.

\includegraphics{figure_1}

\begin{enumerate}[label=(\roman*)]
\item Calculate the equation of the regression line of $t$ on $v$, giving your answer in the form $t = a + bv$. [5]

\item Use this equation to predict the time taken for the kettle to boil when the amount of water which it contains is
\begin{enumerate}[label=(\Alph*)]
\item 0.5 litres,
\item 1.5 litres.
\end{enumerate}
Comment on the reliability of each of these predictions. [4]

\item In the equation of the regression line found in part (i), explain the role of the coefficient of $v$ in the relationship between time taken and volume of water. [2]

\item Calculate the values of the residuals for $v = 0.8$ and $v = 1.0$. [4]

\item Explain how, on a scatter diagram with the regression line drawn accurately on it, a residual could be measured and its sign determined. [3]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI S2 2007 Q1 [18]}}