Moderate -0.8 This is a straightforward S1 regression question requiring standard formula application (calculating regression line from summary statistics), simple substitution for predictions, and basic commentary on extrapolation. The 'show that' in part (a) guides students to the answer, and all parts follow routine textbook procedures with no problem-solving or novel insight required. Easier than average due to its mechanical nature and scaffolding.
Some values, \((x, y)\), of a bivariate distribution are plotted on a scatter diagram and a regression line is to be drawn. Explain how to decide whether the regression line of \(y\) on \(x\) or the regression line of \(x\) on \(y\) is appropriate. [2]
In an experiment the temperature, \(x\) °C, of a rod was gradually increased from 0 °C, and the extension, \(y\), was measured nine times at 50 °C intervals. The results are summarised below.
\(n = 9\) \quad \(\Sigma x = 1800\) \quad \(\Sigma y = 14.4\) \quad \(\Sigma x^2 = 510000\) \quad \(\Sigma y^2 = 32.6416\) \quad \(\Sigma xy = 4080\)
Show that the gradient of the regression line of \(y\) on \(x\) is 0.008 and find the equation of this line. [4]
Use your equation to estimate the temperature when the extension is 2.5 mm. [1]
Use your equation to estimate the extension for a temperature of \(-50\) °C. [1]
Comment on the meaning and the reliability of your estimate in part (c). [2]
\begin{enumerate}[label=(\roman*)]
\item Some values, $(x, y)$, of a bivariate distribution are plotted on a scatter diagram and a regression line is to be drawn. Explain how to decide whether the regression line of $y$ on $x$ or the regression line of $x$ on $y$ is appropriate. [2]
\item In an experiment the temperature, $x$ °C, of a rod was gradually increased from 0 °C, and the extension, $y$, was measured nine times at 50 °C intervals. The results are summarised below.
$n = 9$ \quad $\Sigma x = 1800$ \quad $\Sigma y = 14.4$ \quad $\Sigma x^2 = 510000$ \quad $\Sigma y^2 = 32.6416$ \quad $\Sigma xy = 4080$
\begin{enumerate}[label=(\alph*)]
\item Show that the gradient of the regression line of $y$ on $x$ is 0.008 and find the equation of this line. [4]
\item Use your equation to estimate the temperature when the extension is 2.5 mm. [1]
\item Use your equation to estimate the extension for a temperature of $-50$ °C. [1]
\item Comment on the meaning and the reliability of your estimate in part (c). [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2010 Q3 [10]}}