| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Find unknown values from regression |
| Difficulty | Standard +0.8 This is a multi-part Further Maths statistics question requiring understanding of regression line properties (line passes through means), solving simultaneous equations from the least squares formula, calculating correlation coefficient, and understanding the effect of linear transformations on regression parameters. While systematic, it requires more conceptual depth than standard A-level statistics and involves algebraic manipulation beyond routine application. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression |
| \(x\) | 1 | 2 | 4 | 2 | 6 |
| \(y\) | 2 | 3 | 6 | \(p\) | \(q\) |
The regression line of $y$ on $x$ obtained from a random sample of five pairs of values of $x$ and $y$ is
$$y = 2.5 x - 1.5$$
The data is given in the following table.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | }
\hline
$x$ & 1 & 2 & 4 & 2 & 6 \\
\hline
$y$ & 2 & 3 & 6 & $p$ & $q$ \\
\hline
\end{tabular}
\end{center}
(i) Show that $p + q = 19$.\\
(ii) Find the values of $p$ and $q$.\\
(iii) Determine the value of the product moment correlation coefficient for this sample.\\
(iv) It is later discovered that the values of $x$ given in the table have each been divided by 10 (that is, the actual values are $10,20,40,20,60$ ). Without any further calculation, state
\begin{enumerate}[label=(\alph*)]
\item the equation of the actual regression line of $y$ on $x$,
\item the value of the actual product moment correlation coefficient.
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2011 Q10 OR}}