| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Find unknown values from regression |
| Difficulty | Standard +0.8 This question requires understanding that all points must satisfy the regression line equation AND that the regression line passes through (x̄, ȳ). Students must set up and solve simultaneous equations using these constraints, then calculate PMCC using sums of squares. While systematic, it demands solid understanding of regression theory beyond formula application, making it moderately challenging for Further Maths students. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09c Calculate regression line |
| \(x\) | 1 | 2 | 3 | 4 | 5 |
| \(y\) | 4 | \(p\) | \(q\) | 2 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9(i) | Σ x = 15, Σ y = 7 + p + q, Σ xy = 17 + 2p + 3q | |
| Σ x2 = 55, [Σ y2 = 21 + p2 + q2] | M1 | Find required summations |
| Answer | Marks |
|---|---|
| xy | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| xy xx | M1 A1 | Find p from gradient in eqn. of regression line |
| (7 + p + q) / 5 = – 0⋅5 × 15/5 + 3⋅5 q = 2 | M1 A1 | Find q from means and regression line |
| Answer | Marks | Guidance |
|---|---|---|
| 9(ii) | Σ y = 10, Σ y2 = 26, S = 26 – 102/5 = 6 | |
| yy | M1 | Find S |
| Answer | Marks | Guidance |
|---|---|---|
| xy xx yy | M1 | Find correlation coefficient r |
| r = – 0⋅645[5] [allow – 0⋅646] | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(i) ---
9(i) | Σ x = 15, Σ y = 7 + p + q, Σ xy = 17 + 2p + 3q
Σ x2 = 55, [Σ y2 = 21 + p2 + q2] | M1 | Find required summations
S = 55 – 152 / 5 = 10 and
xx
S = 17 + 2p + 3q – 15 × (7 + p + q) / 5 = – 4 – p
xy | M1 A1
– 0⋅5 = S / S = (– 4 – p) / 10 p = 1
xy xx | M1 A1 | Find p from gradient in eqn. of regression line
(7 + p + q) / 5 = – 0⋅5 × 15/5 + 3⋅5 q = 2 | M1 A1 | Find q from means and regression line
7
--- 9(ii) ---
9(ii) | Σ y = 10, Σ y2 = 26, S = 26 – 102/5 = 6
yy | M1 | Find S
yy
r = S / √(S S ) = – 5 / √(10 × 6)
xy xx yy | M1 | Find correlation coefficient r
r = – 0⋅645[5] [allow – 0⋅646] | A1
3
Question | Answer | Marks | GuidanceA random sample of five pairs of values of $x$ and $y$ is taken from a bivariate distribution. The values are shown in the following table, where $p$ and $q$ are constants.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$y$ & 4 & $p$ & $q$ & 2 & 1 \\
\hline
\end{tabular}
The equation of the regression line of $y$ on $x$ is $y = -0.5x + 3.5$.
\begin{enumerate}[label=(\roman*)]
\item Find the values of $p$ and $q$.
[7]
\item Find the value of the product moment correlation coefficient.
[3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2019 Q9 [10]}}