| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Standard +0.8 This is a multi-part statistics question requiring calculation of regression line, correlation coefficient, and hypothesis testing with critical values. While the calculations are systematic (sums, means, standard formulas), it involves multiple techniques, careful arithmetic with 10 data pairs, and part (iv) requires understanding of correlation testing and working backwards from significance levels—more demanding than typical A-level questions but still within standard Further Maths scope. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.09a Dependent/independent variables5.09c Calculate regression line |
| \(x\) | 4 | 6 | 6 | 8 | 2 | 7 | 12 | 14 | 9 | 5 |
| \(y\) | 2 | 4 | 6 | 8 | 6 | 10 | 9 | 8 | 6 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(S_{xy} = 513 - 73 \times \frac{64}{10} = 45.8\) | ||
| \(S_{xx} = 651 - \frac{73^2}{10} = 118.1\) | ||
| \(b = \frac{S_{xy}}{S_{xx}} = 0.388\) | M1 A1 | Calculate gradient \(b\) in \(y - \bar{y} = b(x-\bar{x})\) |
| \(y = \frac{64}{10} + 0.388\left(x - \frac{73}{10}\right) = 0.388x + 3.57\) or \(\frac{458x+4215}{1181}\) | M1, A1 | Find regression line of \(y\) on \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(S_{yy} = 462 - \frac{64^2}{10} = 52.4\) | ||
| \(r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} = 0.582\) | M1 A1 | Find correlation coefficient \(r\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = 7.45\) | B1 | Find \(y\) when \(x=10\) |
| Not reliable as value of \(r\) is small; or reliable since \(x=10\) is in range / or is near mean | B1 | State valid comment on reliability |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Require one-tail \(r_{N,1\%} < r\ [0.582]\) | M1 | Formulate condition for \(N\) |
| \(15\) or \(16\) (\(\checkmark\) on \(r\)) | A1\(\checkmark\) | Identify critical value near \(r\) using table |
| \(N \geq 16\) | A1 | State set of possible values of \(N\) |
# Question 9:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $S_{xy} = 513 - 73 \times \frac{64}{10} = 45.8$ | | |
| $S_{xx} = 651 - \frac{73^2}{10} = 118.1$ | | |
| $b = \frac{S_{xy}}{S_{xx}} = 0.388$ | M1 A1 | Calculate gradient $b$ in $y - \bar{y} = b(x-\bar{x})$ |
| $y = \frac{64}{10} + 0.388\left(x - \frac{73}{10}\right) = 0.388x + 3.57$ or $\frac{458x+4215}{1181}$ | M1, A1 | Find regression line of $y$ on $x$ |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $S_{yy} = 462 - \frac{64^2}{10} = 52.4$ | | |
| $r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} = 0.582$ | M1 A1 | Find correlation coefficient $r$ |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = 7.45$ | B1 | Find $y$ when $x=10$ |
| Not reliable as value of $r$ is small; or reliable since $x=10$ is in range / or is near mean | B1 | State valid comment on reliability |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Require one-tail $r_{N,1\%} < r\ [0.582]$ | M1 | Formulate condition for $N$ |
| $15$ or $16$ ($\checkmark$ on $r$) | A1$\checkmark$ | Identify critical value near $r$ using table |
| $N \geq 16$ | A1 | State set of possible values of $N$ |
---9 A random sample of 10 pairs of values of $x$ and $y$ is given in the following table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 4 & 6 & 6 & 8 & 2 & 7 & 12 & 14 & 9 & 5 \\
\hline
$y$ & 2 & 4 & 6 & 8 & 6 & 10 & 9 & 8 & 6 & 5 \\
\hline
\end{tabular}
\end{center}
(i) Find the equation of the regression line of $y$ on $x$.\\
(ii) Find the product moment correlation coefficient for the sample.\\
(iii) Find the estimated value of $y$ when $x = 10$, and comment on the reliability of this estimate.\\
(iv) Another sample of $N$ pairs of data from the same population has the same product moment correlation coefficient as the first sample given. A test, at the $1 \%$ significance level, on this second sample indicates that there is sufficient evidence to conclude that there is positive correlation. Find the set of possible values of $N$.
\hfill \mbox{\textit{CAIE FP2 2014 Q9 [11]}}