| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Find unknown values from regression |
| Difficulty | Challenging +1.2 This is a multi-part regression question requiring knowledge of the relationship between regression coefficients and correlation (b₁b₂ = r²), use of means in regression lines, and straightforward substitution. While it involves several steps and algebraic manipulation, the techniques are standard for Further Maths statistics with no novel problem-solving required beyond applying known formulas systematically. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09c Calculate regression line |
| Answer | Marks | Guidance |
|---|---|---|
| \(b \times 0.6331 = 0.9797^2\), \(b = 1.516\) | M1 A1 | Find \(b\) from given gradients and coefficient |
| Answer | Marks | Guidance |
|---|---|---|
| \(46.5/6\ [=7.75] = b \times \bar{x} + 1.306\) or \(b \times (\Sigma x)/6 + 1.306\) | M1 | Find \(p\) from means and regression line of \(y\) on \(x\) |
| \(\bar{x} = 4.25\) or \(\Sigma x = 25.5\) | A1 | (intermediate values may be implied) |
| \(6\bar{x}\) or \(\Sigma x = 21.3 + p\), \(p = 4.2\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((\Sigma x)/6 = 0.6331\ (\Sigma y)/6 + d\ [d = -0.656]\) | M1 | Find \(d\) from means and regression line of \(x\) on \(y\) |
| \(x = 0.6331 \times 8.5 + d = 4.725_{[3]}\) or \(4.73\) | M1 A1 | Estimate \(x\) when \(y = 8.5\) using regression line of \(x\) on \(y\) |
| OR: \(x - (\Sigma x)/6 = 0.6331\{8.5 - (\Sigma y)/6\}\) | (M2) | Combine above into single step |
| \(x = 4.725_{[3]}\) or \(4.73\) | (A1) | |
| SC: \(x = (8.5 - 1.306)/b\ [= 4.7452] = 4.75\) | (B1) | SC: Estimate \(x\) when \(y = 8.5\) using regression line of \(y\) on \(x\) |
## Question 8:
### Part 8(i):
| $b \times 0.6331 = 0.9797^2$, $b = 1.516$ | M1 A1 | Find $b$ from given gradients and coefficient |
**Total: 2 marks**
### Part 8(ii):
| $46.5/6\ [=7.75] = b \times \bar{x} + 1.306$ or $b \times (\Sigma x)/6 + 1.306$ | M1 | Find $p$ from means and regression line of $y$ on $x$ |
| $\bar{x} = 4.25$ or $\Sigma x = 25.5$ | A1 | (intermediate values may be implied) |
| $6\bar{x}$ or $\Sigma x = 21.3 + p$, $p = 4.2$ | M1 A1 | |
**Total: 4 marks**
### Part 8(iii):
| $(\Sigma x)/6 = 0.6331\ (\Sigma y)/6 + d\ [d = -0.656]$ | M1 | Find $d$ from means and regression line of $x$ on $y$ |
| $x = 0.6331 \times 8.5 + d = 4.725_{[3]}$ or $4.73$ | M1 A1 | Estimate $x$ when $y = 8.5$ using regression line of $x$ on $y$ |
| OR: $x - (\Sigma x)/6 = 0.6331\{8.5 - (\Sigma y)/6\}$ | (M2) | Combine above into single step |
| $x = 4.725_{[3]}$ or $4.73$ | (A1) | |
| SC: $x = (8.5 - 1.306)/b\ [= 4.7452] = 4.75$ | (B1) | SC: Estimate $x$ when $y = 8.5$ using regression line of $y$ on $x$ |
**Total: 3 marks**
---8 For a random sample of 6 observations of pairs of values $( x , y )$, the equation of the regression line of $y$ on $x$ is $y = b x + 1.306$, where $b$ is a constant. The corresponding equation of the regression line of $x$ on $y$ is $x = 0.6331 y + d$, where $d$ is a constant. The values of $x$ from the sample are
$$\begin{array} { l l l l l l }
2.3 & 2.8 & 3.7 & p & 6.1 & 6.4
\end{array}$$
and the sum of the values of $y$ is 46.5 . The product moment correlation coefficient is 0.9797 .\\
(i) Find the value of $b$ correct to 3 decimal places.\\
(ii) Find the value of $p$.\\
(iii) Use the equation of the regression line of $x$ on $y$ to estimate the value of $x$ when $y = 8.5$.\\
\hfill \mbox{\textit{CAIE FP2 2018 Q8 [9]}}