| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Relate two regression lines |
| Difficulty | Standard +0.8 This question requires understanding of regression line formulas, correlation coefficients, and the effect of linear transformations on regression equations. Part (i) involves algebraic manipulation of the gradient formula to find n. Parts (ii) and (iii) require standard but careful application of regression formulas. The final part tests conceptual understanding of how scaling affects regression lines. While systematic, it requires multiple techniques and careful algebraic work beyond routine A-level statistics questions. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09c Calculate regression line |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(-\frac{3}{4} = (192 - 24 \times 34/n)/(160 - 24^2/n)\) | M1 A1 | Find \(n\) using gradient |
| \(4992/n = 1248,\ n = 4\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x - 24/4 = (-12/35)(y - 34/4)\) | M1 | Find regression line \(x = Ay + B\); M0 for \(y - 34/4 = -\frac{3}{4}(x - 24/4)\) |
| \(x = -12y/35 + 312/35\) or \(-0.343y + 8.91\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = -12/\sqrt{16 \times 35}\) or \(-\sqrt{\frac{3}{4} \times 12/35}\) | M1 | Find coefficient \(r\) (ignore sign for M1) |
| State regression line (B1 for each term on RHS) | ||
| \(= -0.507\) or \(-3\sqrt{35}/35\) or \(-3/\sqrt{35}\) | A1 |
## Question 9:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $-\frac{3}{4} = (192 - 24 \times 34/n)/(160 - 24^2/n)$ | M1 A1 | Find $n$ using gradient |
| $4992/n = 1248,\ n = 4$ | A1 | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x - 24/4 = (-12/35)(y - 34/4)$ | M1 | Find regression line $x = Ay + B$; M0 for $y - 34/4 = -\frac{3}{4}(x - 24/4)$ |
| $x = -12y/35 + 312/35$ or $-0.343y + 8.91$ | A1 | |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = -12/\sqrt{16 \times 35}$ or $-\sqrt{\frac{3}{4} \times 12/35}$ | M1 | Find coefficient $r$ (ignore sign for M1) |
| State regression line (B1 for each term on RHS) | | |
| $= -0.507$ or $-3\sqrt{35}/35$ or $-3/\sqrt{35}$ | A1 | |
**Total: 9 marks**9 A researcher records a random sample of $n$ pairs of values of $( x , y )$, giving the following summarised data.
$$\Sigma x = 24 \quad \Sigma x ^ { 2 } = 160 \quad \Sigma y = 34 \quad \Sigma y ^ { 2 } = 324 \quad \Sigma x y = 192$$
The gradient of the regression line of $y$ on $x$ is $- \frac { 3 } { 4 }$. Find\\
(i) the value of $n$,\\
(ii) the equation of the regression line of $x$ on $y$ in the form $x = A y + B$, where $A$ and $B$ are constants to be determined,\\
(iii) the product moment correlation coefficient.
Another researcher records the same data in the form $\left( x ^ { \prime } , y ^ { \prime } \right)$, where $x ^ { \prime } = \frac { x } { k } , y ^ { \prime } = \frac { y } { k }$ and $k$ is a constant.\\
Without further calculation, state the equation of the regression line of $x ^ { \prime }$ on $y ^ { \prime }$.
\hfill \mbox{\textit{CAIE FP2 2013 Q9 [9]}}