| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Explain least squares concept |
| Difficulty | Moderate -0.3 This is a straightforward Further Statistics question testing standard regression concepts: explaining least squares (bookwork), calculating a regression line from summary statistics using standard formulae, and applying simple linear transformations. While it's Further Maths content, these are routine computational exercises with no problem-solving insight required, making it slightly easier than an average A-level question overall. |
| Spec | 5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression |
| Answer | Marks | Guidance |
|---|---|---|
| Vertical lines drawn from points | M1 | Or squares with correct sides |
| Best-fit line minimises squares of lengths of these lines | A1 | oe, clearly stated |
| Answer | Marks | Guidance |
|---|---|---|
| \(u = -3.55 + 2.05t\) | M1 | Evidence for correct method for \(b\) (e.g. right answer) |
| A1 | Both numbers right | |
| A1 | All correct including letters |
| Answer | Marks | Guidance |
|---|---|---|
| \(w - 4 = -3.55 + 2.05t/2\) | M1 | Put \(t = v/2,\ u = w - 4\) |
| \(w = 0.45 + 1.025v\) | A1 | Fully correct, simplified to 3 terms |
## Question 3:
### Part (a):
Vertical lines drawn from points | **M1** | Or squares with correct sides
Best-fit line minimises squares of lengths of these lines | **A1** | oe, clearly stated
[2]
### Part (b)(i):
$u = -3.55 + 2.05t$ | **M1** | Evidence for correct method for $b$ (e.g. right answer)
| **A1** | Both numbers right
| **A1** | All correct including letters
[3]
### Part (b)(ii):
$w - 4 = -3.55 + 2.05t/2$ | **M1** | Put $t = v/2,\ u = w - 4$
$w = 0.45 + 1.025v$ | **A1** | Fully correct, simplified to 3 terms
[2]
---3
\begin{enumerate}[label=(\alph*)]
\item Using the scatter diagram in the Printed Answer Booklet, explain what is meant by least squares in the context of a regression line of $y$ on $x$.
\item A set of bivariate data $( t , u )$ is summarised as follows.\\
$n = 5 \quad \sum t = 35 \quad \sum u = 54$\\
$\sum t ^ { 2 } = 285 \quad \sum u ^ { 2 } = 758 \quad \sum \mathrm { tu } = 460$
\begin{enumerate}[label=(\roman*)]
\item Calculate the equation of the regression line of $u$ on $t$.
\item The variables $t$ and $u$ are now scaled using the following scaling.\\
$\mathrm { v } = 2 \mathrm { t } , \mathrm { w } = \mathrm { u } + 4$\\
Find the equation of the regression line of $w$ on $v$, giving your equation in the form $w = f ( v )$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2021 Q3 [7]}}