| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate from summary statistics |
| Difficulty | Moderate -0.3 This is a standard S1 linear regression question requiring routine application of formulas for Sxx, Sxy, and regression coefficients, plus straightforward transformations between coded and original variables. While multi-part with 6 sections, each step follows directly from textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 2.02c Scatter diagrams and regression lines5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(S_{xy} = 1474.1 - \frac{441.5 \times 59.8}{20} = 154.015\) | M1A1 | awrt 154; M1 for one correct expression for \(S_{xy}\) or \(S_{xx}\); 1st A1 for either \(S_{xy}=\) awrt 154 or \(S_{xx}=\) awrt 1520 |
| \(S_{xx} = 11261.25 - \frac{441.5^2}{20} = 1515.1375\) | A1 | awrt 1520; 2nd A1 for both |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(b = \left[\frac{S_{xy}}{S_{xx}}\right] = \frac{\text{"154.015"}}{\text{"1515.1375"}} = [0.10165084]\) | M1 | 1st M1 for correct expression for \(b\) (ft their \(S_{xy} \neq 1474.1\)) |
| \([a = \bar{y} - b\bar{x} \rightarrow]\quad a = \frac{59.8}{20} - b \times \frac{441.5}{20} = [0.7460577...]\) | M1 | 2nd M1 for correct expression for \(a\) (allow use of letter \(b\)) |
| \(y = 0.746 + 0.102x\) | A1 | \(a=\) awrt 0.746 and \(b=\) awrt 0.102; must be in \(y\) and \(x\), no fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{v}{100000} = \text{'0.746'} + \text{'0.102'}\left(\frac{s-50}{10}\right)\) | M1 | M1 for substituting \(y = \frac{v}{100000}\) and \(x = \left(\frac{s-50}{10}\right)\) into their equation in (b) |
| \(v = 23780.34997 + 1016.508403s\) | ||
| \(c =\) awrt 23600–23800 | A1 | 1st A1 \(c=\) awrt 23600–23800 |
| \(d =\) awrt 1020\*\* | A1 | 2nd A1 \(d=1020\)**; answer given so must come from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(v = 23780.34997 + 1016.508403 \times 130 = 155926.44236\), awrt 156000 | M1A1 | M1 for substituting \(s=130\) into their (c) or \(x=8\) into their (b); A1 awrt 156000 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| For each (additional) \(1\text{ m}^2\) in floor size, the value of the house increases by '£1020' | B1 | Must mention \(\text{m}^2\) or floor size and £ or value; allow follow through from regression equation in (c) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([31d =]\) £31511.76, awrt (£)32000 | B1 | awrt (£)32000 |
## Question 2:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $S_{xy} = 1474.1 - \frac{441.5 \times 59.8}{20} = 154.015$ | M1A1 | awrt **154**; M1 for one correct expression for $S_{xy}$ or $S_{xx}$; 1st A1 for either $S_{xy}=$ awrt 154 or $S_{xx}=$ awrt 1520 |
| $S_{xx} = 11261.25 - \frac{441.5^2}{20} = 1515.1375$ | A1 | awrt **1520**; 2nd A1 for both |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $b = \left[\frac{S_{xy}}{S_{xx}}\right] = \frac{\text{"154.015"}}{\text{"1515.1375"}} = [0.10165084]$ | M1 | 1st M1 for correct expression for $b$ (ft their $S_{xy} \neq 1474.1$) |
| $[a = \bar{y} - b\bar{x} \rightarrow]\quad a = \frac{59.8}{20} - b \times \frac{441.5}{20} = [0.7460577...]$ | M1 | 2nd M1 for correct expression for $a$ (allow use of letter $b$) |
| $y = 0.746 + 0.102x$ | A1 | $a=$ awrt 0.746 and $b=$ awrt 0.102; must be in $y$ and $x$, no fractions |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{v}{100000} = \text{'0.746'} + \text{'0.102'}\left(\frac{s-50}{10}\right)$ | M1 | M1 for substituting $y = \frac{v}{100000}$ and $x = \left(\frac{s-50}{10}\right)$ into their equation in (b) |
| $v = 23780.34997 + 1016.508403s$ | | |
| $c =$ awrt **23600–23800** | A1 | 1st A1 $c=$ awrt 23600–23800 |
| $d =$ awrt **1020\*\*** | A1 | 2nd A1 $d=1020$**; answer given so must come from correct working |
### Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $v = 23780.34997 + 1016.508403 \times 130 = 155926.44236$, awrt **156000** | M1A1 | M1 for substituting $s=130$ into their (c) or $x=8$ into their (b); A1 awrt 156000 |
### Part (e)
| Answer/Working | Mark | Guidance |
|---|---|---|
| For each (additional) $1\text{ m}^2$ in floor size, the value of the house increases by '£1020' | B1 | Must mention $\text{m}^2$ or floor size **and** £ or value; allow follow through from regression equation in (c) |
### Part (f)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[31d =]$ £31511.76, awrt (£)**32000** | B1 | awrt (£)32000 |
---2. Paul believes there is a relationship between the value and the floor size of a house. He takes a random sample of 20 houses and records the value, $\pounds v$, and the floor size, $s \mathrm {~m} ^ { 2 }$
The data were coded using $x = \frac { s - 50 } { 10 }$ and $y = \frac { v } { 100000 }$ and the following statistics obtained.
$$\sum x = 441.5 , \quad \sum y = 59.8 , \quad \sum x ^ { 2 } = 11261.25 , \quad \sum y ^ { 2 } = 196.66 , \quad \sum x y = 1474.1$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $S _ { x y }$ and the value of $S _ { x x }$
\item Find the equation of the least squares regression line of $y$ on $x$ in the form $y = a + b x$
The least squares regression line of $v$ on $s$ is $v = c + d s$
\item Show that $d = 1020$ to 3 significant figures and find the value of $c$
\item Estimate the value of a house of floor size $130 \mathrm {~m} ^ { 2 }$
\item Interpret the value $d$
Paul wants to increase the value of his house. He decides to add an extension to increase the floor size by $31 \mathrm {~m} ^ { 2 }$
\item Estimate the increase in the value of Paul's house after adding the extension.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2015 Q2 [13]}}