| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Draw scatter diagram from data |
| Difficulty | Moderate -0.3 This is a standard S1 regression question requiring a scatter diagram, identification of an outlier, calculation of regression line using given summations (formula application), and interpretation of extrapolation. All techniques are routine textbook exercises with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 2.02c Scatter diagrams and regression lines5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Town | Abberton | Bember | Claster | Deller | Edgeton | Figland |
| \(P\) (0.000's) | 3.2 | 7.6 | 5.2 | 9.0 | 8.1 | 4.8 |
| \(T\) (£ 000's) | 11.1 | 12.4 | 13.3 | 19.3 | 17.9 | 11.8 |
| Answer | Marks | Guidance |
|---|---|---|
| [Scatter diagram with points plotted] | B4 | |
| (b) (i) Bember | A1 | |
| (ii) e.g. how near to town centre; size of shop | B2 | |
| \(S_{TT} = 574.25 - \frac{37.9 \times 85.8}{6} = 32.28\) | M1 | |
| \(S_{PP} = 264.69 - \frac{37.9^2}{6} = 25.288\) | M1 | |
| \(b = \frac{32.28}{25.288} = 1.2765\) | M1 A1 | |
| \(a = \frac{85.8}{6} - 1.2765(\frac{37.9}{6}) = 6.2369\) | M1 A1 | |
| \(T = 6.24 + 1.28P\) | A1 | |
| \(P = 6.8\) giving \(T = 14.917 \therefore £14900\) | M1 A1 | |
| \(P = 17.2\) which lies outside the set of values used to obtain the equation | B1 | (17) |
| [Scatter diagram with points plotted] | B4 | |
| (b) (i) Bember | A1 | |
| (ii) e.g. how near to town centre; size of shop | B2 | |
| $S_{TT} = 574.25 - \frac{37.9 \times 85.8}{6} = 32.28$ | M1 | |
| $S_{PP} = 264.69 - \frac{37.9^2}{6} = 25.288$ | M1 | |
| $b = \frac{32.28}{25.288} = 1.2765$ | M1 A1 | |
| $a = \frac{85.8}{6} - 1.2765(\frac{37.9}{6}) = 6.2369$ | M1 A1 | |
| $T = 6.24 + 1.28P$ | A1 | |
| $P = 6.8$ giving $T = 14.917 \therefore £14900$ | M1 A1 | |
| $P = 17.2$ which lies outside the set of values used to obtain the equation | B1 | (17) |Penshop have stores selling stationary in each of 6 towns. The population, $P$, in tens of thousands and the monthly turnover, $T$, in thousands of pounds for each of the shops are as recorded below.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Town & Abberton & Bember & Claster & Deller & Edgeton & Figland \\
\hline
$P$ (0.000's) & 3.2 & 7.6 & 5.2 & 9.0 & 8.1 & 4.8 \\
\hline
$T$ (£ 000's) & 11.1 & 12.4 & 13.3 & 19.3 & 17.9 & 11.8 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Represent these data on a scatter diagram with $T$ on the vertical axis. [4]
\item \begin{enumerate}[label=(\roman*)]
\item Which town's shop might appear to be underachieving given the populations of the towns?
\item Suggest two other factors that might affect each shop's turnover. [3]
\end{enumerate}
\end{enumerate}
You may assume that
$$\Sigma P = 37.9, \quad \Sigma T = 85.8, \quad \Sigma P^2 = 264.69, \quad \Sigma T^2 = 1286, \quad \Sigma PT = 574.25.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the equation of the regression line of $T$ on $P$. [7]
\item Estimate the monthly turnover that might be expected if a shop were opened in Gratton, a town with a population of 68 000. [2]
\item Why might the management of Penshop be reluctant to use the regression line to estimate the monthly turnover they could expect if a shop were opened in Haggin, a town with a population of 172 000? [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q6 [17]}}