| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2003 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.8 This is a straightforward S1 regression question requiring standard calculations (means, Sxx, Sxy) from a small dataset with simple numbers, followed by routine substitution into the regression formula. The multi-part structure and scatter diagram are typical textbook exercises with no conceptual challenges beyond basic recall of the regression method. |
| 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 |
| \multirow{2}{*}{} | Student | ||||||||
| \(A\) | B | \(C\) | D | \(E\) | \(F\) | G | \(H\) | ||
| \multirow{2}{*}{Mark} | \(m\) | 9 | 14 | 13 | 10 | 7 | 8 | 20 | 17 |
| \(p\) | 11 | 23 | 21 | 15 | 19 | 10 | 31 | 26 | |
| Answer | Marks | Guidance |
|---|---|---|
| \(m\) is explanatory variable | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| scales and labels | B1 B2 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\Sigma m = 98; \quad \Sigma p = 156; \quad \Sigma m^2 = 1348; \quad \Sigma mp = 2119\) | ||
| \(S_{mp} = 2119 - \frac{98 \times 156}{8} = 208\) | M1 A1 | |
| \(S_{mm} = 1348 - \frac{98^2}{8} = 147.5\) | A1 | |
| \(\therefore b = \frac{S_{mp}}{S_{mm}} = \frac{208}{147.5} = 1.410169\) | M1 A1 | (awrt 1.41) |
| \(a = \frac{156}{8} - (1.410169\ldots) \times \frac{98}{8} = 2.225429\) | M1 A1 | (awrt 2.23) |
| \(\therefore p = 2.23 + 1.41m\) | A1 ft | (8 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Line on graph | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(p = 2.23 + 1.41 \times 15 = 23.38\) | M1 A1 | (2 marks) |
| (14 marks) |
**(a)**
| $m$ is explanatory variable | B1 | (1 mark) |
**(b)**
| scales and labels | B1 B2 | (3 marks) |
**(c)**
| $\Sigma m = 98; \quad \Sigma p = 156; \quad \Sigma m^2 = 1348; \quad \Sigma mp = 2119$ | | |
| $S_{mp} = 2119 - \frac{98 \times 156}{8} = 208$ | M1 A1 | |
| $S_{mm} = 1348 - \frac{98^2}{8} = 147.5$ | A1 | |
| $\therefore b = \frac{S_{mp}}{S_{mm}} = \frac{208}{147.5} = 1.410169$ | M1 A1 | (awrt 1.41) |
| $a = \frac{156}{8} - (1.410169\ldots) \times \frac{98}{8} = 2.225429$ | M1 A1 | (awrt 2.23) |
| $\therefore p = 2.23 + 1.41m$ | A1 ft | (8 marks) |
**(d)**
| Line on graph | M1 A1 | (2 marks) |
**(e)**
| $p = 2.23 + 1.41 \times 15 = 23.38$ | M1 A1 | (2 marks) |
| | | (14 marks) |\begin{enumerate}
\item Eight students took tests in mathematics and physics. The marks for each student are given in the table below where $m$ represents the mathematics mark and $p$ the physics mark.
\end{enumerate}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{8}{|c|}{Student} \\
\hline
& & $A$ & B & $C$ & D & $E$ & $F$ & G & $H$ \\
\hline
\multirow{2}{*}{Mark} & $m$ & 9 & 14 & 13 & 10 & 7 & 8 & 20 & 17 \\
\hline
& $p$ & 11 & 23 & 21 & 15 & 19 & 10 & 31 & 26 \\
\hline
\end{tabular}
\end{center}
A science teacher believes that students' marks in physics depend upon their mathematical ability. The teacher decides to investigate this relationship using the test marks.\\
(a) Write down which is the explanatory variable in this investigation.\\
(b) Draw a scatter diagram to illustrate these data.\\
(c) Showing your working, find the equation of the regression line of $p$ on $m$.\\
(d) Draw the regression line on your scatter diagram.
A ninth student was absent for the physics test, but she sat the mathematics test and scored 15 .\\
(e) Using this model, estimate the mark she would have scored in the physics test.
\hfill \mbox{\textit{Edexcel S1 2003 Q7 [16]}}