| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.3 This is a standard S1 linear regression question requiring calculation of regression line from summary statistics (which students compute from the data table), plotting points, interpreting coefficients, making predictions, and calculating residuals. While it involves multiple parts and careful arithmetic, all techniques are routine textbook exercises with no novel problem-solving required. Slightly easier than average due to straightforward data and standard question structure. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context |
| Roof \(( \boldsymbol { i } )\) | \(\mathbf { 1 }\) | \(\mathbf { 2 }\) | \(\mathbf { 3 }\) | \(\mathbf { 4 }\) | \(\mathbf { 5 }\) | \(\mathbf { 6 }\) | \(\mathbf { 7 }\) | \(\mathbf { 8 }\) | \(\mathbf { 9 }\) | \(\mathbf { 1 0 }\) | \(\mathbf { 1 1 }\) |
| \(\boldsymbol { x } _ { \boldsymbol { i } }\) | 8 | 11 | 14 | 14 | 16 | 20 | 22 | 22 | 25 | 27 | 30 |
| \(\boldsymbol { y } _ { \boldsymbol { i } }\) | 5.0 | 5.2 | 6.3 | 7.2 | 8.0 | 8.8 | 10.6 | 11.0 | 11.8 | 12.1 | 13.0 |
| Answer | Marks |
|---|---|
| - (a) Plot 4 data pairs (roofs 8–11) on scatter diagram | [2 marks] |
| - (b)(i) Calculate least squares regression line of \(y_i\) on \(x_i\); draw on diagram | [6 marks] |
| - (b)(ii) Interpret gradient and intercept values | [3 marks] |
| - (c) Estimate time to replace 15 tiles | [1 mark] |
| - (d)(i) Calculate \(r_6\) | [2 marks] |
| - (d)(ii) State why \(\sum_{i=1}^{11} r_i\) gives no useful information | [1 mark] |
**Question 4:**
- **(a)** Plot 4 data pairs (roofs 8–11) on scatter diagram | [2 marks]
- **(b)(i)** Calculate least squares regression line of $y_i$ on $x_i$; draw on diagram | [6 marks]
- **(b)(ii)** Interpret gradient and intercept values | [3 marks]
- **(c)** Estimate time to replace 15 tiles | [1 mark]
- **(d)(i)** Calculate $r_6$ | [2 marks]
- **(d)(ii)** State why $\sum_{i=1}^{11} r_i$ gives no useful information | [1 mark]
If you have the actual **mark scheme** pages, please share those and I can extract the content you need.4 Stephan is a roofing contractor who is often required to replace loose ridge tiles on house roofs. In order to help him to quote more accurately the prices for such jobs in the future, he records, for each of 11 recently repaired roofs, the number of ridge tiles replaced, $x _ { i }$, and the time taken, $y _ { i }$ hours. His results are shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Roof $( \boldsymbol { i } )$ & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$ & $\mathbf { 6 }$ & $\mathbf { 7 }$ & $\mathbf { 8 }$ & $\mathbf { 9 }$ & $\mathbf { 1 0 }$ & $\mathbf { 1 1 }$ \\
\hline
$\boldsymbol { x } _ { \boldsymbol { i } }$ & 8 & 11 & 14 & 14 & 16 & 20 & 22 & 22 & 25 & 27 & 30 \\
\hline
$\boldsymbol { y } _ { \boldsymbol { i } }$ & 5.0 & 5.2 & 6.3 & 7.2 & 8.0 & 8.8 & 10.6 & 11.0 & 11.8 & 12.1 & 13.0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item The pairs of data values for roofs 1 to 7 are plotted on the scatter diagram shown on the opposite page.
Plot the 4 pairs of data values for roofs 8 to 11 on the scatter diagram.
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the equation of the least squares regression line of $y _ { i }$ on $x _ { i }$, and draw your line on the scatter diagram.
\item Interpret your values for the gradient and for the intercept of this regression line.
\end{enumerate}\item Estimate the time that it would take Stephan to replace 15 loose ridge tiles on a house roof.
\item Given that $r _ { i }$ denotes the residual for the point representing roof $i$ :
\begin{enumerate}[label=(\roman*)]
\item calculate the value of $r _ { 6 }$;
\item state why the value of $\sum _ { i = 1 } ^ { 11 } r _ { i }$ gives no useful information about the connection between the number of ridge tiles replaced and the time taken.\\[0pt]
[1 mark]\\
\section*{Answer space for question 4}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-11_2385_1714_322_155}
\end{center}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2015 Q4 [15]}}