| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 15 |
| 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 linear regression question requiring standard calculations (Sxx, Sxy, finding a and b) from a small dataset, plotting points, and interpreting the result. All steps are routine textbook procedures with no conceptual challenges or novel problem-solving required. |
| Spec | 5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Customer | \(\mathbf { A }\) | \(\mathbf { B }\) | \(\mathbf { C }\) | \(\mathbf { D }\) | \(\mathbf { E }\) | \(\mathbf { F }\) | \(\mathbf { G }\) | \(\mathbf { H }\) |
| \(\boldsymbol { x }\) | 360 | 140 | 860 | 600 | 1180 | 540 | 260 | 480 |
| \(\boldsymbol { y }\) | 50 | 25 | 135 | 70 | 140 | 90 | 55 | 70 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| 8 or 7 points plotted accurately | B2 | |
| (6 or 5 points plotted accurately) | (B1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Gradient, \(b = 0.114\) to \(0.115\) | B2 | AWFW (0.11469) |
| \((b = 0.11\) to \(0.12)\) | (B1) | |
| Intercept, \(a = 15.9\) to \(16.1\) | B2 | AWFW (16.00824) |
| \((a = 13\) to \(19)\) | (B1) | |
| Attempt at \(\sum x\), \(\sum x^2\), \(\sum y\) and \(\sum xy\) or attempt at \(S_{xx}\) and \(S_{xy}\) | (M1) | 4420, 3230800, 635 and 441300; 788750 and 90462.5 |
| Attempt at correct formula for \(b\); \(b = 0.114\) to \(0.115\); \(a = 15.9\) to \(16.1\) | (m1)(A1)(A1) | AWFW; AWFW |
| Line plotted accurately (Evidence of correct method for \(\geq 2\) points) | B2 (M1) | At least from \(x = 200\) to \(1000\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Res}_H = y_H - Y_H = 70 - (a + b \times 480)\) | M1 | Used; or implied by correct answer; allow for \(Y_H - y_H\) shown |
| \(= -1.5\) to \(-0.5\) | A1 | AWFW \((-1.06)\) |
| Point H is (almost) on / just below the line | B1 | Accept near/close/just above or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(Y = a + b \times 560\) or reading from scatter diagram | M1 | Used |
| \(= 79\) to \(81\) | A1 | AWFW (80.2) |
| \(\text{Cost} = Y \times \frac{12}{60}\) or \(\frac{Y}{5}\) | M1 | Used |
| \(= £15.8\) to \(£16.2\) | A1 | AWFW; ignore units (£16.05) |
# Question 7:
## Part 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| 8 or 7 points plotted accurately | B2 | |
| (6 or 5 points plotted accurately) | (B1) | |
## Part 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Gradient, $b = 0.114$ to $0.115$ | B2 | AWFW (0.11469) |
| $(b = 0.11$ to $0.12)$ | (B1) | |
| Intercept, $a = 15.9$ to $16.1$ | B2 | AWFW (16.00824) |
| $(a = 13$ to $19)$ | (B1) | |
| Attempt at $\sum x$, $\sum x^2$, $\sum y$ and $\sum xy$ or attempt at $S_{xx}$ and $S_{xy}$ | (M1) | 4420, 3230800, 635 and 441300; 788750 and 90462.5 |
| Attempt at correct formula for $b$; $b = 0.114$ to $0.115$; $a = 15.9$ to $16.1$ | (m1)(A1)(A1) | AWFW; AWFW |
| Line plotted accurately (Evidence of correct method for $\geq 2$ points) | B2 (M1) | At least from $x = 200$ to $1000$ |
## Part 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Res}_H = y_H - Y_H = 70 - (a + b \times 480)$ | M1 | Used; or implied by correct answer; allow for $Y_H - y_H$ shown |
| $= -1.5$ to $-0.5$ | A1 | AWFW $(-1.06)$ |
| Point H is (almost) **on / just below** the line | B1 | Accept near/close/just above or equivalent |
## Part 7(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| $Y = a + b \times 560$ or reading from scatter diagram | M1 | Used |
| $= 79$ to $81$ | A1 | AWFW (80.2) |
| $\text{Cost} = Y \times \frac{12}{60}$ or $\frac{Y}{5}$ | M1 | Used |
| $= £15.8$ to $£16.2$ | A1 | AWFW; ignore units (£16.05) |7 [Figure 1, printed on the insert, is provided for use in this question.]\\
Stan is a retired academic who supplements his pension by mowing lawns for customers who live nearby.
As part of a review of his charges for this work, he measures the areas, $x \mathrm {~m} ^ { 2 }$, of a random sample of eight of his customers' lawns and notes the times, $y$ minutes, that it takes him to mow these lawns. His results are shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
Customer & $\mathbf { A }$ & $\mathbf { B }$ & $\mathbf { C }$ & $\mathbf { D }$ & $\mathbf { E }$ & $\mathbf { F }$ & $\mathbf { G }$ & $\mathbf { H }$ \\
\hline
$\boldsymbol { x }$ & 360 & 140 & 860 & 600 & 1180 & 540 & 260 & 480 \\
\hline
$\boldsymbol { y }$ & 50 & 25 & 135 & 70 & 140 & 90 & 55 & 70 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item On Figure 1, plot a scatter diagram of these data.
\item Calculate the equation of the least squares regression line of $y$ on $x$. Draw your line on Figure 1.
\item Calculate the value of the residual for Customer H and indicate how your value is confirmed by your scatter diagram.
\item Given that Stan charges $\pounds 12$ per hour, estimate the charge for mowing a customer's lawn that has an area of $560 \mathrm {~m} ^ { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2007 Q7 [15]}}