| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | June |
| Marks | 12 |
| 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 standard S1 regression question requiring routine calculations: finding regression line from summary statistics, computing residuals, and making predictions. All techniques are textbook procedures with no novel problem-solving required. The multi-part structure and context add length but not conceptual difficulty—this is easier than an average A-level question across all modules. |
| Spec | 5.09a Dependent/independent variables5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Sheet | \(\mathbf { 1 }\) | \(\mathbf { 2 }\) | \(\mathbf { 3 }\) | \(\mathbf { 4 }\) | \(\mathbf { 5 }\) | \(\mathbf { 6 }\) | \(\mathbf { 7 }\) | \(\mathbf { 8 }\) |
| \(\boldsymbol { x }\) | 10 | 16 | 23 | 27 | 31 | 35 | 38 | 44 |
| \(\boldsymbol { y }\) | 2.4 | 3.5 | 3.2 | 4.1 | 4.1 | 5.6 | 4.6 | 5.3 |
| Sheet | \(\mathbf { 1 }\) | \(\mathbf { 2 }\) | \(\mathbf { 3 }\) | \(\mathbf { 4 }\) | \(\mathbf { 5 }\) | \(\mathbf { 6 }\) | \(\mathbf { 7 }\) | \(\mathbf { 8 }\) |
| Residual | - 0.174 | 0.418 | 0.085 | - 0.254 | 0.906 | - 0.157 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Gradient \(b = 0.0848\) OR \(b = 0.084\) to \(0.085\) | B2 (B1) | AWRT; AWFW |
| Intercept \(a = 1.72\) to \(1.73\) OR \(a = 1.7\) | B2 (B1) | AWFW; CAO |
| Attempt at \(\Sigma x\), \(\Sigma x^2\), \(\Sigma y\), \(\Sigma xy\) OR \(S_{xx}\), \(S_{xy}\) | (M1) | 224, 7180; 32.8; 995.4 OR 908; 77 |
| Attempt at correct formula for \(b\) | (m1) | |
| \(b = 0.0848\) | (A1) | AWRT |
| \(a = 1.72\) to \(1.73\) | (A1) | AWFW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Residual \(= y - a - bx\) | M1 | \(\text{Res} = |
| \((\text{Residual})_3 = -0.465\) to \(-0.485\) | A1 (A1) | AWFW; both correct magnitude |
| \((\text{Residual})_7 = -0.335\) to \(-0.365\) | A1 | AWFW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Residuals are small (relative to \(y\)-values); no pattern to residuals | B1 | Except for \((\text{Residual})_6\); any sensible comment; residuals random |
| Fitted equation is appropriate/suitable | B1 | Or equivalent; do not allow "equation is good", "equation is accurate"; allow "equation is suitable" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Total \(=\) Scan \(+\) Transmit \(= y + z\) | M1 | Use of; or equivalent |
| \(T_{15} = 4.45\) to \(4.6\) | A1 | AWFW |
| Reliable as interpolation or small residuals | B1 | Or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(T_{75} = 12.5\) to \(12.7\) | A1 | AWFW |
| Unreliable as extrapolation; cannot get 75 lines of print on A4 page | B1 | Or equivalent |
# Question 4:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Gradient $b = 0.0848$ OR $b = 0.084$ to $0.085$ | B2 (B1) | AWRT; AWFW |
| Intercept $a = 1.72$ to $1.73$ OR $a = 1.7$ | B2 (B1) | AWFW; CAO |
| Attempt at $\Sigma x$, $\Sigma x^2$, $\Sigma y$, $\Sigma xy$ OR $S_{xx}$, $S_{xy}$ | (M1) | 224, 7180; 32.8; 995.4 OR 908; 77 |
| Attempt at correct formula for $b$ | (m1) | |
| $b = 0.0848$ | (A1) | AWRT |
| $a = 1.72$ to $1.73$ | (A1) | AWFW |
## Part (b)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Residual $= y - a - bx$ | M1 | $\text{Res} = |(\text{Obs } y) - (\text{Pred } y)|$ used; allow use of $x=3$ and/or $x=7$ |
| $(\text{Residual})_3 = -0.465$ to $-0.485$ | A1 (A1) | AWFW; both correct magnitude |
| $(\text{Residual})_7 = -0.335$ to $-0.365$ | A1 | AWFW |
## Part (b)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Residuals are small (relative to $y$-values); no pattern to residuals | B1 | Except for $(\text{Residual})_6$; any sensible comment; residuals random |
| Fitted equation is **appropriate/suitable** | B1 | Or equivalent; do **not** allow "equation is good", "equation is accurate"; allow "equation is suitable" |
## Part (c)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Total $=$ Scan $+$ Transmit $= y + z$ | M1 | Use of; or equivalent |
| $T_{15} = 4.45$ to $4.6$ | A1 | AWFW |
| Reliable as interpolation or small residuals | B1 | Or equivalent |
## Part (c)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $T_{75} = 12.5$ to $12.7$ | A1 | AWFW |
| Unreliable as extrapolation; cannot get 75 lines of print on A4 page | B1 | Or equivalent |4 The time taken for a fax machine to scan an A4 sheet of paper is dependent, in part, on the number of lines of print on the sheet. The table below shows, for each of a random sample of 8 sheets of A4 paper, the number, $x$, of lines of print and the scanning time, $y$ seconds, taken by the fax machine.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
Sheet & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$ & $\mathbf { 6 }$ & $\mathbf { 7 }$ & $\mathbf { 8 }$ \\
\hline
$\boldsymbol { x }$ & 10 & 16 & 23 & 27 & 31 & 35 & 38 & 44 \\
\hline
$\boldsymbol { y }$ & 2.4 & 3.5 & 3.2 & 4.1 & 4.1 & 5.6 & 4.6 & 5.3 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the least squares regression line of $y$ on $x$.
\item The following table lists some of the residuals for the regression line.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
Sheet & $\mathbf { 1 }$ & $\mathbf { 2 }$ & $\mathbf { 3 }$ & $\mathbf { 4 }$ & $\mathbf { 5 }$ & $\mathbf { 6 }$ & $\mathbf { 7 }$ & $\mathbf { 8 }$ \\
\hline
Residual & - 0.174 & 0.418 & & 0.085 & - 0.254 & 0.906 & & - 0.157 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\roman*)]
\item Calculate the values of the residuals for sheets 3 and 7 .
\item Hence explain what can be deduced about the regression line.
\end{enumerate}\item The time, $z$ seconds, to transmit an A4 page after scanning is given by:
$$z = 0.80 + 0.05 x$$
Estimate the total time to scan and transmit an A4 page containing:
\begin{enumerate}[label=(\roman*)]
\item 15 lines of print;
\item 75 lines of print.
In each case comment on the likely reliability of your estimate.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2005 Q4 [12]}}