| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | January |
| 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 calculation of summary statistics (Sxx, Sxy) and the regression line from a small dataset, followed by routine interpolation/extrapolation and interpretation. The calculations are straightforward with a calculator, and all steps follow textbook procedures with no problem-solving or novel insight required. |
| Spec | 5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(\boldsymbol { x }\) | 9 | 16 | 22 | 11 | 19 | 26 | 14 | 10 | 11 | 17 |
| \(\boldsymbol { y }\) | 79 | 127 | 172 | 109 | 152 | 214 | 131 | 80 | 94 | 148 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Scatter diagram, 8, 9 or 10 points plotted | B2 | 5, 6 or 7 points plotted — B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(b = 7.49\) to \(7.51\) | B2 | AWFW; accept 7.5 |
| \(a = 14.1\) to \(14.6\) | B2 | AWFW; for attempts at \(\Sigma x\), \(\Sigma x^2 \times 4\) or \(S_{xx} \times 2\) — M1 |
| Regression line (implied) \(\geq 2\) points calculated or use of point \((\bar{x}, \bar{y})\); e.g. \(x=0\ y=14.3\) and \(x=25\ y=201.9\); straight line drawn | M1, A1 | M1 for attempted use of correct formula for \(b\); A1 for answers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y_{15} = 126\) to \(128\) | B1 | AWFW; OE; accept points close to line |
| Reliable as 15 is within (observed) range | E1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y_{35} = 276\) to \(278\) | B1 | AWFW; OE |
| Not reliable as 35 is outside (observed) range | E1 | accept \(y > 4\) hrs so break needed; point off graph \(\Rightarrow\) E0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a\): time to travel to and from area from/to depot | E1 | OE; both correct but reversed \(\Rightarrow\) E1 |
| \(b\): (average) time to deliver a/one parcel (within area) | E1 | OE; proportional to packages \(\Rightarrow\) E0 |
## Question 3:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Scatter diagram, 8, 9 or 10 points plotted | B2 | 5, 6 or 7 points plotted — B1 |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $b = 7.49$ to $7.51$ | B2 | AWFW; accept 7.5 |
| $a = 14.1$ to $14.6$ | B2 | AWFW; for attempts at $\Sigma x$, $\Sigma x^2 \times 4$ or $S_{xx} \times 2$ — M1 |
| Regression line (implied) $\geq 2$ points calculated or use of point $(\bar{x}, \bar{y})$; e.g. $x=0\ y=14.3$ and $x=25\ y=201.9$; straight line drawn | M1, A1 | M1 for attempted use of correct formula for $b$; A1 for answers |
### Part (c)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| $y_{15} = 126$ to $128$ | B1 | AWFW; OE; accept points close to line |
| Reliable as 15 is within (observed) range | E1 | — |
### Part (c)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $y_{35} = 276$ to $278$ | B1 | AWFW; OE |
| Not reliable as 35 is outside (observed) range | E1 | accept $y > 4$ hrs so break needed; point off graph $\Rightarrow$ E0 |
### Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $a$: time to travel to and from area from/to depot | E1 | OE; both correct but reversed $\Rightarrow$ E1 |
| $b$: (average) time to deliver a/one parcel (within area) | E1 | OE; proportional to packages $\Rightarrow$ E0 |3 [Figure 1, printed on the insert, is provided for use in this question.]\\
A parcel delivery company has a depot on the outskirts of a town.
Each weekday, a van leaves the depot to deliver parcels across a nearby area. The table below shows, for a random sample of 10 weekdays, the number, $x$, of parcels to be delivered and the total time, $y$ minutes, that the van is out of the depot.
\begin{center}
\begin{tabular}{ | r | r | r | r | r | r | r | r | r | r | r | }
\hline
$\boldsymbol { x }$ & 9 & 16 & 22 & 11 & 19 & 26 & 14 & 10 & 11 & 17 \\
\hline
$\boldsymbol { y }$ & 79 & 127 & 172 & 109 & 152 & 214 & 131 & 80 & 94 & 148 \\
\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 $x$ and draw your line on Figure 1.
\item Use your regression equation to estimate the total time that the van is out of the depot when delivering:
\begin{enumerate}[label=(\roman*)]
\item 15 parcels;
\item 35 parcels.
Comment on the likely reliability of each of your estimates.
\end{enumerate}\item The time that the van is out of the depot delivering parcels may be thought of as the time needed to travel to and from the area plus an amount of time proportional to the number of parcels to be delivered.
Given that the regression line of $y$ on $x$ is of the form $y = a + b x$, give an interpretation, in context, for each of your values of $a$ and $b$.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2005 Q3 [12]}}