| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| 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 linear regression question requiring standard calculations (finding regression line from data) with routine interpretation. The context provides a clear physical meaning for the intercept (initial candle length = 30cm), making part (a) trivial. The calculations involve standard S1 formulas with simple arithmetic, and the extrapolation in part (iii) is direct substitution. This is easier than average A-level content as it's purely procedural with no problem-solving insight required. |
| Spec | 5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression |
| \(\boldsymbol { x }\) | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 |
| \(\boldsymbol { y }\) | 27 | 25 | 21 | 19 | 16 | 11 | 9 | 5 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = 30\) | B1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| \(b \text{ (gradient)} = -0.64\) | B2 (B1) | CAO; AWFW \(-0.6\) to \(-0.7\); Treat rounding of correct answers as ISW; Written form of equation is not required |
| \(a \text{ (intercept)} = 31\) | B2 (B1) | CAO (31); AWFW \(30\) to \(32\) |
| Attempt at \(\sum x\), \(\sum x^2\), \(\sum y\), \(\sum xy\) \((\sum y^2)\) or Attempt at \(S_{xx}\) & \(S_{xy}\) \((S_{yy})\) | M1 | 225, 7125, 135 & 2415 (2643) all 4 attempted; 1500 & \(-960\) (618) both attempted |
| Attempt at correct formula for \(b\) (gradient) | m1 | |
| \(b \text{ (gradient)} = -0.64\), \(a \text{ (intercept)} = 31\) | A1 A1 | CAO both |
| Answer | Marks | Guidance |
|---|---|---|
| Candle length reduces by 0.64 (cm) per hour | B1, BF1 | OE; must be in context |
| (Length, \(y\), cm) decreases with (time, \(x\), hours) or As (time, \(x\), hours) increases then (length, \(y\), cm) decreases | B1 | OE; context not required; B0 for reference only to correlation |
| Answer | Marks | Guidance |
|---|---|---|
| When \(x = 50\), \(y = (31 \text{ or } 30) - 0.64 \times 50 = -1\) or \(-2\) or When \(y = 0\), \(x = 31 \div 0.64 = 48\) to \(48.5\) or \(30 \div 0.64 = 46.8\) to \(47\) | B1 | CAO; accept correct comparison of 32 with either 30 or 31; AWFW |
| Claim not justified or \(-1\) is impossible or value \(< 50\) | Bdep1 | OE; dependent on previous B1 |
| Claim cannot be answered due to uneven burning or unlikely to burn completely | B1 | Extrapolation required |
# Question 1:
## Part (a)
$a = 30$ | B1 | CAO
## Part (b)(i)
$b \text{ (gradient)} = -0.64$ | B2 (B1) | CAO; AWFW $-0.6$ to $-0.7$; Treat rounding of correct answers as ISW; Written form of equation is **not** required
$a \text{ (intercept)} = 31$ | B2 (B1) | CAO (31); AWFW $30$ to $32$
Attempt at $\sum x$, $\sum x^2$, $\sum y$, $\sum xy$ $(\sum y^2)$ **or** Attempt at $S_{xx}$ & $S_{xy}$ $(S_{yy})$ | M1 | 225, 7125, 135 & 2415 (2643) all 4 attempted; 1500 & $-960$ (618) both attempted
Attempt at correct formula for $b$ (gradient) | m1 |
$b \text{ (gradient)} = -0.64$, $a \text{ (intercept)} = 31$ | A1 A1 | CAO both
## Part (b)(ii)
Candle **length reduces** by **0.64 (cm) per hour** | B1, BF1 | OE; must be in context
(Length, $y$, cm) **decreases** with (time, $x$, hours) **or** As (time, $x$, hours) **increases** then (length, $y$, cm) **decreases** | B1 | OE; context **not** required; B0 for reference only to correlation
## Part (b)(iii)
When $x = 50$, $y = (31 \text{ or } 30) - 0.64 \times 50 = -1$ **or** $-2$ **or** When $y = 0$, $x = 31 \div 0.64 = 48$ **to** $48.5$ **or** $30 \div 0.64 = 46.8$ **to** $47$ | B1 | CAO; accept correct comparison of 32 with either 30 or 31; AWFW
Claim **not** justified **or** $-1$ is impossible **or** value $< 50$ | Bdep1 | OE; dependent on previous B1
Claim cannot be answered due to uneven burning **or** unlikely to burn completely | B1 | Extrapolation required
---1 Bob, a church warden, decides to investigate the lifetime of a particular manufacturer's brand of beeswax candle. Each candle is 30 cm in length.
From a box containing a large number of such candles, he selects one candle at random. He lights the candle and, after it has burned continuously for $x$ hours, he records its length, $y \mathrm {~cm}$, to the nearest centimetre. His results are shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 \\
\hline
$\boldsymbol { y }$ & 27 & 25 & 21 & 19 & 16 & 11 & 9 & 5 & 2 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item State the value that you would expect for $a$ in the equation of the least squares regression line, $y = a + b x$.
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the equation of the least squares regression line, $y = a + b x$.
\item Interpret the value that you obtain for $b$.
\item It is claimed by the candle manufacturer that the total length of time that such candles are likely to burn for is more than 50 hours.
Comment on this claim, giving a numerical justification for your answer.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2013 Q1 [9]}}