| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| 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 application of standard S1 regression formulas with clear data. Parts (a)-(c) involve routine calculations (finding means, Sxx, Sxy, then a and b), standard interpretation, and substitution. Part (d) requires basic understanding of interpolation vs extrapolation reliability. No conceptual challenges or novel problem-solving required—purely procedural work that any well-prepared S1 student should handle comfortably. |
| Spec | 5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(\boldsymbol { x }\) | 9 | 3 | 4 | 10 | 8 | 12 | 7 | 11 | 2 | 6 |
| \(\boldsymbol { y }\) | 11 | 6 | 5 | 11 | 9 | 13 | 9 | 12 | 4 | 7 |
| Answer | Marks | Guidance |
|---|---|---|
| 1(a) | ||
| Gradient, \(b = 0.886\) to \(0.887\) | B2 | AWFW |
| \(b = 0.88\) to \(0.89\) | B1 | AWFW |
| Intercept, \(a = 2.31\) to \(2.33\) | B2 | AWFW |
| \(a = 2.3\) | B1 | AWRT |
| Attempt at \(\Sigma x\), \(\Sigma x^2\), \(\Sigma y\), \(\Sigma xy\) or attempt at \(S_{xx}\), \(S_{yy}\) | M1 | |
| Attempt at a correct formula for \(b\): \(b = 0.886\) to \(0.887\) | m1, A1 | AWFW |
| \(a = 2.31\) to \(2.33\) | A1 | AWFW |
| Accept \(a\) & \(b\) interchanged only if \(y = ax + b\) stated or subsequently used correctly in either (b) or (c) | 4 | |
| 1(b) | ||
| \(a\): average waiting time of 2.32 minutes (139 seconds) when entering empty restaurant | B1 | OE; accept minimum waiting time |
| \(b\): average increase in waiting time of 0.886 minutes (53 seconds) for each customer in restaurant on entry | B1 | OE |
| 2 | ||
| 1(c) | ||
| Use of \(y = a + 5b\) or \(y = a + 25b\) | M1 | |
| 1(i) | ||
| For \(x = 5\): \(y = 6.6\) to \(6.8\) | ||
| 1(ii) | ||
| For \(x = 25\): \(y = 24.3\) to \(24.6\) | A1 | Both; AWFW |
| 2 | ||
| 1(d)(i) | ||
| Reliable as interpolation and small residuals | B1, B1 | Within range OE |
| or Reliable as interpolation but large percentage residuals so inconclusive | B1, B1 | OE |
| or Large percentage residuals so unreliable | B1 | |
| 3 | ||
| 1(d)(ii) | ||
| Unreliable as extrapolation | B1 | Outside range OE |
| 3 |
| **1(a)** |
|----------|
| Gradient, $b = 0.886$ to $0.887$ | B2 | AWFW |
| $b = 0.88$ to $0.89$ | B1 | AWFW |
| Intercept, $a = 2.31$ to $2.33$ | B2 | AWFW |
| $a = 2.3$ | B1 | AWRT |
| Attempt at $\Sigma x$, $\Sigma x^2$, $\Sigma y$, $\Sigma xy$ or attempt at $S_{xx}$, $S_{yy}$ | M1 | |
| Attempt at a correct formula for $b$: $b = 0.886$ to $0.887$ | m1, A1 | AWFW |
| $a = 2.31$ to $2.33$ | A1 | AWFW |
| Accept $a$ & $b$ interchanged only if $y = ax + b$ stated or subsequently used correctly in either (b) or (c) | | 4 |
| **1(b)** |
|----------|
| $a$: average waiting time of 2.32 minutes (139 seconds) when entering empty restaurant | B1 | OE; accept minimum waiting time |
| $b$: average increase in waiting time of 0.886 minutes (53 seconds) for each customer in restaurant on entry | B1 | OE |
| | | 2 |
| **1(c)** |
|----------|
| Use of $y = a + 5b$ or $y = a + 25b$ | M1 | |
| **1(i)** |
|----------|
| For $x = 5$: $y = 6.6$ to $6.8$ | | |
| **1(ii)** |
|----------|
| For $x = 25$: $y = 24.3$ to $24.6$ | A1 | Both; AWFW |
| | | 2 |
| **1(d)(i)** |
|----------|
| Reliable as interpolation and small residuals | B1, B1 | Within range OE |
| or Reliable as interpolation but large percentage residuals so inconclusive | B1, B1 | OE |
| or Large percentage residuals so unreliable | B1 | |
| | | 3 |
| **1(d)(ii)** |
|----------|
| Unreliable as extrapolation | B1 | Outside range OE |
| | | 3 |
**Total for Q1: 11**
---1 At a certain small restaurant, the waiting time is defined as the time between sitting down at a table and a waiter first arriving at the table. This waiting time is dependent upon the number of other customers already seated in the restaurant.
Alex is a customer who visited the restaurant on 10 separate days. The table shows, for each of these days, the number, $x$, of customers already seated and his waiting time, $y$ minutes.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$\boldsymbol { x }$ & 9 & 3 & 4 & 10 & 8 & 12 & 7 & 11 & 2 & 6 \\
\hline
$\boldsymbol { y }$ & 11 & 6 & 5 & 11 & 9 & 13 & 9 & 12 & 4 & 7 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the least squares regression line of $y$ on $x$ in the form $y = a + b x$.
\item Give an interpretation, in context, for each of your values of $a$ and $b$.
\item Use your regression equation to estimate Alex's waiting time when the number of customers already seated in the restaurant is:
\begin{enumerate}[label=(\roman*)]
\item 5 ;
\item 25 .
\end{enumerate}\item Comment on the likely reliability of each of your estimates in part (c), given that, for the regression line calculated in part (a), the values of the 10 residuals lie between + 1.1 minutes and - 1.1 minutes.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2006 Q1 [11]}}