| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Draw scatter diagram from data |
| Difficulty | Easy -1.2 This is a routine, textbook-style linear regression question requiring standard procedures: plotting points, calculating regression coefficients using given summations, and interpreting results. All formulas are standard S1 content with no problem-solving insight needed, making it easier than average A-level maths questions. |
| Spec | 2.02c Scatter diagrams and regression lines5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(x\) | 3 | 5 | 6 | 8 | 10 | 12 | 13 | 15 | 16 | 18 |
| \(y\) | 36 | 50 | 53 | 61 | 69 | 79 | 82 | 90 | 88 | 96 |
| Answer | Marks | Guidance |
|---|---|---|
| Sensible graph scales, labels, shape (scatter diagram plotted correctly) | B1, B1, B1 | 3 marks for scales/labels/shape |
| Answer | Marks | Guidance |
|---|---|---|
| Points lie close to a straight line | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_{xy} = 8354 - \frac{106 \times 704}{10} = 891.6\) | B1 | |
| \(S_{xx} = 1352 - \frac{106^2}{10} = 228.4\) | B1 | |
| \(b = \frac{891.6}{228.4} = 3.903677...\) | M1A1 | awrt 3.9 |
| \(a = \frac{704}{10} - b\frac{106}{10} = 29.021015...\) | M1A1 | awrt 29 |
| Both values 29.02, 3.90 | A1ft | (7) |
| Answer | Marks | Guidance |
|---|---|---|
| For every extra week in storage, another 3.90 ml of chemical evaporates | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) 103.12 (ii) 165.52 | B1B1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Close to range of \(x\), so reasonably reliable | B1, B1 | |
| (ii) Well outside range of \(x\), could be unreliable since no evidence that model will continue to hold | B1 | (4) |
# Question 3:
## Part (a)
| Sensible graph scales, labels, shape (scatter diagram plotted correctly) | B1, B1, B1 | 3 marks for scales/labels/shape |
## Part (b)
| Points lie close to a straight line | B1 | (1) |
## Part (c)
| $S_{xy} = 8354 - \frac{106 \times 704}{10} = 891.6$ | B1 | |
| $S_{xx} = 1352 - \frac{106^2}{10} = 228.4$ | B1 | |
| $b = \frac{891.6}{228.4} = 3.903677...$ | M1A1 | awrt 3.9 |
| $a = \frac{704}{10} - b\frac{106}{10} = 29.021015...$ | M1A1 | awrt 29 |
| Both values 29.02, 3.90 | A1ft | (7) |
## Part (d)
| For every extra week in storage, another 3.90 ml of chemical evaporates | B1 | (1) |
## Part (e)
| (i) 103.12 (ii) 165.52 | B1B1 | (2) |
## Part (f)
| (i) Close to range of $x$, so reasonably reliable | B1, B1 | |
| (ii) Well outside range of $x$, could be unreliable since no evidence that model will continue to hold | B1 | (4) |
---3. A manufacturer stores drums of chemicals. During storage, evaporation takes place. A random sample of 10 drums was taken and the time in storage, $x$ weeks, and the evaporation loss, $y \mathrm { ml }$, are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 3 & 5 & 6 & 8 & 10 & 12 & 13 & 15 & 16 & 18 \\
\hline
$y$ & 36 & 50 & 53 & 61 & 69 & 79 & 82 & 90 & 88 & 96 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item On graph paper, draw a scatter diagram to represent these data.
\item Give a reason to support fitting a regression model of the form $y = a + b x$ to these data.
\item Find, to 2 decimal places, the value of $a$ and the value of $b$.
$$\text { (You may use } \Sigma x ^ { 2 } = 1352 , \Sigma y ^ { 2 } = 53112 \text { and } \Sigma x y = 8354 \text {.) }$$
\item Give an interpretation of the value of $b$.
\item Using your model, predict the amount of evaporation that would take place after
\begin{enumerate}[label=(\roman*)]
\item 19 weeks,
\item 35 weeks.
\end{enumerate}\item Comment, with a reason, on the reliability of each of your predictions.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2006 Q3 [18]}}