| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2003 |
| Session | November |
| Marks | 16 |
| 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 calculation requiring standard formulas (Sxy, Sxx, then a and b) with given summary statistics. All steps are routine S1 procedures with no conceptual challenges—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 line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context |
| \(x\) | 15 | 40 | 27 | 39 | 27 | 15 | 20 | 30 | 19 | 24 |
| \(y\) | 216 | 384 | 234 | 399 | 226 | 132 | 175 | 316 | 187 | 196 |
| Answer | Marks | Guidance |
|---|---|---|
| Part (b) | ||
| \(S_{xy} = 69798 - \frac{256 \times 2465}{10} = 6694\) | B1 | |
| \(256, 2465\) | B1 | |
| \(S_{xy}\) or \(S_{xx}\) | M1 | |
| \(S_{xx} = 7266 - \frac{256^2}{10} = 712.4\) | A1 | |
| \(6694\) | A1 | |
| \(712.4\) | A1 | |
| Part (c)(i) | ||
| \(b = \frac{6694}{712.4} = 9.3964\ldots\) | M1, A1 | |
| \(a = \frac{2465}{10} - \frac{6694}{712.4} \times \frac{256}{10} = 5.95199\ldots\) | B1 | |
| \(\therefore y = 5.95 + 9.40x\) | B1, \(\checkmark\) | Three significant figures |
| Part (c)(ii) | ||
| Line on graph | B1 | |
| Part (d) | ||
| Salary increases by £940 for every 1 point performance increase | B1 | (1 mark) |
| Part (e) | ||
| \(x = 35 \Rightarrow y = 334.95\) | B1 | |
| Salary is £33,495 | B1, \(\checkmark\) | (2 marks) |
**Part (b)** | |
$S_{xy} = 69798 - \frac{256 \times 2465}{10} = 6694$ | B1 |
$256, 2465$ | B1 |
$S_{xy}$ or $S_{xx}$ | M1 |
$S_{xx} = 7266 - \frac{256^2}{10} = 712.4$ | A1 |
$6694$ | A1 |
$712.4$ | A1 |
**Part (c)(i)** | |
$b = \frac{6694}{712.4} = 9.3964\ldots$ | M1, A1 |
$a = \frac{2465}{10} - \frac{6694}{712.4} \times \frac{256}{10} = 5.95199\ldots$ | B1 |
$\therefore y = 5.95 + 9.40x$ | B1, $\checkmark$ | Three significant figures |
**Part (c)(ii)** | |
Line on graph | B1 |
**Part (d)** | |
Salary increases by £940 for every 1 point performance increase | B1 | (1 mark)
**Part (e)** | |
$x = 35 \Rightarrow y = 334.95$ | B1 |
Salary is £33,495 | B1, $\checkmark$ | (2 marks)
---\begin{enumerate}
\item A company wants to pay its employees according to their performance at work. The performance score $x$ and the annual salary, $y$ in $\pounds 100$ s, for a random sample of 10 of its employees for last year were recorded. The results are shown in the table below.
\end{enumerate}
\begin{center}
\begin{tabular}{ | r | r | r | r | r | r | r | r | r | r | r | }
\hline
$x$ & 15 & 40 & 27 & 39 & 27 & 15 & 20 & 30 & 19 & 24 \\
\hline
$y$ & 216 & 384 & 234 & 399 & 226 & 132 & 175 & 316 & 187 & 196 \\
\hline
\end{tabular}
\end{center}
$$\text { [You may assume } \left. \Sigma x y = 69798 , \Sigma x ^ { 2 } = 7266 \right]$$
(a) Draw a scatter diagram to represent these data.\\
(b) Calculate exact values of $S _ { x y }$ and $S _ { x x }$.\\
(c) (i) Calculate the equation of the regression line of $y$ on $x$, in the form $y = a + b x$.
Give the values of $a$ and $b$ to 3 significant figures.\\
(ii) Draw this line on your scatter diagram.\\
(d) Interpret the gradient of the regression line.
The company decides to use this regression model to determine future salaries.\\
(e) Find the proposed annual salary for an employee who has a performance score of 35 .\\
\hfill \mbox{\textit{Edexcel S1 2003 Q1 [16]}}