| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.8 This is a straightforward application of standard linear regression formulas to a small dataset (6 points). Students need to calculate summary statistics (means, Sxx, Sxy), find the regression line equation, compute one residual, and make a prediction with a comment on extrapolation. All steps are routine A-level statistics procedures with no conceptual challenges beyond textbook exercises. |
| Spec | 5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Year since 2000 | 0 | 3 | 6 | 9 | 12 | 15 |
| Turnover ( \(\pounds\) millions) | 2.30 | 2.94 | 3.37 | 3.97 | 4.93 | 6.13 |
**Question 2(i)**
- From the data: $n = 6$, $\Sigma x = 45$, $\Sigma x^2 = 495$, $\Sigma y = 23.64$, $\Sigma xy = 215.88$
- $S_{xy} = 215.88 - \dfrac{45 \times 23.64}{6} = 38.58$ **M1**
- $S_{xx} = 495 - \dfrac{45^2}{6} = 157.5$ **M1**
- $b = \dfrac{S_{xy}}{S_{xx}} = \dfrac{38.58}{157.5} = 0.244(95\ldots) \approx 0.245$ **A1** (cao)
- $\therefore a = \dfrac{23.64}{6} - 0.24495\ldots \times \dfrac{45}{6}$ **M1**
- $= 3.94 - 0.24495\ldots \times 7.5 = 2.10(28\ldots) \approx 2.10$ **A1** ($b = 0.245$ gives $a = 2.1025$. FT *their* $b$.)
**Question 2(ii)**
- For 2009: $r = 3.97 - (2.10 + 0.245 \times 9)$ **M1** ("obs-calc". Allow SC B1 for "calc-obs".)
- $= 3.97 - 4.307(42\ldots)$
- $= -0.337(42\ldots)$ **A1** ($a$ and $b$ to 3sf give $y = -0.335$. FT *their* $a$ and $b$.)
**Question 2(iii)**
- In 2024, $x = 24$, $y = 7.98(17\ldots) \approx 7.98$ (millions) **B1** ($a$ and $b$ to 3sf give $y = 7.98$. FT *their* $a$ and $b$.)
- 2024 estimate is unreliable since it involves extrapolation. **B1** (o.e.)
**Total: 9 marks**2 The table shows the turnover, in millions of pounds, of a small company at 3-year intervals over a period of 15 years, starting in 2000.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Year since 2000 & 0 & 3 & 6 & 9 & 12 & 15 \\
\hline
Turnover ( $\pounds$ millions) & 2.30 & 2.94 & 3.37 & 3.97 & 4.93 & 6.13 \\
\hline
\end{tabular}
\end{center}
(i) For the information in the table find the equation of the least squares regression line of $y$ on $x$, where $x$ is the year since 2000 and $y$ is the turnover in millions of pounds.\\
(ii) Use the equation of the regression line to calculate the residual for 2009.\\
(iii) Use the equation of the regression line to estimate the turnover in 2024, and explain why it is inadvisable to rely on this estimate.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2017 Q2 [9]}}