| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Linearize non-linear relationships |
| Difficulty | Standard +0.3 This is a standard S1 linearization question with straightforward data transformation (x = 1/m), routine regression calculations using given summations, and interpretation. All steps are mechanical applications of formulas with no novel problem-solving required, making it slightly easier than average. |
| 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 |
| \(m\) | 1 | 2 | 3 | 4 | 5 | 6 |
| \(d\) | 102 | 69 | 61 | 58 | 52 | 48 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x\) | 1 | 0.5 |
| \(d\) | 102 | 69 |
| M1 A1 | ||
| [Scatter plot shown] | B3 | |
| \(S_{xd} = 189.733 - \frac{2.45 \times 390}{6} = 30.483\) | M1 | |
| \(S_{xx} = 1.491 - \frac{2.45^2}{6} = 0.490583\) | M1 | |
| \(b = \frac{30.483}{0.490583} = 62.136\) | M1 A1 | |
| \(a = \frac{390}{6} - (62.136 \times \frac{2.45}{6}) = 39.628\) | M1 A1 | |
| \(d = 39.6 + 62.1x\) | A1 | |
| \(m = 13, x = \frac{1}{13}; d = 39.6 + (62.1 \times \frac{1}{13}) = 44.4\), so 44 cases | M2 A1 | |
| not very reliable as it requires extrapolation well outside the data | B1 | (17) |
| $x$ | 1 | 0.5 | 0.333 | 0.25 | 0.2 | 0.167 |
|-----|--------|--------|---------|---------|--------|---------|
| $d$ | 102 | 69 | 61 | 58 | 52 | 48 |
| M1 A1 |
[Scatter plot shown] | B3 |
$S_{xd} = 189.733 - \frac{2.45 \times 390}{6} = 30.483$ | M1 |
$S_{xx} = 1.491 - \frac{2.45^2}{6} = 0.490583$ | M1 |
$b = \frac{30.483}{0.490583} = 62.136$ | M1 A1 |
$a = \frac{390}{6} - (62.136 \times \frac{2.45}{6}) = 39.628$ | M1 A1 |
$d = 39.6 + 62.1x$ | A1 |
$m = 13, x = \frac{1}{13}; d = 39.6 + (62.1 \times \frac{1}{13}) = 44.4$, so 44 cases | M2 A1 |
not very reliable as it requires extrapolation well outside the data | B1 | (17)7. A new vaccine is tested over a six-month period in one health authority.
The table shows the number of new cases of the disease, $d$, reported in the $m$ th month after the trials began.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
$m$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
$d$ & 102 & 69 & 61 & 58 & 52 & 48 \\
\hline
\end{tabular}
\end{center}
A doctor suggests that a relationship of the form $d = a + b x$ where $x = \frac { 1 } { m }$ can be used to model the situation.
\begin{enumerate}[label=(\alph*)]
\item Tabulate the values of $x$ corresponding to the given values of $d$ and plot a scatter diagram of $d$ against $x$.
\item Explain how your scatter diagram supports the suggested model.
You may use
$$\Sigma x = 2.45 , \quad \Sigma d = 390 , \quad \Sigma x ^ { 2 } = 1.491 , \quad \Sigma x d = 189.733$$
\item Find an equation of the regression line $d$ on $x$ in the form $d = a + b x$.
\item Use your regression line to estimate how many new cases of the disease there will be in the 13th month after the trial began.
\item Comment on the reliability of your answer to part (d).
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 Q7 [17]}}