| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 9 |
| Topic | Linear regression |
| Type | Calculate y on x from summary statistics |
| Difficulty | Moderate -0.3 This is a standard linear regression question requiring calculation of the regression line from summary statistics using well-known formulas, followed by routine interpretation. The calculations are straightforward (finding means, then gradient and intercept), and the commentary parts require only basic understanding of interpolation vs extrapolation. Slightly easier than average due to being purely procedural with no conceptual challenges. |
| Spec | 5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression |
| \(t\) | 10 | 20 | 30 | 40 | 50 |
| \(y\) | 24.5 | 24.0 | 21.7 | 18.6 | 12.4 |
**(i)** Obtain $S_{tt} = 5500 - \frac{1}{5}(150)^2 = 1000$, $S_{ty} = 2740 - \frac{1}{5}(150)(101.2) = -296$,
and $\bar{t} = 30$, $\bar{y} = 20.24$ B1
Use $y = \left(\bar{y} + \frac{S_{ty}}{S_{tt}}\bar{t}\right) + \frac{S_{ty}}{S_{tt}}t$ M1
Obtain $y = 29.12 - 0.296t$ A1 **[3]**
**(ii)** 26.2 or better B1
A reasonable but not very accurate approximation or equiv B1 **[2]**
**(iii)** $-0.48$ B1
Extrapolation is invalid beyond a certain time *OR* a negative value is impossible B1 **[2]**
**(iv)** Over the range $10 \leq t \leq 50$ the model appears reasonably accurate B1
As $t$ approaches 100 the model breaks down *OR* may be a non-linear *OR* curvilinear relationship *OR* $y$ levels off B1 **[2]**13 A seed company investigated how well African Marigold seeds germinated when the seeds were past their sell-by date. The table shows the average number of seeds which germinated per packet, $y$, and the number of months past their sell-by date, $t$.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$t$ & 10 & 20 & 30 & 40 & 50 \\
\hline
$y$ & 24.5 & 24.0 & 21.7 & 18.6 & 12.4 \\
\hline
\end{tabular}
\end{center}
The summary data for the investigation were as follows.
$$\Sigma t = 150 \quad \Sigma t ^ { 2 } = 5500 \quad \Sigma y = 101.2 \quad \Sigma y ^ { 2 } = 2146.86 \quad \Sigma t y = 2740$$
(i) Calculate the equation of the regression line of $y$ on $t$.\\
(ii) Use your regression line to calculate $y$ when $t = 10$. Compare your answer with the value of $y$ when $t = 10$ in the table and comment on the result.\\
(iii) Use your regression line to calculate $y$ when $t = 100$. Comment on the validity of this result.\\
(iv) Suggest with reasons whether the regression line provides a good model for predicting the germination of seeds past their sell-by date.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q13 [9]}}