| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| 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 S1 regression question requiring standard formula application for the regression line equation (using provided summary statistics), simple substitution for prediction, and basic interpretation of correlation. All calculations are routine with no conceptual challenges or problem-solving required—easier than average A-level maths. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression |
| \(x\) | 0 | 5 | 10 | 15 | 20 | 25 | 30 | 35 |
| \(y\) | 0.8 | 3.0 | 6.8 | 10.9 | 15.6 | 19.6 | 23.4 | 26.7 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{2685 - 140 \times 106.8}{8}\) or \(2685 - \frac{8 \times 17.5 \times 13.35}{2}\) | M1 | Correct sub in any correct formula for \(b\) (incl. \((x - \bar{x})\) etc) |
| \(= \frac{136}{175}\) or \(0.777\) (3 sfs) | A1 | |
| \(y - 106.8 = 0.777(x - 140)\) | M1 | or \(a = 106.8 \times 0.777 - 140/8\) ft \(b\) for M1 |
| \(y = 0.78x - 0.25\) or better or \(y = \frac{139}{175} x - \frac{1}{4}\) | A1 | \(\geq 2\) sfs sufficient for coeffs; M1: ft their equn; A1: dep const term in equn |
| ii) \(0.78 \times 12 - 0.25 = 9.1\) (2 sfs) | M1 | |
| iii a) Reliable | B1 | |
| iii b) Unreliable because extrapolating oe | B1 | 2 marks |
**i)**
$\frac{2685 - 140 \times 106.8}{8}$ or $2685 - \frac{8 \times 17.5 \times 13.35}{2}$ | M1 | Correct sub in any correct formula for $b$ (incl. $(x - \bar{x})$ etc)
$= \frac{136}{175}$ or $0.777$ (3 sfs) | A1 |
$y - 106.8 = 0.777(x - 140)$ | M1 | or $a = 106.8 \times 0.777 - 140/8$ ft $b$ for M1
$y = 0.78x - 0.25$ or better or $y = \frac{139}{175} x - \frac{1}{4}$ | A1 | $\geq 2$ sfs sufficient for coeffs; M1: ft their equn; A1: dep const term in equn
**ii)** $0.78 \times 12 - 0.25 = 9.1$ (2 sfs) | M1 |
**iii a)** Reliable | B1 |
**iii b)** Unreliable because extrapolating oe | B1 | 2 marks
**Total for Question 5:** 8 marks5 A chemical solution was gradually heated. At five-minute intervals the time, $x$ minutes, and the temperature, $y ^ { \circ } \mathrm { C }$, were noted.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 \\
\hline
$y$ & 0.8 & 3.0 & 6.8 & 10.9 & 15.6 & 19.6 & 23.4 & 26.7 \\
\hline
\end{tabular}
\end{center}
$$\left[ n = 8 , \Sigma x = 140 , \Sigma y = 106.8 , \Sigma x ^ { 2 } = 3500 , \Sigma y ^ { 2 } = 2062.66 , \Sigma x y = 2685.0 . \right]$$
(i) Calculate the equation of the regression line of $y$ on $x$.\\
(ii) Use your equation to estimate the temperature after 12 minutes.\\
(iii) It is given that the value of the product moment correlation coefficient is close to + 1 . Comment on the reliability of using your equation to estimate $y$ when
\begin{enumerate}[label=(\alph*)]
\item $x = 17$,
\item $x = 57$.
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2007 Q5 [8]}}