| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate r from summary statistics |
| Difficulty | Moderate -0.3 This is a straightforward application of the product moment correlation coefficient formula using provided summary statistics, followed by standard conceptual questions about correlation properties and linear regression. All formulas are standard S1 content with no problem-solving insight required—just careful arithmetic and recall of when correlation is invariant under linear transformations. |
| Spec | 2.05f Pearson correlation coefficient2.05g Hypothesis test using Pearson's r5.08a Pearson correlation: calculate pmcc5.09c Calculate regression line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context |
| City | \(x\) | \(y\) |
| Berlin | 52.5 | 58.2 |
| Bucharest | 44.4 | 58.7 |
| Moscow | 55.8 | 53.3 |
| St Petersburg | 60.0 | 47.8 |
| Warsaw | 52.3 | 56.6 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Correct subst in \(\geq\) two \(S\) formulae | M1 | Any correct version |
| \[\frac{14464.1 - \frac{265 \times 274.6}{5}}{\sqrt{\left(14176.54 - \frac{265^2}{5}\right)\left(15162.22 - \frac{274.6^2}{5}\right)}}\] | M1 A1 | or \(\frac{14464.1 - 5 \times 53 \times 54.92}{\sqrt{(14176.54 - 5 \times 53^2)(15162.22 - 5 \times 54.92^2)}}\) or fully correct method with \((x - \bar{x})^2\) etc |
| \(= -0.868\) (3 sfs) | A1 | |
| (ii) No difference oe | B1 | 1 |
| (iii) Choose y on x stated | B1ind | Not just "more accurate" or implied, eg by \(S_y/S_x\) or \(y = ax + b\) |
| Answer | Marks | Guidance |
|---|---|---|
| If state x on y, but wking is y on x: B1 or their \(\frac{-89.7}{131.54}\) seen or \(\frac{14464.1 - 5 \times 53 \times 54.92}{14176.54 - 5 \times 53^2}\) or correct subst into a correct formula \(S_{xy}\), \(S_{xx}\) | M1 | |
| or \(a = \frac{274.6/s_x}{s} - (\text{their} - 0.682) \times \frac{265}{s_x}\), Simplify to 3 terms. Coeffs to \(\geq 2\) sfs | M1ind A1 | 5 |
| \(y - \frac{274.6}{s} = (\text{their} - 0.682)(x - \frac{265}{s})\) or \(y = 91(-, 1) - 0.682(2) x\) or \(49.9\) (3sfs) or \(50\) | A1 A1 | |
| Use of x on y: equiv M mks as above |
(i) Correct subst in $\geq$ two $S$ formulae | M1 | Any correct version
$$\frac{14464.1 - \frac{265 \times 274.6}{5}}{\sqrt{\left(14176.54 - \frac{265^2}{5}\right)\left(15162.22 - \frac{274.6^2}{5}\right)}}$$ | M1 A1 | or $\frac{14464.1 - 5 \times 53 \times 54.92}{\sqrt{(14176.54 - 5 \times 53^2)(15162.22 - 5 \times 54.92^2)}}$ or fully correct method with $(x - \bar{x})^2$ etc | 3 |
$= -0.868$ (3 sfs) | A1 |
(ii) No difference oe | B1 | 1 | Or slightly diff or more acc because of rounding errors when mult by 2.54 oe
(iii) Choose y on x stated | B1ind | Not just "more accurate" or implied, eg by $S_y/S_x$ or $y = ax + b$ |
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**Continued on next page:**
If state x on y, but wking is y on x: B1 or their $\frac{-89.7}{131.54}$ seen or $\frac{14464.1 - 5 \times 53 \times 54.92}{14176.54 - 5 \times 53^2}$ or correct subst into a correct formula $S_{xy}$, $S_{xx}$ | M1
or $a = \frac{274.6/s_x}{s} - (\text{their} - 0.682) \times \frac{265}{s_x}$, Simplify to 3 terms. Coeffs to $\geq 2$ sfs | M1ind A1 | 5 | cao
$y - \frac{274.6}{s} = (\text{their} - 0.682)(x - \frac{265}{s})$ or $y = 91(-, 1) - 0.682(2) x$ or $49.9$ (3sfs) or $50$ | A1 A1 |
Use of x on y: equiv M mks as above |4 The table shows the latitude, $x$ (in degrees correct to 3 significant figures), and the average rainfall $y$ (in cm correct to 3 significant figures) of five European cities.
\begin{center}
\begin{tabular}{ | l | c c | }
\hline
City & $x$ & $y$ \\
\hline
Berlin & 52.5 & 58.2 \\
Bucharest & 44.4 & 58.7 \\
Moscow & 55.8 & 53.3 \\
St Petersburg & 60.0 & 47.8 \\
Warsaw & 52.3 & 56.6 \\
\hline
\end{tabular}
\end{center}
$$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$
(i) Calculate the product moment correlation coefficient.\\
(ii) The values of $y$ in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54 . State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.\\
(iii) It is required to estimate the annual rainfall at Bergen, where $x = 60.4$. Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate.
\hfill \mbox{\textit{OCR S1 2005 Q4 [9]}}