| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate y on x from raw data table |
| Difficulty | Moderate -0.3 This is a straightforward S1 linear regression question requiring standard calculations (PMCC, regression line, prediction) with all summations provided. The only slight variation is part (ii) testing conceptual understanding of correlation invariance under linear transformations, but this is a standard textbook result. Slightly easier than average due to given summations and routine application of formulas. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08b Linear coding: effect on pmcc5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e 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 |
|---|---|---|
| 4(i)(a) | Binomial seen or implied; 0.6228 – 0.3497 = 0.273 (3 sf) | M1, A1 |
| 4(i)(b) | 0.3497 or 0.350 (3 sf) | B1 |
| 4(i)(c) | 6, 2 | B1ft, B1ft |
| 4(ii) | 27 seen; B(27, \(\frac{2}{3}\)) seen or implied; \({}^{27}C_{18}(\frac{1}{3})^9(\frac{2}{3})^{18}\) = 0.161 (3 sf) | B1, M1, A1 |
**4(i)(a)** | Binomial seen or implied; 0.6228 – 0.3497 = 0.273 (3 sf) | M1, A1 | eg 0.6228 seen; Eg 0.0362 or 1–0.5535 or 0.95$^6$ × 0.05$^2$ or 0.95$^7$ × 0.05$^0$ or similar |
**4(i)(b)** | 0.3497 or 0.350 (3 sf) | B1 | NB 0.3498 (from 0.6228 - 0.273) rounds to 0.350 so B1 |
**4(i)(c)** | 6, 2 | B1ft, B1ft | NB 2, 6 B0B0 unless labelled correctly |
**4(ii)** | 27 seen; B(27, $\frac{2}{3}$) seen or implied; ${}^{27}C_{18}(\frac{1}{3})^9(\frac{2}{3})^{18}$ = 0.161 (3 sf) | B1, M1, A1 | not necessarily in a statement; or attempt eg P($X_1 = 6$) × P($X_2 = 6$) × P($X_3 = 6$) = 0.273$^3$ = 0.0203 M0M0A0; $\ge 3$ sets with $X_i+X_2+X_3=18$ (not nec'y added) M1 |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.
\section*{June 2005}
\hfill \mbox{\textit{OCR S1 Q4 [8]}}