| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Calculate PMCC from summary statistics |
| Difficulty | Moderate -0.8 This is a straightforward S1 question requiring direct application of standard PMCC and regression formulas to given summary statistics. All parts involve routine calculations with no problem-solving insight needed—students simply substitute values into memorized formulas. The interpretation parts (ii) and (iii) test basic understanding of correlation vs causation, which is standard bookwork. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09c Calculate regression line5.09e Use regression: for estimation in context |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{7351.12-\frac{86.6×943.8}{12}}{\sqrt{(658.76-\frac{86.6^2}{12})(3163.63-\frac{943.8^2}{12})}}\) = \(\frac{540.03}{\sqrt{33.8 | o\times 9433}}\) = 0.9564... or 0.956 or 0.96 | M1, M1, A1 3 |
| Allow Almost complete relationship (or corr'n or link) between amount spent on advert & profit | B1 1 | or Very positive corr'n or Very reliable relationship or Near perfect relationship between spend on advert & profit oe, in context |
| $\frac{7351.12-\frac{86.6×943.8}{12}}{\sqrt{(658.76-\frac{86.6^2}{12})(3163.63-\frac{943.8^2}{12})}}$ = $\frac{540.03}{\sqrt{33.8|o\times 9433}}$ = 0.9564... or 0.956 or 0.96 | M1, M1, A1 3 | Must see at least 2 sfs. 1st M1 for correct subst in any correct $S$ formula. 2nd M1 for all correct subst'n in any correct $r$ formula. 0.96 or correct better, no working: M1M1A1. eg 0.958 → 0.96 with correct working M1M1A0 without working: M0M0A0 |
| Allow Almost complete relationship (or corr'n or link) between amount spent on advert & profit | B1 1 | or Very positive corr'n or Very reliable relationship or Near perfect relationship between spend on advert & profit oe, in context | Must state or imply "strong" or "good" or equiv & in context but NOT Strong agreement between etc NOT High spend on ads produces high profits NOT The more spent on adverts, the higher the profit NOT Positive corr'n between spend on ads & profits NOT There is a relationship between spend on ads & profit NOT There is a great relationship between etc NOT ans involving "proportional)" Ignore irrelevant or incorrect. If incorrect $r$ (< 0.9) in (i), no ft for ans "weak rel'nship" here; but correct ans here scores B1 even if inconsistent with their $r$ |3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, $\pounds x$ thousand, and the annual profit, $\pounds y$ thousand. A summary of the figures for 12 years is as follows.
$$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$
(i) Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .\\
(ii) Comment briefly on this value in this context.\\
(iii) A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.\\
(iv) Calculate the equation of the regression line of $y$ on $x$.\\
(v) Estimate the annual profit during a year when $\pounds 7400$ was spent on advertising.
\hfill \mbox{\textit{OCR S1 2011 Q3 [12]}}