| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Interpret correlation coefficient value |
| Difficulty | Moderate -0.8 This is a straightforward S1 question testing standard formulas for correlation coefficient and understanding of how linear transformations affect summary statistics. Parts (a)-(c) involve direct formula application with given summary statistics, while (d)-(e) test conceptual understanding that scaling affects Sxx but not correlation. All steps are routine with no problem-solving or novel insight required. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08b Linear coding: effect on pmcc |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_{yy} = 4305 - \frac{181^2}{8} = 209.875\) (awrt 210) | M1, A1 | Allow one slip e.g. 4350 for 4305 |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = \frac{(-) 23726.25}{\sqrt{3535237.5 \times "209.875"}} = -0.87104...\) (awrt -0.871) | M1, A1 | Follow through their answer to (a). Condone no "−". Allow M1 for ± 0.87 with no working. (−0.871 is M1A1) |
| Answer | Marks | Guidance |
|---|---|---|
| Higher towns have lower temperature or temp. decreases as height increases | B1 | Must mention temperature (o.e.) and height (above sea level) and interpret the relationship between them. Must be a correct and sensible comment. e.g. "As temperature increases the height of the sea decreases" is B0. BUT simply stating "As temperature increases the height decreases" is B1 although "As height increases the temperature decreases" would be better. Treat mention of 0.87... as ISW |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_{hh} = 3.5352375\) (awrt 3.54) (condone 3.53) | B1 | Accept awrt 3.54 and condone 3.53 (i.e truncation) |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = -0.87104...\) (awrt -0.871) | B1ft | For awrt −0.871 or ft their final answer to part (b) to the same accuracy (or 3 sf) provided −1 < r < 1. Answer to part (e) must be a number "it's the same" is B0 |
## (a)
$S_{yy} = 4305 - \frac{181^2}{8} = 209.875$ (awrt 210) | M1, A1 | Allow one slip e.g. 4350 for 4305
## (b)
$r = \frac{(-) 23726.25}{\sqrt{3535237.5 \times "209.875"}} = -0.87104...$ (awrt -0.871) | M1, A1 | Follow through their answer to (a). Condone no "−". Allow M1 for ± 0.87 with no working. (−0.871 is M1A1)
## (c)
Higher towns have lower temperature or temp. decreases as height increases | B1 | Must mention temperature (o.e.) and height (above sea level) and interpret the relationship between them. Must be a correct and sensible comment. e.g. "As temperature increases the height of the sea decreases" is B0. BUT simply stating "As temperature increases the height decreases" is B1 although "As height increases the temperature decreases" would be better. Treat mention of 0.87... as ISW
## (d)
$S_{hh} = 3.5352375$ (awrt 3.54) (condone 3.53) | B1 | Accept awrt 3.54 and condone 3.53 (i.e truncation)
## (e)
$r = -0.87104...$ (awrt -0.871) | B1ft | For awrt −0.871 or ft their final answer to part (b) to the same accuracy (or 3 sf) provided −1 < r < 1. Answer to part (e) must be a number "it's the same" is B0
---On a particular day the height above sea level, $x$ metres, and the mid-day temperature, $y$°C, were recorded in 8 north European towns. These data are summarised below
$S_{xx} = 3\,535\,237.5 \quad \sum y = 181 \quad \sum y^2 = 4305 \quad S_{yy} = -23\,726.25$
\begin{enumerate}[label=(\alph*)]
\item Find $S_{yy}$. [2]
\item Calculate, to 3 significant figures, the product moment correlation coefficient for these data. [2]
\item Give an interpretation of your coefficient. [1]
\end{enumerate}
A student thought that the calculations would be simpler if the height above sea level, $h$, was measured in kilometres and used the variable $h = \frac{x}{1000}$ instead of $x$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Write down the value of $S_{hh}$ [1]
\item Write down the value of the correlation coefficient between $h$ and $y$. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2011 Q1 [7]}}