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 - Edexcel - A-Level Maths Statistics - Question 1 - 2011 - Paper 2

The product moment correlation coefficient r is calculated using:

$r = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}$

We have values:

• $S_{xy} = -23.72625$\
• $S_{xx} = 353.2375$\
• $S_{yy} = 209.875$

Thus: $r = \frac{-23.72625}{\sqrt{353.2375 \times 209.875}}\ \approx \frac{-23.72625}{\sqrt{74033.086}}\ \approx \frac{-23.72625}{271.37} \approx -0.087104.$

Therefore, to 3 significant figures, $r \approx -0.0871$.

The correlation coefficient of approximately -0.0871 suggests a very weak negative correlation between height above sea level and mid-day temperature. This implies that as the height increases, there tends to be a slight decrease in temperature, though the relationship is not strong.

Given the transformation $h = \frac{x}{1000}$, the value of $S_{hh}$ can be calculated as:

$S_{hh} = S_{xx} \cdot \left(\frac{1}{1000}\right)^2\ = 353.2375 \times \left(\frac{1}{1000}\right)^2\ = 353.2375 \times 10^{-6} \approx 0.3532375.$

Thus, $S_{hh} \approx 0.3532.$

The correlation coefficient between $h$ and $y$ can be calculated using the previously found value for $S_{hy}$:

r_{hy} = \frac{S_{hy}}{\sqrt{S_{hh} \cdot S_{yy}}} \approx \frac{-23.72625}{\sqrt{0.3532375 \cdot 209.875}}.\

Since $S_{hh}$ is scaled down by a factor of 1000 (from $x$), the correlation coefficient remains the same. Therefore, $r_{hy} \approx -0.0871.$