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 formula for the product moment correlation coefficient, $r$, is:

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

Where:

• $S_{xx} = 353.2375$
• $S_{yy} = 4305 - \frac{(181^2)}{8}$ (Calculate for $S_{yy}$)

Thus,

$S_{yy} = 4305 - \frac{(181^2)}{8} = 4305 - 4090.25 = 214.75$

Finally, substituting into the correlation formula gives: $r = \frac{-23.72625}{\sqrt{353.2375 * 214.75}} = -0.87104$

To 3 significant figures, this results in: $r \approx -0.871$

The correlation coefficient of $r = -0.871$ indicates a strong negative correlation between the height above sea level and the mid-day temperature. This implies that, generally, as the height increases, the temperature tends to decrease.

To find $S_{hh}$, we need to apply the transformation of the variable:

$h = \frac{x}{1000}$

Thus,

$S_{hh} = S_{xx} \times \left(\frac{1}{1000^2}\right) = 353.2375 \times \frac{1}{1000000} = 0.0003532375$

Since $h$ is a linear transformation of $x$, the correlation coefficient between $h$ and $y$ remains the same. Therefore, the correlation coefficient between $h$ and $y$ is:

$r = -0.871$