A clothes shop manager records the weekly sales figures, £ S, and the average weekly temperature, t °C, for 6 weeks during the summer - Edexcel - A-Level Maths Statistics - Question 1 - 2017 - Paper 1

$S_{s} = \frac{784}{6} = 130.67$, $S_{t} = \frac{119}{6} = 19.83$.

The product moment correlation coefficient, $r$, is calculated as: $r = \frac{\sum (w_i - \bar{w})(t_i - \bar{t})}{\sqrt{\sum (w_i - \bar{w})^2 \sum (t_i - \bar{t})^2}}$ Substituting the values, we find that $r \approx -0.801.$

The value of $r \approx -0.801$ indicates a strong negative correlation between sales and temperature, suggesting that as the temperature increases, sales tend to decrease. This supports the manager's belief about the relationship between the two variables.

To find the regression line, we use: $w = a + bt$ Where $b = \frac{\sum wt - n \cdot \bar{w} \cdot \bar{t}}{\sum t^{2} - n \cdot \bar{t}^{2}}$ and $a = \bar{w} - b \cdot \bar{t}$. Using the calculated values, we find: $b = -0.065\ and \ a = 20.000 - 0.065(19.83) = 20.000 - 1.289 = 18.711.$ Thus, $w = 18.711 - 0.065t$.

Using the previous result, we find: $s = c + d \cdot t$ where $d = 0.065 \times 1000 = 65$ and $c = 18.711 \times 1000 = 18711.$ So, the regression equation is: $s = 18711 - 65t.$

From the equation $s = 18711 - 65t$, a 1°C increase in average weekly temperature corresponds to a decrease of £65 in weekly sales, indicating that higher temperatures tend to reduce sales significantly.