A student is investigating the relationship between the price ($y$ pence) of 100g of chocolate and the percentage ($x$ %) of cocoa solids in the chocolate - Edexcel - A-Level Maths Statistics - Question 3 - 2007 - Paper 2

Given that $S_{xy} = 28,750 - \frac{(315)(620)}{8} = 4337.5$ holds true. To find $S_{xx}$, use the formula:

$S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n}$
Where $\sum x^2 = 15,225$, $\sum x = 315$, and $n = 8$.
Substituting these values gives:
$S_{xx} = 15,225 - \frac{(315)^2}{8} = 1,875.$

From the linear regression equations, we have:

$b = \frac{S_{xy}}{S_{xx}} = \frac{4337.5}{1875} \approx 2.31.$
For $a$, we calculate:
$a = \frac{\sum y - b \sum x}{n} = \frac{620 - (2.31)(315)}{8} \approx 17.0.$
Thus, $a \approx 17.0$ and $b \approx 2.3$.

Using the values of $a$ and $b$ obtained, plot the regression line on the scatter diagram with the equation $y = 17 + 2.3x$. This line should ideally pass through the cluster of points representing the chocolate brands.

i) Brand D is overpriced as it lies above the regression line significantly.
ii) A fair price for Brand D can be calculated using the regression equation:
For $x = 35$ (the cocoa percentage for Brand D), the price is approximately:
$y = 17 + 2.3(35) = 69.5$ pence. Therefore, a fair price to suggest would be around 70 pence.