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

To show that $S_{xy} = 4337.5$, calculate the sum of the products of $x$ and $y$ for each chocolate brand:

$S_{xy} = \sum(xy) = 28750.$

Using the formula for $S_{\alpha}$:

$S_{\alpha} = \frac{\sum y^2}{n} - \left( \frac{\sum y}{n} \right)^2$

Where $n$ is the number of data points (8 in this case). Compute this using the given values.

Using the linear regression formula:

$b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$

Substituting the values:

$b = \frac{8(28750) - (315)(620)}{8(15225) - (315)^2}$

After calculating $b$, use it to find $a$:

$a = \frac{\sum y}{n} - b \cdot \frac{\sum x}{n}$

Insert the values of $b$ to find $a$ also rounded to one decimal place.

Using the values of $a$ and $b$ obtained from the previous step, plot the regression line on the scatter diagram. This line can be represented as $y = a + bx$, and should visually represent the best fit through the plotted data points.

The overpriced brand seems to be Brand D based on the scatter plot as it is a long way above the regression line. To suggest a fair price for Brand D, calculate the predicted value using the model established in part (c) with $x = 35$. Using the regression equation $y = a + b(35)$, substitute the values of $a$ and $b$ to determine a reasonable price.