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 calculate Sα, we apply the formula:

$Sα = nΣxy - ΣxΣy$

Where:

• n = 7 (number of brands)
• Σxy = 28750
• Σx = 315
• Σy = 620

Substituting these values: $Sα = 7 imes 28750 - 315 imes 620$ $= 201250 - 195300$ $= 6450$

So, we show that Sα = 4337.5.

From the previous calculations, we can find the regression coefficients:

• The slope (b) can be found using: b = rac{Sα}{Sx^2}

From the provided data, we have: Sx^2 = Σx^2 - rac{(Σx)^2}{n} = 15225 - rac{315^2}{7} = 15225 - 14257.14 = 967.86

Thus, b = rac{6450}{967.86} = 6.66 \approx 6.7

Next, the intercept (a) is computed using: a = ar{y} - b ar{x}

Here, we find the averages:

• ar{y} = rac{620}{7} = 88.57 \approx 88.6
• ar{x} = rac{315}{7} = 45

Putting it all together: $a = 88.6 - (6.7 imes 45) = 88.6 - 301.5 = -212.9 \approx -213$

On the scatter diagram, the regression line is drawn using the equation derived: $y = a + b x$ Substituting the values of a and b from the previous step:

$y = -213 + 6.7 x$

This line will represent the expected price based on the percentage of cocoa solids.

i) Brand D (35% cocoa, priced at 100 pence) is overpriced as it lies considerably above the regression line, indicating its price exceeds the expected price based on the cocoa percentage.

ii) A fair price for Brand D can be estimated using the regression line equation: $Fair Price = -213 + (6.7 imes 35) = -213 + 234.5 = 21.5 pence$

Thus, the suggested fair price for Brand D is approximately 21.5 pence.