A manufacturer stores drums of chemicals - Edexcel - A-Level Maths Statistics - Question 3 - 2006 - Paper 1

The points lie close to a straight line, indicating a linear relationship between the time in storage and evaporation loss. This justifies the use of a linear regression model.

Using the given formulas:
Given data:
$S_{xy} = 8354, ext{ } S_{y} = 53112, ext{ } S_{x} = 1352$
The formula for b is:
b = rac{S_{y} - S_{x} imes rac{S_{y}}{n}}{S_{x} - rac{S_{x}^2}{n}}
Substituting the values into the formulas:
$b = 3.90$
Now, substituting to find a:
a = $y - b$x:
Based on the calculations, we find:
$a = -29.02$
Thus, values are:
a = -29.02 and b = 3.90.

The value of b indicates that for each additional week in storage, the evaporation loss increases by approximately 3.90 ml.

To predict evaporation at 19 weeks, substitute into the model:
y = a + bx
$y = -29.02 + 3.90(19) = 57.87$
Thus, the predicted evaporation is approximately 57.87 ml.

To predict evaporation at 35 weeks, substitute into the model:
y = a + bx
$y = -29.02 + 3.90(35) = 108.17$
Thus, the predicted evaporation is approximately 108.17 ml.

The prediction for 19 weeks is reliable as it falls within the range of x values used to create the model. However, the prediction for 35 weeks is less reliable as it lies outside the original data range, and there is no evidence to suggest that the linear model will hold true for longer periods.