Question

ocessing of raw coal involves "washing," in which coal ash (nonorganic, incombustible material) is removed. The article "Quantifying Sampling Precision for Coal Ash Using Gy’s Discrete Model of the Fundamental Error" (Journal of Coal Quality, 1989:33–39) provides data relating the percentage of ash to the volume of a coal particle. The average percentage of ash for six volumes of coal part was measured, and represented below:Volume (cm3) 0.01 0.06 0.58 2.24 15.55 276.02% Ash 3.32 4.05 5.69 7.06 8.17 9.36Compute the least square line for predicting percent ash from in volume.

Answer 1

y = 5.5765*x + 0.0142

Explanation:

Let's consider the volume as the explanatory variable x and the percentage of ash as the response variable y

We construct the following table

`\large \left[\begin{array}{ccccc}&x&y&x^2&xy\\&0.01&3.32&0.0001&0.032\\&0.06&4.05&0.0036&0.243\\&0.58&5.69&0.3364&3.3002\\&2.24&7.06&5.0176&15.8144\\&15.55&8.17&241.8025&127.0435\\&276.02&9.36&76187.0404&2583.5472\\\sum&294.46&37.65&76434.2006&2729.9815\end{array}\right]`

Once this table is built, the rest is easy.

The least squares line is given by

y = mx + b

where

`\large m=((\sum y)(\sum x^2)-(\sum x)(\sum xy))/(n(\sum x^2)-(\sum x)^2)`

`\large b=(n(\sum xy)-(\sum x)(\sum y))/(n(\sum x^2)-(\sum x)^2)`

where n is the number of observations taken, in this case 6.

Computing all these values we have our least squares line

`\large \boxed{y=5.5765x+0.0142}`

And that's it!