Question

'The article "Reduction in Soluble Protein and Chlorophyll Contents in a Few Plants as Indicators of Automobile Exhaust Pollution" (International Journal of Environmental Studies [1983]: \(239-244\) ) reported the following data on \(x=\) distance from a highway (in meters) and \(y=\) lead content of soil at that distance (in parts per million): a. Use a statistical computer package to construct scatterplots of \(y\) versus \(x, y\) versus \(\log (x), \log (y)\) versus \(\log (x)\) and \(\frac{1}{y}\) versus \(\frac{1}{x}\). b. Which transformation considered in Part (a) does the best job of producing an approximately linear relationship? Use the selected transformation to predict lead content when distance is \(25 \mathrm{~m}\).

Short answer

After constructing the scatterplots and assessing the linearity of the relationships, best transformation will be identified. Using the equation of the best fit line associated with it, the lead content in the soil when the distance from the highway is 25 meters will be predicted.

Step-by-step solution

Construct the scatterplots

Using a statistical computer package, construct the 4 scatterplots. The scatterplots will be \(y\) versus \(x\), \(y\) versus \(\log (x)\), \(\log (y)\) versus \(\log (x)\) and \(\frac{1}{y}\) versus \(\frac{1}{x}\). This will give a visual representation of the relationship between distance from the highway and the lead content of the soil.

Determine the most linear relationship

Analysis of the 4 scatterplots created in step 1 is done to visually determine which transformation gives the most linear relationship between \(x\) and \(y\). This can be done by looking at the 'spread' of points from any determined line of best fit - the more closely the points lie to the line, the more linear the relationship.

Prediction using the best transformation

Once the best transformation has been identified, it is then used to predict the lead content in soil when the distance from the highway is 25 meters. This can be done by plugging 25 into the equation of the best fit line related to the optimal transformation.