Predicting the total PCB level in fish. Polychlorinated biphenyls (PCBs) are synthetic compounds, called congeners, that are particularly toxic to fetuses and young children. Although PCBs are no longer produced in the United States, they are still found in the environment. Since human exposure to these PCBs is primarily through the consumption of fish, the Environmental Protection Agency (EPA) monitors PCB levels in fish. Unfortunately, there are 209 different congeners and measuring all of them in a fish specimen is an expensive and time-consuming process.

You’ve been asked to see if the total amount of PCBs in a specimen can be estimated with only a few, easily quantifiable congeners.26 If this can be done, costs can be greatly reduced.

Consider the following variables: PCB (the total amount of PCB)

and four congeners, PCB52, PCB118, PCB138, and PCB180.

DATADATA DATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATA DATADATADATADATADATA DATA FILE PCB

(a) Using numerical and graphical summaries, describe the distribution of each of these variables.

(b) Using numerical and graphical summaries, describe the relationship between each pair of variables in this set.

(c) Use the four congeners, PCB52, PCB118, PCB138, and PCB180, in a multiple regression to predict PCB.

(d) Examine the residuals. Do they appear to be approximately Normal? When you plot them versus each of the explanatory variables, are any patterns evident?

(e) There may be two outliers, one with a high residual and one with a low residual. Because of safety issues, we are more concerned about underestimating total PCB in a specimen than about overestimating. Give the specimen number for each of the two suspected outliers. Which one corresponds to an overestimate of PCB?

(f) Rerun the analysis with the two suspected outliers deleted, summarize these results, and compare them with those you obtained in part (c).