PCBs make up a family of hazardous chemicals that are often dumped, illegally, by industrial plants into the surrounding streams, rivers, or bays. The table below reports the annual concentrations of PCBs (measured in parts per billion) in water samples collected for two consecutive years from 37 U.S. bays and estuaries. An official from the Environmental Protection Agency wants to model the year 2 PCB concentration (y) of a bay as a function of the previous year’s PCB concentration (x).

a. Fit the first-order model, E(y) = β_o + β₁x, to the data. Give the least-squares prediction equation.

b. Is the model adequate for predicting y? Explain.

c. Construct a residual plot for the data. Do you detect any outliers? If so, identify them.

d. Refer to part c. Although the residual for Boston Harbor is not, by definition, an outlier, the EPA believes that it has strong influence on the regression because of its large y value. Remove the observation for Boston Harbor from the data and refit the model. Has model adequacy improved?

e. An alternative approach is to use the natural log transformations y* = ln(y + 1) and x* = ln(x + 1), and fit the model E(y*) = β_o + β₁x*. Fit this model, then conduct a test for model adequacy and perform a residual analysis. Interpret the results. In particular, comment on the residual value for Boston Harbor.

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BAYPСВ PCB Concentration PCB Concentration Bay State Year 1| Year 2 Bay State Year 1 Year 2 Casco Bay ME 95.28 77.55 Mississippi River Delta LA 34 30.14 Merimack River МА 52.97 29.23 Barataria Bay LA Salem Harbor МА 533.58 403.1 San Antonio Bay TX Corpus Christi Bay San Diego Harbor Boston Harbor МА 17,104.86 736 TX Buzzards Bay MA 308.46 192.15 СА 422.1 531.67 Narragansett Bay RI 159.96 220.6 San Diego Bay СА 6.74 9.3 East Long Island Sound West Long Island Sound NY 10 8.62 Dana Point CA 7.06 5.74 NY 234.43 174.31 Seal Beach СА 46.71 46.47 Raritan Bay NJ 443.89 529.28 San Pedro Canyon CA 159.56 176.9 Delaware Bay DE 2.5 130.67 Santa Monica Bay СА 14 13.69 Lower Chesapeake Bay VA 51 39.74 Bodega Bay CA 4.18 4.89 Pamlico Sound NC Coos Bay OR 3.19 6.6 Charleston Harbor SC 9.1 8.43 Columbia River Mouth OR 8.77 6.73 Sapelo Sound GA Nisqually Beach WA 4.23 4.28 St. Johns River FL 140 120.04 Commencement Bay WA 20.6 20.5 Tampa Bay FL Elliott Bay WA 329.97 414.5 Apalachicola Bay FL 12 11.93 Lutak Inlet AK 5.5 5.8 Mobile Bay AL Nahku Bay AK 6.6 5.08 Round Island MS