An article in the Journal of Environmental Engineering

(Vol. 115, No. 3, 1989, pp. 608–619) reported the results of a study on the occurrence of sodium and chloride in surface streams in central Rhode Island. The data shown are chloride concentration y (in mg/1) and roadway area in the watershed x (in %).

(a) Draw a scatter diagram of the data. Does a simple linear regression model seem appropriate here?

(b) Fit the simple linear regression model using the method of least squares.

(c) Estimate the mean chloride concentration for a watershed that has 1% roadway area.

(d) Find the fitted value corresponding to x 0.47 and the associated residual.

(e) Suppose we wish to fit a regression model for which the true regression line passes through the point (0, 0). The appropriate model is Y x . Assume that we have n pairs of data (x1, y1), (x2, y2), . . . , (xn, yn). Show that the least squares estimate of is

(f) Use the results of part

(e) to fit the model Y x to the chloride concentration–roadway area data in this exercise.
Plot the fitted model on a scatter diagram of the data and comment on the appropriateness of the model.

Transcribed Image Text:

y 4.4 6.6 9.7 10.6 10.8 x 0.19 0.15 0.57 0.70 0.67 y 10.9 11.8 12.1 14.3 14.7 X 0.63 0.47 0.70 0.60 0.78 y x + 15.0 17.3 19.2 23.1 27.4 0.81 0.78 0.69 1.30 1.05 y 27.7 31.8 39.5 x 1.06 1.74 1.62