An article in the Journal of the Environmental Engineering Division [“Least Squares Estimates of BOD Parameters” (1980, Vol. 106, pp. 1197–1202)] took a sample from the Holston River below Kingport, Tennessee, during August 1977. The biochemical oxygen demand (BOD) test is conducted over a period of time in days. The resulting data follow:

(a) Assuming that a simple linear regression model is appropriate, fit the regression model relating BOD (y) to the time (x). What is the estimate of σ²?

(b) What is the estimate of expected BOD level when the time is 15 days?

(c) What change in mean BOD is expected when the time changes by three days?

(d) Suppose that the time used is six days. Calculate the fitted value of y and the corresponding residual.

(e) Calculate the fitted y_i for each value of xi used to fit the model. Then construct a graph of y_i versus the corresponding observed values y_i and comment on what this plot would look like if the relationship between y and x was a deterministic (no random error) straight line. Does the plot actually obtained indicate that time is an effective regressor variable in predicting BOD?