Using the data of Exercise 100,

(a) Create a new variable, x₁x₂.

(b) Fit a surface of the form

(c) Find the correlation matrix of the four independent variables. Is there evidence of multi collinearity? (d) Standardize each of the three independent variables x₁, x₂, and x₃, and create a new variable that is the product of the standardized values of x₁ and x₂.

(e) Fit a curved surface of the same form to the standardized variables. Compare the goodness of fit of this surface to that of the linear surface fitted in Exercise 100.

(f) Find the correlation matrix of the four standardized independent variables and compare with the results of part (c).

Exercise 100

The following data represent more extended measurements of monthly water usage at the plant referred to in Exercise 98 over a period of 20 months:

(a) Use an appropriate computer program to fit a linear surface to these data.

(b) Use a computer program to make a normal-scores plot of the residuals. Does the assumption of normality appear to be satisfied at least approximately?

(c) Plot the residuals against  and determine whether the pattern is random.

(d) Check for excessive multi collinearity among the independent variables.

Exercise 98

(a) Use an appropriate computer program to fit a plane to the following data relating the monthly water usage of a production plant (gallons) to its monthly production (tons), mean monthly ambient temperature (?F), and the monthly number of days of plant operation over a period of 12 months.

(b) Estimate the water usage of the plant during a month when its production is 90.0 tons, the mean ambient temperature is 65?F, and it operates for 20 days.