Here are some data taken from the article “Chemithermomechanical Pulp from Mixed High Density Hardwoods” by Miller, Shankar, and Peterson (Tappi Journal, 1988). Given are the percent NaOH used as a pretreatment chemical, x1, the pretreatment time in minutes, x2, and the resulting value of a specific surface area variable, y (with units of cm2/g), for nine batches of pulp produced from a mixture of hardwoods at a treatment temperature of 75°C in mechanical pulping

% NaOH, x1	Time, x2	Specific Surface Area, y
3	30	5.95
3	60	5.60
3	90	5.44
9	30	6.22
9	60	5.85
9	90	5.61
15	30	8.36
15	60	7.30
15	90	6.43

a) Fit the approximate relationship y ≈ β0 + β1x1 + β2x2 to these data via least squares. Interpret the coefficients b1 and b2 in the fitted equation. What fraction of the observed raw variation in y is accounted for using the equation?

b) Compute and plot residuals for your fitted equation from a). Discuss what these plots indicate about the adequacy of your fitted equation. (At a minimum, you should plot residuals against all of x1, x2, and yˆ and normal-plot the residuals.)

c) Make a plot of y versus x1 for the nine data points and sketch on that plot the three different linear functions of x1 produced by setting x2 first at 30, then 60, and then 90 in your fitted equation from a). How well to fitted responses appear to match observed responses?

d) What specific surface area would you predict for an additional batch of pulp of this type produced using a 10% NaOH treatment for a time of 70 minutes? Would you be willing to make a similar prediction for 10% NaOH used for 120 minutes based on your fitted equation? Why or why not?

e) There are many other possible approximate relationships that might be fitted to these data via least squares, one of which is y ≈ β0 + β1x1 + β2x2 + β3x1x2. Fit this equation to the preceding data and compare the resulting coefficient of determination to the one found in a). On these basis of these alone, does the use of a more complicated equation seem necessary?

f) For the equation fit in part e), repeat the steps of part c) and compare the plot made here to the 1 one made earlier.

g) What is an intrinsic weakness of this real published data set?

h) What terminology (for data structures) introduced in section 1.2 describes this data set? It turns out that since the data set has this special structure and all nine sample sizes are the same (i.e., all are 1), some special relationships hold between the equation fit in a) and what you get by separately fitting linear equations in x1 and x2 to the y data. Fit such one-variable linear equations and compare coefficients and R2 values to what you obtained in a). What relationships exist between these?