In mining engineering, holes are often drilled through rock, using drill bits. As a drill hole gets deeper, additional rods are added to the drill bit to enable additional drilling to take place. It is expected that drilling time increases with depth. This increased drilling time could be caused by several factors, including the mass of the drill rods that are strung together. The business problem relates to whether drilling is faster using dry drilling holes or wet drilling holes. Using dry drilling holes involves forcing compressed air down the drill rods to flush the cuttings and drive the hammer. Using wet drilling holes involves forcing water rather than air down the hole. Data have been collected from a sample of 50 drill holes that contains measurements of the time to drill each additional 5 feet (in minutes), the depth (in feet), and whether the hole was a dry drilling hole or a wet drilling hole. The data are organized and stored in Drill . (Data extracted from R. Penner and
D. G. Watts, “ Mining Information,” The American Statistician, 45, 1991, pp. 4– 9.) Develop a model to predict additional drilling time, based on depth and type of drilling hole (dry or wet). For (a) through (j) do not include an interaction term.
a. State the multiple regression equation.
b. Interpret the regression coefficients in (a).
c. Predict the mean additional drilling time for a dry drilling hole at a depth of 100 feet. Construct a 95% confidence interval estimate and a 95% prediction interval.
d. Perform a residual analysis on the results and determine whether the regression assumptions are valid.
e. Is there a significant relationship between additional drilling time and the two independent variables (depth and type of drilling hole) at the 0.05 level of significance?
f. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data.
g. Construct a 95% confidence interval estimate of the population slope for the relationship between additional drilling time and depth.
h. Construct a 95% confidence interval estimate of the population slope for the relationship between additional drilling time and the type of hole drilled.
i. Compute and interpret the adjusted r2.
j. What assumption do you need to make about the slope of additional drilling time with depth?
k. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model.
l. On the basis of the results of (f) and (k), which model is most appropriate? Explain.
m. What conclusions can you reach concerning the effect of depth and type of drilling hole on drilling time?