Chapter 14, Problem 42PS

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 increase with depth. This increased drilling time could be caused by several factors, including the mass of the drill roads 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 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.

Develop a model to predict additional drilling time, based on depth the type of drilling hole (dry or wet). For (a) through (k) 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 model 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 inde- pendent 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 type of hole drilled.

i. Compute and interpret the adjusted $r^2.$

j. Compute the coefficients of partial determination and interpret their meaning.

k. What assumption do you need to make about the slope of additional drilling time with depth?

l. Add an interaction team to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model.

m. On the basis of the results of (f) and (I), which model is most appropriate? Explain.

n. What conclusions can you reach concerning the effect of depth and type of drilling hole or drilling time?