Natural gas is found in rock formations underground. In order to extract the gas, a procedure known as hydraulic fracturing, or

‘‘fracking,’’ is often used. In this procedure, fluid mixed with sand is pumped into the gas well and forced through cracks in the rock. The sand holds open the cracks, allowing the gas to flow through. A study was performed at the Colorado School of Mines to determine whether increasing the amounts of fluid or sand would increase the production of a well. Varying amounts of fluid and sand were pumped into 255 gas wells, and the amount of gas recovered was measured. The amount of gas produced is measured in units of 100 cubic feet per each foot of well depth, fluid is measured in thousands of gallons per foot, and sand is measured in thousands of pounds per foot. The following MINITAB output presents the multiple regression equation Production = b0 + b1Fluid + b2 Sand.

a. Predict the gas production for a well treated with 3 thousand gallons of fluid and 7 thousand pounds of sand.

b. Two wells have the same amount of sand pumped in, and the amount of fluid differs by 2 thousand pounds. By how much should we predict their productions to differ?

c. Is the model useful for prediction? Explain. Use the α = 0.01 level.

d. How much of the variation in the gas production is explained by the model?

e. Let β1 be the coefficient of fluid. Test H0: β1 = 0 versus H1: β1 ≠ 0 at the α = 0.01 level. What do you conclude?

f. Let β2 be the coefficient of sand. Test H0: β2 = 0 versus H1: β2 ≠ 0 at the α = 0.05 level. What do you conclude?

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The regression equation is Production 1.0729 + 1.1752 Fluid - 0.024886 Sand = Predictor Coef StDev T P Constant 1.0729 0.23348 4.5954 0.000 Fluid 1.1752 0.24609 4.7753 0.000 Sand -0.024886 0.096792 -0.25711 0.797 S = 2.0197 R-Sq = 29.1% R-Sq (adj) = 28.5% Analysis of Variance Source DF SS MS F P Regression 2 421.70 210.85 51.689 0.000 Residual Error 252 1028.0 4.0792 Total 254 1449.7