Refer to the IHS Journal of Hydraulic Engineering (September, 2012) study of the repair and replacement of water pipes, Exercise 10.8. Recall that a team of civil engineers used regression analysis to model y = the ratio of repair to replacement cost of commercial pipe as a function of x = the diameter (in millimeters) of the pipe. Data for a sample of 13 different pipe sizes are reproduced in the accompanying table. In Exercise 10.8, you fit a straight-line model to the data. Now consider the quadratic model, E(y) = β_o + β₁x + β₂x². A MINITAB printout of the analysis follows.

a. Give the least squares prediction equation relating ratio of repair to replacement cost (y) to pipe diameter (x).

b. Conduct a global F-test for the model using α = .01. What do you conclude about overall model adequacy?

c. Evaluate the adjusted coefficient of determination, R_a² for the model.

d. Give the null and alternative hypotheses for testing if the rate of increase of ratio (y) with diameter (x) is slower for larger pipe sizes.

e. Carry out the test, part d, using α = .01.

f. Locate, on the printout, a 95% prediction interval for the ratio of repair to replacement cost for a pipe with a diameter of 240 millimeters. Interpret the result

Data from Exercise 10.8

Pipes used in a water distribution network are susceptible to breakage due to a variety of factors. When pipes break, engineers must decide whether to repair or replace the broken pipe. A team of civil engineers used regression analysis to estimate y = the ratio of repair to replacement cost of commercial pipe in the IHS Journal of Hydraulic Engineering (September 2012). The independent variable in the regression analysis was x = the diameter (in millimeters) of the pipe. Data for a sample of 13 different pipe sizes are provided in the table.

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WATERPIPE DIAMETER RATIO 80 6.58 100 6.97 125 7.39 150 7.61 200 7.78 250 7.92 300 8.20 350 8.42 400 8.60 450 8.97 500 9.31 600 9.47 700 9.72