Refer to the Engineering Structures (July 2013) study of the characteristics of fresh concrete, Exercise 10.38. Recall that the researchers investigated the linear effect of time needed for pressure stabilization (x) on each of three different dependent variables: y₁ = initial setting time (hours), Y₂ = final setting time (hours), and y₃ = maturity index (ºC - hours). The data on these variables for n = 8 fresh concrete lateral pressure tests are reproduced in the table (next column).

a. Apply Spearman’s rank correlation test to determine which of the three dependent variables is most strongly positively associated with pressure stabilization time.

b. Consider the linear model, E(y_i) = β₀ + β₁x, i = 1, 2, 3. Apply Theil’s nonparametric procedure to determine which of the three slopes are significantly greater than 0. (Test using α = .05.)

Data from Exercise 10.38

Engineering Structures (July 2013) published a study of the characteristics of fresh concrete. One key variable studied was the time (in hours) needed for pressure stabilization of the concrete. The researchers investigated the effect of time needed for pressure stabilization (x) on each of three different dependent variables: y₁ = initial setting time (hours), y₂ = final setting time (hours), and y₃ = maturity index (ºC-hours). The data on these variables for fresh concrete lateral pressure tests are listed in the table.

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CONCRETE2 Test Y1 Y2 Y3 X Al 4.63 7.17 385.81 12.03 A2 4.32 6.52 358.44 11.32 A3 4.54 6.31 292.71 9.51 A4 4.09 6.19 253.16 8.25 A5 4.56 6.81 279.82 9.02 A6 4.48 6.98 318.74 9.97 A7 4.35 6.45 262.14 8.42 A8 4.23 6.69 244.97 7.53