Civil engineers often use the straight-line equation, E(y) = β
0 + β
1x, to model the relationship between the mean shear strength E(y) of masonry joints and precompression stress, x. To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for n = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the regression analysis.
Sum of
Mean
Source
DF
Squares
Square
F Value
Prob > F
Model
1
2.39555
2.39555
47.732
0.0010
Error
5
0.25094
0.05019
C Total
6
2.64649
Root MSE
0.22403
R-square
0.9052
Dep Mean
2.32857
Adj R-sq
0.8862
C.V.
9.62073
Parameter Estimates
Parameter
Standard
T for HO:
Variable
DF
Estimate
Error
Parameter=0
Prob > |T|
INTERCEP
1
1.191930
0.18503093
6.442
0.0013
X
1
0.987157
0.14288331
6.909
0.0010
Give a practical interpretation of R
2, the coefficient of determination for the least squares model. Express R
2 to the nearest whole percent.
◦ About 91% of the total variation in the sample of y-values can be explained by (or attributed to) the linear relationship between shear strength and precompression stress.
◦ In repeated sampling, approximately 91% of all similarly constructed regression lines will accurately predict shear strength.
◦ We expect about 91% of the observed shear strength values to lie on the least squares line.
◦ We expect to predict the shear strength of a triplet test to within about .91 ton of its true value.
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