The article “The Undrained Strength of Some Thawed Permafrost Soils” (Canadian Geotechnical J., 1979:

420–427) contains the following data on undrained shear strength of sandy soil (y, in kPa), depth (x1, in m), and water content (x2, in %).

y x1 x2 y y ˆ

y e ˆ *

1 14.7 8.9 31.5 23.35 8.65 1.50 2 48.0 36.6 27.0 46.38 1.62 .54 3 25.6 36.8 25.9 27.13 1.53 .53 4 10.0 6.1 39.1 10.99 .99 .17 5 16.0 6.9 39.2 14.10 1.90 .33 6 16.8 6.9 38.3 16.54 .26 .04 7 20.7 7.3 33.9 23.34 2.64 .42 8 38.8 8.4 33.8 25.43 13.37 2.17 9 16.9 6.5 27.9 15.63 1.27 .23 10 27.0 8.0 33.1 24.29 2.71 .44 11 16.0 4.5 26.3 15.36 .64 .20 12 24.9 9.9 37.8 29.61 4.71 .91 13 7.3 2.9 34.6 15.38 8.08 1.53 14 12.8 2.0 36.4 7.96 4.84 1.02 The predicted values and residuals were computed by fitting a full quadratic model, which resulted in the estimated regression function y
151.36  16.22x1 13.48x2 .094x2 1

 .253x2 2 .492x1x2

a. Do plots of e* versus x1, e* versus x2, and e* versus yˆ

suggest that the full quadratic model should be modified?

Explain your answer.

b. The value of R2 for the full quadratic model is .759. Test at level .05 the null hypothesis stating that there is no linear relationship between the dependent variable and any of the five predictors.

c. It can be shown that V(Y)
2
V(Yˆ) V(Y Yˆ). The estimate of  is ˆ
s
6.99 (from the full quadratic model). First obtain the estimated standard deviation of Y Yˆ, and then estimate the standard deviation ofYˆ (i.e.,

ˆ

0 ˆ

1x1 ˆ

2x2 ˆ

3x2 1 ˆ

4x2 2 ˆ

5 x1x2) when x1
8.0 and x2
33.1. Finally, compute a 95% CI for mean strength. [Hint: What is (y  yˆ)/e*?]