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 The predicted values and residuals were computed by fitting a full quadratic model, which resulted in the estimated regression function 2 .253x 2 2 1 .492x 1x 2 y 5 2151.36 2 16.22x 1 1 13.48x 2 1 .094x 1 2

a. Do plots of e* versus x1, e* versus x2, and e* versus 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 .
The estimate of s is (from the full quadratic model). First obtain the estimated standard deviation of , and then estimate the standard deviation of (i.e., )
when and . Finally, compute a 95%
CI for mean strength. [Hint: What is ?]

d. Fitting the first-order model with regression function results in .
Test at level .05 the null hypothesis that states that all quadratic terms can be deleted from the model.