85. The article The Undrained Strength of Some Thawed Permafrost Soils (Canad. Geotech. 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 %).

Obs y x1 x2 e*
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 tting a full quadratic model, which resulted in the estimated regression function

a. Do plots of e* versus x1, e* versus x2, and e* versus suggest that the full quadratic model should be modi ed? 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 ve predictors.

c. Each of the null hypotheses H0: bi0 versus Ha:
bi  0, i  1, 2, 3, 4, 5, is not rejected at the 5%
level. Does this make sense in view of the result in (b)? Explain.

d. It is shown in Section 12.8 that V(Y)  s2 
. The estimate of s is 
s  6.99 (from the full quadratic model). First obtain the estimated standard deviation of Y 
, and then estimate the standard deviation of (i.e., ) when x1  8.0 and x2  33.1. Finally, compute a 95% CI for mean strength. [Hint:
What is /e*?]

e. Sometimes an investigator wishes to decide whether a group of m predictors (m 1) can simultaneously be eliminated from the model. The null hypothesis says that all b s associated with these m predictors are 0, which is interpreted to mean that as long as the other k  m predictors are retained in the model, the m predictors under consideration collectively provide no useful information about y. The test is carried out by rst tting the full model with all k predictors to obtain SSE(full) and then tting the reduced model consisting just of the k  m predictors not being considered for deletion to obtain SSE(red).
The test statistic is The test is upper-tailed and based on m numerator df and n(k1) denominator df. Fitting the rst-order model with just the predictors x1 and x2 results in SSE  894.95. State and test at signi cance level .05 the null hypothesis that none of the three second-order predictors (one interaction and two quadratic predictors) provides useful information about y provided that the two rst-order predictors are retained in the model.