  Linear regression models are among the models most used in practice, although
the practitioners are often not sure whether their assumed linear regression
model is at least approximately true. In such situations, only designs for
which the linear model can be checked are accepted in practice. For important
linear regression models such as polynomial regression, optimal designs do not
have this property. To get practically attractive designs, we suggest the
following strategy. One part of the design points is used to allow one to carry
out a lack of fit test with good power for practically interesting
alternatives. The rest of the design points are determined in such a way that
the whole design is optimal for inference on the unknown parameter in case the
lack of fit test does not reject the linear regression model. To solve this
problem, we introduce efficient lack of fit designs. Then we explicitly
determine the $\mathbf{e}_k$-optimal design in the class of efficient lack of
fit designs for polynomial regression of degree $k-1$.
