In a regression situation, suppose that the observed values Xj and Yj of the independent and dependent variable differ from certain true values X

j and Y

j by errors Uj , Vj which are independently normally distributed with zero means and variances σ2 U and σ2 V . The true values are assumed to satisfy a linear relation: Y

j = α+βX

j . However, the variables which are being controlled, and which are therefore constants, are the Xj rather than the X

j . Writing xj for Xj , we have xj = X

j + Uj , Yj = Y

j + Vj , and hence Yj = α + βxj + Wj , where Wj = Vj − βUj . The results of Section 7.6 can now be applied to test that β or

α + βx0 has a specified value.