Exercise 3.6

The least squares line is given by:

\[\hat{y} = \hat{\beta_0} + \hat{\beta_1} \times x\]

where \(\hat{\beta_0}\) and \(\hat{\beta_1}\) are the least squares coefficient estimates for simple linear regression.

By definition, \(\hat{\beta_0}\) is:

\[\hat{\beta_0} = \bar{y} - \hat{\beta_1} \times \bar{x}\]

where \(\bar{y}\) and \(\bar{x}\) are the average values of \(y\) and \(x\), respectively.

Since we want to know if the least squares line always passes through the point (\(\bar{x}\), \(\bar{y}\)), all we have to do is to substitute (\(\bar{x}\), \(\bar{y}\)) into the first equation above and see if the condition is satisfied. We get:

\[\bar{y} = \hat{\beta_0} + \hat{\beta_1} \times \bar{x}\]

and substituting the expression above for \(\hat{\beta_0}\), we obtain:

\[\bar{y} = \bar{y} - \hat{\beta_1} \times \bar{x} + \hat{\beta_1} \times \bar{x}\]

Since this is always true, we conclude that the least squares line always passes through the point (\(\bar{x}\), \(\bar{y}\)).