Chapter 11.4, Problem 21E

Under the assumptions of Exercise 11.20, find $\text{Cov}({\hat{\beta }}_0,{\hat{\beta }}_1)$. Use this answer to show that ${\hat{\beta }}_0$ and ${\hat{\beta }}_1$ are independent if ${\displaystyle {\sum }_{i=1}^n{x}_i=0}$. [Hint: $\text{Cov}({\hat{\beta }}_0,{\hat{\beta }}_1)=\text{Cov}(\bar{Y}-{\hat{\beta }}_1\bar{x},{\hat{\beta }}_1)$. Use Theorem 5.12 and the results of this section.]

11.21 Under the assumptions of Exercise 11.20, find $\text{Cov}({\hat{\beta }}_0,{\hat{\beta }}_1)$. Use this answer to show that ${\hat{\beta }}_0$ and ${\hat{\beta }}_1$ are independent if ${\displaystyle {\sum }_{i=1}^n{x}_i=0}$. [Hint: $\text{Cov}({\hat{\beta }}_0,{\hat{\beta }}_1)=\text{Cov}(\bar{Y}-{\hat{\beta }}_1\bar{x},{\hat{\beta }}_1)$. Use Theorem 5.12 and the results of this section.]