Chapter 11, Problem 105SE

a.

To determine

Show that ${\widehat{\beta }}_1=r\sqrt{\frac{S_{yy}}{S_{xx}}}$.

b.

To determine

Prove that under $H_0:\rho =0$ $\frac{{\widehat{\beta }}_1\sqrt{\left(n-2\right)S_{xx}}}{\sqrt{S_{yy}\left(1-r^2\right)}}$ has a t distribution with $\left(n-2\right)$ df.

c.

To determine

Check whether the test statistic $T=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$ has the same distribution unconditionally when conditional on $X_i=x_i\text{ for }i=1,2,\cdots ,n$.