Correlation and \(t\)-Statistics. Use the following steps to establish a relationship between the correlation coefficient and the \(t\)-statistic for the slope.

a. Use algebra to check that \[R^{2}=1-\frac{n-2}{n-1} \frac{s^{2}}{s_{y}^{2}}\]

b. Use part (a) to establish the following quick computational formula for \(s\),\[s=s_{y} \sqrt{\left(1-r^{2}\right) \frac{n-1}{n-2}}\]

c. Use part (b) to show that\[t\left(b_{1}\right)=\sqrt{n-2} \frac{r}{\sqrt{1-r^{2}}} .\]