a. Show that [s_{y}^{2}=frac{1}{n-1} sum_{i=1}^{n}left(y_{i}-bar{y} ight)^{2}=frac{1}{n-1}left(sum_{i=1}^{n} y_{i}^{2}-n bar{y}^{2} ight) .] b. Follow the same steps to show

a. Show that

\[s_{y}^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}=\frac{1}{n-1}\left(\sum_{i=1}^{n} y_{i}^{2}-n \bar{y}^{2}\right) .\]

b. Follow the same steps to show \(\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)\left(x_{i}-\bar{x}\right)=\sum_{i=1}^{n} x_{i} y_{i}-\) \(n \bar{x} \bar{y}\).

c. Show that

\[b_{1}=\frac{\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)\left(x_{i}-\bar{x}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}} \text {. }\]

d. Establish the commonly used formula

\[b_{1}=\frac{\sum_{i=1}^{n} x_{i} y_{i}-n \bar{x} \bar{y}}{\sum_{i=1}^{n} x_{i}^{2}-n \bar{x}^{2}} .\]

(R) EMPIRICAL Filename is "WiscNursingHome"