Show algebraically that a. (sum_{i=1}^{n}left(x_{i}-bar{x} ight)^{2}=left(sum_{i=1}^{n} x_{i}^{2} ight)-n bar{x}^{2}) b. (sum_{i=1}^{n}left(x_{i}-bar{x} ight)left(y_{i}-bar{y} ight)=left(sum_{i=1}^{n} x_{i} y_{i} ight)-n bar{x}

Show algebraically that

a. \(\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=\left(\sum_{i=1}^{n} x_{i}^{2}\right)-n \bar{x}^{2}\)

b. \(\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)=\left(\sum_{i=1}^{n} x_{i} y_{i}\right)-n \bar{x} \bar{y}\)

c. \(\sum_{j=1}^{n}\left(x_{j}-\bar{x}\right)=0\)