For multiple regression, the definitions of $R^2$ and adjusted $R^2$ are: where $k$ is the number of betas and $n$ is the sample size. Assume that $n>k$.

Part a)
Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted$R^2$ is $\le R^2$.
Choose the single best item

A. $1-{(n-1){\hat\sigma}^2\over \sum_i y_i^2}$
B. $1-{SS(Res)\over SS(Total)}-{(k-1)SS(Res)\over (n-1)SS(Total)}$
C. $1-{SS(Res)\over SS(Total)}+{(k-1)SS(Res)\over (n-1)SS(Total)}$
D. $1-{(n-k)SS(Res)\over (n-1)SS(Total)}$
E. $1-{(n-1)SS(Res)\over (n-k)SS(Total)}$
F. $1-{{\hat\sigma}^2\over s_y^2}$
G. $1-{SS(Res)\over SS(Total)}-{(k-1)SS(Res)\over (n-k)SS(Total)}$
H. None of the above


Part b)
Which of the following is a correct version of adjusted $R^2$ and shows clearly that adjusted$R^2$ increases as ${\hat\sigma}$ decreases.
Choose the single best item

A. $1-{{\hat\sigma}^2\over s_y^2}$
B. $1-{(n-1){\hat\sigma}^2\over \sum_i y_i^2}$
C. $1-{SS(Res)\over SS(Total)}+{(k-1)SS(Res)\over (n-1)SS(Total)}$
D. $1-{SS(Res)\over SS(Total)}-{(k-1)SS(Res)\over (n-1)SS(Total)}$
E. $1-{(n-k)SS(Res)\over (n-1)SS(Total)}$
F. $1-{SS(Res)\over SS(Total)}-{(k-1)SS(Res)\over (n-k)SS(Total)}$
G. $1-{(n-1)SS(Res)\over (n-k)SS(Total)}$
H. None of the above

Hint: