Multiple regression and hypothesis test $H_0$:$\beta_2=0$ Multiple regression model
$H_0$:$\beta_2=0$, $H_1$:$\beta_2 \neq 0$
where $\beta_2$ is the vector of elements ($\beta_2, \beta_3, \dots, \beta_k$) and $\beta$ is slope of regression line.
Why it is equivalent to a test based on the statistic
$$\frac{R^2/(k-1)}{(1-R^2)/(t-k)}$$
where $R^2$ is the square of the multiple correlation coefficient of the equation.  
I don't know how to solve this. Please give me any suggestion or hint.
 A: The statistic of the $F$ test is defined by 
$$
F_{stat} = \frac{[SSR(\text{fm}) -SSR(\text{rm})]/[df(\text{fm})-df(\text{rm})] }{SSE(\text{fm})/[n-df(\text{fm})-1]}
$$
where $\text{fm}$ and $\text{rm}$ are the full (unrestricted model) and the restricted model, respectively. $df$ denote the degrees of freedom of the corresponding model, where $SSR$ are the sum of the squared regression, i.e.,  $\sum(\hat{y}_{i} - \bar{y})^2$,  and $SSE$ are the errors, $\sum(y_i - \hat{y}_i)^2$. Recall the orthogonal decomposition of the $SST=\sum (y_i - \bar{y})^2$, i.e., $SST = SSE + SSR$, and the definition of $R^2$, which is $R^2 = \frac{SSR}{SST}$. So, your restricted model is the null model with $df=1$ such that $SSR(\text{rm}) = 0$, because $\hat{y}=\bar{y}$, and your full model with $df=k$. Hence, by dividing both the nominator and the denominator by $SST$ you get 
\begin{align}
F_{Stat} &= \frac{(SSR(\text{fm})/SST)/(k-1)}{[1-SSR(\text{fm})/SST]/(n-k-1)}\\
         &= \frac{R^2/(k-1)}{(1-R^2)/(t-k)}
\end{align}
where $n-k=t$ and $SSE = SST- SSR$. 
