Applications of Cochran Theorem 
I am trying to show that SST/$\sigma^2$ $\approx$ $\chi^2$ using Cochran's Theorem above. I tried using regression but it didn't make sense.
 A: The first step is to show that there are DF(Treatment) iid standard normally distributed RVs that sum to SS(Treatment). This may
involve some linear algebra to do an orthogonal transformation
to the appropriate axes.
The absolute simplest version of this is show that for
$Z_1$ and $Z_2$ independently standard normal, their
$(n - 1)S^2 \sim Chisq(n - 1),$ where $n = 2$ and $S$ is
the sample variance of $Z_1$ and $Z_2$. This begins with
a 45 degree rotation showing that $\bar Z$ is a function
of $Z_1 + Z_2$, and that $S$ is a function of $Z_1 - Z_2$.
Thus $\bar Z$ and $S$ are uncorrelated, and by normality,
independent. Then appropriate univariate transformations of $\bar Z$ and
of $S^2$ are independently standard normal and $Chisq(1)$,
respectively.
The next step up from there is to show that for
a random sample $X_i$ of size $n$ from $Norm(\mu, \sigma)$,
you have $(n -1)S^2/\sigma^2 \sim Chisq(n-1).$ Here one
dimension after transformation corresponds to $\bar X$
and $(n-1)$ dimensions correspond to $S^2.$
In a balanced one-factor ANOVA, with $g$ treatment groups
and $n$ replications of each: one dimension corresponds
to the grand mean in your model, $g - 1$ dimensions
correspond to differences of the $g$ group means from the
grand mean (from which you get SS(Treatment)), and the rest of the $gn$ dimensions correspond SS(Error).
