Variance of OLS estimator with binary treatment I know that in general, given a (stacked) regression of the form
$ y = X \beta + \epsilon$, where $\mathbb{V}(\epsilon_i) = \sigma^2 \forall i$, then letting $\hat{\beta}$ denote the OLS estimate of $\beta$,
\begin{equation}
\mathbb{V}(\hat{\beta}) = \sigma^2 (X'X)^{-1}
\end{equation}
This I understand. In my econometrics class, we are studying randomized trials with binary treatment, so $X \in \{0,1\}^n$, where $n$ is the number of observations. I am told that in this simple setup,
\begin{equation}
\mathbb{V}(\hat{\beta}) = \frac{\sigma^2}{p(1-n)p}
\end{equation}
where $p$ is the proportion of observations $i$ such that $X_i = 1$.
I have no idea how to derive the second equation from the first. Any hints?
 A: In the case of simple linear regression
$$\underbrace{\begin{bmatrix}
    y_1 \\
    y_2 \\
    \vdots\\
    y_n 
\end{bmatrix}}_{Y} = \underbrace{\begin{bmatrix}
    1 & x_1 \\
    1 &  x_2 \\
    \vdots\\
     1 & x_n 
\end{bmatrix}
\begin{bmatrix}
    \beta_0  \\
    \beta_1 
\end{bmatrix}}_{X\beta}.$$
Taking $X'X$ gives
$$
\begin{bmatrix}
    n & \sum_{i=1}^{n}x_i \\
    \sum_{i=1}^{n}x_i  &  \sum_{i=1}^{n}x_i^2  
\end{bmatrix}.
$$
After we take the inverse (i.e., $(X'X)^{-1}$) , we would find that the element at $(2,2)$ is
$$\frac{1}{ \sum_{i=1}^{n}(x_i-\bar{x})^2 }.$$
So $\mathbb{V}(\beta_1) = \frac{\sigma^2}{ \sum_{i=1}^{n}(x_i-\bar{x})^2 }$.
Next, we need to make $\sum_{i=1}^{n}(x_i-\bar{x})^2$ look like $np(1-p)$.
Since $x$ is binary, we know $\bar{x} = \frac{\sum_{i=1}^{n}x_i}{n} = p$. Therefore, we can rewrite the denominator of $\mathbb{V}(\beta_1)$ as
$\sum_{i=1}^{n}(x_i-p)^2$.
Expanding gives
$$
\begin{align}
\sum_{i=1}^{n}(x_i-p)^2 &= \sum_{i=1}^{n}(x_i^2 - 2px_i + p^2)\\
&= np - 2np^2 + np^2\\
&= np - np^2\\
&= np(1-p).\\
\end{align}
$$
Therefore, $\mathbb{V}(\beta_1) = \frac{\sigma^2}{np(1-p)}$, which corresponds to $\mathbb{V}(\beta)$ in your question.
