This question is based on How to derive a regression formula
Start with a regression equation: $s_{ij} = \alpha + \beta \bar{s_j} + \epsilon_{ij},$
The goal is to get to this. *Update. I figured this part out. ***The remaining question is listed at the very bottom.
$\hat{\beta} = \frac{\sum_j \sum_i s_{ij} (\bar{s_j} - \bar{s})}{\sum_j n_j (\bar{s_j} - \bar{s})^2} = \frac{\sum_j (\bar{s_j} - \bar{s})(n_j \bar{s_j})}{\sum_j n_j( \bar{s_j} - \bar{s})^2} = 1 $
First, get $\hat{\alpha}$:
\begin{equation} argmin_{\hat{\alpha}}~~ \hat{\epsilon_{ij}^2} = argmin_{\hat{\alpha}}~ \sum_j \sum_i (s_{ij} - \hat{\alpha} - \hat{\beta} \bar{s_j})^2 \end{equation}
\begin{equation} \rightarrow \sum_j \sum_i (s_{ij} - \hat{\alpha} - \hat{\beta} \bar{s_j}) = 0 \end{equation}
\begin{equation} \rightarrow \sum_j \sum_i \hat{\alpha} = \sum_j \sum_i (s_{ij} - \bar{s_j} \hat{\beta}) \end{equation}
\begin{equation} \rightarrow \sum_j n_j \hat{\alpha} = \sum_j n_j \bar{s_j} - \hat{\beta} \sum_j n_j \bar{s_j} \end{equation}
\begin{equation} \rightarrow \hat{\alpha} = (1 - \hat{\beta}) \frac{\sum_j n_j \bar{s_j} }{\sum_j n_j} = (1 - \hat{\beta}) \bar{s} \end{equation}
Next, get $\hat{\beta}$.
\begin{equation} argmin_{\hat{\beta}}~~ \hat{\epsilon_{ij}^2} = argmin_{\hat{\beta}}~ \sum_j \sum_i (s_{ij} - \hat{\alpha} - \hat{\beta} \bar{s_j})^2 \end{equation}
\begin{equation} = argmin_{\hat{\beta}}~ \sum_j \sum_i (s_{ij} - (1 - \hat{\beta}) \bar{s} - \hat{\beta} \bar{s_j})^2 \end{equation}
\begin{equation} = argmin_{\hat{\beta}}~ \sum_j \sum_i ((s_{ij} - \bar{s}) - \hat{\beta}( \bar{s_j} - \bar{s}))^2 \end{equation}
\begin{equation} \rightarrow \sum_j \sum_i ((s_{ij} - \bar{s}) - \hat{\beta}( \bar{s_j} - \bar{s}))( \bar{s_j} - \bar{s})= 0 \end{equation}
\begin{equation} \rightarrow \sum_j \sum_i (s_{ij} - \bar{s})( \bar{s_j} - \bar{s}) = \sum_j \sum_i \hat{\beta}( \bar{s_j} - \bar{s})( \bar{s_j} - \bar{s}) \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i (s_{ij} - \bar{s})( \bar{s_j} - \bar{s}) } {\sum_j \sum_i ( \bar{s_j} - \bar{s})( \bar{s_j} - \bar{s})} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i (s_{ij} - \bar{s})( \bar{s_j} - \bar{s}) } {\sum_j \sum_i ( \bar{s_j} - \bar{s})^2} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i (s_{ij} - \bar{s})( \bar{s_j} - \bar{s}) } {\sum_j n_j ( \bar{s_j} - \bar{s})^2} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i (s_{ij} - \bar{s})( \bar{s_j} - \bar{s}) } {\sum_j n_j ( \bar{s_j} - \bar{s})^2} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i s_{ij}(\bar{s_j} - \bar{s}) - \sum_j \sum_i \bar{s}(\bar{s_j} - \bar{s}) } {\sum_j n_j ( \bar{s_j} - \bar{s})^2} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i s_{ij}(\bar{s_j} - \bar{s}) - \sum_j \sum_i \bar{s}(\bar{s_j} - \bar{s}) } {\sum_j n_j ( \bar{s_j} - \bar{s})^2} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i s_{ij}(\bar{s_j} - \bar{s}) - \sum_j \sum_i \bar{s}\bar{s_j} - \bar{s}^2 } {\sum_j n_j ( \bar{s_j} - \bar{s})^2} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i s_{ij}(\bar{s_j} - \bar{s}) - \bar{s}\sum_j n_j(\bar{s_j} - \bar{s}) } {\sum_j n_j ( \bar{s_j} - \bar{s})^2} \end{equation}
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j \sum_i s_{ij}(\bar{s_j} - \bar{s}) } {\sum_j n_j ( \bar{s_j} - \bar{s})^2} \end{equation}
because $\sum_j n_j(\bar{s_j} - \bar{s}) = 0$
\begin{equation} \rightarrow \hat{\beta} = \frac{\sum_j (\bar{s_j} - \bar{s})(n_j \bar{s_j})}{\sum_j n_j( \bar{s_j} - \bar{s})^2} \end{equation}
Why does the last equality produce 1 in:
\begin{equation} \hat{\beta} = \frac{\sum_j (\bar{s_j} - \bar{s})(n_j \bar{s_j})}{\sum_j n_j( \bar{s_j} - \bar{s})^2} = 1 \end{equation}
