Let $y_{i j}$ for $i=1, \ldots, b$ and $j=1, \ldots, k$ be the measurement for the unit assigned to the $j^{\text {th }}$ treatment in the $i^{t h}$ block.

a. Show that

\[S S_{T}=S S_{\text {Treat }}+S S_{\text {Blocks }}+S S_{E}\]

where

\[\begin{gathered}S S_{T}=\sum_{i=1}^{b} \sum_{j=1}^{k}\left(y_{i j}-\overline{y . .}\right)^{2}, \\S S_{\text {Treat }}=b \cdot \sum_{i=1}^{k}\left(\overline{y_{\cdot j}}-\overline{y_{. \cdot}}\right)^{2}, \\S S_{\text {Blocks }}=k \cdot \sum_{j=1}^{b}\left(\overline{y_{i \cdot}}-\overline{y_{. .}}\right)^{2} \text {, and } \\S S_{E}=\sum_{i=1}^{b} \sum_{j=1}^{k}\left(y_{i j}-\overline{y_{i \cdot}}-\overline{y_{\cdot j}}+\overline{y_{. .}}\right)^{2} .\end{gathered}\]

b. Show that blocking reduces the variability not due to treatment among units within each treatment group.