A split-plot design has blocks, and each block has experimental units. For every block, the levels of a treatment factor $W$ are assigned at random to the units in the block, with independent randomizations for the blocks. Each experimental unit is divided into sub-units. For every unit, the levels of a treatment factor $S$ are assigned at random to the sub-units in the unit, with independent randomizations for the $\times$ units.

Part a) How many observations (runs) does the experiment have?

Part b) How many ways can the levels of $W$ be randomized in one block?

Part c) How many ways can the levels of $S$ be randomized in one unit?

Part d) Take the usual model for a split-plot design with the usual assumptions about the random effects. In particular, the units effects are assumed to be independent ${\mbox{N}}(0, \sigma^2_\rho)$, the sub-unit effects are assumed to be independent ${\mbox{N}}(0, \sigma^2_\epsilon)$, and the unit effects are independent of the sub-unit effects. The model is fit using ${\tt R}$ commands like $$\texttt{aov.out <- aov(y ~ Block + Error(Unit) + W + S + W:S, data = a.dataframe)}$$ $$\texttt{summary(aov.out)}$$ Here, ${\tt a.dataframe}$ is an ${\tt R}$ data frame with 5 columns:
* ${\tt Block}$ containing levels labelled ${\tt B1}$ through ${\tt B }$;
* ${\tt Unit}$ containing levels labelled ${\tt U1}$ through ${\tt U }$;
* ${\tt W}$ containing levels labelled W1 through ${\tt W }$;
* ${\tt S}$ containing levels labelled S1 through ${\tt S }$; and
* ${\tt y}$ containing the values of the response variable.

In the ${\tt R}$ output two values of MS(Residuals) are reported: under ${\tt Error: Unit}$ and under ${\tt Error: Within}$.

(i) Give an estimate of $\sigma^2_\epsilon$ to 2 decimal places.

(ii) Give an estimate of $\sigma^2_\rho$ to 2 decimal places.