= Solution
Write $t_{ij}$ for stirring rate of observation $i$ in furnace $j$. The <independent random-intercept and random-slope model> is
$$
Y_{ij}=\beta_0+\beta_1t_{ij}+u_j+v_jt_{ij}+\varepsilon_{ij},
\qquad
\begin{pmatrix}u_j\\v_j\end{pmatrix}\overset{\mathrm{iid}}\sim
N_2\!\left(0,\begin{pmatrix}\tau_0^2&0\\0&\tau_1^2\end{pmatrix}\right),
\qquad\varepsilon_{ij}\overset{\mathrm{iid}}\sim N(0,\sigma^2).
$$
Errors and <random effects> are independent. The separate R terms `(1 | furnace)` and `(0 + stir | furnace)` impose independent <random intercepts> and <random slopes>; `(1 + stir | furnace)` would instead estimate their covariance as well. The estimates are
$$
\boxed{\widehat\beta_0=277.6636,\quad\widehat\beta_1=0.5601,\quad
\widehat\tau_0^2=48.73800,\quad\widehat\tau_1^2=0.01316,\quad
\widehat\sigma^2=36.29950.}
$$
The corresponding estimated standard deviations are $6.9813$, $0.1147$, and $6.0249$; they are not additional parameters.
For a furnace with predictor vector $t_j$, the marginal formulation, obtained by integrating the Gaussian <random effects>, is
$$
Y_j\sim N\!\left(\beta_0\mathbf1+\beta_1t_j,
\tau_0^2\mathbf1\mathbf1^T+\tau_1^2t_jt_j^T+\sigma^2I\right),
$$
independently across furnaces. In particular,
$$
\operatorname{Cov}(Y_{ij},Y_{kj})=\tau_0^2+\tau_1^2t_{ij}t_{kj}+\sigma^2\mathbf1_{\{i=k\}}.
$$
In stacked notation $Y=X\beta+Zb+\varepsilon$, $b\sim N(0,D)$, giving the <Gaussian linear mixed model> marginal law $N(X\beta,V)$ with $V=ZDZ^T+\sigma^2I$.
The <residual-versus-fitted plot> on page 17 has residuals of both signs over the fitted range, with no convincing smooth curvature or clear fan shape. Its <quantile-quantile plot> is roughly straight in the central region, with some tail departures and a few large negative residuals. \b[There is no decisive visible violation], but investigate those observations. These plots mainly address conditional observation errors; they do not validate the distribution of furnace <random effects> or <independence> within furnaces. With only three furnaces, <normal distribution>[normality] and the <variance> of the <random effects> are especially difficult to assess.
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