= Solution
Let $\mathcal F_t$ contain observations through time $t$. In the Gaussian <moving-average process of order one>, the innovations in
$$
Y_{t+7}=\mu+\varepsilon_{t+7}+\theta\varepsilon_{t+6}
$$
are independent of $\mathcal F_t$. Hence the <conditional expectation> is the constant mean. For known parameters, the forecast and its <mean squared prediction error> are
$$
\boxed{\widehat Y_{t+7\mid t}=\mu,\qquad \operatorname{MSPE}=\sigma_\varepsilon^2(1+\theta^2).}
$$
The mean prediction error itself is zero; the requested uncertainty is naturally interpreted as expected squared error. Plugging in the fit gives forecast $15.0133$, mean squared error $6.731002$, and root mean squared error about $2.5944$.
The usual <naive time-series forecast> repeats $Y_t$. Its unconditional expected squared error is
$$
\mathbb E(Y_{t+7}-Y_t)^2=2\gamma(0)-2\gamma(7)=2\sigma_\varepsilon^2(1+\theta^2),
$$
since $\gamma(7)=0$. Thus
$$
\boxed{\operatorname{MSPE}_{\rm naive}\simeq13.462004,\qquad \operatorname{MSPE}_{\rm model}\simeq6.731002.}
$$
Conditional on $\mathcal F_t$, the naive error is $\gamma(0)+(Y_t-\mu)^2$, explaining directly why the conditional-mean forecast is better under <squared-error loss>. These are fitted-model calculations that ignore parameter-estimation uncertainty. Estimating the unknown mean from the past adds its estimation mean squared error to the seven-step forecast risk; the printed summary does not provide a complete assessment of that extra uncertainty. This is distinct from claiming a signed mean error as a measure of forecasting accuracy.
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