= Solution
The <orthogonal projections> of the <Gaussian noise> onto the column space of $X$ and its <orthogonal complement> are <independent>. Consequently $\widehat\beta_0$ and the residual sum of squares are <independent>, and
$$
Z=\frac{\widehat\beta_0-\beta_0}{\sigma\sqrt{V_{00}}}\sim N(0,1),\qquad
U=\frac{\nu s^2}{\sigma^2}\sim\chi^2_\nu.
$$
Thus under the alternative,
$$
\boxed{T=\frac{Z+\delta}{\sqrt{U/\nu}},\qquad\delta=\frac{\beta_0}{\sigma\sqrt{V_{00}}},\quad Z\perp U.}
$$
This is the <noncentral t-distribution>. The rejection region is fixed by the design, $\nu$ and the chosen level, while this <probability distribution> depends on $\beta$ only through $\beta_0$. Therefore \b[the <statistical power> does not depend on the other regression coefficients], with the design and $\sigma$ fixed.
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