= Solution
Let $\nu=n/2-p$. Under $H_0$, the two residual sums of squares are independent and satisfy
$$
\frac{\lVert Y_i-X_i(X_i^TX_i)^{-1}X_i^TY_i\rVert^2}{\sigma^2}
\sim\chi^2_\nu,
\qquad i=1,2.
$$
Hence
$$
\boxed{T\sim F_{\nu,\nu}\quad\text{under }H_0}.
$$
Under $H_1$, the numerator is scaled by the smaller variance, so small values provide evidence against the null. If $F^{-1}_{\nu,\nu}(\alpha)$ denotes the lower $\alpha$ quantile, a size-$\alpha$ test rejects when
$$
\boxed{T<F^{-1}_{\nu,\nu}(\alpha)}.
$$
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