= Solution
Write $S_{xx}=\sum_i x_i^2$. When $S_{xx}>0$, minimizing $\sum_i(Y_i-bx_i)^2$ gives the strictly convex quadratic normal equation $bS_{xx}=\sum_ix_iY_i$. The <least-squares estimator> is therefore
$$
\boxed{\widehat\beta=\frac{\sum_i x_iY_i}{\sum_i x_i^2}.}
$$
This is <linear regression through the origin>, with no fitted intercept. For $n\geq2$, let $\mathrm{SSE}=\sum_i(Y_i-\widehat\beta x_i)^2$ and $s^2=\mathrm{SSE}/(n-1)$. An orthonormal change of coordinates in the <Gaussian vector> $Y$ separates the unit design direction $x/\sqrt{S_{xx}}$ from its $n-1$ orthogonal residual directions. The first coordinate is normal with mean $\beta\sqrt{S_{xx}}$ and variance $\sigma^2$; the remaining coordinates are independent centered normals with that variance. Consequently
$$
\frac{(\widehat\beta-\beta)\sqrt{S_{xx}}}{\sigma}\sim N(0,1),
\qquad\frac{\mathrm{SSE}}{\sigma^2}\sim\chi^2_{n-1},
$$
and these variables are independent. Thus under the null, the <Student t test for regression through the origin> statistic is
$$
\boxed{T=\frac{\widehat\beta\sqrt{S_{xx}}}{s}\sim t_{n-1}.}
$$
For a prescribed significance level $\eta$, reject the two-sided null when $|T|>t_{n-1,1-\eta/2}$. The two-sided p-value is $2[1-F_{t_{n-1}}(|T|)]$. There is one fitted coefficient, so the variance degrees of freedom are $n-1$, not $n-2$.
The assumptions $S_{xx}>0$ and $n\geq2$ matter. If all $x_i=0$, the distribution contains no information about $\beta$ and no informative test is possible. With one nonzero design observation there is no residual variance degree of freedom, so the stated unknown-variance t test cannot be constructed.
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