 Solution

Write $S_{xx}={\sum }_i{x}_i^2$. When $S_{xx}>0$, minimizing ${\sum }_i(Y_i-b{x}_i{)}^2$ gives the strictly convex quadratic normal equation $b{S}_{xx}={\sum }_i{x}_i{Y}_i$. The least-squares estimator is thereforeThis is linear regression through the origin, with no fitted intercept. For $n\ge 2$, let $\mathrm{SSE}={\sum }_i(Y_i-\hat{\beta }{x}_i{)}^2$ 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. Consequentlyand these variables are independent. Thus under the null, the Student t test for regression through the origin statistic isFor 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 $n\ge 2$ 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.