 Student t test for regression through the origin

For independent $Y_i\sim N(\beta x_i,{\sigma }^2)$ with $n\ge 2$ and $\sum x_i^2>0$, the least-squares estimator is $\hat{\beta }=\sum x_i{Y}_i/\sum x_i^2$. The projection of the isotropic Gaussian vector onto the design direction is independent of its orthogonal residual; hence $\hat{\beta }\sqrt{\sum x_i^2}/\sigma$ is standard normal under $\beta =0$ and $\mathrm{SSE}/{\sigma }^2$ is independent chi-squared with $n-1$ degrees of freedom. The displayed statistic therefore has Student's t-distribution with $n-1$ degrees of freedom. A two-sided test rejects at the corresponding upper/lower quantiles. The zero design has no information about the slope; with one observation there is no residual variance degree of freedom.