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Under the null hypothesis, the standardized intercept $Z=\sqrt{n}(\hat{\alpha }-{\alpha }_0)/\sigma$ is standard normal, and $V=\mathrm{RSS}/{\sigma }^2$ is independent with chi-squared distribution of $\nu =n-2$ degrees of freedom. Consequently the Student t test for a simple-regression intercept usesChoose a significance level $\gamma \in (0,1)$ before observing the test result. For the two-sided alternative, reject $H_0$ when $|T|>t_{n-2,1-\gamma /2}$, where the subscript denotes the indicated quantile of the Student t-distribution. The null rejection probability is exactly $\gamma$, independent of the nuisance slope and variance. Equivalently report the p-value $2[1-F_{t_{n-2}}(|T_{\mathrm{obs}}|)]$. The test assumes the normal independent-error model and a positive residual variance estimate. In the intercept-only case with $n>1$, use $s^2=\mathrm{RSS}/(n-1)$ and $t_{n-1}$ instead.

Assume the explanatory variables are not all equal, so $S_{xx}={\sum }_i(x_i-\bar{x}{)}^2>0$. In the Gaussian linear model, the log-likelihood isFor fixed ${\sigma }^2$, maximum likelihood estimation therefore minimizes the residual sum of squares. Differentiating in $\alpha ,\beta$ gives the least-squares normal equationsThe first gives $\hat{\alpha }=\bar{Y}-\hat{\beta }\bar{x}$; substituting it into the second gives $\hat{\beta }=S_{xy}/S_{xx}$, with $S_{xy}={\sum }_i(x_i-\bar{x})(Y_i-\bar{Y})$. Let $\mathrm{SSE}={\sum }_i(Y_i-\hat{\alpha }-\hat{\beta }{x}_i{)}^2$. Differentiating the profiled log-likelihood in ${\sigma }^2$ givesFor the variance estimator this is the interior maximum when $\mathrm{SSE}>0$, which occurs almost surely under the model when $n>2$.

The coefficient estimators are linear functions of the independent normal errors. In particular,Their multivariate normal distribution has mean $(\alpha ,\beta {)}^T$ and covariance matrixThe fitted vector is the orthogonal projection onto the two-dimensional span of $(1,\dots ,1{)}^T$ and $(x_1,\dots ,x_n{)}^T$. The residual vector is the complementary projection of the spherical normal error vector. Orthogonal normal projections are independent, and in an orthonormal basis of that complementary space its $n-2$ coordinates are independent $N(0,{\sigma }^2)$. Consequentlyindependently of $(\hat{\alpha },\hat{\beta })$. This also gives the unbiased estimator $s^2=\mathrm{SSE}/(n-2)$ rather than the variance MLE.

Under the null hypothesis $\alpha =0$, for $n>2$ the Student t test for a simple-regression intercept usesHere $\eta$ is the prescribed significance level. The normal numerator divided by its true standard deviation is independent of $(n-2)s^2/{\sigma }^2$, which proves the stated Student t-distribution. If $S_{xx}=0$, the two coefficients are not separately identifiable; if $n\le 2$, this residual-variance-based test is unavailable.