Assumptions such as linearity, independence, and normally distributed residuals must be satisfied for the t-test on $$\beta$$ to be valid.

A very low p-value indicates that the observed sample result is highly unlikely if $$H_0$$ were true.

The alternative hypothesis in testing the slope might state $$H_a: \beta \neq 0$$ or $$H_a: \beta > 0$$, depending on the test.

Under the null hypothesis $$H_0: \beta = 0$$, the t-statistic is computed with df = $$n-1$$.