Question. Given the data in the table below we want to compute a 95% confidence interval for the change in the expected value of the salary, given decrease in GPA by 0.3. In other words, we want a 95% CI for $\Delta\mathbb{E}(y)$ given that $\Delta x=-0.3$.
Attempt at an answer. My first thought is that since $\Delta\mathbb{E}(y)=\beta_1\cdot\Delta x$, where $(1.86076,3.48089)$ is a 95% CI for $\beta_1$, then perhaps we could just multiply that by $-0.3$ to obtain $(-1.04427,-0.55823)$. However, this is just a conjecture and I can't prove it, and moreover it seems suspiciously too easy.
Any ideas?
Thanks!
