Suppose I have two random variables, $X$ and $Y$. Suppose $X$ is normally distributed, and therefore I know how to compute a 95% confidence interval (CI) estimator for $X$. Suppose that $Y$ is not normally distributed, but that I have an unbiased 95% CI estimator for $Y$.
Given that I know how to compute CIs for $X$ and $Y$ separately, how can I compute a 95% CI estimator for the quantity
$$W = a \cdot X + b \cdot Y,$$
where $a$ and $b$ are real constants?