Suppose I have realizations $x_1, ..., x_n$ for random variable $X$ and I have realizations $y_1, ... , y_m$ for random variable $Y$. $n<<m$ are small numbers.
Let $ Z = a*X + b*Y $, where $a,b$ constants. How can I derive a confidence interval for the mean $\mu_Z$?
Can I simply use bootstrapping? That is, suppose $F_n$ and $H_m$ are the empirical distribution functions obtained from the two sets of realizations. I could, for example, generate many bootstrap samples $z_i^* = x_i^* + y_i^*$, $i = 1, ...$ , compute the empirical distribution of (studentized) means and then the confidence interval.
Is this a reasonable procedure or am I making a conceptual mistake somewhere? By the way, this question is motivated by a real application. For that I am actually interested in a sum of random variables with many terms, all of which have a different number of observations.