Summing bootstrap-derived confidence intervals?

Question

I have a couple things to add together- two different pools of material, and I need to know the total material. Because of various things related to the sampling, I would like to bootstrap the sampling data from each pool. So I generate a list of 100 masses for one pool, and 50 masses for the second pool. Plus, I get the confidence intervals built off the bootstrap. I can just add the mean, to get the mean total mass, but what about the CIs?

I'm still learning bootstrapping, obviously, so sorry if it's a simple question. At first, I figured I could combine them similar to SD's, like sqrt(x^2 + y^2), but that doesn't work. Does simple addition work? Or do I need to build the CI's from scratch using the standard errors from each pool?

Answer 1

Correct me if i am wrong but it sounds to me like your problem is simply to get a confidence interval for the sum of two random variables $X$ and $Y$ with $X$ is the total mass from the sample for the first pool and Y is the total mass for the sample from the second pool. I think for this problem you can use either the sum or the average. To get a confidence interval for X+Y you would need to get the sampling distribution of $Z=X+Y$. For a normal distribution a $95\%$ confidence interval for the population mean equates to the

$$\overline{x} \pm 1.96 \cdot s$$

where $\overline{x}$ denotes the sample mean and $s$ is the standard deviation. But this relationship between confidence level does not hold for other distributions. If you assume that $X$ and $Y$ are independent with a known parametric distribution then you can solve the problem without bootstrapping. If you don't you can generate a bootstrap confidence interval for $Z$. Get a bootstrap sample from the sample that generated $X$ and do the same for $Y$. Take the bootstrap estimate for $X$ and add it to the bootstrap estimate for Y to get a bootstrap estimate for $Z$. Repeat this say 10000 times and you will be able to construct a histogram of the $Z$s. The percentile method bootstrap is one simple way to generate approximate confidence intervals. Let $Z_1$ be the value at which $25$ values are at it or below and let $Z_2$ be the values at which $25$ values are at it or above. Then $[Z_1, Z_2]$ is an approximate $95\%$ bootstrap confidence interval for the mean of the distribution for $Z$.

The interval is approximate and there are other types of bootstrap confidence intervals that can be more accurate for the given sample size.