I am looking for some help on confidence intervals calculated using the residual bootstrapping technique.

For confidence intervals calculated according to multipling the t stastistic by the standard error of the mean, there is a clear link between the width of the interval and sample size (quadrupling sample size halves width). I am wanting to know whether any such link exists between sample size and width when the confidence interval is instead calculated by residual bootstrapping (note I am not talking about the number of resamples during bootstrapping, but the number of samples in the original dataset). Basically, I want to estimate how the width of a confidence interval would reduce according to changes in sample size. This is for a continuous, rather than categorial, dataset.

Thank you in advance!