If the bootstrapping procedure and the formation of the confidence interval were performed correctly, it means the same as any other confidence interval. From a frequentist perspective, a 95% CI implies that if the entire study were repeated identically ad infinitum, 95% of such confidence intervals formed in this manner will include the true value. Of course, in your study, or in any given individual study, the confidence interval either will include the true value or not, but you won't know which. To understand these ideas further, it may help you to read my answer here: Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean?
Regarding your further questions, the 'true value' refers to the actual parameter of the relevant population. (Samples don't have parameters, they have statistics; e.g., the sample mean, $\bar x$, is a sample statistic, but the population mean, $\mu$, is a population parameter.) As to how we know this, in practice we don't. You are correct that we are relying on some assumptions--we always are. If those assumptions are correct, it can be proven that the properties hold. This was the point of Efron's work back in the late 1970's and early 1980's, but the math is difficult for most people to follow. For a somewhat mathematical explanation of the bootstrap, see @StasK's answer here: Explaining to laypeople why bootstrapping works . For a quick demonstration short of the math, consider the following simulation using R:
# a function to perform bootstrapping
boot.mean.sampling.distribution = function(raw.data, B=1000){
# this function will take 1,000 (by default) bootsamples calculate the mean of
# each one, store it, & return the bootstrapped sampling distribution of the mean
boot.dist = vector(length=B) # this will store the means
N = length(raw.data) # this is the N from your data
for(i in 1:B){
boot.sample = sample(x=raw.data, size=N, replace=TRUE)
boot.dist[i] = mean(boot.sample)
}
boot.dist = sort(boot.dist)
return(boot.dist)
}
# simulate bootstrapped CI from a population w/ true mean = 0 on each pass through
# the loop, we will get a sample of data from the population, get the bootstrapped
# sampling distribution of the mean, & see if the population mean is included in the
# 95% confidence interval implied by that sampling distribution
set.seed(00) # this makes the simulation reproducible
includes = vector(length=1000) # this will store our results
for(i in 1:1000){
sim.data = rnorm(100, mean=0, sd=1)
boot.dist = boot.mean.sampling.distribution(raw.data=sim.data)
includes[i] = boot.dist[25]<0 & 0<boot.dist[976]
}
mean(includes) # this tells us the % of CIs that included the true mean
[1] 0.952
First of all, I agree with the comments left by heropup. I'll add some details.
The reason why your simulation breaks down may be a little subtle. At least I spend some time reading your code to find the source of the problem. Please notice, that you only simulate once for each of the cases. Then your CIs functions resample this initial data set. This clearly gives a lot of dependence between all of the samples. For instance, if you draw a sample of 1000 of the original data set, there is only one way to do this. If you draw a sample of 999 an overwhelming majority of the data set will still be the same between resamples. You'll need to do independent resampling. Otherwise, the 100 samples are essentially the same when you let $n$ get large.
Turning to your question, a confidence interval as the ones you do above are based on a distributional assumption, for instance your observations are normally distributed. If that's the case the confidence interval will be 'centered' in the sense you talk about when you construct the confidence interval symmetrically. This is evident from the symmetry of the distribution and of the procedure of constructing a confidence interval.
In the above, you also calculate a confidence interval when the distributional assumption you make is not correct. Then an confidence interval need not be centered, even if the distribution assumed is symmetric. This can be seen simulated observations from a chi squared distribution and calculating confidence intervals based on a normal distribution.
However, using a central limit theorem we can argue that the mean of the chi squared observations will be approximately normally distributed for large enough sample sizes.
Finally, I just want to note that a confidence interval (or more generally, a confidence set) is basically and loosely speaking just some subset of a parameter set such that when you calculate this set a lot of times (hypothetically) it will contain the true parameter value for example 95% of the times. There's no claim of this set being 'centered' or symmetric around a parameter estimate. It can be chosen to have all sorts of strange forms. This is just not very intuitive and most of the time not very helpful.
Best Answer
I like to think of CIs as some way to escape the Hypothesis Testing (HT) framework, at least the binary decision framework following Neyman's approach, and keep in line with theory of measurement in some way. More precisely, I view them as more close to the reliability of an estimation (a difference of means, for instance), and conversely HT are more close to hypothetico-deductive reasoning, with its pitfalls (we cannot accept the null, the alternative is often stochastic, etc.). Still, with both interval estimation and HT we have to rely on distribution assumptions most of the time (e.g. a sampling distribution under $H_0$), which allows to make inference from our sample to the general population or a representative one (at least in the frequentist approach).
In many context, CIs are complementary to usual HT, and I view them as in the following picture (it is under $H_0$):
that is, under the HT framework (left), you look at how far your statistic is from the null, while with CIs (right) you are looking at the null effect "from your statistic", in a certain sense.
Also, note that for certain kind of statistic, like odds-ratio, HT are often meaningless and it is better to look at its associated CI which is assymmetrical and provide more relevant information as to the direction and precision of the association, if any.