We will assume that the parameters \( \beta \) follow a normal distribution. We can then define the confidence interval. Here we will be using as shorthands \( \mu_{\beta} \) for the above mean value and \( \sigma_{\beta} \) for the standard deviation. We have then a confidence interval
$$ \left(\mu_{\beta}\pm \frac{z\sigma_{\beta}}{\sqrt{n}}\right), $$where \( z \) defines the level of certainty (or confidence). For a normal distribution typical parameters are \( z=2.576 \) which corresponds to a confidence of \( 99\% \) while \( z=1.96 \) corresponds to a confidence of \( 95\% \). A confidence level of \( 95\% \) is commonly used and it is normally referred to as a two-sigmas confidence level, that is we approximate \( z\approx 2 \).
For more discussions of confidence intervals (and in particular linked with a discussion of the bootstrap method), see chapter 5 of the textbook by Davison on the Bootstrap Methods and their Applications
In this text you will also find an in-depth discussion of the Bootstrap method, why it works and various theorems related to it.