4.14 Confidence Interval

Another idea is to report a \((1-\alpha)\%\) (e.g., 95%) confidence interval.

The interpretation of a confidence interval is a bit subtle. It is this: if we collected a large number of samples, and computed a confidence interval each time, 95% of these would contain the true value. This is subtly different than: there is a 95% probability that \(\theta\) (the population parameter of interest) falls within the confidence interval — this second interpretation doesn’t make sense because \(\theta\) is non-random.

A 95% confidence interval is given by

\[ CI_{95\%} = \left[\hat{\theta} - 1.96 \ \textrm{s.e.}(\hat{\theta}), \hat{\theta} + 1.96 \ \textrm{s.e.}(\hat{\theta})\right] \]

For the particular case where we are interested in \(\mathbb{E}[Y]\), this becomes

\[ CI_{95\%} = \left[ \bar{Y} - 1.96 \ \textrm{s.e.}(\bar{Y}), \bar{Y} + 1.96 \ \textrm{s.e.}(\bar{Y}) \right] \]