95% confidence interval

If the estimator of any parameter, \(\theta\), is approximately unbiased and normally distributed, and if we can evaluate an approximate standard error, then an interval calculated with the formula

\[ \hat{\theta}-1.96 \times \se(\hat {\theta}) \quad \text{ to } \quad \hat{\theta}+1.96 \times \se(\hat {\theta}) \]

has approximately probability 0.95 of including the true value of \(\theta\). We call the resulting interval from a single random sample a 95% confidence interval for \(\theta\) and say that we have 95% confidence that it will include the actual value of the parameter.

We explained this earlier for estimators based on discrete random sample, but it also holds for random samples from continuous distributions. In particular, this method works for maximum likelihood estimators, due to their asymptotic properties.

Other confidence levels

Intervals with different confidence levels can be obtained by replacing "1.96" by other quantiles of the standard normal distribution. For example, a 90% confidence interval for \(\theta\) is

\[ \hat{\theta}-1.645 \times \se(\hat {\theta}) \quad \text{ to } \quad \hat{\theta}+1.645 \times \se(\hat {\theta}) \]