Suppose we have $\boldsymbol{\hat{\mu}}-\boldsymbol{\mu_0}$~$N(0,\Sigma)$ and we want to find a 95% confidence interval for $\boldsymbol{\mu_0}$
The way we would do this in the 1 dimensional case, is we would standardise it by dividing by the standard deviation and then use the cdf of the standard normal to find the Confidence interval.
But how do you find a 95% CI for $\mu_0$ in the 2 dimensional case?
You have an elliptical confidence region of the form $$ C_{\mu} = \{\mu \in \mathbb{R}^2 | ( \bar{X} - \mu)^TS^{-1}(\bar{X}-\mu) \le \frac{2}{n-2} F_{1-\alpha; 2, n-2} \}, $$ where $$ \bar{X} = (\hat{\mu}_1, \hat{\mu}_2)^T, $$ and $$ S = \frac{1}{n}\sum_{i=1}^n (x_i -\hat\mu_1)(x_i -\hat\mu_2)^T. $$