# Different methods, different confidence intervals

A definition of a confidence interval could be:

A confidence interval for the parameter θ, with confidence level or confidence coefficient γ, is an interval with random endpoints ($u(X)$, $v(X)$), determined by the pair of random variables $u(X)$ and $v(X)$, with the property:

${\Pr}_{\theta,\varphi}(u(X)<\theta<v(X))=\gamma\text{ for all }(\theta,\varphi).$

Here $Pr(θ,φ)$ indicates the probability distribution of X characterised by $(θ, φ)$.

Implicit in this definition is the notion that one can build different confidence intervals using different pairs of methods $\left(u, v\right)$, for the same confidence coefficient $\gamma$.

What would be a good example to show this? Say we have a 100 sample from a 1D normal distribution, and we seek two 95% confidence intervals of the mean obtained through two different methods $\left(u_1, v_1\right)$ and $\left(u_2, v_2\right)$. What methods could we use here?

Second 95% Confidence Interval: Upper 1-sided: [$-\infty$, sample mean + 1.645 * (standard error)]
• Instead of dividing up the 5% "non-confidence" evenly on the two sides, put the 5% entirely on one side, and put 0% on the other side by going all the way to + or - $\infty$, as the case may be. – Mark L. Stone Jan 18 '16 at 0:44