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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?

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First 95% Confidence Interval: Symmetric 2-sided, so sample mean +/- 1.960 * (standard error)

Second 95% Confidence Interval: Upper 1-sided: [$-\infty$, sample mean + 1.645 * (standard error)]

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  • $\begingroup$ Thanks! How do you derive the 1-sided confidence interval? I thought the sampling distribution of the sample mean (the estimator) was symmetric around the point estimate (the sample mean), i.e. it's a normal distribution, so one can't build a 1-sided confidence interval for that estimator. Or am I completely misunderstanding how one constructs a confidence interval? $\endgroup$ – Josh Jan 18 '16 at 0:38
  • $\begingroup$ 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. $\endgroup$ – Mark L. Stone Jan 18 '16 at 0:44
  • $\begingroup$ Just because the sampling distribution is symmetric doesn't mean the confidence interval needs to be. I frequently use one-sided confidence intervals in conjunction with performance requirements. We want to make sure the performance is good enough, but not get dinged because the performance is too much better than required. $\endgroup$ – Mark L. Stone Jan 18 '16 at 0:52

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