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Let $\theta$ be an unknown population characteristic (say average height).

A confidence interval written as $P(\hat \theta - \delta < \theta < \hat \theta + \delta) = 1 - \alpha$ makes perfect sense as $\hat \theta$ and $\delta$ are random variables.

If we do, however, write the result, say for estimated $\hat \theta = 180$, $\delta = 13$, does the following syntax (which is the standard in the books I'm studying) make sense, since the numbers are constant and $\theta$ is an unkown characteristic (and therefore a constant in frequentist probability)?

$P(167 < \theta < 193) = 0.95$

mreq
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  • Additional discussion of this and related issues can be found through a site search on confidence interval. – whuber Apr 20 '15 at 17:11
  • This not a problem of "syntax", it's a conceptual defect of frequentist CI's. $P(167 < \theta < 193)$ is either $0$ or $1$ depending on the value of $\theta$. Congratulations for perceiving this by yourself! – Zen Apr 20 '15 at 17:11
  • You are wrong, Hubber. There is a quantifier "for all $\theta$" in the definition of the frequentist CI as $P(L(X)<\theta<R(X)=1-\alpha$. But here $L(X)$ and $R(X)$ are random variables and everything is fine. After you have data, $L(x)$ and $R(x)$ are real numbers. Hence, the event ${167<\theta<193}$ is either $\Omega$ or $\emptyset$, depending on the value of $\theta$, which you don't know (it's a parameter; hence unobservable). – Zen Apr 20 '15 at 17:17
  • Thank you and sorry for the duplicate. That clears it out. – mreq Apr 20 '15 at 17:57
  • @Zen It is fundamental that probabilities refer only to measurable subsets of a sample space. In this context, that sample space consists of possible values of $\hat\theta$. The notation you are using, "$167\lt\theta\lt 193$" is, at best, ambiguous concerning what measurable set it refers to, and therefore the meaning of statements like "$P(167\lt\theta\lt 193)$" is problematic. – whuber Apr 20 '15 at 18:30
  • There are no measure theoretic problems here. It's a known conceptual problem with the interpretation and use of CIs. The sets ${\omega\in\Omega:\text{any false statement}}=\emptyset$, and ${\omega\in\Omega:\text{any true statement}}=\Omega$ are both measurable, and the indexed family of events ${\omega\in\Omega:167<\theta<193}_{\theta\in\Theta}={\emptyset,\Omega}$. – Zen Apr 21 '15 at 03:04

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