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I just started learning confidence intervals and I read online the definitions of a confidence interval.
Defn 1: http://stattrek.com/statistics/dictionary.aspx?definition=confidence_interval
Defn 2: My notes

Now, I feel as if those two are contradicting. The first defn (which I'll extract) states:

"... This means that if we used the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter to fall within the interval estimates 95% of the time. "

However, to my knowledge, isn't this one of the common misconceptions of a confidence interval? And that the actual interpretation is that if we gather $n$ amount of these confidence intervals, there is a $95\%$ probability that the $n$ collected intervals all contain the true parameter?

My notes give this definition:
Let $L:= L(X_1,\ldots,X_n)$ and $U:= U(X_1,\ldots,X_n)$ be such that for all $\theta \in \Theta$,
$$\mathbb{P}(L < \theta \leq U) \geq 1- \alpha$$ and then it says, that this is the probability ($1-\alpha$) that the true parameter lies within this interval . Which one is correct?

Twenty-six colours
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1 Answers1

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No, the probability is for a single drawing.

The probability that $n$ drawings give a "right" answer would be $(1-\alpha)^n$, a much smaller number.

  • Is the second definition (my notes) correct in saying that then?
    "the probability that the true parameter lies within this interval is $95%$"?     – Twenty-six colours Jun 07 '17 at 09:13
  • @Twenty-sixcolours: yes it is. Your interpretation with all $n$ is wrong. –  Jun 07 '17 at 09:14
  • Thank you. I'm not sure what the difference is then, to this:
    "“There is a 95% chance that the true population mean falls within the confidence interval.” (FALSE)"
    source: http://www.statisticssolutions.com/misconceptions-about-confidence-intervals/
    and it seems to say that only 95% of the confidence intervals calculated with contain that true parameter     – Twenty-six colours Jun 07 '17 at 09:16
  • @Twenty-sixcolours: I disagree with this statement. –  Jun 07 '17 at 09:43
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    Does it have anything to do with the interval (L,U) being random, BUT once we have data, it's fixed? – Twenty-six colours Jun 07 '17 at 09:54