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I'm wondering about the correctness of the Boston University School of Public Health's web page on confidence intervals, particularly the interpretations. For example, specific interpretation sections all have the form of...

We are 95% confident that the difference in mean systolic blood pressures between men and women is between -25.07 and 6.47 units.

In their definition of the CI they seem to get it correct mostly...

Strictly speaking a 95% confidence interval means that if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true mean value (μ). In practice, however, we select one random sample and generate one confidence interval, which may or may not contain the true mean. The observed interval may over- or underestimate μ. Consequently, the 95% CI is the likely range of the true, unknown parameter. The confidence interval does not reflect the variability in the unknown parameter. Rather, it reflects the amount of random error in the sample and provides a range of values that are likely to include the unknown parameter. Another way of thinking about a confidence interval is that it is the range of likely values of the parameter (defined as the point estimate + margin of error) with a specified level of confidence (which is similar to a probability).

But then end with...

Suppose we want to generate a 95% confidence interval estimate for an unknown population mean. This means that there is a 95% probability that the confidence interval will contain the true population mean.

Is that incorrect? Whether incorrect or correct, why?

John
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  • Why does it seem incorrect? The last phrase just restates the previous passage, I don't see how it's different. – sashkello Jun 17 '13 at 01:15
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    This seems to me like a mish-mash of partly correct & partly incorrect understandings (it reminds me of many students, alas). Concepts in frequentist inference are just really *hard* and counter- / un-intuitive. Of course, you are familiar w/ this thread: [Why does a 95% CI not imply a 95% chance of containing the mean?](http://stats.stackexchange.com/questions/26450/). – gung - Reinstate Monica Jun 17 '13 at 02:41
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    The question contains a question. I realize there are several highly related CI questions. However, CI's are difficult to understand and convey correct interpretations. There are many pitfalls on the way (as evidenced by the answers so far). It is often difficult for the kinds of people that need answers from the CrossValidated to convert those answers to what they read elsewhere. This particular example struck me as one with much right and much wrong but I was looking for the best way to articulate which was which. – John Jun 17 '13 at 12:53
  • "*The question contains a question*" -- well, yes it does now. – Glen_b Jun 17 '13 at 13:10

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The third paragraph is all right if you interpret it as talking about probability prior to realization of the random variables $L$ & $U$; so that $\Pr(L < \theta < U)=95\%$, where $\theta$ is the unknown parameter. Given the use of 'will', & the discussion in the previous paragraph, this interpretation doesn't seem unduly charitable.

The second paragraph is all right if you cross out every clause containing 'likely'. If 'likely' is a synonym for probable those statements are incorrect (or rather the implication that they follow from the definition of a CI is incorrect); if it's not it requires definition to make sense of it. There is more discussion of this point at Are all values within a 95% confidence interval equally likely?.

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Scortchi - Reinstate Monica
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