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A 95% confidence interval under Neyman-Pearson is defined as the interval upon which if we took many samples of size n from the population, 95% of the intervals formed around the sample means would contain the population mean.

In the circumstance where you have knowledge of the population variance, this interval will have the same range for each sample, assuming each sample is of size n.

However, in the circumstance where you don't have knowledge of the population variance, each sample of size n will use its sample standard deviation and therefore the interval range will vary across the samples as a result.

With this in mind, I am struggling to see the material benefit, as a part of a piece of analysis, to provide a confidence interval when the population variance isn't known. It feels as though I am presenting a metric which a) requires the reader to consider an almost-abstract number of samples, b) has a range which is going to vary across those samples.

Are there any benefits to presenting a confidence interval formed using the sample variance?

Edit - to clarify - I am mainly focusing on presenting to non-statistical audiences, such as decision makers within an organisation.

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    $\begingroup$ In practice, you never know population variance. So when you are asking if there are any benefits to presenting CIs constructed using sample variance -- benefits as compared to what? $\endgroup$ – amoeba says Reinstate Monica Feb 27 '15 at 18:10
  • $\begingroup$ It is a good point - perhaps I am asking the marginal benefit of providing a confidence interval built on the sample variance alongside other metrics/analysis, as opposed to only providing the other metrics/analysis. Does it genuinely enhance the ability of an individual to make decisions? I do appreciate this may move into the territory of opinion however. $\endgroup$ – NickB2014 Feb 27 '15 at 18:18
  • $\begingroup$ But what other "metrics/analysis" are you referring to? It's important. Perhaps you do have some metrics that convey similar information to CI. But perhaps you don't. One metric is the sample mean itself. What else? $\endgroup$ – amoeba says Reinstate Monica Feb 27 '15 at 18:31
  • $\begingroup$ I think you mean Neyman-Pearson? $\endgroup$ – StatsStudent Feb 27 '15 at 20:11
  • $\begingroup$ @StatsStudent I agree, at the very least, certainly Neyman that should be credited with work on confidence intervals ... I've edited NickB's question to correct "Neyton" to "Neyman". $\endgroup$ – Glen_b -Reinstate Monica Feb 28 '15 at 1:00
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As @amoeba said in a comment, you never can know the population variance, so that part of the question is not really practical.

I find confidence intervals useful. You've measured something in a sample, and want to know about the population. Given some assumptions, you can be 95% sure that your calculated interval contains the true population mean. It gives you a measure of precision. If the CI is very wide, you don't know much. If it is quite narrow, you know a lot. (Of course, "wide" and "narrow" must be in the context of whatever you are measuring).

In many situations, the goal of statistics is to let you make inferences about a population from values in a sample. The confidence interval does this in an easy to understand way.

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