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Lets say I have confidence interval at 85% with 4% margin of error, and I get a value from my analysis equal to $x$.

Does this mean that if I were to perform the test 100 times, then at least 85 times we would expect to see a value for $x$ ranging from $.96x$ to $1.04x$?

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No.

It is possible that just by chance your value of X is very high or low compared to the population value. If so, you'd expect few (or at least fewer than 85%) repetitions to have values in your confidence interval.

The correct interpretation is that if you repeated the experiment and calculations many times, you'd expect 85% of those confidence intervals to include the true population value, and for the other 15% to exclude that true population value. Unless you are doing simulations, you won't know the true population value, so will never know if a particular experiment is part of the 85% or part of the 15%. (This interpretation is based on a bunch of standard assumptions about sampling, and the population distribution.)

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  • $\begingroup$ I guess I don't fully understand. Let's say I have 5000 population, and I sample 1176 of them (necessary size for 95% confidence interval and 2.5% margin of error if I am not mistaken). Then what are we technically saying if we get an average value of x for something among those 1176? $\endgroup$
    – Remy F
    Commented Jan 30, 2014 at 16:48
  • $\begingroup$ Now you switched from 85% confidence interval to 95%, which is more typical. You can be 95% confident that the range you computed includes the true population value. I say "value" which is vague, because your question is vague. Are you computing a confidence interval for a mean, a median, a proportion, a percent survival....? $\endgroup$
    – Harvey Motulsky
    Commented Jan 30, 2014 at 16:55
  • $\begingroup$ I don't understand what any of this means, would you be able to express things more mathematically? What does it mean to be "95% confident that the range I computed includes the true value?" What is the range, and 95% of what technically? (and let's say x = a mean) I don't understand what a confidence interval / "true population mean" implies going forward after doing an analysis. Does that mean if we performed infinitely many samples, then 95% of the recorded values would be between .95x and 1.05x, and 5% would be outside that range? $\endgroup$
    – Remy F
    Commented Jan 30, 2014 at 16:59
  • $\begingroup$ The "range" is what you included in the question: .96x to 1.04x if you repeated the experiment and calculations many times, you'd expect 95% of those confidence intervals to include the true population mean, and for the other 5% to exclude that true population mean. Unless you are doing simulations, you won't know the true population mean, so will never know if a particular experiment is part of the 85% or part of the 15%. You'll need to read elsewhere about the fundamental concept of analyzing a sample of data to make inferences about the population it was drawn from. $\endgroup$
    – Harvey Motulsky
    Commented Jan 30, 2014 at 23:34

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