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I wanted to understand difference between Prediction and Confidence intervals and found an explanation on the internet at enter link description here (i saw similar explanations for that terms in other sources as well):

In general, if we would repeat our sampling process infinitely, 95% of such constructed confidence intervals would contain the true mean hemoglobin concentration.

In general, if we would repeat our sampling process infinitely, 95% of the such constructed prediction intervals would contain the new hemoglobin concentration measurement.

I was confused whether these explanations are in agreement with multiple testing correction concept? I think that someone will get much less Prediction and Confidence intervals containing a true value according the multiple testing correction concept. Could you correct me please and provide an explanation?

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    $\begingroup$ Main difference is that CI uses SE of sample mean, $S/\sqrt{n}.$ but PI must account for variance of new value not included in current sample mean, so uses $S(\sqrt{1 + 1/n}).$ $\endgroup$ – BruceET Mar 14 at 16:17
  • $\begingroup$ Thank for your reply! Is S a standard deviation? What is SE? Could you clarify that please? $\endgroup$ – Denis Mar 14 at 16:43
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    $\begingroup$ $S$ is standard deviation. SE is standard error of the mean $SD(\bar X) =\sigma/\sqrt{n},$ estimated by $S/\sqrt{n},$ when $\sigma$ not known. $\endgroup$ – BruceET Mar 14 at 20:37
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To understand the difference between a confidence interval and a prediction interval you need to understand what each is trying to do.

Imagine you have a target population you would like to learn something about. This population could be patients at a large hospital.

You conduct a study in which you select 100 patients at random from this target population. For each patient in your random sample, you measure their whole blood hemoglobin concentration (to stay with the example you mentioned). These measurements will become your data

Your first study goal will be to estimate the average value of the whole blood hemoglobin concentrations for ALL patients in the target population. To this end, you will compute a 95% confidence interval for that average. This interval will give you the likely range of values of this average. (Another word for average is mean.)

Your second study goal will be to predict the whole blood hemoglobin concentration for A NEW randomly selected patient from the target population. To this end, you will compute a 95% prediction interval. (This would be patient # 101, which was not part of your original sample of 100 randomly selected patients from the target population.)

Both intervals are computed from the same data, but are used to guess different unknown values.

The 95% confidence interval will help you guess the average value of what you are interested in - whole blood hemoglobin concentration - for ALL the patients in your target population of patients.

The 95% prediction interval will help you guess the value of what you are interested in - whole blood hemoglobin concentration - for a single patient in your target population of patients, randomly selected from the target population but originally not included in your study sample of 100 patients.

Both of these intervals have long-run properties. If you repeated your study a large number of time, each time selecting a different sample of 100 randomly selected patients from the target population and computing your 95% confidence interval and 95% prediction interval following the same statistical methodology, you would expect that:

  • 95% of the confidence intervals will include the average value of whole blood hemoglobin concentration for ALL the patients in your target population of patients;

  • 95% of the prediction intervals will include the value of whole blood hemoglobin concentration for a new, randomly selected patient in your target population of patients.

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  • $\begingroup$ Thank you so much for the excellent explanation! Could you clarify for me please one more point regarding intervalas long-run properties? If there is 5% chance to make error (incorrect interval) in each individual repetition why we will still obtain only 5% incorrect intervals after numerous repetitions? $\endgroup$ – Denis Mar 15 at 9:22
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    $\begingroup$ You’re welcome, @Denis! Think of it this way: If you repeat your study 100 times as described in my answer and build a 95% confidence interval for the mean value of whole blood hemoglobin concentration, you expect 5% of your intervals to NOT capture that mean value. What this means is that in reality you might observe that only 3 intervals in your set of 100 would actually NOT capture that value. If you were to repeat your study again 100 times with another 100 confidence intervals, this time you might observe that 5 of those intervals would NOT capture the value of interest. $\endgroup$ – Isabella Ghement Mar 16 at 2:32
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    $\begingroup$ If you had a large number of sets of 100 repetions of your study and averaged the number of 95% confidence intervals which did NOT capture the value of interest across all sets, you would expect the average to be equal to 5. But each particular set would give you a number that would fluctuate about 5. $\endgroup$ – Isabella Ghement Mar 16 at 2:36

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