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I need to quantify the relative error of a process that computes the volume of some objects. I also have a sample ($n=400$) of objects that are known to be made of a homogeneous material. Since the density is fixed and known, I figured one could estimate the CI of the density to conclude bounds for the relative error of the volume (95% of the time, relative error will be bounded by B), but I am not sure that this is the correct interpretation for what a CI interval is or that CI intervals are the right approach to the problem.

Alexis
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statsnewbie
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    if the density is fixed and known you know everything. – utobi Oct 24 '23 at 19:14
  • It seems like what you want to do is compute the density of the objects using a weight that you measure and the volume you compute with your process of interest, and then compare this empirical density to the actual known density. Is this right? If so, this is not clear from the text of your question. – dherrera Oct 24 '23 at 20:24
  • Your question isn't clear. What do you know about the 400 objects? What do you want to quantify? Try to avoid terms like confidence interval and relative error and clearly state what you already know and what you want to find out (by calculation) – Harvey Motulsky Oct 24 '23 at 20:46
  • Thank you for the responses. To clarify:
    • The mass of each object is known.
    • The true density of all objects in the population is not known
    • I want to characterize the error in volume for a single measurement, i.e.: to be able to say that, for a single measurement of volume of a random object, there is a 95% likelihood that true volume is within [l,u] of the measured value, where l,u

    are the parameters I want to estimate, which represent lower and upper bounds for the relative error. Is this just a straightforward application of percentiles?

     – statsnewbie Oct 24 '23 at 21:58
  • "I want to characterize the error in volume for a single measurement," --- that's not a confidence interval. – Glen_b Oct 24 '23 at 22:13

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If you calculate a 95% confidence interval for the mean using your n=400 sample then most of the values in that sample will lie outside of the interval. The interval is scaled by the square root of the sample size, and it is meant to communicate the uncertainty about the true mean. You know the true mean, so presumably that is not what you want.

To quantify the errors you should probably convert the measurements to errors by subtracting the known density from them. Then a confidence interval of those errors will give you a range for the mean error from a sample of n=400. Presumably that will be quite small; much smaller than the probable error from any single measurement.

If you want to know about how far any single measurement might be from the known density then the distribution of the errors in your n=400 sample will help. The upper and lower bounds of that distribution (or the 2.5th to 97.5th percentiles) will have a straightforward meaning. Depending on who you wish to communicate that with, a histogram of the errors might be better.

Michael Lew
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