Hypotheses testing a sample vs an unknown sample size


New Member
How do you test the nul hypotheses that two samples are equal if only one of the population sizes are known?

The situation appears when you test a measurement instrument by measuring on a certified reference sample with a given nominal value and measurement uncertainty (=standard deviation).

In my case I have made 30 repeated measurements on a certified reference sample , so my own sample size is known.
I have calculated the average and standard deviation of my sample of 30 measurements.

I now want to check the nul hypotheses that the instrument measures correctly versus the alternative hypotheses that they are not equal.
That is, that the mean of my sample is the same as the nominal value of the reference sample vs the alternative hypotheses that they are not equal and the instrument is measuring incorrectly...

How can I do this?
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New Member
Thanks for helping out!
But how is then the uncertainty of the constant taken into account?
In the physical world there are no "constants", all measurements has an uncertainty.

And I think it should be two sided even if it was a constant.
As an example, measure a known "meter long standard" with a ruler. The ruler can measure the "constant meter" both to be longer than and shorter than one meter.


TS Contributor
I agree with @fed2. The measurement uncertainty will be part of your 30 measurements, and the reference sample should be known to a greater level of precision and resolution than your 30 measurements. A 1-sample t-test should work. Yes, a 2-tailed test is appropriate.

BTW, if your interest is more in measurements, uncertainty, etc. and less in statistical test, I recommend posting this question at the Elsmar Cove Quality forum. I recommend posting in either the General Metrology sub-forum, or the Measurement Uncertainty sub-forum.
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Well-Known Member
a certified reference sample with a given nominal value and measurement uncertainty (=standard deviation).
This sounds to me like you are given the reference sample mean and the standard error of this mean (rather than the SD of the sample measurements.) If this is true, then you can do an informal (but reasonably accurate) test.
Find the difference in the means Diff = large mean - small mean
Find the SE of your sample =SD/sqrt(30)
Find the SE of the difference SEDiff = sqrt(your SE^2 + ref SE^2)
Find the margin of error on Diff MoE = 2xSEDiff.
If Diff>MoE then there is a significant difference, or else No sig diff.
But we need to appreciate that No sig diff is simply a face saving way of saying "We don't know if there is a difference or not." You can probably get No sig diff if you want one just by taking a small enough sample. Perhaps what you need is an equivalence test.