Martin -

Well, that seems rather haphazard to have been designed, something that needs to be done before collecting data.  Not my subject matter area, but I take it that by "sample" you might mean something like a biological specimen - chosen at random from within a certain category? - from which you have taken 2 or 3 observations each.   These repeat observations do not represent more than one member of what a statistician would call the overall "sample" (sample in statistics-speak, apparently not your biology-speak).  You were merely trying to control for data reliability problems.  If you got different results for your repeated observations, and have to "average" to get the 'official' observation for given biology-speak "samples," to get members of your statistics-speak sample (when I say "sample" after this, I'll mean statistics-speak sample), then you have a data reliability problem which will degrade the usefulness of results to an unknown degree.  But if you can come away with your needed yes/no (binomial) result you noted, for each member of the sample (each biology-speak sample), then you have overall_n=49, stratified into 8 categories, if I read this correctly.  If within each category (stratum here), you think there are no distributional differences, and none between each of the three categories on one hand, and five on the other, you might get away with estimating the proportion of "yeses" in each of the two final groups, and the variances, and estimate a confidence interval around the difference in the two proportions.

You might look at this:

 https://onlinecourses.science.psu.edu/stat414/node/209

But if each category differs substantially, and not all such strata even have enough sample members to obtain a stratum variance estimate (it is not possible to estimate variance in a statistical sample of size n = 1), then you are stuck either making the assumption that the members are close enough to random, or you might need more data.  In that case, before doing so, you need to check a text and get help with biostatistics.  Not really my experience, but I'm sure you would be told to plan ahead with considerable effort. 

Best wishes - Jim 

PS -  You said that the "...test is somewhat prone to false negatives..." so instead of "averaging" three results, would it make more sense to say if you get even one of three (or two) "positive" results, then you just say positive for that member of the statistical sample?  Sounds like it. 

I think I agree with James Knaub.

But... If you really just want to know if there is a difference between the 3-locations and the 5-locations, I would probably just make a simple 2 x 2 contingency table with counts, and perform a chi-square test of independence, or related test.

Note: Hypothesis tests still have the same problem with the groupings described.  Also, beware of the misleading use generally made of p-values. They represent incomplete information. See https://www.researchgate.net/publication/262971440_Practical_Interpretation_of_Hypothesis_Tests_-_letter_to_the_editor_-_TAS

Also note:  Whoever down voted the question is just being foolish and unpleasant/unkind, and not useful.  That seems in fashion, but it's just wrong. 

Article Practical Interpretation of Hypothesis Tests - letter to the...

Dear Jim, Dear Salvatore,

Thanks a lot!! I did as recommended by Jim: If there is at least one positive (true) result with one of my biological samples in the 3 tests, then i just say positive. This means my contingency table and the test Salvadore recommended, will look like that:

5-Location True 29

5-Location False1

3-Location True 0

3-Location False 16

Treatment True False

Five    29     1

Three     0     16

Pearson's Chi-squared test with Yates' continuity correction

Chi squared equals 37.805 with 1 degrees of freedom.

The two-tailed P value is less than 0.0001

I might come back to you, in case the reviewers tell me that's nonsense!

Thanks again!!

As noted earlier, p-values are generally not very useful.  I still recommend that you check out the Penn State link.   Also, your results are biased by the way they were collected.  See above.  

The data you present do suggest there is an association, which you would probably see no matter how you analyzed the data.

Since the data collection and summary is... unusual... there could be criticism from reviewers.  I think the best approach is to keep things clear and simple.

Dear Jim and Dear Salvadore

Thanks for your support!!

The 46 samples  I am testing, have been produced between 1975 and 2000 by five persons, working sometimes alone sometimes together (8 combinations / alias locations). This historical dimension makes me unable to change anything about the experimental design. All i can do is testing the existing samples.

With respect to the test results, the eight combinations of workers seem to fall into two distinct categories. I have 3 combinations where a  person X was involved (these are the 16 samples that produce false values when tested), and i have 5 combinations where person X was not involved. These are the 30 samples that produce 29 true and 1 false values when tested.

Looking at the data i also think that there is a association. Reviewers in biology tend to love p-values and it seems i can nw deliver one. Will check out Jims link and hope to learn something!

Thanks again!

http://amstat.tandfonline.com/doi/abs/10.1080/00031305.2015.1069760