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Many questions seem to ask something similar, but I could find none that seemed to really match. Suppose there are two potentially distinct archaeological cultures "East" and "West". Both East and West produce statues that are either red or blue. Five eastern sites are known, and six western sites are known. The percentage of statues that are blue at each site are as follows:

Eastern site 1: 91.7% (n=24)
Eastern site 2: 75.0% (n = 24)
Eastern site 3: 88.5% (n=61)
Eastern site 4: 81.3% (n = 16)
Eastern site 5: 100% (n = 34)
Mean frequency of blue statues at Eastern sites (equal weighting): 87.3%

Western site 1: 47.8% (n=23)
Western site 2: 66.7% (n=15)
Western site 3: 73.3% (n=75)
Western site 4: 73.7% (n=19)
Western site 5: 92.3% (n=26)
Western site 6: 77.2% (n=22)
Mean frequency of blue statues at Western sites (equal weighting): 71.8%

Suppose an archaeologist believes there is a difference in the mean blue statue percentage between eastern and western sites. How could I investigate this belief statistically? What statistical test could be used to test for a significant difference in the mean site-wide percentages in eastern and western sites? Because these are percentages, values are bounded between 0 and 100% so the data are not normal. And because there are not many sites, the distributions cannot really be observed. Is there a test I could use that would not require the normality assumption? Or would it still be fine to use a t-test anyway? I read that when the normality assumption is violated, the t-test can be too conservative. I don't want a test that is too conservative because then I would be unfairly concluding that the archaeologist's belief is ill-founded. A Welch's t-test comes out at p = 0.065.

I would like to weigh sites equally, rather than give greater weight to sites with a greater sample size, because the choice of colour of statues at one site are perhaps linked. In the extreme, for example, one artisan could choose to produce 50 blue statues at one site over a week. These 50 blue statues would ultimately come down to a single choice made by one artisan. So 50 blue statues at one site cannot really be assumed to be 50 separate datapoints.

What statistical test could be used for this?

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    $\begingroup$ “I would like to weigh sites equally, rather than give greater weight to sites with a greater sample size“ To me, this requires considerable justification. It seems like you have the counts of total blue in each region and total statues in each region, and then you can do a standard two-sample proportion test like a chi-squared test. $\endgroup$
    – Dave
    May 2, 2023 at 20:45
  • $\begingroup$ @Dave The other archaeologist is trying to argue that East and West are two distinct cultures and I am trying to see whether this is defendable. This blue-red statue example is made up, but in many cases the n can be 10 times greater at each site (but the number of sites stays the same). Using proportion tests like you suggest, there would be statistically significant differences absolutely everywhere. It might then be argued that the 11 sites above might as well be 11 different cultures, which in practice no archaeologist would claim. Perhaps statistical tests are not the right tool for this. $\endgroup$
    – Pertinax
    May 2, 2023 at 21:09
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    $\begingroup$ A chi-squared approach is straightforward and easy. The hard part would be justifying the implicit assumption that these sites are random samples of cultures and the assumption that you question; namely, that the colors of the statues might be independent. If that really is questionable (and I think it is), you need to come up with some model of the dependence. In effect, your problem is tantamount to figuring out who made each statue. Is that even possible? $\endgroup$
    – whuber
    May 2, 2023 at 21:39
  • $\begingroup$ @whuber I am aware of the first assumption, but it is hard to investigate and is thus rarely discussed. $\endgroup$
    – Pertinax
    May 2, 2023 at 22:04
  • $\begingroup$ @whuber I'm afraid figuring out who made each statue is not possible. One can, however, group the artefacts at each site depending on which archaeological layer it was found in, so-called "assemblages". In the above I say "site" for simplification but I mean assemblage. Each assemblage at a site may represent 200 years or more, so it is assumed that subsequent assemblages were produced by different people. But often, assemblages from different periods at the same site tend to be more similar than assemblages from the same period at different sites, suggesting assemblages are not independent. $\endgroup$
    – Pertinax
    May 2, 2023 at 22:05

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