Consider, for example, that I demand that every farmer gives me 10 of their best apples from each of the tens of apple orchards that I own. I detest worms in my apples, but know that some will worm their way into some of them in any case.
The farmers know exactly how many of their apples contained worms and they give me a proportion relating the number of spoilt fruit to the number of all apples produced in their orchard. The orchards are of different sizes and are known.
Despite warnings, I find some worms in my treats. How can I know whether the farmers tried to cheat me or this was a sincere mistake on their part?
In other words, how can I know the proportions of worm-eaten apples in my small samples are indeed smaller than the general proportions in their orchards? Note, that I would like a general assessment of fairness of my farmers. That is, I am not looking to single out a farmer and its fruit garden, but instead have a general sense of whether (most of) the farmers are trying to cheat me or not.
My thoughts and questions:
- Is a paired t-test (or a non-parametric alternative) an appropriate analysis here given that one of the samples is actually the whole population (all of the apples produced in an orchard)?
- What can the correlation between the proportion of wormy fruit in my sample and the proportion in the corresponding orchard tell me? Should I expect this correlation to be zero if the farmers are fair?
- Are the sizes of orchards of importance here? Given that my sample is always of fixed size, should I take the sizes of the populations into account somehow?
- Should I drop the idea of proportions altogether and do something with raw number of apples?
I was trying to find related questions and this one tried to compare proportions. My example is different (or doesn't it matter?), however, because I have population estimate on one hand.