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Me and my wife had bought and planted ~20 plants in ~8 pots in our garden. the plants are of the same specie and were all very much alive when we bought them. The number of plants per pot varies as the pots are of different size. I've notice that some many of the pots are dying/dead and I think that the proportion of dead is different in different pots which would suggest that something in the pot (different watering/soil/virus in the soil etc) is what is killing the plants. My wife sticks with H0 "there is not difference in the proportion of dead plants between the pots" how would you go about this? should I assume same probability per pot (and count the total dead/total as P)? and use chi-squared to show that the number of dead in each pot does not fit this distribution? I have only a few (1-4) plants per pot so I guess ANOVA is out of the question...your help is appreciated.

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Since the expected cell count is small, instead of $\chi^2$ you may do a Fisher exact test.

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  • $\begingroup$ While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review $\endgroup$ – Antoine Nov 8 '17 at 9:47
  • $\begingroup$ @Antoine well... it's not quite link-only, and Fisher's exact test will remain googlable even if the linked page changes. I believe short answers are in the spirit of the model for all these sites, StackOverflow. $\endgroup$ – Bjørn Kjos-Hanssen Nov 8 '17 at 22:30
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20 plants in 8 pots are few plants and a lot of pots. This is not likely to bring a $p < .05$. First see, if you can group some pots together (maybe some where in the shadow, others in bright light? Some planted with compost, some not)? Or maybe you can restrict your analysis to a few pots with many plants in them? Of course, you would have to do that before looking at the observations.

If that does not work, change to Baysian for this time. In Frequentist statistics, when $p > \alpha$, nothing is gained whilst in Baysian, you at least gain some information about the true rate of dead plants in small pots.

Given, that there is a handy conjugate prior for the Binomial distribution, this comes easy with no need to learn any new software: $Beta(1, 1)$ is a flat prior. If you find 3 living plants and 1 dead plant in a pot, the posterior probability distribution of the survival chance of each plant is $Beta(1+3, 1+1) = Beta(4,2)$. All these beta-Distributions will give you and your wive a good basis to argue upon data, instead of a $p$-value that is basically useless, if not smaller than $\alpha$.

Here is an easy example that may take you even a bit further: https://lingpipe-blog.com/2009/10/13/bayesian-counterpart-to-fisher-exact-test-on-contingency-tables/

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