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A coin may be fair or it may be biased so that $p = P(\mathtt{Heads}) = 2/3.$ [Notice that I've gotten rid of the possibly confusing twist of letting $p$ be the probability of Tails.] We want to test $H_0: p = 1/2$ against $H_1: p = 2/3.$ [This situation in which $H_0$ and $H_1$ each specify only one value is called 'simple vs. simple'.] Data for the test ...


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Based on your first graph, your problem is granularity; there is a limited number of values that happen often. It is impossible to (meaningfully) fix that using a transformation. So, the second graph suggests to me that you made an error when applying the arcsine transformation. Appart from the granularity, the original distribution does not look bad, so I ...


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The test you are looking for is called a test of marginal homogeneity (the table you give are the two margins of a 4x4 cross-classification, and you want to test if these are equal, hence the name). There is at least one R-packages for the Stuart-Maxwell test, which is a Wald test. A bit more accurate for small samples may be a likelihood ratio test using ...


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