We want to compare different counts or proportions to a theoretical uniform distribution. Hence, we are setting up a goodness-of-fit test.
This can be calculated as an exact procedure using a multinomial test, or with an asymptotic approximation, using a chi-square goodness-of-fit. Although both tests yield a significant result, the difference between the exact multinomial test and the chi square is quite remarkable.
In R, the two calls are:
observed <- c(A = 26, B = 8, C = 6); prob <- rep(1/length(observed), 3)
For the multinomial test
multinomial.test(observed, prob, useChisq = FALSE, MonteCarlo = FALSE)
Exact Multinomial Test, distance measure: p
Events pObs p.value
861 0 0.0002
chisq.test(observed, p = prob, correct = F)
Chi-squared test for given probabilities
X-squared = 18.2, df = 2, p-value = 0.0001117
These are "omnibus" tests, and post-hoc comparisons are needed. In this regard, this post was useful, and already indirectly discussed by @Glen_b here. The idea is to compare each one of the proportions to the aggregate of the other two groups with a binomial test, and compensate with the Bonferroni correction.
In this case, since we are doing $3$ comparisons, and with a significance level of $0.05$ the correction is: $0.05/3 = 0.01666667$.
But we really won't need it, because the results are quite obvious after running the first test, which doesn't even come close to being questionable:
binom.test(26, sum(observed), 1/3, alternative="greater")
Exact binomial test
data: 26 and sum(observed)
number of successes = 26, number of trials = 40, p-value =
alternative hypothesis: true probability of success is greater than 0.3333333
95 percent confidence interval:
probability of success