# Tests on binomial distribution

I have the observations $X(n)$, where $X(n)$ is the realization of a binomial random variable with probability of success $p(n)$, and with $Y(n)$ trials. The observations are independent across $n$. I would like to test the null hypothesis H0: $p(1)=(2)=\cdots=p(N)=0.5$. Is there a standard recommended test? An approach would be to perform a multiple comparison test with a correction of the significance level, but I wonder if othe methods would be possible. If $Y(n)=$const, I could have used a goodness-of-fit test, but this doesn't apply here. Suggestions welcome!

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This is a question of testing if several proportions are equal and equal to a specific value. This is quite standard, and you can do this by a likelihood-ratio test or a $\chi^2$-test. In R, the $\chi^2$-test can be computed using prop.test, and you can specify that you want the vector of proportions to be equal to the vector $(0.5, \ldots, 0.5)$. The computations are, however, not complicated.