# Solved – n error in the one-sided binomial test in R

I am not sure, but I think R is doing not what it is supposed to do.
Using `binom.test()` my understanding of the parameter `alternative="greater"` is the following hypothesis.
(I mean it says greater and not greater or equal).

P-Value then should be calculated as follows:
\$\$p(c) = P_{H_0}(Z>c) = 1 – P_{H_0}(Zle c)\$\$

Here is the equivalent R-Code

``binom.test(8, 10, alternative="greater", p=0.5)\$p.value # 0.0546875  1-pbinom(8,10,0.5) # 1-P(c<=Z) # 0.01074219  1-pbinom(8-1,10,0.5) # 1-P(c<Z) # 0.0546875 ``

So where is my mistake? Or is R just a little imprecise? And if I am right and R is realy testing the "wrong" (="not exactly what one would expect") hypothesis: What is with the other tests? Does "greater" always mean >= ?

Contents

An effect of this is that a test at significance level \$alpha=0.05\$ rarely has type I error rate (size) equal to \$0.05\$. The actual size of the one-sided binomial test for different \$pin(0,1)\$ and \$n\$ is shown in the figure below. As you can see, the size oscillates quite a lot. It is however bounded by \$alpha\$ for all \$p\$. 