# Can I test for inequality in H0 using chi square test?

Let's say I want to test whether an $$n$$-sided dice is not too unfair. In the standard chi-square test we test the zero-hypothesis $$H_0\colon (p_1,\dots p_n) = (1/n,\dots,1/n) ,\quad\text{i.e.,}\quad H_0\colon \forall j\colon p_j = 1/n$$

Whereas I would like to test something like $$H_0\colon \forall j\colon p_j \geq 1/2n .$$

My "naive" and "wrong" idea would be to change the test value from $$nN\sum_{j=1}^n \bigr(\frac{c_j}{N}-\frac1n\bigr)^2$$ to $$nN\sum_{j=1}^n \min\bigl\{0,\frac{c_j}{N}-\frac{1}{2n}\bigr\}^2$$, wheren $$N$$ is the number of samples and $$c_1,\dots,c_n$$ the counts of hits. This would simply disregard any $$j$$'s such that $$c_j/N>1/2n$$.

However, I understand that statistics are not this simple :-). So please, is there a way how to do this properly?

• This would be related to equivalence and noninferiority testing. – Glen_b Mar 11 at 12:33