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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?

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    $\begingroup$ This would be related to equivalence and noninferiority testing. $\endgroup$ – Glen_b Mar 11 at 12:33

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