Suppose $X=6$ is observed where $X \sim \mathsf{Binom}(36,p)$ in a test of $H_0: p=1/4$ vs $H_1: p \ne 1/4$ along with a second binomial experiment.

Question

When compared to the first experiment, find the value $y$ such that $Y=y$ provides less evidence against $H_o$ and $Y=y+1$ provides more evidence against $H_o$.

I have no idea where to start- any guidance in the right direction would be appreciated! Also, how do I decide whether to reject $H_o$ or not if the question doesn't specify the level of significance? Thanks

Answer 1

You don't need a level of significance because you aren't being asked whether or not to reject $H_o$. You just need to find the value of $y+1$ that provides more evidence than $x$.

To do this, first find the $p$-value for $x=6$, then find the value of $y$ for which the $p$-value for $Y=y$ is higher than the $p$-value for $x$, but the $p$-value for $Y=y+1$ is lower.