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How do you determine the highest level of significance (in decimal form) at which the null hypothesis will not be rejected?

Let's say for instance I got a p-value of $0.001$ and a test statistic of $3.273$, then is $0.0005$ the highest level of level of significance at which $H_{0}$ will not be rejected?

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The "p-value" can be viewed as an empirical highest level of significance, i.e., in your case is $0.001$.

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