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Suppose I created a null hypothesis and calculated the p-value and it turned out to be less than the significance level. So, I reject the null hypothesis. In this case I get the type one error to be the significance level.

My question is whether I need to worry about type two error since I am already rejecting the null hypothesis so there isn't any chance of failing to reject it given the null is false?

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You don't, for just that reason: you're rejecting the null hypothesis, so you can't possibly be making a type-II error.

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    $\begingroup$ To add a bit more, see the graphic at i.stack.imgur.com/oAJ5l.png. If you reject the null hypothesis, then you can only be in the bottom row of the table of realities, specifically a type 1 error or an accurate finding. $\endgroup$ Sep 25, 2016 at 21:36

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