# How tailed area decided for hypothesis testing?

I have a doubt in basic hypothesis testing. Suppose I have a testing setup as below. I have assumed null hypothesis that my sampling distribution has $$\mu = 60$$, and I get a sample set with sample mean at $$\overline{x} = 62.75$$ Now to evaluate my hypothesis, if discrete distribution, I would simply look up at $$p(\overline{X} = 62.75)$$ and if that is less than $$\alpha$$ we reject null or otherwise.

Since it is continuous, $$p(\overline{X} = 62.75)$$ is 0. But why instead of doing a continuity correction around 62.75, we take all probabilities above 62.75? What is the justification for that?

I am not able to justify why we leave the whole blue area. I understand my alternate hypothesis is $$\mu > 60$$, but not $$\mu > 62.75$$ or something like that. So its confusing.

Kindly clarify.

• How would you apply a countinuity correction on a continuous scale? Infinitesimally small intervals will give you infinitesimally small values. Large intervals just make no sense, there is no justification for them. Oct 4 '18 at 14:05
• So we apply right tail area because continuity correction is not possible? Oct 4 '18 at 14:18
• about continuity correction: people.stern.nyu.edu/jsimonof/classes/1305/pdf/contcorr.pdf Oct 4 '18 at 14:22