We want to make an hypothesis test for the mean value $\mu$ of a normal population with known variance $\sigma^2=13456$, using a sample of size $n=100$ that has sample mean value equal to $562$.
Calculate the p-value.
Make the test with significance level 1% about if the population mean value from which the sample comes from is greater than 530 using the p-value.
For the first one, about the p-value, do we have to calculate $P\left (\frac{530-562}{\frac{\sigma}{\sqrt{n}}}\right ) $ ?
And for the second we have to check the p value with the significance level, right?
Yes. correct.
If you calculate $\Phi(\frac{530-562}{\sqrt{134.56}})=\Phi(-2.76)\approx 0.29\%$
this p-value is highly significant so the test is significant at 1% and the hypothesis that the mean is 530 is rejected.
Lower is the p-value and higher is the significance of the test. This because the p-value is the area of the queue. A very low value of p indicates that the quantile (your observed mean) is very far from the centered mean (null hypothesis)
This drawing is left queue but the brainstorming is the same with right queue...as you can see, the lower is the p-value (the shadowed area) the higher is the distance between $H_0$ and your observed data