I'm not a mathematician and I need to prove what this theorem says. I think is easy and I know how it works, but definitively I'm not too much rigorous to make a demonstration. Can anybody help me?
I enounce it:
If the test statistic has a continuous distribution, then under H0 : θ = θ0, the p-value has a Uniform[0, 1] distribution. Therefore, if we reject H0 when the p-value is less than α, the probability of a type I error is α.
Therefore, when H0 is true, the p-value is like a random draw from a Uniform[0, 1].
On the other hand, if H0 is not true, the distribution of the p-value will tend to concentrate closer to 0.
A large p-value can occur for two reasons:
- H0 is true, or
- H0 is false, but the test has low power
Do not confuse the p-value with P(H0 | data). The p-value is not the probability that the null hypothesis is true.
Thank you so much!