today i would like to get a deep understanding of meaning of p value in hypothesis testing, idea is that a lot of book say-if p value is less then $\alpha$ reject null hypothesis, but if not dont reject, i dont want to give students such kind of memorization, instead i would like to explain them in detailed why we are rejecting,so let me explain my idea
we have two type of errors Type 1 and Type 2 error
A type I error occurs if you reject the null hypothesis when it is true.
A type II error occurs if you do not reject the null hypothesis when it is false.
$\alpha$ controls probability of A type I, so if $\alpha$ is 5%, that means that there is $5$% percent of reject the null hypothesis when it is true.now let us consider the following example
null hypothesis is that $\mu$=50 , while alternative hypothesis is that $\mu$>50, we collected data, we get sample mean 52 and we estimated p value which is 0.013, which means that probability of getting test statistic 52 or more is 0.013 while null hypothesis is true, because it is less then alpha we are rejecting null hypothesis, but here is my question : what is the idea of rejecting? why we are rejecting? lets dont talk in terms of formulated laws, just in a simple manner , if null hypothesis is true(that means if mean is 50) and probability of getting 52 or more is less then 5%, that means we should not expect too much to that sample mean will be different then 50 right? in other words, sample mean should give the evidence of 50 right?probability of getting more then 50 is less then 5%, am i right?
Following your example, if you believe the population mean to be $50$ and you obtain a sample mean of $52$, the small p value is indicating that this value is a very unlikely occurrence and may be evidence that the mean is actually greater than $50$.
So on the basis of this sample alone, we are rejecting the null hypothesis that the mean is $50$.