Given that what a p-value tells us is the probability of observing the data assuming the null is true, should we ever use null hypothesis significance testing (NHST) when we would rather accept the null hypothesis?
Take the Shapiro-Wilk test through which we might want to establish the normality of our data. Given a p-value of .36 and a conventional $\alpha$ of .05, we would fail to reject the null hypothesis. And the typical follow up is the suggestion that our data are normal.
However, the specific interpretation of $p=.36$ is: assuming our data are normal, the probability of observing these data is 36%. Considering this, it does not seem like good practice to act like failure to reject the null is equivalent to accepting the null. Am I correct?
Absence of evidence is not evidence of absence. Why then is the NHST framework often used when we would want to accept the null?