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I'm new to P-Values and Hypothesis testing and I noticed that p-values are explained differently everywhere I go. In my lecture notes it has a method for hypothesis testing and the last part says that "if the p-value is not small enough to rule out chance, we fail to reject null hypothesis". A p-value is stated as: the probability of observing a value of the test statistic at least as contradictory to the null hypothesis as the one computed from the sample data"
Now, what doesn't make sense to me is that if p-value tells me the probability of observing a value of a test statistic at least as contradictory to the null hypothesis, then surely a low p-value(less than significance) would mean that it isn't very contradictory and thus we fail to reject but in fact they do it the other way around. I.e if p-value is less than significance, they actually reject null hypothesis?
Some sites say that the p-value is the strength of the evidence from the test statistic. Any clarificaiton?