##hatp## : the estimated value of the parameter ##p## based on your sample.
##p_0## : the real value of ##p## under the null hypothesis.
p-value : formally, the probability, under the null hypothesis of obtaining the observed value or any other value wich brings more evidence to reject ##H_0##.
As an example let’s say you want to test if a coin toss is actually fair or heads has a larger probability. In this case you would have:
##H_0##: ##p = p_0 = 0.5## vs ##H_1##: ##p > 0.5##, where p is the probability of the “heads” results.
Then you calculate your ##hatp## wich in this case will be the proportion of heads in your sample. Let’s say you got a 0.65 proportion in your sample.
Then your p-value will be the probability under ##H_0## (this means you calculate this probability assuming it’s a fair coin) of having a proportion of 0.65 or more in your sample, because the larger the proportion, we get more evidence that “heads” has more probability.
Small p-values leads us to reject ##H_0##.
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