What is the differences among p hat, po, and the p value ?

##\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##.