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QUESTION

# 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.