According to Goodman (2008) "A p-value means only one thing (although it can be phrased in a few different ways), it is: The probability of getting the results you did (or more extreme results) given that the null hypothesis is true."
So let's use an example of a coin. We take p1 = 0.5 and make n1=1000000 (a very large number) and take p2=1 n2=4. And perform the following hypothesis test:
H0: p1-p2=0
H1: p1-p2!=0
We get Z=-2 and a p-value of 0.0455.
But, according to Goodman, the p-value be equal to P(x>=4) which, using binomial distribution gives P(x>=4)=(4!/(4!*(4-4)!))*0.5^4*0.5^0. Which gives 0.0625 (or 0.5^4).
So my question is why is the p-value 0.0455 and not 0.0625?
Thank you in advance!
Goodman SN, Royall R (1988). Evidence and scientific research. American Journal of Public Health 78(12):1568–1574