P-Values:

A p-value is a probability which is associated with a particular null and alternative hypothesis. Consider the following example:

e.g. A sample is collected and the mean is found to be 12.2. The experimenter wishes to test the following hypotheses:

H0:mu=12.5
Ha:mu<12.5

The p-value associated with these hypotheses is the probability of obtaining a sample for which ybar is as, or more exteme, than that observed (i.e.12.2) given that H0 is true.

We can compute this p-value because we know that the sampling distribution of ybar is approximately normal with mean mu and standard deviation SD/sqrt(n).

Let SD be 1. If n is 26 then the sample is small and we can compute the t-value: 

		t = (12.2-12.5)/(1/5)
		  = -0.3/0.2
		  = -1.5
Since we have 25 degrees of freedom the required p-value is between 0.1 and 0.05. This p-value is not significant and hence we have insufficient evidence to reject H0.

So, you should think of the p-value as providing an objective measure of the strength of evidence which the data supplies in favour of the null hypothesis. A small p-value provides evidence against the null hypothesis, because the probability of getting the data observed is small given that the null hypothesis is correct. Thus it is reasonable to reject the null hypothesis when the p-value is small.