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Suppose that you've collected data from two samples of animals treated with different drugs. You've measured an enzyme in each animal's plasma, and the means are different. You want to know whether that difference is due to an effect of the drug – whether the two populations have different means.

Observing different sample means is not enough to persuade you to conclude that the populations have different means. It is possible that the populations have the same mean (i.e., that the drugs have no effect on the enzyme you are measuring) and that the difference you observed between sample means occurred only by chance. There is no way you can ever be sure if the difference you observed reflects a true difference or if it simply occurred in the course of random sampling. All you can do is calculate probabilities.

The first step is to state the null hypothesis, that really the treatment does not affect the outcome you are measuring (so all differences are due to random sampling).

The P value is a probability, with a value ranging from zero to one, that answers this question (which you probably never thought to ask):

In an experiment of this size, if the populations really have the same mean, what is the probability of observing at least as large a difference between sample means as was, in fact, observed?

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