Misconceptions about p-value ::

(1) p-value ≠ 1 - Φ(|z|)

(2) p^-value ≠ 2 × (p-value)

p-value made EASY!
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For a two-tailed z test, p-value is calculated as

p-value = 2*P(Z>|z|)

Φ(z) = P(Z ≤ z)

where Φ is the Cumulative Distribution Function for the normal distribution

P(Z>|z|) = 1 − P(Z ≤ z) = 1 − Φ(|z|)

p-value = 2*[1 - Φ(|z|)]

1. For a continuous distribution, there are an infinite number of possible-event/values within a range. Hence a probability is defined as the integral of f(x) from a to b, with f(x) being your probability density function.

So you might ask what is the probability of one point, say 'a'? a could be 3.35246 for example.

We apply the definition, which in integral from a to a of f(x). What is that? It is F(a)+C -F(a) +C = 0. If you look at it, as b approaches a from the right, the integral from a to b gets closer and closer to 0.

Another way is to use another definition of probability which is (number of outcomes of interest)/(total number of possible outcomes)

You will get 1/infinity, which you can analyze by doing 1/b as b->inf. You will see that it approaches 0.

Now while the probability is 0, it is not that 3.35246 is not a possible event, it's just very unlikely you will pick that exact number in a random selection.

An impossible event will have a probability of 0, but a probability of 0 does not imply an impossible event.

Same goes with a probability of 1. It doesn't mean it will happen definitely. Let's prove it.

What is the probability of NOT selecting 3.35246? It will be 1-Pr(selecting 3.35246). Which is 1-0=1

But we do know it is possible to select 3.35246, thus even though the probability is 1, it's not guaranteed.

Like all statistical construct, the meaning and value of any 'statistic' merely rest upon it's definition and intended usage.

2. I hope you really read my long explanation.

p-value is NOT the probability of an event. p-value is what what you use to compare against the alpha value.

When you look up your normal/z-score table, please read everything on the page and see that it is defined as P(Z<z). Same goes with the question, see if they are giving you the p-value or the probability in the form of P(X<some number)

This post has been edited by mumeichan: Dec 10 2013, 11:15 PM