What does p-value represent?

The P is a probability that ranges from zero to one that measures the chance that the sample mean is different, given that the population have the same mean. To be more specific, the p-value can be MAINLY classified into 2 categories: 1-tail p-value and 2-tail p-value. In fact, p-values have many tails.

(Note: When you use p-value, you are assuming that the population mean is the SAME.In the A' level syllabus, we are assuming that the population mean are the same, and are calculating the probability that the the sample mean is different by chance.)

1-tail p-value is used when the difference you observed can only be in ONE direction. (This is used in our calculations, when we say that the alternative hypothesis is more/less than our null hypothesis)

2-tail p-value is used when the difference you observed can be in BOTH direction. (When we observe that the alternative hypothesis IS NOT EQUAL to our null hypothesis.)

In MOST cases, people tend to use 2-tail p-values instead of 1-tail p-value MAINLY because when the p-value is LARGE in the opposite direction, one HAVE TO conclude that the difference is due to CHANCE, and it IS NOT statistically significant. This results in a dilemma, because the LARGE difference tells you that difference is NOT due chance. Hence, to prevent such a situation, people tend to use 2-tail p-values.

In hypothesis testing, the p-value is the probability of obtaining a result as extreme as the one observed. We use the p-value as a guage to determine if our null hypothesis should be rejected or not. Your threshold p-value(or your level of significance, or confidence level) should be set to a value based on the consequence of missing a true difference, or to falsely finding a difference. However, conventionally, it is set to 0.05. After we get the p-value, we compare it to our threshold p-value, and compare the difference and conclude whether or not to reject our null hypothesis.

To conclude on your value of p-value, you can either comment that it is by coincidence(with the chance equal to your p-value) that the difference is as large as you observed even though your population mean is the same, or that your population mean is different from the start.

The p-value IS NOT the probability that the null hypothesis is true, because we are conducting the test based on the assumption that the difference is by chance ALONE.(pls notify me if there's any error here) It is also not the probability of falsely rejecting the null hypothesis. This is further explained here

A simple example of calculation of p-value can be found here

Source:

http://www.graphpad.com/articles/pvalue.htm

http://en.wikipedia.org/wiki/P-value

http://www.isixsigma.com/dictionary/P-Value-301.htm

http://economics.about.com/od/termsbeginningwithp/g/pvaluedef.htm

Other source:

p-value in science

This is interesting