Understanding the P-value

Question

I'm having difficulty understanding the p-value. It is said to reject the null hypothesis when the p-value is small. Smaller than the significance level.

So does that mean in a hypothesis test, the p-value represents the area of the null hypothesis? Therefore because the p-value is small, it would imply the probability of the null hypothesis being unlikely?

Answer 1

The $p$-value represents the probability that an event as unlikely at the observed one could have happened under the assumption that the null hypothesis is true.

It doesn't really represent the 'area of the null hypothesis', because $p$-values are specific to given observations.

Answer 2

In statistics, the p-value is the probability that, using a given statistical model, the statistical summary (such as the sample mean difference between two compared groups) would be the same as or more extreme than the actual observed results.

Less technical, lets say the null hypothesis is actually true. With p-value we calculate the probability that the statistic would be the same as or more extreme than the value we calculate from the sample(e.g. sample mean). So we can interpret p-value as how much our null hypothesis supports our data. If that probability is lower than a pre-determined level, we conclude that it is unlikely that null hypothesis is actualy true.

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

Answer 3

Consider certain observed event $E$ with assumption the null hypothesis is true. The null hypothesis should give you the way to calculate probability of event observation which is more casual than $E$. It is done first by estimating of test statistics $x$ corresponding to $E$ and then by calculation of $Pr(X < x)$ - probability of event more casual than observed one. How exactly you would calculate $x$ and $Pr(X < x)$ is prescribed by null hypothesis.

Then probability of event more or equal in rarity than $E$ is $1 - P(X < x) = P(X >= x)$ - and this is defined as p-value of observed event $E$.

Once p-value of event $E$ is found you can decide is it really that extraordinary to have event $E$ occurrence. It is done by comparing your p-value with magic number like 0.05.

If p-value you have is <= 0.05 you may say - well, what I see is either big luck (5% chance) or my assumptions (null hypothesis) are wrong - then you reject null hypothesis. If p-value you have is > 0.05 you may say - it doesn't look like something rare given my assumptions - so you accept null hypothesis.

To answer you question about 'area of null hypothesis': p-value defined as $Pr(X >= x)$ is the area of right tail of distribution $Pr(X < x)$ which is part of your null hypothesis.