# Meaning of the p-value

Suppos that we have a null-hypothesis $$H_0: \ \theta=\theta_0$$. Our alternative hypothesis could be for example $$H_1: \ \theta\ne \theta_0$$. We want to test the null-hypothesis so we construct a test-statistic $$T$$ which has some probability distribution. Based on the data we compute the numerical value of the test statistic to be $$t$$.

For some reason, we define $$p:=2 \mathrm{min}\left\{\mathbb{P}\left(T\ge t\mid H_0 \right),\mathbb{P}(T \le t\mid H_0)\right\}$$ and if the p-value is very small, we abandon the null-hypothesis. Could someone please clarify, how does the p-value imply that the null-hypothesis is incorrect? And also in cases when we have $$H_1: \ \theta>\theta_0$$ or $$H_1: \ \theta < \theta_0$$. I think it would be more convenient to check the probability of having $$T\in \left(t-\varepsilon, t+\varepsilon \right)$$ assuming that the null-hypothesis holds.

## 2 Answers

Just to clarify the problem look at the following example (it is only an example...)

Suppose you have a distribution like the one in the picture (it's a std gaussian) and suppose that your critical value is $$z$$. The area in the queue left to $$z$$ is your significance level, say $$\alpha$$.

If your test statistic (t stat) is the one I showed, it is clear that you are in the rejection area, very far from the center of the distribution and the p-value is the probability expressed by the purple area; It is evident that in this situation

$$\text{p-value}<\alpha$$

and this means: abandon the null hypothesis.

Now I think it is clear that "the lower is the p-value, the less is good $$H_o$$"

If the test is "two sided" the pvalue must be multiplied by 2...it is the area of both extreme queues

The observed $$t$$ is a realization of a random variable $$T$$ which we would expect to have values in a certain region if $$H_0$$ holds. To be precise: We define an interval such that $$T$$ has values in this interval with a probability of $$1-p$$ if indeed the null hypothesis $$H_0$$ is true.

So if we observe that $$T$$ does not lie in this interval, it is very unlikely that this data was generated under $$H_0$$.

As you noticed, this is a conditional probability concerning the observed data and not a probability of the hypothesis $$H_0$$ itself (if you want to assign probabilities to hypotheses you need the framework of Bayesian statistics).

• Is such an interval unique? – mathslover Jun 25 at 9:59