fisher’s hypothesis test

We consider a single hypothesis that compares a population parameter $theta $ (总体参数) to a given null value $theta_0$. This hypothesis will be denoted by $H_0$ and is called the null hypothesis.

Our goal is to find statistical evidence that allow us to reject $H_0$.

$$H_{0} : theta leq theta_{0} quad text { or } quad H_{0} : theta geq theta_{0}$$

$$H_0 :theta = theta_{0}$$

The P-value is an upper bound of the probability of obtaining the data if $H_0$ is true.

If $D$ represents the statistical data,

$$P = P[D | H_0]$$

and we will reject $H_0$ if the value is small.

We say that we reject $H_0$ at the [P-value] level of significance.

A small P-value 不能推出 $H_0$ is false.

In hypothesis test, we want to argue that

$$text{If P then Q; Q is unlikely, therefore P is unlikely}$$

which is obviously wrong.