Recently I’m reading some papers related to the sequential probability ratio test(SPRT). Here is some notes about the concepts from statistical hypothesis testing.
I don’t have any background knowledge about statistics. If you find any mistake in this notes please tell me :)

Statistical Hypothesis Testing

After reading the definition and lots of examples, in my opinion, actually the statistical hypothesis testing is a testing by contradiction. The hypothesis of the statistical hypothesis testing is a hypothesis of
During the testing process, we will follow these basic steps:

Set a pair of hypothesis: null hypothesis and alternative hypothesis.
Do the testing and analysis the consequence
Compute the probability
Make a conclusion

Null Hypothesis & Alternative Hypothesis

Rivals on testing

I really love the definition of these 2 hypotheises on the Wikipedia:Alternative hypothesis, which is easy to understand and sounds like Liverpool vs Everton :P

In statistical hypothesis testing, the alternative hypothesis (or maintained hypothesis or research hypothesis) and the null hypothesis are the two rival hypotheses which are compared by a statistical hypothesis test.

The null hypothesis is symbolised as:
$$ H{0} $$
The alternative hypothesis is symbolised as:
$$ H{1} $$

Actually, the null hypothesis is the hypothesis which we want to proof it is wrong. Then because the null hypothesis is wrong, the alternative hypothesis is true. From the symbol of these 2 hypothesises we can also find this (0=false, 1=true). I think it is the basic idea of statistical hypothesis and that’s why I consider it looks like a contradiction.

Type I & II Error

Briefly, type I is false positive, type II is false negative.
The definition of type I error on wiki is very clear:

The type I error rate or significance level is the probability of rejecting the null hypothesis given that it is true.

We use the $$ \alpha $$ to sybolise the type I error or significance level. Ronald Fisher set the significance level to 0.05 (with no reason) in 1926.

The table on Wikipedia:Type I and type II errors is good enough, the following table is modified from it:

	H0 is True	H0 is False
Reject H0	Type I Error(FP)	correct
Not Reject H0	correct	Type II Error (FN)

Likelihood-Ratio Test (LRT)

The sequential probability ratio test (SPRT)

Notes of Statistical Hypothesis Testing