How is the p-value of a hypothesis test related to type I and type II error?

The p-value of a hypothesis test is directly related to the probabilities of type I and type II errors.

To understand this, let's first define type I and type II errors:

- Type I error: This occurs when you reject a null hypothesis that is actually true. In other words, you incorrectly conclude that there is a significant effect or relationship when there is none. The probability of making a type I error is denoted by alpha (α), also known as the significance level. It is typically set before conducting the hypothesis test, and common values are 0.05 or 0.01.

- Type II error: This occurs when you fail to reject a null hypothesis that is actually false. In other words, you fail to identify a significant effect or relationship when it actually exists. The probability of making a type II error is denoted by beta (β), and it is related to the power of the test (1 - β). Power is the probability of correctly rejecting a false null hypothesis.

Now, let's bring in the concept of p-value:

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming that the null hypothesis is true. It provides a measure of the strength of evidence against the null hypothesis.

The relationship between p-value, type I error, and type II error can be summarized as follows:

- If the p-value is less than the significance level (α), you reject the null hypothesis in favor of the alternative hypothesis. This means that you have evidence to conclude that there is a significant effect or relationship. In this case, the probability of making a type I error is less than or equal to α.

- If the p-value is greater than the significance level (α), you fail to reject the null hypothesis. This means that there is not enough evidence to conclude that there is a significant effect or relationship. In this case, the probability of making a type II error remains high, as you may be failing to identify a true effect or relationship.

In summary, the p-value of a hypothesis test is directly related to the probabilities of type I and type II errors. A smaller p-value increases the evidence against the null hypothesis and decreases the probability of a type I error. On the other hand, a larger p-value decreases the evidence against the null hypothesis and increases the probability of a type II error.