What is the relationship between probability of type 1 error and probability of type 2 error?

A) As α increases, β decreases

B) As α decreases, β decreases

C) The relationship depends on the context of the hypothesis test

D) Both α and β are independent of one anothe

A) As α increases, β decreases

Step-by-step explanation:

I am assuming that the probabilities of both kind of error are positive, otherwise there is no reason to make an hypothesis test.

Both events arent possible simultaneously, so they cant be independent because the probability of both kind of errors is greater than 0.

If the p-value increases, then by definition, the probability of type 1 error increases, in this case, you are more likely to reject the null hypothesis, therefore you are less likely to make the mistake of not rejecting the null hypothesis when you have to. Thus, the probability of type 2 error decreases.

On the other way around, if you are less likely to reject the null hypothesis (hence, the probability of type 1 error decreases), then you will be more likely to not reject the null hypothesis even on weird scenarios, therefore the probability of type 2 error increases.

There is no formula to determine the relatioship between both errors, but the errors are related: when the probability of type 1 error increases, then the probability of type 2 error decreases. The correct answer is A)