Question

probability question. hypothesis testing, we know that rejecting the null hypothesis when the null hypothesis is in fact true is called a type I error. The probability of a type I error is alpha. my question is, what is the probability of rejecting the null hypothesis when the null hypothesis is false. i dont see this in the book

Answer 1

You are looking, I think, for \(1-\beta\).

\(\beta\) is used to indicate the probability of the failure of rejecting the null hypothesis when the null hypothesis is false. I say that \(1-\beta\) is the probability of rejecting the null hypothesis when the null hypothesis is false on the basis of the following reasoning: You can, presumedly, only succeed or fail at rejecting the null hypothesis when the null hypothesis is false. Since \(\beta\) is some probability for the first condition, basic probability theory indicates that the probability of the other condition is \(1-\beta\). (A classic example of this reasoning is the following: If a coin can land only on heads or tails, then--if the probability of landing on heads is \(\frac{1}{2}\)---the probability of landing on tails is \(1-\frac{1}{2}\), or \(\frac{1}{2}\). In this case, we have that \(\beta=\frac{1}{2}\).)