If I have an example as follows with 280 cases studied of a new test developed on animals:
| true state of the animal | |||
|---|---|---|---|
| non-infected by virus X | infected by virus X | ||
| non-infected by virus X | 131 | 15 | |
| results of the new test | infected by virus X | 9 | 125 |
If the null hypothesis is: the animal is not infected by virus X, how do I find the probability of the Type 2 error?
I know that Type 2 error is accepting a false null hypothesis so does that mean I am trying to find the probability that the animal is not infected by virus X then? I saw online that all examples of finding the Type 2 error are using the Normal distribution, but what do I do in this case?
Probability of type II error in this case is the proportion of infected animals which the test fails to detect as infected. In this case, total number of infected animals is $(15 + 125)$, while the number of infected animals the test declares as non-infected is $15$. So, the type II error is $15/(15 + 125) = 15/140$. As you see, the only information used is what available from the table, and nothing else. If one knows the underlying truth as given in the table (true state of the animal) then one needs no further assumption on the distribution, like assuming normal distribution.