How do you calculate the probability of a type II error?

[IntegrateMe]

I'm finding Type I and Type II very difficult to understand. We received a packet today that had a type II probability problem and i had no idea what they were explaining. Can anyone explain to me how we find the probability of a type II error?

 

[Mark44]

I don't remember whether there is a way to calculate this type of probability, and I'm not sure there is a way to do so. In case you don't understand these errors, here's a refresher.

In inferential statistics you are presented with two choices, usually called the null hypothesis and the alternate hypothesis. The goal in these two problems is to use some statistical data to decide to reject the null hypothesis (and thereby accept the alternate hypothesis) or fail to reject the null hypothesis (and thereby accept it).

There are four possibilities based on what you decide:

1. You reject the null hypothesis when it is in fact false.
2. You fail to reject the null hypothesis when it is false.
3. You reject the null hypothesis when it is true.
4. You fail to reject the null hypothesis when it is true.

The 1st and 4th possibilities represent correct choices. The 2nd and 3rd represent incorrect choices.

An analogy with a criminal court case might be helpful. Consider a jury trial that aims to establish the guilt or nonguilt of a person accused of a crime. There are four possible outcomes based on a theoretical ability to see into the accused person's mind to know whether he or she actually did the crime.
These outcomes are laid out to be parallel to the four outcomes listed above, where "not guilty" corresponds to the null hypothesis. In all cases, I'm assuming we have some all-knowing power to be able to determine whether a person committed a crime or not.

1. The person is convicted, and actually committed the crime.
2. The person is acquitted but actually did commit the crime. This is an error.
3. The person is convicted, but did not perpetrate the crime. This also is an error.
4. The person is acquitted, and did not perpetrate the crime.

Again, the second and third possibilities are errors. I forget which one is Type I and which is Type II.

We can eliminate errors of type #2 (in list above), by convicting everyone who comes to court, but that drastically increases type #3 errors. We have fewer criminals running around the streets, but at the cost of locking up innocent people. Likewise, we can eliminate type #3 errors by acquitting everyone who comes to court, but this increases the type #2 errors. We have fewer wrongly convicted people in jail, but at the cost of more criminals running the streets.

Hope that helps.