Hi, this tutorial covers type I and type II errors. All right, let's start with an example. A local prison is trying a new program to reduce crimes committed by recently released prisoners. Nationally, the proportion of prisoners that return to jail on less than one year is 0.61 or 61%.
The local prison takes a random sample of 42 released prisoners and found that 16 returned to prison within a year. So if we look at the sample proportion there, it would be 16 out of 42. So based on the local prison sample, they claim their program is working. You have been hired to test whether or not the prison should continue to fund their program.
OK, so we're going to try to evaluate this and see if they should fund the program or not fund the program. All right, so let's start by defining p. So we're going to let p equal the true proportion of released prisoners from the local prison that return within a year. OK, so remember, p is our symbol for a population proportion. So we're going to assume that p is the population proportion for this local prison.
All right, so if we write some hypotheses, remember that our parameter for our null hypothesis needs to be-- we need a parameter first, and then the null hypothesis is always equality. So we want to-- what we're going to do is assume that the local prison has kind of a release proportion the same as it is nationally. So we're going to assume that the population proportion here is 0.61 or 61%.
Now, let's think about what the alternate hypothesis is. We want to show that the local prison's program works. So if it works, that means we will-- that it lowers kind of this proportion of prisoners that return. So what we want to do is show that p is less than 0.61. So that's going to be our alternate hypothesis or our alternative hypothesis.
So when we run the tests, remember, we have two conclusions. We're either going to reject the null hypothesis. If we reject the null hypothesis, we have evidence for the alternate hypothesis. So if we have evidence for the alternate, that means our testable will tell us that the program is working.
If we fail to reject the null hypothesis-- that's the second possible conclusion that we could make. If we fail to reject the null hypothesis, we have no evidence for the alternate, so in that case, the test would show that we probably-- the program is probably not working. So since hypothesis tests are about considering the probability of a result from a sample that differs statistically-- in a statistically significant way from the assumed null values, errors in the outcome of the hypothesis test can occur. So basically, it's possible that our test is wrong, OK? So what we want to do is look at the conclusion and kind of evaluate it.
So remember, our two possible conclusions are reject the null hypothesis or fail to reject the null hypothesis. Then based on the conclusion that we make, we can test that against whether the null is actually true or whether it's false. So there's really four outcomes here.
So we want to, first of all, think about what the good outcomes are. So if we start with reject the null hypothesis, do we want to reject a true null hypothesis or reject a false null hypothesis? Well, of course, we want to reject something that's false, so this is a good outcome.
Now, if we look at fail to reject the null hypothesis, we want to fail-- do we want to fail to reject something that's true or something that's false? Well, we want to fail to reject something that's true. So this is our other good outcome here. Which makes these two conclusions errors. So these are the negative-- kind of the negative evaluations, OK?
So this is an error here, and this is an error here. Now, we have specific names for each of these types of errors, and they are just simply a type I error. So rejecting a true null hypothesis is called a type I error. And failing to reject a false null hypothesis is what's called a type II error.
So again, these are the four different possibilities. Two of these are errors, two of them are good. All right, so just to kind of formalize our different types of errors, a type I error is an error that occurs when a true null hypothesis is rejected. A type II error is an error that occurs when a false null hypothesis is not rejected. So then depending on the situation, one type of error may be easier to deal with, or one type of error just might be worse.
So let's take a look at, going back to the present context, and look at the implications of the error. So if we think about a type I error, a type I error would be to reject a true null hypothesis. So if the null hypothesis is true-- remember, here were our hypotheses, that the null was p equals 0.61, the alternate is that p is less than 0.61.
So if we reject a true null hypothesis, if the null hypothesis is true, this is basically saying that the program really isn't working, but if we reject that notion, we're saying that it is working. So if we make a type I error, what we're going to be doing is we're going to fund an ineffective program. OK, so we're going to be dumping money and resources into a program that's not working. So the prisoners are still coming back at the same rate, but we're wasting a lot of time and resources.
Now, if we're thinking about a type II error, we're failing to reject a false null hypothesis. So if the null hypothesis is false, it means the program is actually working, but the test says that we're going to fail to reject it, so that means that we're not going to support the alternate hypothesis. So type I-- the implications of a type II error here is to not fund an effective program. So kind of the consequences here is that a lot of the prisoners won't be able to have access to those rehabilitation services, so there might be more crimes that are committed, the prisons might be become overcrowded, things like that. So these are kind of the two different implications.
It's kind of hard in this case to say which one is worse. I mean, I think from kind of a worldly perspective, this is going to be worse, because we're not going to be helping people that could be helped through this program. If you are the accountant for the prison, you might be a little more wary of the type I error, because you'd be wary of kind of wasting that money. But you can see that the two different types of errors have different implications, and a lot of times, it's important to consider those implications before making your conclusion. All right, this has been your tutorial on type I and type II errors. Thanks for watching.