Examples identifying Type I and Type II errors | AP Statistics | Khan Academy

We are told a large nationwide poll recently showed an unemployment rate of nine percent in the United States. The mayor of a local town wonders if this national result holds true for her town. So, she plans on taking a sample of her residents to see if the unemployment rate is significantly different than nine percent in her town. Let p represent the unemployment rate in her town.

Here are the hypotheses she'll use. Her null hypothesis is that the unemployment rate in her town is the same as for the country, and her alternative hypothesis is that it is not the same. Under which of the following conditions would the mayor commit a type 1 error?

So pause this video and see if you can figure it out on your own. Now let's work through this together.

So let's just remind ourselves what a type 1 error even is. This is a situation where we reject the null hypothesis even though it is true. Reject the null hypothesis even though our null hypothesis is true. In general, if you're committing either a type 1 or a type 2 error, you're doing the wrong thing. You're doing something that somehow contradicts reality, even though you didn't intend to. In this case, that would be rejecting the hypothesis that the unemployment rate is nine percent in this town, even though it actually is nine percent in this town.

So, let's see which of these choices match up to that. She concludes the town's unemployment rate is not nine percent when it actually is. Yeah, in this situation, in order to conclude that the unemployment rate is not nine percent, she would have to reject the null hypothesis even though the null hypothesis is actually true. Even though the unemployment rate actually is nine percent, so I'm liking this choice.

But let's read the other ones just to make sure. She concludes the town's unemployment rate is not nine percent when it actually is not. Well, this wouldn't be an error. If the null hypothesis isn't true, it's not a problem to reject it. So this one wouldn't be an error.

She concludes the town's unemployment rate is nine percent when it actually is. Well, once again, this would not be an error. This would be failing to reject the null hypothesis when the null hypothesis is actually true. Not an error.

Choice D: She concludes the town's unemployment rate is nine percent when it actually is not. So, this is a situation where she fails to reject the null hypothesis even though the null hypothesis is not true. So this one right over here, this one would actually be an error, but this is a type 2 error.

So one way to think about it is, first you say, okay, am I making an error? Am I rejecting something that's true, or am I failing to reject something that's false? Rejecting something that is true, that's type one. Failing to reject something that is false, that is type two.

With that in mind, let's do another example. A large university is curious if they should build another cafeteria. They plan to survey a sample of their students to see if there is strong evidence that the proportion interested in a meal plan is higher than 40 percent, in which case they will consider building a new cafeteria.

Let p represent the proportion of students interested in a meal plan. Here are the hypotheses they'll use. So the null hypothesis is that 40 or fewer of the students are interested in a meal plan. Well, the alternative hypothesis is that more than 40 are interested.

What would be the consequence of a type 2 error in this context? So once again, pause this video and try to answer this for yourself.

Okay, now let's do it together. Let's just remind ourselves what a type two error is. We just talked about it. Failing to reject the null hypothesis even though it is false.

This would be a scenario where this is false, which would mean that more than 40 actually do want a meal plan, but you fail to reject this. So what would happen is that you wouldn't build another cafeteria because you'd say, hey, you know, there's not that many people who are interested in the meal plan. But actually, there are a lot of people who are interested in the meal plan, so you probably won't have enough cafeteria space.

And so this says they don't consider building a new cafeteria when they should. Yeah, this is exactly right. They don't consider building a new cafeteria when they shouldn't. Well, this would just be a correct conclusion.

They consider building a new cafeteria when they shouldn't, and so this is a scenario where they do reject the null hypothesis even though the null hypothesis is true. So this right over here would be a type 1 error. Type 1 error because if they're considering building a new cafeteria, that means they rejected the null hypothesis even when they shouldn't. That means that the null hypothesis was true, so type 1.

They consider building a new cafeteria when they should. Well, once again, this wouldn't be an error at all. This would be a correct conclusion. This one and this one are correct conclusions.

This is the concept. A and C are the consequences of a type 2 and a type 1 error, respectively.