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## Statistics and probability

### Course: Statistics and probability > Unit 12

Lesson 2: Error probabilities and power- Introduction to Type I and Type II errors
- Type 1 errors
- Examples identifying Type I and Type II errors
- Type I vs Type II error
- Introduction to power in significance tests
- Examples thinking about power in significance tests
- Error probabilities and power
- Consequences of errors and significance

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# Examples identifying Type I and Type II errors

AP.STATS:

UNC‑5 (EU)

, UNC‑5.A (LO)

, UNC‑5.A.1 (EK)

, UNC‑5.A.2 (EK)

Examples identifying Type I and Type II errors.

## Want to join the conversation?

- I thought null hypotheses only have statements of equality, wouldn't statement of equality be enough for this example ?(6 votes)
- In example 2, if
`p`

is less than 0.40, you would still not want to build the cafeteria. After all, it could be the case that 30% or 10% or even 0% of the people are interested in the meal plan. If you were to set`H_0: p = 0.40`

, then you would ignore all these less than options, so we need the less than or equal sign.(2 votes)

- An interesting example this is. Suppose that building a cafeteria entails profits if more that 40 percent of the students make a purchase (interested = would purchase) a meal plan. Which is more serious

1) Type 1 error: lose the opportunity to make profits?

2) Type 2 error: bear the cost and the loss if a cafetria is built?(1 vote) - Apart from decreasing alpha how else can I lower the chances of making a type 1 error?(1 vote)
- Asking for a communication research methods class:

I was told the null hypothesis is always a statement of "no difference" in my statistics class? Is this true? In this example it seems as its phrased in a more complicated manner. But essentially the equal sign is still saying, there is no difference, right? I'm confused by how this question is framed.(1 vote) - I read that statistical significance means that the result didn't happen by chance, isn't random, therefore something else caused it (not the population we already have). Is this correct? If a p-value is statistically significant, does it mean it is smaller than alpha and we reject the null hypothesis?(0 votes)
- Yes, if p is less than the set alpha level you reject the null hypothesis. Thus you have evidence for your alternative hypothesis.(0 votes)

- When do we know the null hypothesis is true or false?(0 votes)
- When we are talking about type 1 error, we are assuming

null hypothesis is true.(0 votes)

## Video transcript

- [Instructor] We are told, "A large nationwide poll recently showed "an unemployment rate of
9% 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 9% in her town. "Let P represent the
unemployment rate in her town. "Here are the hypotheses she'll use." So, her null hypothesis is that, hey, 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 I 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 I error even is. This is a situation where we reject the null hypothesis,
even though it is true. Reject null hypothesis, even though, even though our null hypothesis is true. And in general, if you're
committing either a Type I or a Type II error, you're
doing the wrong thing, you're doing something that
somehow contradicts reality, even though you didn't intend to. And so, in this case, that would be rejecting the hypothesis that the unemployment
rate is 9% in this town, even though it actually
is 9% in this town, so let's see which of these
choices match up to that. "She concludes the
town's unemployment rate "is not 9% when it actually is." Yeah, in this situation, in order to conclude that
the unemployment rate is not 9%, she would have to reject the null hypothesis, even
though the null hypothesis is actually true, even
though the unemployment rate actually is 9%. 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 9% "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 9% "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 9% "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, this is an error. This is an error, but
this is a Type II error. So, one way to think about it, 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?" And the rejecting something that is true, that's Type I, and failing
to reject something that is false, that is Type II. And so, 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%, 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, while the
alternative hypothesis is that more than 40% are interested. "What would be the consequence "of a Type II 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 II error is, we just talked about it. So, failing, failing to reject, in this case, our null hypothesis, even though it is false. So, 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, is that you wouldn't
build another cafeteria 'cause you'd say, "Hey, no,
there's not that many people "who are interested in the meal plan," but you wouldn't, but,
actually there are a lot of people who are
interested in the meal plan, and so you probably wouldn'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 I error. Type I 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 I. "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. A and C are the consequences of a Type II and a Type
I error, respectively.