7.06 Type I and Type II errors

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Do curso por University of Amsterdam

Basic Statistics

1795 classificações

University of Amsterdam

Basic Statistics

1795 classificações

Curso 3 de 5 em Specialization Métodos e Estatística Aplicados às Ciências Sociais

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

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Significance Tests

In this module we’ll talk about statistical hypotheses. They form the main ingredients of the method of significance testing. An hypothesis is nothing more than an expectation about a population. When we conduct a significance test, we use (just like when we construct a confidence interval) sample data to draw inferences about population parameters. The significance test is, therefore, also a method of inferential statistics. We’ll show that each significance test is based on two hypotheses: the null hypothesis and the alternative hypothesis. When you do a significance test, you assume that the null hypothesis is true unless your data provide strong evidence against it. We’ll show you how you can conduct a significance test about a mean and how you can conduct a test about a proportion. We’ll also demonstrate that significance tests and confidence intervals are closely related. We conclude the module by arguing that you can make right and wrong decisions while doing a test. Wrong decisions are referred to as Type I and Type II errors.

7.06 Type I and Type II errors4:59

7.07 Example4:42

Conheça os instrutores

Matthijs Rooduijn
Dr.
Department of Political Science
Emiel van Loon
Assistant Professor
Institute for Biodiversity and Ecosystem Dynamics

Think about a courtroom trial.

The point of departure is that the accused is innocent.

The prosecutor tries to convince the jury or judge that the accused is guilty.

The burden of proof to convince jury or judge is on the prosecutor.

The defendant is found guilty only if the prosecutor presents strong evidence

against the defendant's presumed innocence.

In a trial, there are four possible outcomes.

First, the defendant could be convicted while he or she is indeed guilty.

That's a correct decision.

Second, the accused could be judged not guilty, while he or

she is indeed innocent.

That's a correct decision, too.

Third, however,

it's also possible that the defendant is convicted who is actually innocent.

That's a wrong decision.

And fourth, an accused who is, in fact, guilty could also be set free.

That's a wrong decision, too.

This is exactly what might also happen if we conduct a significance test.

The point of departure that the accused is innocent is analogous

to the null hypothesis being true, and a situation where the defendant is guilty

is the equivalent of the null hypothesis being false.

Convicting the defendant is analogous to rejecting the null hypothesis, and

setting him or her free is the equivalent of not rejecting the null hypothesis.

This leads to four possible scenarios.

In two of them, you make a correct decision.

This is the case if the null hypothesis is true,

and you don't reject it, or if the null hypothesis is false, and

you do reject it, but it is also possible that you make a wrong decision.

This is the case if the null hypothesis is true, but you decide to reject it, or

if the null hypothesis is false, but you decide not to reject it.

The first wrong decision is what we call a Type I error, or a false positive.

Aand the second one is what we call a Type II error, or a false negative.

Let me give you an example.

Imagine your null hypothesis is that, in the American population of certified

scuba divers, 50% have more than 35 hours of diving experience.

In other words, pi equals 0.5.

The alternative hypothesis is that it is another percentage.

In other words, pi is not 0.5.

You ask a simple random sample of 500 American divers

how much diving experience they have, and

you find that a proportion of 0.56 has more than 35 hours of diving experience.

Now, suppose that your null hypothesis is actually true.

A Type I error occurs if you decide on the basis of your sample data

to reject the true null hypothesis.

If the null hypothesis is true, the sampling distribution looks like this.

If your significance level, alpha, is equal to 0.05,

this is your rejection region.

In the z table, you can find the z scores of -1.96 and

1.96 from your critical values.

Now, if your test statistic falls within the rejection region and is,

in other words,

a value more extreme than the critical values, you reject the null hypothesis.

The probability that this happens is 0.025 plus 0.025 equals 0.05.

This means that the probability of a Type I error

is equal to the significance level, or, in other words,

the P of a Type I error given that the null hypothesis is true equals alpha.

It might, therefore, seem to be tempting to just decrease the significance level.

This is, however, not necessarily a good idea.

If you decrease the probability of rejecting the null hypothesis

while it is actually true, you increase the probability of not rejecting a null

hypothesis that is actually false.

In other words, by decreasing alpha, you decrease the probability of making

a Type I error, but you increase the probability of making a Type II error.

The probability of making a Type II error is what we call beta.

It's complicated to compute beta.

It depends on various factors, such as the value of alpha, the sample size, and

the true value of the parameter.

For that reason, we won't compute the value of beta here, but

it is important that you realize that, when we try to decrease the probability of

one type of error, the probability of the other type increases.

When the null hypothesis is false,

and you are conducting a test, you want the power of the test to be high.

The power of a test is the probability of rejecting the null hypothesis,

given that it is false, or, in other words,

the power of a test equals 1 minus the probability of a Type II error.

That's the same as 1 minus beta.

Why is power so important?

Well, before you've conducted a study,

it can help you determine how many participants you need.

After you've conducted a study, it can help you, for

instance, to make sense of results that are not statistically significant.

One final note, in practice,

you will never know if a decision was correct or not.

The only thing we can do is control the probability of making an incorrect

decision.

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