7.04 Step-by-step plan

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Del curso dictado por University of Amsterdam

Basic Statistics

2068 calificaciones

University of Amsterdam

Basic Statistics

2068 calificaciones

Curso 3 de 5 en SpecializationMethods and Statistics in Social Sciences

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.

De la lección

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.

7.04 Step-by-step plan7:11

7.05 Significance test and confidence interval4:43

Conoce a los instructores

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

Compare the following two expectations.

1, you expect that more than half of all certified divers in America have more than

35 hours of diving experience.

And 2, the mean number of hours of diving experience of

all certified American divers is more than 35 hours.

At first sight, these two expectations look very similar.

But, in the first case, you're dealing with proportion.

You're interested in the proportion of the certified divers with more than 35

hours of diving experience.

And, in the second case, with the mean.

You want to know the mean number of hours.

So, when conducting a significance test,

you should think very carefully about your approach.

In this video, I will guide you through a step-by-step plan.

Suppose you've asked a simple random sample of 500 certified divers how many

hours of experience they have.

Suppose you find that when it comes to your sample of 500,

a proportion of 0.57 has more than 35 hours of diving experience,

and a mean number of hours of experience is 35.5.

The standard deviation is 8.

In our sample,

the distribution of the variable hours of diving experience is approximately normal.

The first step is, assess if you're dealing with a proportion or with a mean.

We already discussed that.

In the first case, we're dealing with a proportion, and

in the second case with a mean.

Step 2, formulate your hypotheses.

In the case of a proportion, your null hypothesis looks like this.

Pi equals pi 0.

In the case of a mean, it is mu equals mu 0.

We can have three types of alternative hypotheses.

Pi or mu does not equal pi or mu 0, here you do a two-sided test,

pi or mu is larger than pi or mu 0, here you do a one-sided right tail test,

and pi or mu is smaller than pi or mu 0, here you do a one-sided left tailed test.

Our null hypotheses are, pi equals 0.5 and mu equals 35.

The alternative hypotheses are, pi is larger than 0.5, and mu is larger than 35.

We thus have to conduct right tailed tests.

Step 3, check if your assumptions are met.

In both cases, randomization is of essential importance.

Your data must have been collected by means of a random sample, or

a randomized experiment.

In the case of a proportion, an additional assumption is that the product of your

sample size in the population proportion, according to your null hypothesis, and

the product of your sample size

and 1 minus the population proportion, according to your null hypothesis,

must be equal to or larger than 15.

If you're dealing with the mean,

your population distribution should be approximately normal.

However, in practice, this is only of importance if your sample size is small

and you do a one-tailed test.

Regarding our example, all assumptions are met.

We have a simple random sample, our n is large enough, and the sample distribution

of the variable hours of diving experience is approximately normal.

Step 4, determine your significance level alpha.

The most common significance level is 0.05.

Our test will be based on an alpha of 0.05.

Step 5, compute your test statistic.

In the case of a proportion, this is the relevant formula, and

in the case of the mean, this is the formula we use.

Note that, in the case of a proportion, we use the z distribution, and

in the case of a mean, the t distribution.

In our examples we get values of 0.57 minus 0.5 divided by the square root of

0.5 times 0.5 divided by 500, that equals 3.13,

and 35.5 minus 35 divided by 8 divided by

the square root of 500. That equals 1.40.

Step 6, draw the relevant sampling distribution and

show the null hypothesis value and the test statistic

supplemented with the rejection region and the corresponding critical value.

In the case of our proportion, this is the distribution.

The critical value corresponding to our right tailed test with a significance level

of 0.05 can be found in the z table.

It is 1.64, so it looks like this.

In the case of our mean, this is the distribution.

We look up the critical value in the t table. That's 1.66. That looks like this.

Step 7, assess if your test statistic is located in the rejection region or not.

In our first example, this is the case.

Our test statistic, which is 3.15,

is further removed from the mean of the sampling distribution than

the critical value, which is 1.64. But in the second example, this is not the case.

Our test statistic, which is 1.40,

lies close to the mean than our critical value, which is 1.66.

Note that, in the case of a proportion,

you can also look at the p value in a z table.

The p value is the probability that our test statistic takes a value like

the observed test statistic, or even more extreme, given the null hypothesis.

Our test statistic is 3.13, the corresponding p value is 0.0009.

This value is much smaller than 0.05,

which also means that our test statistic is located in the rejection region.

Statistical packages can also give you your exact p value when you use

a t distribution.

It is impossible, however, to look it up in the t table.

The reason is that t table is not specific enough.

Since the shape of the distribution changes with the degrees of freedom,

we would need a separate table for each number of degrees of freedom.

That's why, for each number of degrees of freedom,

only the most important t values are listed, that is, those that correspond

to the most commonly used significance levels for one and two tailed tests.

The next step, number 8, decide if the null hypothesis should be rejected.

The answer is yes when it comes to our proportion example, and

no regarding our mean example.

Finally, step 9, interpret your findings substantively.

We can conclude that more than half of all certified divers in America

have more than 35 hours of diving experience.

However, we can not conclude that the mean number of hours of diving experience

is more than 35 hours.

Before I conclude, one warning, the decision not to reject your null

hypothesis does not imply that you accept your null hypothesis.

In our second example, we did not reject the null hypothesis that the mean number of

hours of diving experience equals 35.

But that doesn't mean that we can conclude that the mean number of hours of diving

experience is exactly 35 hours.

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