7.03 Test about mean

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En provenance du cours de University of Amsterdam

Basic Statistics

1705 notes

University of Amsterdam

Basic Statistics

1705 notes

Cours 3 sur 5 dans Specialization Methods 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.

À partir de la leçon

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.

7.01 Hypotheses5:20

7.02 Test about proportion7:37

7.03 Test about mean4:46

Rencontrer les enseignants

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

This video is about significance tests about a population mean.

How long can scuba divers stay underwater?

Well, that depends on the size of their oxygen tank, their experience,

the depths of the dive and many more things.

Suppose you have very good reasons to expect that

experienced American divers who dive with an average tank at an average depth,

will stay underwater for more than 60 minutes.

Suppose you have also approached 100 experienced American scuba divers and measured how

long they could stay underwater with an average tank at an average depths.

You find that the mean time these divers spent underwater is 62 minutes.

The standard deviation is five minutes.

You expect that the divers will stay underwater for more than 60 minutes.

That leads to the following null hypothesis.

Mu equals 60.

The alternative hypothesis is mu is larger than 60.

We conduct a significance test about a population mean like this.

We assess if it's likely that the sample we have collected, actually comes from

a population with a mean that equals the value formulated in the null hypothesis.

So this is the distribution we're interested in.

It is the sampling distribution of the sampled mean, with a mean of 60,

which is the null hypothesis value.

How likely is a sampled mean of 62 if the population mean is 60?

To answer that question, we compute a test statistic.

That's the number of standard errors the sample mean is removed from the population

mean according to the null hypotheses.

You might remember that to compute the standard error, we need to know the value

of the population standard deviation. But because we don't know that value,

we have to estimate it using the sample standard deviation.

Because this implies that we introduced extra error,

we employ the t distribution instead of the z distribution.

Our test statistic can be computed with the following formula.

It is the sample mean minus the null hypothesis mean

divided by the standard error of the sample mean.The standard error equals

the sample standard deviation divided by the square root of the sample size.

Let's first compute the standard error.

That's 5 divided by the square root of 100, that's 0.5.

That makes 62 minus 60 divided by 0.5 equals 4.

Our test statistic has a t score of 4.

Is that enough proof to reject the null hypothesis?

Well, that depends on the significance level.

Let's employ the most common significance level of alpha equals 0.05.

We do a one-tail test, so this is where our rejection region is.

The critical value here can be found in a t table, it is 1.67.

Note that we look at 60 degrees of freedom because we have 99 degrees of freedom, and

60 is the closest lower value in the table.

We look at t 90% because we want a cumulative probability

of 0.05 in the right tail of the distribution.

You need to remember that t 90% score stands for the confidence level of 90%,

which assumes that we have 10% in both tails of the distribution together.

That means that we have to take into account that,

there's also 0.05 in the left tail.

This is the result.

Our test statistic falls within the rejection region, which means that we

reject the null hypothesis that the population mean is 60 minutes.

We can conclude that on average, experienced American divers

with an average tank at an average depth stay underwater for more than 60 minutes.

What if our expectation is not that experienced divers who dive

with an average tank at an average depth stay underwater more than 60 minutes, but

that they stay underwater either more or less than 60 minutes.

Our alternative hypothesis then becomes mu is not 60.

Now, we have to do a two-tailed test.

Say we set the significance level at 0.01.

Our sampling distribution then looks like this.

We have a cumulative probability of 0.005 here and here.

If we look up the critical values we find -2.66 and 2.66.

Our test statistic is 4.

So we still reject the null hypothesis and

conclude that our finding is statistically significant.

Because we did a two tail test our substantive conclusion now is that

the mean time experienced American divers with an average

tank at an average depth stay underwater is not 60 minutes.

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