6.05 Confidence levels

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來自 University of Amsterdam 的課程

统计基础

1714 個評分

University of Amsterdam

统计基础

1714 個評分

課程 3（共 5 門，Specialization 社会科学的方法和统计）

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.

從本節課中

Confidence Intervals

We can distinguish two types of statistical inference methods. We can: (1) estimate population parameters; and (2) test hypotheses about these parameters. In this module we’ll talk about the first type of inferential statistics: estimation by means of a confidence interval. A confidence interval is a range of numbers, which, most likely, contains the actual population value. The probability that the interval actually contains the population value is what we call the confidence level. In this module we’ll show you how you can construct confidence intervals for means and proportions and how you should interpret them. We’ll also pay attention to how you can decide how large your sample size should be.

6.04 CI for proportion5:41

6.05 Confidence levels6:51

與講師見面

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

The 95% confidence interval,

tells us that we can be 95% confident that our point estimate,

which could be a mean or a proportion, falls within our confidence interval.

Or in other words, it tells us that if we would draw an infinite number of samples,

similar to our extra sample and for every sample,

we would compute a 95% confidence interval with a similar margin of error.

In 95% of the cases,

the population value would fall within this confidence interval.

This, of course, also means that in 5% of the cases, this method

will produce an interval that does not contain the actual population parameter.

If you would like to reduce the chance of an incorrect inference, you could go for

a larger confidence interval, such as, for instance, 99%.

In this video, I will tell you how you can change your confidence level.

And what the consequences are, of doing so.

Imagine you asked a sample of 100 new parents,

if the babies like to answer nature's call, during the diaper changing process.

17% reported that, this is the case.

Our sample, proportion p, thus equals 0.17.

The formula to compute a 95% confidence interval for a proportion is, p plus and

minus the z score for the 95% confidence level, times the standard error,

which equals the square root of p, multiplied with 1, minus p divided by n.

You can look up a z score for a 95% confidence level, in the z table.

Look at this standard normal distribution here.

When you have a 0.95 probability, that your value falls within z standard

errors from the mean, that means, that 0.025 probability falls in the two tails.

If you look up the z scores, which are displayed here in the z table,

we find values of plus or minus 1.96.

You can see that here.

We can now easily compute the interval, that's 0.17 plus and

minus 1.96 times the standard error which is the square root of 0.17,

times 0.83, divided by 100.

This leads to a confidence interval with the end point 0.10 and 0.24.

You can now imagine, that it is not so

difficult to construct intervals with other confidence levels.

Let's first look at the 99% confidence interval.

This is the formula.

p plus or

minus the z score for the 99% confidence level, times the standard error.

The only difference is, the different z score.

Look at this standard normal distribution.

When you have a 0.99 probability that your value falls within z standard errors

from the mean, that means, that 0.005 probability falls in the two tails.

If we look up the z scores, which are indicated here.

We find values of plus and minus 2.58.

You can see that here.

We can now compute the interval.

That is 0.17, plus and minus 2.58, times the standard error.

Which was 0.038.

This leads to a confidence interval with the endpoints 0.07 and 0.27.

For the 90% confidence level, we find a z score of 1.645.

This leads to a confidence interval of 0.17, plus and

minus 1.645, times 0.038.

That makes a confidence interval with the endpoints of 0.11 and 0.23.

I have here displayed confidence intervals graphically.

You can see that a higher confidence level leads to a wider confidence interval.

In other words, the more confident we are that we draw a correct inference,

the larger of margin of error.

That means, that we have to compromise between confidence and precision.

As one gets better, the other gets worse.

We never settle for a 100% confidence interval, because the margin of error then

is far too large, which means that our conclusions are not very informative.

In most cases, the 95% confidence interval is used.

We can also use other confidence intervals when we construct a confidence interval

to estimate a population mean.

Suppose, we've asked a sample of 30 new parents in Amsterdam,

how much hours of sleep they've lost after the first child was born.

The mean is 2.6 hours per night.

And the standard deviation is 0.9 hours per night.

This is the formula we use, to construct a 95% confidence interval.

x-bar, plus and minus the t score for the 95% confidence level,

times the standard error.

Which equals the sample standard deviation,

divided by the square root of the sample size.

Now, what is the t score for 95% confidence level?

That's dependent on the degrees of freedom, which equals n minus 1.

That is, 30 minus 1 is 29.

In the t table, we should look in the column of the 95% confidence level and

in the row of 29 degrees of freedom.

That gives a t score of 2.045.

The confidence interval becomes 2.6 plus and minus 2.045, times 0.9,

divided by the square root of 30.

That gives an interval from 2.26 to 2.94.

If we would want to construct an interval with a confidence level of 99%,

we simply replace the t score for the 95% level with the t score for the 99% level.

You can look it up in the table, it's 2.756.

The confidence interval is 2.6 plus and minus 2.756,

times 0.9, divided by the square root of 30.

That leads to an interval from 2.15 to 3.05.

You can also easily do that for other confidence levels.

Let me conclude this video by giving you a step by step plan for

constructing a confidence interval.

First, decide which confidence level you want to use.

For instance, do you settle for the regular 95% level?

Or do you want to be more confident and less precise?

Or more precise and less confident?

Second, decide if you're dealing with a proportion or a mean.

If you're interested in a proportion, you work with the z distribution, and

if you're interested in a mean, you have to use the t distribution.

So, in the case of a proportion, you look up the relevant z score

and in the case of a mean, you look up the relevant t score.

Don't forget, that if you're interested in a mean,

you should also compute the degrees of freedom which is equal to n minus 1.

Third, compute the endpoints of the confidence interval.

And finally, four, interpret the results substantively.

That's it.

If you're not 95% confident now that you can construct a confidence interval

yourself, rewatch the last couple of videos.

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