6.02 CI for mean with known population sd

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来自 University of Amsterdam 的课程

统计基础

2057 个评分

University of Amsterdam

统计基础

2057 个评分

课程 3（共 5 门，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.

从本节课中

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.

6.01 Statistical inference3:56

6.02 CI for mean with known population sd5:44

6.03 CI for mean with unknown population sd7:37

与讲师见面

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

New parents, like me, usually lose a lot of sleep.

In fact, often new parents become very much like nursing zombies.

But despite my own zombieness, I still want to know the numbers.

To what extent do other new parents also lose sleep?

Suppose that to answer the question, I asked a sample of 60 new parents how much

sleeping hours they have lost after their first child was born.

Now imagine that the mean lost sleeping hours per night is 2.6 hours.

The standard deviation in the sample is 0.9 hours.

And suppose you also know the standard deviation in the population,

it is 1.1 hours.

In practice, it is very unlikely that you'll know the value of this parameter,

but for now, just suppose you do.

In this video I will show you how you can construct a confidence interval

based on the information from your sample and the population's standard deviation.

I'll explain how such a confidence interval should be interpreted.

And I will also show you why being a new dad could have

disastrous consequences for one's ability to.

Focus.

>> Focus.

To construct a confidence interval,

we make use of the sampling distribution of the mean.

We're dealing, after all, with a sample from a population.

We know that as long as our sample is sufficiently large,

the sampling distribution is normally distributed

with a mean that is equal to the population mean mu

and the standard deviation that's equal to the population standard deviation divided

by the square root of n.

We also know that the probability of finding a sample mean of less than

about two standard deviations from the mean is 0.95.

More precisely, if we look up the z scores which correspond to

this probability, we'll find values of minus 1.96 and 1.96.

This means that we have a 95% chance that our sample mean will fall

within 1.96 standard deviations of population mean mu.

This distance of 1.96 standard deviations is what we call the margin of error.

The margin of error tells us how accurately our sample mean x-bar is likely to

estimate a population mean mu.

Now, the formula of the 95% confidence interval is the following.

It is the point estimate or the sample mean plus and minus the margin of error,

which equals 1.96 standard deviations.

Note that we're dealing with the sampling distribution of the sample mean here.

So the standard deviation equals sigma divided by the square root of n.

Now pay close attention because this is a little complicated.

Suppose you draw a sample.

The mean of the sample is represented by this dot.

The lines here represent the margins of error on both sides of the mean.

Together they form a 95% confidence interval.

If the sample mean falls within the red area,

then the confidence interval contains the population mean mu.

That's the case here.

However if the sample mean does not fall within the red area,

the confidence interval does not contain population parameter mu.

That's the case here.

We're talking about a 95% confidence interval.

That means that the probability that the confidence interval of a randomly selected

sample contains the population parameter is 0.95.

The probability that it does not contain the population mean is 0.05.

In other words,

if we would draw an infinite number of samples from our population,

in 95% of the cases, our confidence interval would contain population mean mu.

Let's go back to our example.

The sample mean of lost sleeping hours is 2.6.

The population standard deviation is 1.1.

And the sample size is 60.

We now have all the necessary ingredients to compute our confidence interval.

This is the formula, x-bar plus and minus 1.96 times sigma x-bar.

We know that sigma x-bar equals sigma divided by the square root of n.

Let's first compute sigma x-bar.

That's 1.1 divided by the square root of 60.

That makes 0.142.

Now we compute the margin of error.

That's 1.96 times 0.142.

That's about 0.28.

The sample mean equals 2.6, so the 95% confidence interval is

the interval between 2.6 minus 0.28 and 2.6 plus 0.28,

which is the interval between 2.32 and 2.88.

We say that we have 95% confidence that this

interval contains the actual population mean.

More precisely, if we would draw an infinite number of samples with n equals

60 from our population, and for every sample,

we would compute the confidence interval with this margin of error,

in 95% of the cases, the population value would fall within the confidence interval.

Okay, if this situation of new parents would last for a year,

we would have 95% confidence that they would lose between 2.32 times 365 and

2.88 times 365 hours of sleep in that year.

That is, between 846.8 hours and 1,051.2 hours.

Or, in other words, between

35.3 and 43.8 full days.

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