6.04 CI for proportion

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Из курса от партнера University of Amsterdam

Basic Statistics

2059 оценки

University of Amsterdam

Basic Statistics

2059 оценки

Курс 3 из 5 — 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.

Из урока

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

Познакомьтесь с преподавателями

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

In the last couple of weeks I have learned that newborn babies like to poo.

They like to poo a lot.

And it happens to be the case that my daughter Lois especially likes to poo

under specific circumstances.

That is, precisely when I'm changing her diaper.

Seriously, once she decided to answer nature's call six times.

I repeat, six times during the diaper changing process.

Is that normal?

I'm not sure, but I know how I could check it.

Suppose that I asked a simple random sample of 100 new parents

if their baby likes to defecate while his or her diaper is being changed.

In this video, I'll tell you how we can, on the basis of such a study,

construct a confidence interval to estimate a population proportion.

Say that 17% of my 100 respondents reported their

babies like to poo while the diapers is being changed.

83% reported their babies don't do that.

We thus have a proportion of 0.17,

whose babies like to poo while the diaper is being changed.

When we construct a confidence interval for

a proportion, we employ the sampling distribution of the sample proportion.

We know that, as long as the sample is large enough, this sampling distribution

is normally distributed with a mean that is equal to the population proportion, pi.

And a standard deviation that is equal to the square root of pi

multiplied with one minus pi divided by n.

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

about two standard deviations of the mean, which is the population proportion,

is 0.95.

More precisely, if we look up the z score which corresponds to this probability,

we'll find a value of 1.96.

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

within 1.96 standard deviations of our population proportion.

This is what we call the margin of error.

The formula with which we can compute a 95% confidence interval looks like this.

p plus and minus 1.96 times the standard deviation of

the sampling distribution of the sample proportion.

1.96 is the z score that corresponds to the 95% confidence level.

So we could also write, p plus and

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

deviation of the sampling distribution of the sample proportion.

We're talking about a 95% confidence level here, that means that we can say that

if we would draw an infinite number of samples from a population in 95% of

the cases, our confidence interval would contain population proportion pi.

However, as you might have noticed,

we don't know the value of population proportion pi.

So it is impossible to compute a standard deviation of the sampling distribution of

the sample proportion.

We therefore substitute the population parameter pi with an estimate, and

this estimate is our sample statistic, P.

This leads to the following formula.

p plus and minus the z score for the 95% confidence level times the estimated

standard deviation of the sample distribution of the sample proportion.

Just like when we constructed the confidence interval for a mean, we call

this estimated standard deviation of the sampling distribution the standard error.

In contrast with the confidence interval for a mean, when it comes to constructing

a confidence interval for a proportion, we don't make use of the t distribution.

And just stick with the standard normal distribution.

However, your data need to satisfy one essential assumption.

You should have at least 15 successes and 15 failures.

In other words, n times p and

n times 1 minus p need to be larger than or equal to 15.

If this is not the case,

you cannot compute a confidence interval on the basis of the discussed formula.

Okay, let's go back to our example.

We have a proportion of 0.17 that

reports that the baby poos while the diaper is being changed.

This is the formula we use.

Let's first compute the standard error.

It's the square root of 0.17 times 0.83 divided by 100.

That makes about 0.038.

The margin of error then is 1.96 times 0.038.

That's about 0.07.

0.17 minus 0.07 equals 0.10 and

0.17 plus 0.07 equals 0.24.

So our confidence interval ranges from 0.10 to 0.24.

This means that we can be 95% confident that the population proportion falls

between 0.10 and 0.24.

Or, in other words, if we would draw an infinite

number of samples with n equals 100 from a population, and for

every sample we would compute confidence intervals with this margin of error,

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

This 95% confidence interval demonstrates that most babies

don't like to poo while their diaper is being changed.

On the other hand, it's not that exceptional if they do poo.

We can be 95% confident that between 10 and

24% does poo during the changing process.

Thank God, nothing is wrong with my little baby, Lois.

In fact, my theory is that my daughter's pooing habits are good news.

That might well indicate that she will be toilet trained soon.

After all, she doesn't like to poo in the diaper.

Now, let's hope that later on, she does like to poo in the toilet.

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