Statistics Introduction to Sampling Distributions

Welcome to educator.com.0000

We are going to be talking about introducing the concept of sampling distribution.0001

Here is the roadmap we have been talking about probability distributions and that is what we call distributions of probabilities from discrete outcomes.0007

We are going on to sampling distributions and that is what we call distributions of outcomes that are continuous.0022

For now we can apply the same fundamental logic from probability distribution directly to sampling distributions.0030

We could treat them roughly similarly, but I do want to connect sampling and research methods to topics we have covered before with sampling distribution.0038

We are going to finally talk about actually generating sampling distributions.0049

First let us talk about the difference between probability and sampling distributions.0053

In probability distributions we are always looking at discrete outcomes, finite or uncountable numbers of outcomes.0060

For example in binomial distributions, if you have 10 trials, you have 0 to 10 different outcomes as possibilities and that is 11.0068

They are discrete, finite, accountable, no problem.0080

In probability distributions, what we are looking for is the probabilities of those discrete outcomes.0085

We get a list of all these probabilities and we can actually make a list because it is a finite number.0097

Let us talk about sampling distribution.0105

Sampling distributions are roughly the same idea.0109

You have the sample space and you want to know how likely each outcome is,0111

the probability of each outcome and the set of all those probabilities that is called the sampling distribution.0118

Here is the big difference, in standard discrete outcomes we are talking about continuous outcomes.0124

Before we ask what is the probability that 2 out of 4 people for random people that you pick from the United States have a bachelor’s degree.0136

Now, we might say if you pick 4 college students at random and we are not looking for the 2 anymore, that is discrete.0146

We are looking for things like what is the average GPA.0157

There is an infinite number of average GPA that you could potentially have.0164

In the way it is not finite anymore, but it is infinite number of outcomes and uncountable number of outcomes.0169

Now that is problematic because here we had like a list of all the different probabilities.0186

Can you list all these outcomes?0196

It is impossible, they are infinite.0199

By definition, not listable, not put in a table of all.0202

That is an issue with sampling distribution but sampling distributions get their power from other sources.0208

We do not have to worry as much about that but I do want to know that this is a big difference between probability distributions and sampling distributions.0216

But still we are going to be trying to find things like how we find the expected value of these distributions?0224

How do we find the probability of some outcome?0231

The same logic overall logic is going to apply for now.0233

I just go over the logic of the original probability distribution.0241

Basically we use probability to find this known population, we had known populations like things like fair coins, roll of two fair dice, or whatever something like that.0249

And from that we generate a probability distribution.0266

A whole bunch of different values for this random variable x and then the probabilities of those x.0271

Okay once we have that, then we have this sample and it is actually from an unknown population.0284

We get the sample but is it from that kind of population or from that population?0304

We do not know.0312

What we do is we take a sample and we compare it to the probability distribution and we look at is the sample very likely or very unlikely.0314

From that we judge whether these known and unknown population is similar to each other or not right.0325

Is it likely, unlikely?0333

That is roughly bad idea.0337

We went over a couple of specific examples, one of the ones that we went over in great detail0342

was the one with 2 dice where the random variable is the sum of the two dice.0352

The logic of sampling distribution is roughly similar, so we have some known population.0360

We generate a sampling distribution this time instead of a probability distribution we will talk about how to generate those later.0370

We get samples from unknown population and we compare it and we say is it likely or unlikely?0378

Same underlying logic.0386

The differences are going to be in the step how we generate is going to be different.0388

How we judge whether it is likely or unlikely this step is also going to be a little bit different in terms of the nitty-gritty like how we actually do it,0395

but the concept is the same.0407

We had this known stuff, we have this unknown stuff, we compare the unknown staff to the known stuff.0408

Let us go over that example that we knew really well.0416

The known population here was 2 fair dice that we brought and we generate it as probability distribution.0421

Here we have the probability distribution and the probability distribution where x the random variable is the sum of 2 dice.0441

We also generated all these probabilities for each sum.0454

We have the sum of the two dice, we have all the probabilities for each of those sums and each of these sums is discrete and countable.0459

There is 10, 11 of them right.0465

What we would do is let us say we rolled 2 dice and we do not know if they are fair dice or not some shady guy give it to us.0474

If we are going to roll such as 1-1, that is what our sample is then we can say okay what is the probability that x=2?0491

That is a pretty small probability, it looks like .025 or something.0510

Because it is a small probably we will say this is unlikely sample.0515

Let us say we got a sample that was something like 3-4.0523

We might compare that to this probability where x =7 and that is pretty likely there is like 16% chance.0533

We would say this is likely.0551

This sample, we would probably say if we got this it is likely that it came from fair dice.0556