Statistics Sampling Distribution of Sample Proportions

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Today we are going to be talking about sampling distribution of sample proportions.0001

First thing, we are going to do is just introduce ourselves to the concept of sampling distribution of sample proportions.0006

This is just me this is not like everybody in statistics but I am going to call it SDOS for short.0014

We do not have to keep writing out sampling distribution of sample proportions.0019

Then we are going to go through some notation and then finally we are going to compare and contrast the SDOS to the SDOM.0024

They are both sampling distribution, but one is that the mean and the others of sample proportions.0034

We are going to compare and contrast the binomial distribution.0040

The probability distribution that we looked at a couple of lessons back with the SDOS.0045

What is this thing?0050

What is the SDOS?0054

First this concept is going to come into play whenever we collect some sort of categorical data.0056

For instance, we might ask a sample of citizen do you approve or disapprove of the president and0063

That would be a categorical response.0070

They are not saying I approve this much, but they are just saying I approve or disapprove.0073

At the end of that data collection what you get is not a mean or a median0079

but you get something like a proportion of citizens who believe the president is doing a good job.0086

Something like 43%, 64%, 29%.0091

Or another example might be proportion of students who plagiarized on a paper before.0097

Here we are getting proportions.0104

They are not means or medians.0107

They are just percentages of the entire sample.0109

Finally another one that is been talked about a lot these days is the proportion of people covered under healthcare.0112

When we collect this categorical data oftentimes we want to use that in order to estimate the proportion0119

of the population that actually is covered by health care or with plagiarized before or who believe the president is doing a good job.0129

We want to estimate the population level parameter.0137

However, samples are very variables.0142

Samples are variable and that means that our estimate would not always be very good.0147

It will be good sometimes, but it would not always be very good.0157

It would help us out if we knew the entire distribution of potential samples.0161

Would not it be handy?0175

That is called the sampling distribution.0179

And because we are not sampling and finding a mean, instead we are sampling0182

and finding a proportion it is called a sampling distribution of sample proportions.0187

This is the idea found here the entire distribution of potential samples, but once we get each sample what do we do to it?0196

We do not find the mean, we find the sample proportion and we plot those on a distribution.0206

Some things that are helpful for us to get straight.0213

When we talk about the population proportion that parameter we will just call it p.0220

When we talk about a sample proportion we are going to call it p hat.0226

We have seen that notation before when we talk about regression when we had y for the actual data but we had y hat for the predicted data.0232

You can think about it like this.0243

The real one from the world is going to be not have the hat.0245

This is sort of the truth that we are trying to find and this is how we are going to estimate that truth.0254

We are going to use this to estimate that but we want to know how good is our estimate?0262

Is it any good?0270

And is it reliable or not?0270

Think about it like this.0273

Here the distribution of the population is binary.0275

There is 1 or the other.0281

I am just going to draw the entire population as just a bar and pretend this bar had the value of 1.0, 100%.0282

Some proportion of this is p and the other is not p.0292

Some proportion of these people approve of the job he's doing as a president and the other proportion does not.0299

How should we represent that in a picture and algebraic form?0312

Well I'm just going to draw a line here and say this part is going to be my proportion p.0316

That is my proportion of those that agree that the president is doing a good job.0324

Then what would be but this little area here?0330

Well, how would we represent that algebraically?0335

That would simply be 1 – p because the whole thing is p.0338

This segment is p so this segment must be 1 – p.0344

When you add p + 1 -p what you get is 1.0350

Notice that we did not draw like a normal distribution or anything because it is not that people have different values.0355

It is not that some people are low, some people are high.0363

It is just yes or no?0365

Have you plagiarized or not?0367

Have you gone bungee jumping or not?0370

Are you covered by health care or not?0375

They are just these binary characteristics that we are interested in.0378

That is what the population is like.0383

Now from population we draw a sample of size n just like always.0385

We are always drawing sample size n.0402

When we look at that little sample of the population what does it look like?0405

Instead of the whole thing you drew a little sample, you drew a subset of those people0412

and presumably this little sample should most likely reflect the population that it came from.0421

These should be radically different from this.0430

It can be sometimes but for the most part this sample should reflect the population that it came from.0434

The entire sample this thing =1 and in this entire sample we have some probability p hat and that is the proportion in our sample that agree.0442

This would be represented by 1 – p hat those are the people in our sample that disagree.0466

You might be thinking how is this whole thing 1 and how is this whole thing 1 because this one looks smaller?0473

When we say 1 we are talking about proportions.0480

We are really think 100%.0484

When we are saying 1 here it represent 100% of the population.0486

When we day 1 down here we are saying 100% of the sample.0490