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Lecture Comments (3)

1 answer

Last reply by: Professor Son
Mon Jul 11, 2016 3:56 PM

Post by Martin Lau on June 25, 2014

Hi Dr. Son,

Could you please explain:
why, for binomial distribution,
as n increases, standard
deviation increases, not decreases?

Thank you in advance.

0 answers

Post by Alexandra Vazquez on December 4, 2012

you're a lifesaver. best teacher ever.

Sampling Distribution of Sample Proportions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Roadmap 0:06
    • Roadmap
  • Intro to Sampling Distribution of Sample Proportions (SDoSP) 0:51
    • Categorical Data (Examples)
    • Wish to Estimate Proportion of Population from Sample…
  • Notation 3:34
    • Population Proportion and Sample Proportion Notations
  • What's the Difference? 9:19
    • SDoM vs. SDoSP: Type of Data
    • SDoM vs. SDoSP: Shape
    • SDoM vs. SDoSP: Center
    • SDoM vs. SDoSP: Spread
  • Binomial Distribution vs. Sampling Distribution of Sample Proportions 19:14
    • Binomial Distribution vs. SDoSP: Type of Data
    • Binomial Distribution vs. SDoSP: Shape
    • Binomial Distribution vs. SDoSP: Center
    • Binomial Distribution vs. SDoSP: Spread
  • Example 1: Sampling Distribution of Sample Proportions 26:07
  • Example 2: Sampling Distribution of Sample Proportions 37:58
  • Example 3: Sampling Distribution of Sample Proportions 44:42
  • Example 4: Sampling Distribution of Sample Proportions 45:57

Transcription: Sampling Distribution of Sample Proportions

Hi and welcome to

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

That is the distinction we want to make.0496

Once you do this then you get this p hat.0499

And once you have that p hat then you can plot it on a sampling distribution.0504

Here is what the sampling distribution of sample portions looks like.0512

The lower bound and upper bound on this have to be 0 and 1.0516

You can never have a p hat that is less than 0 and you can never have a p hat that is greater than 100%.0522

You are inevitably stuck between 0 and 1.0530

Those are the only sample proportions you could possibly get.0534

Whatever we get here we plot here.0537

Soon we will build up a sampling distribution of sample proportions.0540

Whatever it looks like that will be our sampling distribution of sample portions.0544

Let us contrast the SDOM versus the SDOS.0556

In fact we are going to learn more about the SDOS by learning about the SDOM and how it relates.0565

There is one key difference between these two and that is the biggest thing you really need to keep in mind.0573

When we are talking about the SDOM we are finding a mean.0579

You cannot find a mean between agree and disagree.0582

Those are categorical data.0585

Here is what we do know is we need the data where you can find the mean and those data are continuous data.0589

Sometimes these are also called measurement data because you actually got this by measuring something.0599

When you have continuous data for instance how many miles do you drive per day?0609

Getting a sample of you know what the average number of miles people in California drive each day?0619

That would be a continuous measure if you get data like that you can actually average it together.0624

But if we ask the question like do you drive every day?0633

Yes or no?0638

That would be categorical data.0639

The type of data we are talking about here happens to be binary because it is either you are in one category or you are in the other.0642

There is not like three categories.0656

It is like there might be you agree with the president or you disagree with the president or you feel neutral.0659

What we would do in order to look at it as SDOS is to lump people together.0666

It might be agree versus disagree or do not care.0674

We might lump those two people together to just call them not agrees.0678

The shape of the SDOM what is nice about it is that as n increases, what happens to the shape?0683

The shape approximates normal.0694

As our sample size increases the shape is more and more reliably normal.0703

The nice thing about the SDOS is that the same principle applies.0710

You could just draw a little link there as an increases, shape approximates normal.0715

Because as we draw sample sizes of size n, as n gets bigger even for SDOS we are actually seeing normal like distributions.0732

Let us talk about center.0748

If you remember the central limit theorem, that is where the shape, center, and spread stuff comes from.0752

Center if you remember, the population mu equals the center of the SDOM which is mu sub x bar.0759

It is a whole bunch of little x bar.0771

There is a similar idea here, but there is a difference.0774

Basically when we talk about center here remember that we do not have the population mu.0786

We do not have a population mu.0794

We do not have a population mean.0795

Instead what we have is more like a population proportion.0796

We had that p.0805

We want to know what is the relationship between p and p hat?0806

In this case, what we see is that the mu that we want to see is going to be equal to the proportion.0814

And if you think about it, let us say you have 60% of your population is approving of the president.0831

If you are draw just 1 person, 1 person from that population what is the chance that that 1 person approved the president?0845

And that 1 person have a 60% chance of approving the president.0851

When you draw 2 people, those 2 people also have a 60% chance of approving of the president.0859

If you draw 3 people you still have 60% chance of approving of the president.0867

The population p is equal to the mu because remember now we have a mean because we have a distribution of p hat.0872

Here is the idea.0888

Get all these p hats, the entire distribution p hats.0890

Once you have those, if you find a mean of that, that is equal to the population.0894

That is the nice thing about the center.0902

Remember this number is between 0 and 1 because you cannot have lower than 0 higher than 1.0906

This value is also between 0 and 1.0915

Another way to think about it is that the rate in the population will be the mean of all the rates that you get in your samples.0921

Let us talk about spread.0935

When we talk about spread before we often look at standard deviation.0938

Obviously you can also look at variance and the equal sides of standard deviation.0943

Here when we talked about standard deviation of the SDOM we called it sigma sub x bar because it is the standard deviation of a bunch of x-bars,0949

a bunch of means and that is equal to sigma.0961

The real population standard deviation divided by √n your sample size.0969

Here as n increases what happens to the standard error?0976

We should also call it, standard error.0988

What happens to standard error?0990

Standard error goes down, decreases.0992

As n goes up standard error goes down because as n gets bigger and bigger and bigger, this whole thing gets smaller and smaller.1002

Let us talk about the spread in the SDOS.1011

Just like here we did not have a population mean.1016

We do not have a population standard deviation.1024

There is no variability there.1027

Instead we use a different formula.1031

First, let us talk about variance here.1034

In order to write variance you call it sigma and instead of sigma sub x bar you call it sigma sub p hat.1038

Just like mu sub p hat.1050

You are constantly saying this is the sigma of the whole bunch of sample proportions.1052

And because we are talking about variance you want to square that.1059

That is going to also be p × 1 - P ÷ n.1064

When you look at this you see that this still holds for both of these.1082

As n increases what happens to the value of the spread?1088

Spread goes down.1097

As n increases spread goes down.1099

Imagine squeezing it.1102

If you wanted to find standard deviation what you would see a sigma sub p hat =vp×1-p / n.1104

We will talk a little bit more about where this comes from in the next segment.1120

But what I want you to see here is that there is this principle as n increases,1125

as your sample size increases your sampling distribution spread goes down.1131

It becomes less variable.1143

We see a lot of similarities across the SDOM and SDOS.1146

Let us talk about the binomial distribution and SDOS.1154

Hopefully remember the binomial distribution from few lessons ago, there we are also talking about categorical data.1163

Not only that we are talking about binary categorical data.1173

Remember we are talking about how many successes, K number of successes out of n.1180

You take a sample of size n and your counting how many number of successes and plotting all of that on a distribution.1197

Here is also categorical and we are also looking at binary choices 1 or the other.1206

Here we are not looking at k number of successes we are looking at sample proportions.1215

I want to stop here to briefly remind you what we are talking about the SDOM the lowest number1223

that p hat could be a 0 and the highest number is 1.1235

Those are the limits.1241

When we talk about a binomial distribution the lowest number that this could be is 0 and the highest number over here is n.1243

It is because we are plotting k on this distribution.1256

0 number of successes 1, 2, 3, 4, 5 all the way up to n number of successes and out of n.1260

What is the shape here when we do not necessarily know.1267

It does not have to be normal.1273

It could be different kinds of shapes.1274

It must be skewed, it could be different shapes.1279

We do not necessarily know.1282

We do not know the shape.1284

Here we know as n increases more normal.1286

Here we do know the shape as long as we have a large enough n.1293

What about center?1300

Here when we talk about center we had looked at sort of how many n would we normally see?1303

What would be the average k?1319

Before in the binomial distribution our notion of center was largely guided by the probability of success × n.1323

You can think of it like this, here is our little sample of n and here some proportion of the sample p1342

of this sample is going to be a success whatever the successes is.1354

Some proportion is going to be the success and how many is that p?1363

To get the raw value I do not want in terms of percent.1369

I want it in raw value here.1373

To get that proportion what I would say is that the center of the binomial distribution is p × n because this whole thing is size n.1375

It is only p(n).1393

If it was 100% then it would be 1 × n, all of them.1396

If it was 75% it would be .75 × n and that will give you only 75% of n.1401

If it was 10% of n it would be .1 × n.1409

This is our definition of center.1414

Here we saw that the definition of center.1417

All we did is basically divide this by n because we no longer want k number of successes we want to know what is that proportion?1421

We do not care what the n is.1431

We care what the actual k is.1433

We just want the proportion.1436

Life becomes easier and mu sub p hat is actually just p.1438

Life is simple.1447

Let us talk about spread.1448

If you remember spread way back in the day here, this is standard deviation so you know why I am square rooting.1452

The standard deviation of n × p × 1 – p.1469

You could see sort of the similarity between this and the standard deviation sigma sub p hat where we have vp×1-p.1476

But instead of multiplying by my √n we are dividing by √n.1491

Let us think about the implications of that.1499

Here as n increases, what happens to the standard deviation?1502

It gets wider and wider and wider because remember if n increases we are stretching out the space.1509

There are more room for variation.1517

Standard deviation increases.1521

However, here you are always limited to 0 and 1.1530

You can never go about that even if you increase your n you try to get more and more people in a sample.1536

It does not matter.1544

You are always stuck between 0 and 1.1545

As n increases the standard deviation decreases.1548

Here there are some definite similarities but there are moments of contrast that are important.1554

Let us go on to some examples.1565

The ethnicity of about 92% of the population of China is Han Chinese, so there are a lot of other ethnic minorities in China, but not a lot only 8%.1570

Suppose you take a random sample of 1,000 Chinese what is the probability of getting 90% or fewer pun Chinese in your sample?1582

What is the probability of getting 925 pun Chinese or more?1592

Well, one thing that helps is for us to realize here if we wanted to we could use binary distributions1597

because we can easily translate from 90% to 900 Hun Chinese.1611

But we can also use the sampling distribution of sample means because we can easily change 925 into 92.5%.1617

We can choose either path we want.1630

I am going to go with the SDOS because that was the lesson is about.1633

First we know that the population I am just going to draw a fake population here, just so that we can remember.1636

Here is my population of China and 92%.1645

My real p= 92% and so my 1 - p =.08.1649

8% is non-Hun Chinese, 92% is Hun Chinese.1663

Now, given this let us say I sample a whole bunch of times and every time I sample I get a sample proportion and I plot that.1669

Because we have a fairly large sample size I can assume that we have a normal distribution.1677

I know that my limits are 0 and 1 and this whole thing this is really p hat.1684

The question is what is the probability of getting 90% or fewer Hun Chinese in your sample?1696

First, it would be helpful to know what this middle is.1705

Actually it is not exactly going to be symmetrical.1710

It is 50%.1715

Here it should really be 92% because the mu(p hat) = p and that is 92%.1717

The upper limit here is 1.0 and the limit down here is 0.1733

What is the probability of getting 90% or fewer Hun Chinese?1742

In order to figure out where 90% is, it would be helpful for us to know the standard error1752

or the standard deviation of the sampling distribution.1761

What is my standard error?1765

This is sigma sub p hat and in order to find that that is going to be the vp×1-p /n.1767

That would be 92% × .08 ÷ 1,000 and take the square root of all of that.1781

Feel free to do it on a calculator I am just going to show it to you one Excel.1794

We have v92% × 8% / 1, 000.1800

Remember order of operations does not really matter for multiplying and dividing.1818

They can be done simultaneously, so it does not matter if they do this first or this first.1825

We see that we have a tiny standard deviation .0086.1830

Even though 90% does not seem like that far away actually is quite far away.1848

How do we find how far away .90 is?1856

You have to think and say this is the normal distribution and there is something we know about normal distribution.1862

We could find these areas in terms of z score.1875

We knew the z score we can find that area.1878

These are my p hats but I'm going to start a row for z scores.1882

Z scores I know the middle is going to be 0 and 1 standard deviation out this is the .0086 distance that is -1.1890

How many of this .0086 is away am I?1904

I could use my notion of z scores.1911

My z score is 4.90 looks something like this.1915

What is the distance between the middle and the score that I'm interested in?1920

That is just 90 -.92.1927

That is going to give me that distance but I do not want that distance in terms of percentages.1931

I want it in terms of my standard error in terms of these little jumps.1937

I'm going to say divide by .0086 to give me how many of these points are 6 jumps away am I if I am at 90?1942

Let us put back into our calculators.1952

I need a parenthesis as order of operations we need to do the subtraction before the division and Excel will not know that.1959

.9 -.92 / .0086 = -2.33.1970

Here is -2 and apparently this is -2.33.1991

Okay, now that we have that z score are we done?1999

No, we need to know what is the probability of getting 90% or fewer Han Chinese in your sample?2013

What we want to know is this area here.2020

This is 90% or fewer are Han Chinese in sample.2022

That is the area we want to know.2032

At this point because you have the z score you could look it up in the back of the book using your z tables.2035

Just to show you I am going to use Excel to find this and I will leave my z score there because it will come in handy.2042

Remember normsdist and it asks me to put in the z and once it does that I know that this proportion should be very small and its only 1% of this.2057

1% is our answer.2073

We should expect what is the probability of getting 90% or fewer Han Chinese in our sample.2080

It is 1%.2086

We want to find out what is the probability of getting 925 Hun Chinese or more.2088

In this case, why do we do the same thing but 92.5 so that would be somewhere past here.2102

925 where is that?2115

Let us find the z score so that we could be exact.2123

Z score of .925 is the distance between 925 and .92 divided by the little jumps, the standard errors .0086.2125

When I do that what do I get .925 -.9 /.0086 = 2.990.2141

That did not look right to me because this should be a smaller z score than this one because this should be farther out.2161

It is .58 is our z score.2190

I wrote this one at a wrong place.2196

.925 is somewhere here.2200

That is .58 that is our z score.2206

In order to find the area let me shade that in so you know in order to find the area because2216

we are looking for is the probability of getting this score or more, that area should be 50%.2228

It should be much more than this and actually I have put it in my normal distribution but remember this will give you what is on this left side or the negative side.2235

We need to look at 1 - that normal distribution.2253

This is 28%.2258

What is the probability of getting 929 Han Chinese or more that is going to be.28.2262

Example 2, college freshmen from a wide variety of colleges across the US participate in a survey2274

where 61% reported that they are attending college that was their first choice.2285

If you took a random sample of 100 freshmen how likely is it that at least 50 of those students are attending their first choice college?2291

Saying at least 50 that is a good thing to keep in mind for later.2299

Let us try this population.2305

Here is my population of college freshmen and 61% a little more than half.2307

61% is our p and 1 - p is not quite 40% but is 39.2315

The other 39 they are not attending their first choice college.2324

Imagine taking out of that population of random sample of 100 freshmen and looking at2328

the sample proportion and plotting that on the SDOS.2340

100 is still a pretty large n so I am going to go with that normal distribution.2344

I know that my SDOS mu.2356

Mu sub p hat this should equal p and that is 61%.2363

What is my standard deviation of this SDOS because I'm not just looking at who is in here.2372

I am looking at it if I took a sample of 100 students how good is my sample?2380

Whenever you hear that, like how good is the sample then you know you need a sampling distribution.2386

I should probably find my standard error because standard error because it is a sampling distribution.2393

Here is vp×1-p /n that is going to be v.61 × .39 ÷ 100.2408

I will just look that up here so v.61 × .39 ÷ 100 = .0488.2428

This little jumper here is .0488 that is how big does little jumps are.2456

I'm looking for how likely is it that at least 50 of these students are attending their first choice college.2481

I can turn this into a percentage by looking at 50/100.2493

My p hat that I have been given is 50/100 and that is .5 and I want to know how likely is this p hat.2499

It is nice to find out where the p hat is and this is the raw proportion.2512

It would be nice to find the z score and the z score of .5 should be the distance between .5 and the mean divided by the little jumps.2520

How big are my jumps in order to find how many jumps away.2540

Let us put that in our calculator, .5 - .61 ÷ .0488 = -2.25.2545

Here we are somewhere like this -2.25 and this is 4.5.2569

We want to know how likely is it that at least 50 of those students are attending that first choice college.2583

When we say at least this is the lower limit.2592

We are looking for this whole thing.2598

You can look that up in the back of your book or you could say the proportion that p hat will be greater than or equal to .5.2603

I do not know if you remember this notation here we want to know, I remember will give us the negative side,2618

so we have the 1 - this little piece.2634

1 – norms s in order for standardized that is how we get that z and we put in our z and we should get .9879.2639

Very close to survey .9879.2660

Almost 99% of our sample should have at least 50% of those students attending their first choice college.2667

Third example, about 75% of the US population owns a cell phone and that is growing.2681

On average, what proportion of people would you expect to have a cell phone in a sample of 10, 20 or 40?2691

This is talking about the average proportion.2698

We are looking at the mu sub p hat on average, what proportion of people would you expect?2702

For 10 people it should be 75% for n=10.2711

What about n=20?2721

Even for that the sampling distributions mean should be 75%.2725

What about n=40?2732

This should be 75%.2735

What it is getting at is that no matter how big or little your sample size your mean of the sampling distribution2740

does not really change and that is similar to what we saw in the sampling distribution of the mean as well.2751

Final example, that 60% of married women are employed.2757

If you select 75 married women, what is the probability that between 30 and 40 women are employed?2763

Here we need to know that our actual population and these are all married ladies and 60% are employed.2770

That is our p and 1 - p is 40%.2793

Imagine now taking samples of 75 so this is SDOS for n=75.2798

75 is still fairly large so I will assume normal distribution.2808

What is the probability that between 30 and 40 women are employed?2815

We know that mu sub p hat is 60% we also know that the standard deviation of p hat is the v.60 × .40 /n(75)2822

and all of that under the square root sign.2842

I will just quickly put this into my calculator.2845

v.6 × .4 ÷ 75 = .0566.2850

What is the probability that between 30 and 40 women are employed?2873

First of all it helps me to figure out what percentage is 30 women out of 75 and what percentage 40 women out of 75?2883

Let us call that p hat sub 30 that is 30 ÷ 75 and also p hat sub 40 that is 40 ÷ 75.2892

If I want to get it in decimals 40 ÷ 75, 30 ÷ 75 that is .53 and 4.2905

I am going to know that these 2 slickers.2922

About the distance in between here is about 6%.2936

I will go about 6 down so this first one and another 6, so this would be roughly .54.2942

Let us actually find the z scores of this.2957

Z(.4) = these are the p hats.2966

These are the z scores is .4 - .6 all divided by the little jumps.2976

And these little jumps are .0566.2992

.4 - .6 we need a parenthesis here divided by .0566.3001

That is the z score of -3 .5.3025

What about the score of .53 I'm just going to forget about the repeating part.3048

It will just be something like .5 - .6 ÷.0566 and that is -1.2.3061

Here is my big problem, first we need to know this area but there is no table that will tell us just that area.3081

Here is what we will have to do, we will have to take everything below this and then subtract out everything below that3109

because then we will get this entire area including this infinite tail and then take out a tiny little bit of it to top that part off3131

to get it in between this part and this part.3141

In order to do that, I will use my normsdist and remember that will give me the negative side.3144

Let me put in my bigger number first and then subtract, that is my entire area below z= -1.2.3161

That is the entire area.3181

I am going to subtract out the tiny sliver way over here.3184

Area below z= -3.5.3188

I can just normsdist -3.5 and that should be a really tiny, tiny, tiny number.3196

I need to subtract this area out of this.3208

I take this whole thing and subtract this little sliver and I get roughly very similar number.3212

.119 that is my area.3221

This area here we can call it the probability where p hat is greater than or equal to .4 and less than or equal to .53 repeating is roughly .119.3227

It is about 11.9% is the probability that between 30 and 40 women are employed.3257

That is the end of sampling distribution of sample proportion.3269

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