For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

### 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
- Roadmap
- Intro to Sampling Distribution of Sample Proportions (SDoSP)
- Notation
- What's the Difference?
- Binomial Distribution vs. Sampling Distribution of Sample Proportions
- 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
- Example 2: Sampling Distribution of Sample Proportions
- Example 3: Sampling Distribution of Sample Proportions
- Example 4: Sampling Distribution of Sample Proportions

- 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

### General Statistics Online Course

### Transcription: Sampling Distribution of Sample Proportions

*Hi and welcome to www.educator.com. *0000

*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 and *0063

*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 median *0079

*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 proportion *0119

*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 sampling *0182

*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 people *0412

*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 number *1223

*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 p *1342

*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 distributions *1597

*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 error *1752

*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 because *2216

*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 survey *2274

*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 at *2328

*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 distribution*2740

*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 that *3109

*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 off*3131

*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

*Thanks for using www.educator.com.*3275

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.

Martin

0 answers

Post by Alexandra Vazquez on December 4, 2012

you're a lifesaver. best teacher ever.