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Lecture Comments (3)

0 answers

Post by Terry Kim on October 20, 2015

why are we adding df and variances when we are actually calculating the DIFFERENCE? H(null): mu_(x-y) = 0 here it is 0 because it is the difference
but I don't get why we add the dfs and variances if its S_(x-y) isn't it also should be sqrt(s^2_(x)-s^2(y))??

0 answers

Post by Professor Son on November 12, 2014

Just for students who happen to have a class with me, I don't emphasize s-pool a lot because typically it's more conservative to assume that they are separate. If you take a more advanced statistics class, you could learn about hypothesis testing that allows us to infer whether we can pool standard deviations together.

0 answers

Post by Professor Son on November 12, 2014

In the section about s-pool, I accidentally refer to SE as "sample error" but what I meant to say was "standard error."

Confidence Intervals for the Difference of Two Independent Means

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 0:14
    • Roadmap
  • One Mean vs. Two Means 1:17
    • One Mean vs. Two Means
  • Notation 2:41
    • A Sample! A Set!
    • Mean of X, Mean of Y, and Difference of Two Means
    • SE of X
    • SE of Y
  • Sampling Distribution of the Difference between Two Means (SDoD) 7:48
    • Sampling Distribution of the Difference between Two Means (SDoD)
  • Rules of the SDoD (similar to CLT!) 15:00
    • Mean for the SDoD Null Hypothesis
    • Standard Error
  • When can We Construct a CI for the Difference between Two Means? 21:28
    • Three Conditions
  • Finding CI 23:56
    • One Mean CI
    • Two Means CI
  • Finding t 29:16
    • Finding t
  • Interpreting CI 30:25
    • Interpreting CI
  • Better Estimate of s (s pool) 34:15
    • Better Estimate of s (s pool)
  • Example 1: Confidence Intervals 42:32
  • Example 2: SE of the Difference 52:36

Transcription: Confidence Intervals for the Difference of Two Independent Means

Hi and welcome to www.educator.com.0000

Today we are going to talk about confidence intervals for the difference of two independent means.0002

It is pretty important that there are for independent means because later we are going to go to non-independent or error means.0007

We have been talking about how to find confidence intervals and hypothesis testing for one mean.0013

We are going to talk about what that means for how we go about doing that for two means.0023

We are going to talk about what two means means?0029

We are going to talk a little bit about mu notation and we are going to talk about sampling distribution of the difference between two means.0032

I am going to shorten this, this is just means this is not like official or anything as SDOD 0041

because it is long to say assembling distribution of the difference between two means, but that is what I mean.0048

We will talk about the rules of the SDOD and those are going to be very similar to the CLT (the central limit theorem) with just a few differences.0055

Finally, we all set it all up so that we can find and interpret the confidence interval.0066

One mean versus two means.0075

So far we have only looked at how to compare one mean against some population, but that is not usually how scientific studies go.0081

Most scientific studies involve comparisons.0091

Comparisons either between different kinds of water samples or language acquisition for babies versus babies who did not.0093

Scores from the control group versus the experimental group.0102

In science we are often comparing two different sets of the two different samples.0106

Two means really means two samples.0112

Here in the one mean scenarios we have one sample and we compare that to an idea in hypothesis testing 0120

or we use that one sample in order to derive the potential population means.0132

But now we are going to be using two different means.0140

What do we do with those two means?0143

Do we just do the one sample thing two times or is there a different way?0145

Actually, there is different and more efficient way to go about this.0152

Two means is a different story.0155

They are related but different story.0159

In order to talk about two means and two samples, we have to talk about some new notation.0162

This is totally arbitrary that we use x and y.0170

You could use j and k or m and n, whatever you want.0176

X and y is the generic variables that we use.0182

Feel free to use your favorite letters. 0189

One sample will just be called x and all of its members in the sample will be x sub 1, x sub 2, x sub 3.0191

When we say x sub I, we are talking about all of these little guys.0203

The other sample we do not just call it x as well because we will get confused. 0208

We cannot call it x2 because x sub 2 has a meaning.0216

What we call it is y.0221

Y sub i now means all of these guys.0224

We could keep them separate.0229

In fact this x and y is going to follow us from here on out.0232

For instance when we talk about the mean of x we call it the x bar.0236

What would be the mean of y?0241

Maybe y bar right. 0243

That makes sense.0246

And if you call this b, this will be b bar.0247

It just follows you. 0253

When we are talking about the difference between two means we are always talking about this difference. 0256

That is going to be x bar - y bar. 0264

Now you could also do y bar - x bar, it does not matter.0267

But definitely mean by the difference between two means.0271

We could talk about the standard error of all whole bunch of x bars, standard error of x, standard error of y.0274

You could also talk about the variance of x and the variance of y.0285

You can have all kinds of thing they need something to denote that they are little different.0292

That standard error of x sort and another way you could write it is that we are not just talking about standard error.0298

When we say standard error, you need to keep in mind if we double-click on it that means the standard deviation of a whole bunch of means.0312

Standard deviation of a whole bunch of x bars.0322

Sometimes we do not have sigma so we cannot get this value.0328

We might have to estimate sigma from s and that would be s sub x bar.0334

If we wanted to know how to get this that would just be s sub x.0345

Notice that is different from this, but this is the standard error and this is the actual standard deviation of your sample ÷ √n.0353

Not just n the n of your sample x.0367

In this way we could perfectly denote that we are talking about the standard error of the x, the standard deviation of the x, and the n(x).0372

You could do the same thing with y.0387

The standard error of y, if you had sigma, you can just call it sigma sub y bar because it is the standard deviation of a whole bunch of y bars.0390

Or if you do not have sigma you could estimate sigma and use s sub y bar.0402

Instead of just getting the standard deviation of x we would get the standard deviation of y and divide that by √n Sub y.0411

It makes everything a little more complicated because now I have to write sub x and sub y after everything.0423

But it is not hard because the formula if you look remains exactly the same.0430

The only thing that is different now is that we just add a little pointer to say we are talking 0438

about the standard deviation of our x sample or standard deviation of our y sample.0446

Even this looks a little more complicated, deep down at the heart of the structure it is still the standard error equals standard deviation of the sample ÷√n.0452

Let us talk about what this means, the sampling distribution of the difference between two means. 0466

Let us first start with the population level.0477

When we talk about the population right now we do not know anything about the population.0480

We do not know if it is uniform, the mean, standard deviation.0491

Let us call this one x and this one y.0500

From this x population and this y population we are going to draw out samples and 0507

create the sampling distribution and that is the SDOM (the sampling distribution of the mean).0514

Here is a whole bunch of x bars and here is a whole bunch of y bars.0522

Thanks to the central limit theorem if we have big enough n and all that stuff then we know that we could assume normality.0530

Here we know a little bit more than we know about the population.0540

We know that in the SDOM, the standard error, I will write s from here because 0545

we are basically going to assume real life examples when we do not have the population standard deviation.0557

The only time we get that is like in problems given to you in statistics textbook.0565

We will call it s sub x bar and that can be the standard deviation of x/√n sub x.0570

We know those things and we also know the standard error of y and that is going to be the standard deviation of y ÷ √n sub y.0585

Because of that you do not write s sub y again because that would not make sense that 0601

the standard error would equal the standard error over into something else.0607

That would not quite make sense. 0612

You want to make sure that you keep this s special and different because standard error 0614

is talking about entirely different idea than the standard deviation.0621

Now that we have two SDOM if we just decided to do this then we would not need to know anything new about creating a confidence interval of two means.0625

You what just create two separate confidence intervals like you consider that x bar, 0638

consider that y bar, construct a 95% confidence interval for both of these guys.0644

You are done.0649

Actually what we want is not a sampling distribution of two means and get two sampling distributions.0650

We would like one sampling distribution of the difference between two means.0661

That is what I am going to call SDOD.0668

Here is what you have to imagine, in order to get the SDOM what we had to do is go to the population and draw out samples of size n and plot the means.0671

Do that millions and millions of times.0682

That is what we had to do here.0685

We also have to do that here, we want the entire population of y pulled out samples and plotted the means until we got this distribution of means.0687

Imagine pulling out a mean from here randomly and then finding the difference of those means and plotting that difference down here.0699

Do that over and over again.0715

You would start to get a distribution of the difference of these two means. 0718

You would get a distribution of a whole bunch of x bar - y bar.0727

That is what this distribution looks like and that distribution looks normal. 0734

This is actually one of the principle of probability distributions that we have covered before.0742

I think we have covered it in binomial distributions.0747

I know this is not a binomial distribution but the same principles apply here where if you draw from two normally distributed population0749

and subtract those from each other you will get a normal distribution down here.0764

We have this thing and what we now want to find is not just the mu sub x bar or mu sub y bar, that is not what we want to find.0769

What we want to find is something like the mu of x bar - y bar because this is our x bar - y bar and we want to find the mu of that.0783

Not only that but we also want to find the standard error of this thing.0796

I think we can figure out what that y might be.0800

At least the notation for it, that would be the standard error.0807

Standard error always have these x bar and y bar things.0812

This is how you notate the standard deviation of x bar - y bar and that is called 0817

the standard error of the difference and that is a shortcut way of saying x bar - y bar. 0829

We could just say of the difference.0837

You can think of this as the sampling distribution of a whole bunch of differences of means. 0839

In order to find this, again it draws back on probability principles but actually let us go to variance first.0845

If we talk about the variance of this distribution that is going to be the variance of x bar + the variance of y bar.0856

If you go back to your probability principles you will see why.0869

This from this we could actually figure out standard error by square rooting both sides.0874

We are just building on all the things we have learned so far. 0881

We know population. 0888

We know how to do the SDOM.0889

We are going to use two SDOM in order to create a sampling distribution of differences.0891

Let us talk about the rules of the SDOD and these are going to be very, very similar to the CLT.0898

The first thing is this, if SDOM for x and SDOM for y are both normal then the SDOD is going to be normal too.0909

Think about when these are normal?0919

These are normal if your population is normal.0922

That is one case where it is normal.0924

This is also normal when n is large.0927

In certain cases, you can assume that the SDOM is normal, and if both of these have met those conditions, 0929

then you can assume that the SDOD is normal too.0939

We have conditions where we can assume it is normal and they are not crazy. 0942

There are things we have learned.0949

What about the mean?0951

It is always shape, center, spread.0953

What about the mean for the SDOD?0956

That is going to be characterized by mu sub x bar - y bar.0959

That is the idea.0972

Let us consider the null hypothesis and in the null hypothesis usually the idea is they are not different like nothing stands out.0975

Y does not stand out from x and x does not stand out from y.0987

That means we are saying very similar.0991

If that is the case we are saying is that when we take x bar – y bar and do it over and over again, on average, the difference should be 0.0994

Sometimes the difference will be positive. 1009

Sometimes the difference will be negative.1012

But if x and y are roughly the same then we should actually get a difference of 0 on average.1014

For the null hypothesis that is 0.1022

The so what would be the alternative hypothesis?1027

Something like the mean of the SDOD is not 0. 1031

This is in the case where x and y assume to be same.1037

That is always with the null hypothesis.1051

They assume to be the same. 1055

They are not significantly different from each other.1056

That is the mean of the SDOD.1058

What about standard error?1062

In order to calculate standard error, you have to know whether these are independent samples or not.1064

Remember to go back to sampling, independent samples is where you know that these two 1073

come from different populations and the picking one does not change the probabilities of picking the other.1079

As long as these are independent samples, then you can use these ideas of the standard error. 1089

As we said before, it is easier when I think about the variance of the SDOD first because that rule is quite easy.1096

The variance of SDOD, so the variance is going to be just the variance of the SDOM + the variance of the SDOM for the other guy.1105

And notice that these are the x bars and the y bars.1121

These are for the SDOM they are not for the populations nor the samples.1131

From here what you can do is sort of justice derive the standard error formula.1137

We can just square root both sides.1149

If you wanted to just get standard error, then it would just be the square root of adding each of these variances together.1153

Let us say you double-click on this guy, what is inside of him?1168

He is like a stand in for just the more detailed idea of s sub x / n sub x.1175

Remember when we talk about standard error we are talking about standard error = s / √n.1193

The variance of the SDOM =s2 /n.1205

If you imagine squaring this you would get s/n but we need the variance.1210

We need to add the variances together before you square root them.1220

Here we have the variance of y / n sub y.1224

You could write it either like this or like this.1235

They mean the same thing. 1240

They are perfectly equivalent.1242

You do have to remember that when you have this all under the square root sign, 1244

the square root sign acts like a parentheses so you have to do all of this before you square root.1253

That is standard error.1261

I know it looks a little complicated, but they are just all the principles we learned before, 1265

but now we have to remember does it come from x or does come from y distributions.1273

That is one of the few things you have to ask yourself whenever we deal with two samples.1279

Now that we know the revised CLT for this sampling distribution of the differences, 1287

now we need to ask when can we construct a confidence interval for the difference between two means?1298

Actually these conditions are very similar to the conditions that must be met when we construct an SDOM.1306

There are a couple of differences because we are dealing with two samples.1314

The three conditions have to be met.1318

All three of these have to be checked.1321

One is independence, the notion of independence. 1323

The first is this, the two samples we are randomly and independently selected from two different populations.1329

That is the first thing you have to meet before you can construct this confidence interval.1340

The second thing is this, this is the assumption for normality.1348

How do we know that the SDOD is normal. 1355

It needs to be reasonable to assume that both populations that the sample comes from the population are normal or your sample size is sufficiently large.1358

These are the same ones that apply to the CLT.1372

This is the case where we can assume normality for the SDOM but also the SDOD.1376

In number 3, in the case of sample surveys the population size should be at least 10 times larger than the sample size for each sample.1384

The only reason for this is we talked before about replacement, a sampling with replacement versus sampling not with replacement.1397

Well, whenever you are doing a sample you are technically not having replacement 1409

but if your population is large enough then this condition actually makes it so that you could assume that it works pretty much like with replacement.1413

If you have many people then it does not matter.1427

That is the replacement rule.1430

Finally, we could get to actually finding the confidence interval.1433

Here is the deal, with confidence interval let us just review how we used to do it for one mean.1444

One mean confidence interval.1450

Back in the day when we did one mean and life was nice and what we would do is often take the SDOM 1455

and assume that the x bar, the sample mean is at the center of it and then we construct something like 95% confidence interval.1466

These are .025 because if this is 95% and symmetrical there is 5% leftover but it needs to be divided on both sides.1484

What we did was we found these boundary values by using this idea, this middle + or – how many standard errors you are away.1496

We used either t or z.1525

I’m just going to use t from now on because usually we are not given the standard deviation of the population × the standard error.1529

That was the basic idea from before and that would give us this value, as well as this value.1530

We could say we have 95% confidence that the population mean falls in between these boundaries.1537

That is for one mean.1545

What about two means?1548

In this case, we are not going to be calculating using the SDOM anymore.1549

We are going to use the SDOD.1560

If this mean is going to be x bar, this sample mean then you can probably assume that 1562

it might be something as simple as a difference between the two means.1575

That is what we assume to be the center of the SDOD.1580

Just like before, whatever level of confidence you need.1583

If it is 99% you have 1% left over on the side.1593

You have to divide that 1% in half so .5% for the side and .5% for that side.1598

In this case, let us just keep the 95%.1603

What we need to do is find these borders.1611

What we can to just use the exact same idea again.1618

We could use that exact same idea because we can find the standard error of this distribution.1624

We know what that is.1629

Let me write this out.1631

We will write s sub x bar.1640

We can actually just translate these ideas into something like this. 1645

That would be taking this, adding or subtracting how many jumps away you are, like the distance you are away.1652

That would be something like x bar - y bar but instead of just having x in the middle we have this thing in the middle.1661

+ or – the t remains the same, t distributions but we have to talk about how to find degrees of freedom for this guy.1670

The new SE, but now this is the SE of the difference.1680

How do we write that?1691

X bar - y bar + or - the t × s sub x bar = y bar.1694

If we wanted to we could take all that out into the square root of variance of the SDOM for x and variance of SDOM for y.1707

We could unpack all of this if we need to but this is the basic idea of the confidence interval of two means.1719

In order to do this I want you to notice something.1727

Here we need to find t and because we need to find t we need to find degrees of freedom 1732

but not just any all degrees of freedom because right now we have 2 degrees of freedom. 1740

Degrees of freedom for x and degrees of freedom for y.1744

We need a degrees of freedom for the difference.1747

That is what we need.1751

Let us figure out how to do that.1753

We need to find degrees of freedom.1756

We know how to find degrees of freedom for x, that is straightforward. 1760

That is n sub x -1 and degrees of freedom for y is just going to be n sub y -1.1764

Life is good.1771

Life is easy.1772

How do we find the degrees of freedom for the difference between x and y?1773

That is actually going to just be the degrees of freedom for x + degrees of freedom for y.1778

We just add them together.1790

If we want to unpack this, if you think about double-clicking on this and get that.1792

N sub x - 1 + n sub y -1.1797

I am just putting that parentheses as you could see the natural groupings but obviously you could 1804

do them in any order because you could just do them straight across this adding and subtracting. 1810

They all have the same order of operation.1816

That is degrees of freedom and once you have that then you can easily find the t.1820

Look it up in the back of your book or you can do it in Excel.1830

Let us interpret confidence interval. 1833

We have the confidence interval let us think about how to say what we have found.1837

I am just going to briefly draw that picture again because this picture anchors my thinking.1844

Here is our difference of means.1852

When you look at this t, think of this as the difference of two means.1858

I guess I could write DOTM but that would just be DOM.1863

Here what we found, if we find something like a 95% confidence interval that means we have found these boundaries.1869

We say something like this. 1887

The actual difference of the two means of the real population, of the population x and y.1891

The real population that they come from should be within this interval 95% of the time or something like 1919

we have 95% confidence that the actual difference between means of the population of x and population of y should be within this interval.1939

That comes from that notion that this is created from the SDOM.1950

Remember the SDOM, the CLT says that their means or the means of the population.1955

We are getting the population means drop down to the SDOM and from the SDOM we get this.1962

Because of that we could actually make a conclusion that goes back to the population.1970

Let us think about if 0 is not in between here.1980

Remember the null hypothesis when we think about two means is going to be something like this.1987

That the mu sub x bar – y bar is going to be equal to 0. 1993

This is going to mean that on average when you subtract these two things the average is going to be 0.1998

There is going to be no difference on average.2004

The alternative hypothesis should then be the mean of these differences should not be 0.2006

They are different.2015

If 0 is not within this confidence interval then we have very little reason to suspect that this would be true.2016

It is a very little reason to think that this null hypothesis is true.2026

We could also say that if we do not find 0 in our confidence interval that we might in my hypothesis testing be able to also reject the null hypothesis.2030

But we will get to that later.2040

I just wanted to show you this because the confidence interval here is very tightly linked to the hypothesis testing part.2042

They are like two side of the same coin.2050

That universe is fairly straightforward but I feel like I need to cover one other thing because sometimes this is emphasized in some books.2052

Some teachers emphasize this over other teachers and so I'm going to talk to you about SPOOL because this will come up.2065

One of the things I hope you noticed was that in order to find our estimate of SDOM, 2076

in order to find the SDOD sample error what we did was we took the variance of one SDOM 2085

and added that to the variance of the other SDOM and square root the whole thing.2106

Let me just write that here. 2110

The s sub x bar - y bar is the square root of one the variances + the variance of the other SDOM.2111

Here what we did was let us just treat them separately and then combine them together.2129

That is what we did.2137

Although this is an okay way of doing it, in doing this we are assuming that they might have different standard deviations.2138

The two different populations might have two different standard deviations.2154

Normally, that is a reasonable assumption to make.2159

Very few populations have the exact standard deviation.2162

For the vast majority of time because we just assumed if you come from two different population you probably have two different standard deviations.2166

This is pretty reasonable to do like 98% of the time.2177

The vast majority of time.2182

But it is actually is not as good as the estimate of this value then, if you had just used up a POOL version of the standard deviation.2184

Here is what I mean.2198

Now we are saying, we are going to create the standard deviation of x.2198

You are going to be what we used to create the standard deviation of y.2206

Just of not make that explicit.2210

I am going to write this out so that you could actually see the variance of x and the variance of y.2213

We use x to create this guy and we use y to create that guy and they remain separate. 2228

This is going to take a little reasoning.2235

Think back if you have more data then your estimate of the population standard deviation is better, more data more accurate. 2239

Would not it be nice if we took all the guys from the x pool and all the guys from the y pull and put them together.2253

Together let us estimate the standard deviation.2262

Would not that be nice?2267

Then we will have more data and more data should give us a more accurate estimate of the population.2268

You can do that but only in the case that you have reason to think that the population of x has a similar standard deviation to the population of y.2278

If you have a reason to think they are both normally distributed.2293

Let us say something like this.2299

If you have reason to believe that the population x and y have similar standard deviation 2303

then you can pull samples together to estimate standard deviation.2324

You can pull them together and that is going to be called spull.2347

There are very few populations that you can do this for.2351

One thing something like height of males and females, height tends to be normally distributed and we know that.2357

Height of Asians and Latinos or something, but there are a lot of examples that come to mind where you could do this.2365

That is why some teachers do not emphasize it but I know that some others do so. 2374

That is why I want to definitely go over it. 2378

How do you get spull and where does it come in?2380

Here is the thing, in order to find Spull, what we would do is we would substitute in spull for s sub x and s sub y.2384

Instead of two separate estimates of standard deviations use Spull.2396

We will be using Spull2.2408

How do we find Spull2?2411

In order to find Spull2, what you would do is you would add up all of the sum of squares.2415

The sum of squares of x and sum of squares of y, add them together and then divide by the sum of all the degrees of freedom.2432

If I double-click on this, this would mean the sum of squares of x + the sum of squares of y ÷ degrees of freedom x + degrees of freedom y.2442

This is what you need only to do in order to find Spull and then what you would do is substitute in s(x)2 and s sub y2.2457

That is the deal.2469

In the examples that are going to follow, I am not going to use Spull because there is very little reason usually to assume that we can use Spull.2471

And but a lot of times you might hear this phrase assumption of homogeneity of variance.2483

If you could assume that these guys have a similar variance, if you can assume 2490

they have similar homogeneous variance then you can use Spull.2502

For the most part, for the vast majority of time you cannot assume homogenous variance.2508

Because of that we will often use this one. 2514

However, I should say that some teachers do want you to be able to calculate both.2517

That is the only thing.2525

Finally I should just say one thing. 2528

Usually this works just as well as pull.2531

It is just that there are sometimes we get more of a benefit from using this one.2536

If worse comes to worse, and after the statistics class you are only remember this one.2543

If not all you are pretty good to go.2548

Let us go on to some examples.2551

A random sample of American college students was collected to examine quantitative literacy.2556

How good they are in reasoning about quantitative ideas.2562

The survey sampled 1,000 students from four-year institutions, this was the mean and standard deviation.2565

800 from two-year institutions, here is the mean and standard deviations.2571

Are the conditions for confidence intervals met?2576

Also construct a 95% confidence interval and interpret it.2581

Let us think about the confidence interval requirements.2586

First is independent random samples.2593

It does say random sample right and these are independent populations.2596

One is for your institutions, one is to your institutions. 2603

There are very few people going to both of them at the same time.2606

First one, check.2609

Second one, can we assume normality either because of the large n or because we know that both these populations are originally normally distributed?2612

Well, they have pretty large n, so I am going to say number 2 check.2622

Number 3, is this sample roughly sampling with replacement?2627

And although 1000 students seem a lot, there are a lot of college students.2635

I am pretty sure that this meets that qualification as well.2640

Go ahead and construct the 95% confidence interval.2643

Well, it helped to start off with the drawing of SDOD just to anchor my thinking.2648

And this mu sub x bar - y bar we could assume that this is x bar - y bar.2656

That is what we do with confidence intervals. 2667

We use what we have from the samples to figure out what the population might be.2670

We want to construct a 95% confidence interval.2678

That is going to be .025 and then maybe it will help us to figure out the degrees of freedom so that we will know the t value to use.2685

Let us figure out degrees of freedom.2703

It is going to be the degrees of freedom for x and I will call x the four-year university guys and the degrees of freedom for y the two-year university guys.2706

That is going to be 999 + 799 and so it is going to be 1800 - 2 = 1798.2718

We have quite large degrees of freedom and let us find the t for this place.2747

We need to find is this and this.2755

Let us find the t first. 2760

This is the raw score, this is the t, and let me delete some of the stuff.2765

I will just put x bar - y bar in there and we can find that later.2772

The t is going to be the boundaries for this guy and the boundaries for this guy.2782

What is our t value?2788

You can look it up in the back of your book or you could do it in Excel.2790

Here we want to put in the t in because we have the probability and remember this one 2799

wants two tailed probability .05 and the degrees of freedom which is 1798 = 1.896.2806

We will put 1.961 just to distinguish it.2819

Let us write down our confidence interval formula and see what we can do.2831

Confidence interval is going to be x bar - y bar.2838

The middle of this guy + or - t × standard error of this guy.2844

That is going to be s sub x bar - y bar.2854

It would be probably helpful to find this thing.2858

X bar - y bar.2862

X bar - y bar that is going to be 330 – 310.2868

Let us also try to figure out the standard error of SDOD which is s sub x bar - y bar.2883

What I'm trying to do is find this guy.2911

In order to find that guy let us think about the formula. 2918

I'm just writing this for myself. 2921

The square root of the variance of x bar + the variance of y bar .2925

We do not have the variance of x bar and y bar.2937

Let us think about how to find the variance of x bar.2943

The variance of x bar is going to be s sub s2 ÷ n sub x.2947

The variance of y bar is going to be the variance of y2 ÷ n sub y.2959

I wanted to write all these things out just because I need to get to a place where finally I can put in s.2977

Finally, I can do that.2986

This is s sub x and this is s sub y.2988

I can put in 1112 ÷ n sub x which is 1000 and I could put in the standard deviation of y2 ÷ 800.2990

I have these two things and what I need to do is go back up here and add these and square root them.3017

Square root this + this.3028

I know that this equal that.3034

We have our standard error, which is 4.49 and this is 20 + or - 1.961. 3038

Now I could do this.3064

I will going to take that in my calculator as well.3066

The confidence interval for the high boundary is going to be 20 + 1.961 × 4.49 3069

and the confidence interval for the low boundary is going to be that same thing.3085

I am just going to change that into subtraction.3097

11.20.3101

Let me move this over.3105

It is going to be 28.8.3110

Let me get the low end first.3117

The confidence interval is from about 11.2 through 28.8.3121

We have to interpret it.3127

This is the hardest part for a lot of people.3130

We have to say something like this.3133

The true difference between the population means 95% of the time is going to fall in between these two numbers.3136

Or we have 95% confidence that the true difference between the two population means fall in between these two numbers.3146

Let us go to example 2.3154

This will be our last example.3157

If the sample size of both samples are the same, what would be the simplified formula for standard error of the difference?3159

If in addition, the standard deviation of both samples are the same, what would be the simplified formula for standard error of the difference?3167

This is just asking depending on how similar the two examples are can we simplify a formula for standard error.3175

We can.3183

Let us write the actual formula out so that would just x bar – y bar = square root of the variance of x bar + variance of y bar.3184

If we double-click on these guys that would give the variance of x / n sub x + the variance of y / n sub y.3207

It is asking, what if the sample size for both samples are the same?3223

What would be the simplified formula?3230

That is saying that if n sub x = n sub y then what would be this?3231

We can get the variance of x + variance of y / n.3240

Because the n for each of them should be the same.3251

This would make it a lot simpler.3254

If in addition a standard deviation of both samples are the same right then this would mean that 3260

because the standard deviation is the same then the variances are the same.3272

That would be that case.3276

If in addition this was the case, then you would just get 2 × s2 whatever the equal variances /n.3279

That would make it a simple formula.3294

That would make life a lot easier but that is not always the case.3298

If it is you know that it will be simple for you. 3303

That is it for the confidence intervals for the difference between two means.3307

Thank you for using www.educator.com.3312