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Lecture Comments (1)

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Post by Brijesh Bolar on August 22, 2012

I really like the way you simplified hypothesis testing..

Confidence Intervals & Hypothesis Testing for the Difference of Two Paired 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 0:09
    • Roadmap
  • The Goal of Hypothesis Testing 1:27
    • One Sample and Two Samples
  • Independent Samples vs. Paired Samples 3:16
    • Independent Samples vs. Paired Samples
    • Which is Which?
  • Independent SAMPLES vs. Independent VARIABLES 7:43
    • independent SAMPLES vs. Independent VARIABLES
  • T-tests Always… 10:48
    • T-tests Always…
  • Notation for Paired Samples 12:59
    • Notation for Paired Samples
  • Steps of Hypothesis Testing for Paired Samples 16:13
    • Steps of Hypothesis Testing for Paired Samples
  • Rules of the SDoD (Adding on Paired Samples) 18:03
    • Shape
    • Mean for the Null Hypothesis
    • Standard Error for Independent Samples (When Variance is Homogenous)
    • Standard Error for Paired Samples
  • Formulas that go with Steps of Hypothesis Testing 22:59
    • Formulas that go with Steps of Hypothesis Testing
  • Confidence Intervals for Paired Samples 30:32
    • Confidence Intervals for Paired Samples
  • Example 1: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means 32:28
  • Example 2: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means 44:02
  • Example 3: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means 52:23

Transcription: Confidence Intervals & Hypothesis Testing for the Difference of Two Paired Means

Hi and welcome to

We are going to talk about confidence interval and hypothesis testing for the difference of two paired means.0002

We have been talking about independent samples so far, one example, two independent samples.0008

We are going to talk about paired samples.0017

We are going to look at the difference between independent samples and paired samples.0020

We are also going to try and clarify the difference between independent sample0025

and independent variables because paired samples still use independent variables.0029

We are going to talk about two types of t-tests.0035

One that we covered or also called hypothesis testing and one that we covered so far with independent samples.0039

The new one that was cover with paired samples.0046

We are going to introduce some notation for paired samples, go through the steps of hypothesis testing0050

for paired samples and adjust or add on to the rules of SDOD that we already have looked at.0058

Finally we are going to go over the formulas that go with the steps of hypothesis testing for independent as well as paired samples.0069

We are going to briefly cover confidence interval for paired samples.0081

Here is the goal of hypothesis testing.0085

Remember, with one sample our goal was to reject the null when we get a sample0091

that significantly different from the hypothesized population.0098

When we talk about two-tailed hypotheses we are really saying the0102

hypothesized population might be significantly higher or significantly lower.0107

Either way, we do not care.0114

The sample is too low or too high, it is too extreme in some way.0116

If that is the case, we reject the null.0123

In two samples, what we do is we reject the null when we get samples that0125

are significantly different from each other in some way.0132

Either one is significantly lower than the other or the other is significantly lower than the one.0135

It does not matter.0141

Our null hypothesis becomes this idea that x - y either = 0 because they are the same0142

and the alternative is that it does not equal 0 because they are different from each other.0152

If they are the same that is considered the null hypotheses and when they are different that considered the alternative hypotheses.0159

Remember another way you could write this is by adding y to each side and then you get x=y.0168

X = y they are the same.0174

In that way you know that you are covering the entire space of all the differences and the end of the day0176

we can figure out whether they are the same or we do not think that the they are the same.0186

Let us talk about independent samples versus paired samples because from here on out,0195

we are totally going to be dealing with paired samples.0203

It would help to know what those are.0205

Independent samples, the scores are derived separately from each other.0208

For instance they came from separate people, separate schools, separate dishes.0212

The samples are independent from each other.0219

My getting of the sample had nothing to do with my getting of this other sample.0222

In dependent, another word for paired, in dependent or paired samples the scores are linked in some way.0227

For instance, they are linked by the same person so my score on the math test and my score on the english test are linked because they both come from me.0236

Maybe we are one married couple, we ask one spouse how many children would you like to have0248

and you ask the other spouse how many children would you like to have?0258

In that way, although they come from different people these scores are linked because they come from the same married couple.0262

Another thing might be a pre and post tests of the class.0269

Maybe a statistics class might do a pre and post test.0276

Maybe 10 different statistics classes from all over the United States picked to do a pre and post test.0279

Those tests are linked because the same class did the first test and the second test.0287

10 different classes did the pairs.0295

It is not just a hodgepodge of pretests scores and a hodgepodge of posttest scores, it is more like a neat line0298

where the pretests scores for this guy, but for this class is lined up with the pretests scores for that class.0309

They are all lined up next to each other.0317

We know these definitions, let us see if we can pick them out.0319

Which of these is which?0327

The test scores from Professor x’s class versus test scores from professor y class.0329

Will these be independent samples because they just come from different classes?0336

They are not each score is not linked in any particular way.0341

River samples from 8 feet deep versus 16 feet deep.0346

This also does not really seem like paired samples unless they went through0350

some procedure to make sure it is the same spot in the river.0355

That is probably an independent sample.0360

Male heights versus female heights, they just a jumble of heights over here and a jumble of heights over here.0364

They are not like match to each other.0370

They are independent samples.0372

Left hand span versus right hand span will in this case basically these two spans came from the same person.0375

It is not a hodgepodge like left hand right hand from person 1, left hand right hand for person2 or person 3.0384

I would say this is a paired sample.0392

Productive vocabulary of two-year-old infant often raised by bilingual parents versus monolingual parents.0395

It is a bunch of scores here and a bunch of scores here.0402

They are not lined up in any way.0406

I would say independent.0408

Productive vocabulary of identical twins, twin 1, twin 2.0410

Here we see paired samples.0417

Scores on an eye gaze by autistic individual and age matched controls.0420

Autistic individuals often have trouble with eye gaze and in order to know that you0427

would have to match them with people who are the same age who are not autistic.0432

Here we have autistic individual lined up with somebody who is their same age is not autistic.0438

They are these nice even pairs and each pair has eye gaze scores.0445

I would say these are paired samples.0452

Hopefully that give you a better idea of some examples of paired samples.0457

What about independence samples versus independent variables?0462

What you will also see is IV.0469

In multi sample statistics like 2, 3, 4 samples we are often trying to find some0471

predictive relationship between the IV and the DV.0477

The independent variable and the dependent variable.0481

Usually this is often called the test or the score.0484

The independent variable is seen as the predictor and the dependent variable0488

is the thing that is been predicted the outcome.0495

We might be interested in the IV of parent language and you might have two levels of bilingual and monolingual.0498

You might be interested in how that impacts the DV of children’s vocabulary.0519

Here we have these two groups, bilingual and monolingual.0534

We have these scorers from children and these are independent samples because0542

although we have two groups these scores are not linked to each other in any particular way.0550

They are just a hodgepodge of scores here and a hodgepodge of scores here.0556

On the other hand, if our IV is something like age of twin.0560

We have slightly older like a couple of minutes or seconds, and younger.0572

We want to know is that has an impact on vocabulary.0582

We will have a bunch of scorers for older twins versus younger twins, but these scores are not just in a jumble.0593

They are linked to each other because these are twins.0611

They are identical.0615

This is the picture you could draw and the IV tells you how you determine these groups. The paired parts tells you whether these groups scores are linked to some scores0617

in the other group for some reason or another.0640

Here they are linked but here they are not linked.0642

In all t tests, we are calling them hypothesis testing.0646

We are going to have other hypothesis tests but so far we are using t test.0657

T tests always have some sort of categorical IV so that you can create different groups0662

and in t-tests it is always technically two groups, two means, paired means.0668

The DV is always continuous.0674

The reason that the dependent variable or the scores always continuous is because you need to calculate means in order to do a t test.0678

We are comparing means too and looking at standard error and you can compute mean0687

and standard error for categorical variables.0694

If you have a categorical variables such as you know, yes or no, you cannot quite compute a mean for that.0697

Or if you have a categorical variable like red or yellow, you cannot compute a standard error for that.0707

If you did have a categorical DV and a categorical IV, you would use what it is called the logistic test.0713

We are actually not going to cover that.0721

That does not usually get covered in descriptive and inferential statistics. T0723

Usually you have to graduate level work or higher level statistics courses.0727

There are two types of t test given all of this.0735

Remember all t tests have this.0740

These are all t tests.0742

Both of these t tests are going to use categorical IV and continuous DV.0743

The first kind of t test is what we have been talking about so far, independent samples t tests.0750

The second type is what we are going to cover today called paired or dependent samples.0762

Both of these have categorical IV and continuous DV.0769

Let us have some notations for paired samples.0778

Just like before, with two sample independent sample t test, for one example,0784

you might call it x so that its individual members are x sub 1, x sub 2, x sub 3.0792

Remember each sample is a set of numbers.0797

It is not just one number but a set of numbers.0800

Second sample, you might call y.0803

I did not have to pick x and y though.0807

I could pick other letters.0809

Y could just mean another sample.0810

You could have picked w or p or n.0816

We usually try to reserve n, t, f, d, k for other things in statistics, but it is mostly by culture more than we have to do it by rules.0820

Here is the third thing you need to know for paired samples.0837

With paired samples remember x sub 1 and y sub 1 are somehow linked to each other.0842

They either come from the same person or the same married couple or0848

they are a set of twins or it is an autistic person and age matched control.0853

All these reasons why these are linked to each other in some way.0859

And because of that you can actually subtract these scores from each other and get a set of different scores.0865

That is what we call d.0872

D is x sub 1 – y sub 1.0874

What is the difference between these two scores?0877

What is the difference between these two scores and what is the difference between these two scores?0882

These are paired differences.0888

Let us think about this.0891

If the mean of x is denoted as x bar and the mean of y is denoted as y bar, what do you think the mean of d might be?0894

I guess d bar and that is what it is.0902

If you got the mean of this entire set that would be d bar.0907

Once you have d bar, you could imagine having a sampling distribution made of d bars.0912

It is not x bars anymore, sampling distribution of the mean is the sampling distribution of the mean of a whole bunch of differences.0924

That is a new idea here.0942

Imagine getting a sample of d, calculating the mean d bar and placing it somewhere here.0945

You will get a sampling distribution of d bars.0959

That is what we are going to learn about next.0964

These are means of a bunch of linked differences.0966

When we go through the steps of hypothesis testing for paired samples it is going0971

to be very similar to hypothesis testing for independent samples with just a few tweaks.0979

First you need to stay to hypothesis and often our null hypothesis is that the two groups of scores, the two samples x and y are the same.0985

Usually that is the null hypothesis.0997

You put the significance level, how weird does our sample has to be for us to reject that null hypothesis.1004

We set a decision stage and we draw here the SDOD d bar.1013

We identify the critical limits and rejection regions and we find the critical test statistic.1020

From here on out I am going to assume that you are almost never going to1027

be given the actual standard deviation of the population.1033

From here on out I am usually going to be using t instead of z.1038

Then we use the actual sample differences and SDOD in order to compute the mean differences.1041

We are not dealing with just the means, we are dealing with mean differences, test statistics, and p value.1053

We compare the sample to the population and we decide whether to reject the null or not.1061

Things are very similar so far.1069

It is going to make us figure out what SDOD is all about.1073

The rules of SDOD we are now adding on to sampling distribution of1083

the differences between means that we talked about before you.1093

We are going to add onto that.1100

The SDOM for x and y are normal then the SDOD is normal too.1103

That is the same here.1109

The mean for the null hypotheses now looks like this.1111

Remember the SDOD with the bar, the mean here is no longer called the mu sub x bar - y bar because it is actually x bar - y bar.1116

A whole bunch of them and then you find the mean of them.1132

That is called d bar.1136

That is the new notation for the differences of paired samples.1137

Here the mu of d bar for the null hypotheses equal 0.1147

Remember for independent samples = that for mu sub x bar - y bar that = 0.1153

It is very similar.1162

For standard error for independent samples when various is not homogenous, which is largely the case,1164

what we would use is s sub x bar - y bar.1174

Instead here for paired samples, we would use s sub d bar.1182

Here what we would do is take the square root of the variance of1188

the standard error from x and the standard error variance of y bar and add that together.1194

If you wanted to write that out more fully, that would be s sub x2 the variance of x / n sub x + variance of y / n sub y.1207

That is what you would do if life was easy and you have independent samples.1228

That is what we know so far.1238

What about for paired samples?1240

For paired samples you have to think about the world differently.1242

You have to think first we are getting a whole bunch of differences then we are finding the standard error of those differences.1245

Here is that we are going to do.1253

Here we would find standard error of those differences by looking at1256

the standard deviation of the differences ÷ how many differences we have.1263

This is a little crazy, but when I show you it, it will be much more easy to understand.1272

I think a lot of people have trouble understanding what is difference between this and this?1281

I cannot keep track all these differences.1287

We have to draw SDOD.1291

You have to remember it is made up of a whole bunch of d bars.1302

He is made up of a whole bunch of these.1312

You have to imagine pulling out samples, finding the differences,1314

averaging those differences together, then plotting it here.1324

Each single sample it has a standard deviation made up of differences.1328

Once you plot a whole bunch of these d bars on here, this is going to have a standard deviation and that is called standard error.1337

Here we have mu sub d bar and this standard error is standard error sub d bar.1347

Standard deviations of d bar whereas this is just for one sample.1359

This guy is for entire sampling distribution.1367

Let us talk about the different formulas that go with the steps of hypothesis testing.1378

Hopefully we can drive home the difference between SDOD from before and SDOD now, we will call it SDOD bar.1385

For independent samples, first we had to write down a null hypothesis and alternative hypothesis.1398

Often a null hypothesis was that the mu sub x bar - y bar = 0 or mu sub x bar - y bar does not equal 0 as the alternative.1408

In paired samples our hypothesis looks very similar except now we are not dealing with x bar - y bars but we are dealing with difference bars.1421

The average of differences.1438

The mean differences.1440

This is the differences of means.1442

This is mean of differences.1448

We will get into the other one.1453

Mu sub d bar does not =0.1457

This so far it seems like okay.1463

Here difference of means and d bar is the mean of a whole bunch of differences.1467

We get a whole bunch of differences first, then we find the mean of it.1484

Here we find the means first and we find the difference between the means.1489

This part is actually the same.1495

It is alpha =.05 usually two tailed.1500

Step 2, we got that.1510

Significant level, we get it.1515

Step 3 is where we draw the SDOD here.1517

Here we draw the SDOD bar.1521

Thankfully you could draw it in similar ways, but conceptually they are talking about different things.1530

Here how we got it was we pulled a bunch of x.1538

We got the mean then we pulled a bunch of y then we got the mean and subtracted those means and plotted that here.1543

We did that millions and millions of time with a whole bunch of that.1550

We got the entire sampling distribution of differences of means.1554

Here what we did was we pull the sample of x and y.1560

We got a bunch of the differences and then we average those differences and then we plot it back.1568

Here this is the sampling distribution of the mean of differences.1579

Where the mean go in the order is really important.1591

Here we get mu sub x bar - y bar, but here we get mu sub d bar.1599

In order to find the degrees of freedom for the differences here what we did was1607

we found the degrees of freedom for x and add it to it the degrees of freedom for y.1615

We are going to do something else in order to find the degrees of freedom for1620

the difference we are going to count how many differences we average together and subtract 1.1626

This is how many n sub d – 1.1637

Finally we need to know the standard error of the sucker.1644

The standard error of differences here we called it s sub x bar - y bar and that1650

was the standard error of x, the variance of x bar + the variance of y bar.1659

The variance of these two things added together then take the square root.1670

This refers to this distribution with the spread of this distribution.1676

This difference here is actually going to be called s sub d bar and that is1688

going to be standard deviation of your sample of differences ÷ √n of those differences.1696

Last thing, I am leaving off step 5 because step 5 is explanatory.1707

Step 4, now we have to find the sample t.1719

Our sample is really two independent samples.1723

We have a sample of x and a sample of y.1732

Because of that we need to find the difference between those two means.1734

We find the mean of this group first, the mean of this group and we subtract.1741

We find the means first then we subtract - the mu sub x bar - y bar.1747

I want you to contrast this with this new sample t.1756

Here we get a bunch of x and y, we have two samples.1761

We find the differences first then we average.1766

Here we find the average first and find a different.1773

Here we find the differences then we find the average.1776

That is going to be d bar.1782

D bar – mu sub d bar.1784

This is getting a little bit cramped.1790

We divide all of that by the standard error of the difference and you could substitute that in.1796

Divide all that by the standard error of the differences.1803

You see how here it really matters when you take the differences.1811

Here you find the differences first and then you just deal with the differences.1820

Here, you have to keep finding means first then you find the differences between those means.1824

Let us talk about the confidence interval for these paired samples.1830

The confidence intervals are going to be very similar to the confidence intervals that you saw before with independent samples.1841

I am just covering it very briefly.1849

Let us think about independent samples.1851

In this case, the confidence interval was just going to be the difference of means and + or - t × the standard error.1854

You need to put in the appropriate standard error and use the appropriate degrees of freedom as well.1877

In confidence intervals for paired samples it is going to look very similar except instead of having the differences of means1884

you are going to put in the mean difference d bar + or - t × the standard error.1897

Remember standard error here is going to mean s sub x bar - y bar.1906

The standard error here is going to be s sub d bar.1914

In order to find degrees of freedom you have to take the degrees of freedom for x and add that to the degrees of freedom for y.1918

In order to find degrees of freedom you have to find the degrees of freedom for d1928

your sample of differences and that equals how many differences you have -1.1935

Let us talk about examples.1945

There is a download available for you and says this data set includes the highway1953

and city gas mileage for random sample of 8 cars.1958

Assume gas mileage is normally distributed.1962

It says that because we could see your sample is quite small so we do not have1965

a reason to assume that normal distribution of the SDOM.1970

Construct and interpret the confidence interval and also conduct an appropriate t test to check your confidence interval interpretation.1974

Here I have my example and going to example 1.1984

Here we have 8 models of cars, their highway miles per gallon, as well as their city miles per gallon.1989

You can see that there is a reason to consider these things as linked.2004

They are linked because they come from the same model car.2010

Let us construct the confidence interval.2013

Remember in confidence interval what we are going to do is use our sample in order to predict something about our population.2018

Here we will use our sample differences to say something about the real difference between these two populations.2028

Here is the big step of difference when you work with paired samples.2036

You have to first find the paired differences so the set of d.2042

That is going to be one of these will take highway - the city.2048

That x1 – y1, x2 – y2, x sub 3 – y sub 3.2054

Here are all our differences and we can now find the average differences.2062

We can find the standard deviation of these differences and all the stuff.2067

Let us find confidence interval and this helps me to say what I need is my d bar + or - t × the standard error.2071

In order to find my t but in order to do that I need to find my degrees of freedom.2090

My degrees of freedom is just going to be the degrees of freedom of the d.2098

How many differences I have -1.2107

That is count how many differences they should have the same number of differences as cars -1 =7.2110

Once I have that, I could find my t.2121

I also need to find d bar.2126

Let us find t.2130

I need to find t and t inverse and I probably am going to assume a 95% confidence interval.2134

My two tailed probability is .05 and my degrees of freedom is down here and so that will be 2.36.2146

Those are my outer boundaries and let us also find d bar, the average.2157

I almost have everything I need.2165

I just need standard error.2172

Standard error here is going to be s sub d ÷ the square root of how many differences I have.2174

That is going to be the standard deviation of my differences ÷ the square root of 8 because I have 8 differences.2187

Once I have that, then I can find the confidence interval.2206

The upper boundary will be the d bar + t × standard error and the lower boundary is the same thing, except that this - t × standard error.2209

My upper boundary is that 10.6.2244

My lower boundary is that 7.6.2249

To interpret my confidence interval I would say the real difference between highway miles per gallon2253

and city miles per gallon I have 95% confidence that the real difference in the population is between 10.6 and 7.6.2264

Notice that 0 is not included in here in this confidence interval.2274

It would be 0 if highway and city miles per gallon could be equal to each other by chance.2280

There is less than 5% chance of them being equal to each other.2288

Because of that, I would guess that we would also reject the null because it does not include 0.2295

Let us do hypothesis testing to see if we do really reject the null because it does not include 02304

I would predict that we would reject the null.2312

Let us go straight into hypothesis testing here.2314

First things first.2317

The step 1, the null hypothesis this should be that the mu of d bar.2320

Here let us do hypothesis testing.2332

The first step is mu sub d bar is equal to 0.2344

Highway and city gas mileage are the same but the alternative is that one of them is different from the other.2356

That they are different from each other in some way.2366

It is significantly stand out.2369

This difference stands out.2371

That would be that mu sub d bar does not equal 0.2373

Step 2, my significance level, the false alarm rate is very low .05 and two tailed.2378

Let us set our decision stage.2392

I need to draw an SDOD bar and here I put my mu as 0 because the mu sub d bar will be 0.2397

Let us also find the standard error here.2418

The standard error here is going to be s sub d bar and that is really the standard deviation of the d / √n sub d.2421

That I could compute here.2434

Actually, we already computed that because we have the standard deviation of the d bars / the square root of how many d I have.2439

That is .64.2449

What is my degrees of freedom?2455

That is 7 because that is how many differences I have -1.2458

Based on that I can find my t and my t is going to be + or - 2.36.2466

Let us deal with our sample.2476

When we talk about the sample t, what we really mean is what the x bar of our sample differences that would be d bar.2483

I would just put x bar sub d because it is a simpler way of doing it.2502

- the mu which is 0 / the standard error which is .64.2505

I could just put this here so I can skip directly to step 4 and I will compute my sample t.2512

I should say this is my critical t so that I do not get confused.2527

My sample t is going to be d bar - mu / standard error.2533

That is d bar - mu which is 0 ÷ standard error = 14.3.2546

I can also find the p value and I'm guessing my p value is probably be tiny.2564

Here 14.3 is really small.2573

My p value is going to be t dist because I want my probability.2577

I put in my t, my degrees of freedom which is 7, and I have a two-tailed hypotheses.2586

That is going to be 2 × 10-6.2593

Imagine .000002 given this tiny p value much smaller than .05 we should say at step 5 reject the null.2610

We had predicted that we would reject the null because the CI, the confidence interval did include 0.2630

Good job confidence interval and hypothesis testing working together.2636

Example 2, see the download again, this data set shows the average salary earned by first-year college graduates.2641

Graduated at the bottom or top 15% of their class for random sample of 10 colleges ranked in the top 100 public colleges in the US.2650

Is there a significant difference in earnings that is unlikely to have occurred by chance alone?2661

We want to know is there a difference between these top 15% folks and the bottom 15% folks.2667

They are linked to having graduated from the same college.2674

We would not necessarily want to compare people from the top 15% of one college that might be really good to one2678

to the bottom percentage of people from a college that might be not as great.2687

We would really want from the same college does not matter if you are in the top 15 or bottom 15%.2693

If you go to example 2, you will see these randomly selected colleges and the earnings in dollars per year, salary per year for the bottom 15%, as well as the top 15%.2699

Because it is a paired sample what we want to do is start off with d or set up d.2718

What is the difference between bottom and top?2724

We are going to get probably a whole bunch of negative numbers assuming that top earners earn more than bottom.2729

Indeed we do, we have a bunch of negative numbers.2738

If you wanted to turn these negatives into positives, you just have to remember2740

which one you decided as x and which one you decided to be y.2745

I will call this one x and I will call this one y.2750

It will help me remember which one I subtracted from which.2759

I am going to reverse all of these and it is just going to give me the positive versions of this.2764

Here is my d.2771

Let us go ahead and start with hypothesis testing.2773

This part I will do by hand.2777

Step 1, the null hypothesis says something that the top 15% folks and the bottom 15% folks are the same.2783

Their difference is going to be 0.2796

The mu sub d bar should be 0 but the alternative is that they are different.2800

We are neutral as to how they are different.2807

We do not know whether one earns more than the other.2811

Whether they are top earns more than bottom or bottom earns more than the top.2813

We can use our common sense to predict that the top ranking folks might earn more, but right now we are neutral.2818

Step 2, is our alpha level .05 or significance level.2827

Let us say two details.2834

Step 3, drawing the SDOD, the mean differences and here we will put 0.2837

And let us figure out the standard error.2850

The standard error here would be s sub d bar and that would be the standard deviation of d / √(n ) sub d.2857

We also want to figure out the degrees of freedom so that is going to be n sub b -1 and we also want to find out the t.2871

These are all things you can do in Excel.2881

Step 3, standard error is going to be s sub d bar and that will be s sub d ÷ √n sub d.2884

That will be the standard deviation of our sample of d ÷ the square root of how many there are and there is 10.2903

Here is our standard error.2923

What is our degrees of freedom?2926

That is going to be 10-1 =9.2930

What is our critical t?2935

We know it is a critical t because we are still in step 3 the decision stage.2939

We are just setting up our boundaries.2944

That is going to be t inverse because we already know the probability .05 two-tailed,2947

degrees of freedom being 9 and we get 2.26.2953

It is + or -2.26 those are our boundaries of t.2959

Step 4, this will say what is our sample t?2966

And that is going to be our d bar – mu / standard error/2973

I will write step 4 here and so I need to find t which is d bar – mu/ standard error.2981

I need to find the bar for sure and standard error.2994

My d bar is the average of all my differences and that is about $12,000 - $13,000 a year.3000

That is just right after college.3016

I need to find the d bar - 0 ÷ the standard error to give me my sample t.3018

That is the difference between sample t and critical t.3033

8.05 is actually the average of the differences.3041

The top 15% are on average earning $13,000 more than the bottom 15%.3056

The sample t gives us how far that differences from 0 in terms of standard error.3065

We know that is way more extreme than 2.26.3075

Let us find the p value.3080

We put it in t dist because we want to know the probability.3083

Put in our t, degrees of freedom, and we have a two-tailed hypotheses.3087

That would be 2 × 10-5.3094

Our p value = 2 × 10-5 which is a very tiny, tiny number, much smaller than the alpha.3100

We would reject the null hypotheses.3113

Is there a significant difference in earnings that is unlikely to have occurred by chance alone?3118

There is always going to be a difference in earnings between these two groups of people, the top 15 and the bottom 15%.3125

Is this difference greater than would be expected by chance?3131

Yes it is because we are rejecting the model that they are equal to each other.3135

Example 3, in fitting hearing aids to individuals, researchers wanted to examine whether3141

there was a difference between hearing words in silence or in the presence of background noise.3151

Two equally difficult wordless are randomly presented to each person.3156

One less than silence and the other with white noise in a random order for each person.3160

This means that some people get silence than noise, other people get noise and silence.3166

Are the hearing aid equally effective in silence or with background noise?3171

First conduct the t test assuming that these are independent samples then conduct the t test assuming that these are paired samples.3178

Which is more powerful?3185

The independent sample t-test or paired samples t test?3188

We need to figure out what it means by more powerful.3192

I need some scratch paper here because the problem was so long I am just going to divide the space in half.3196

This top part I am going to use for assuming independent samples.3205

They are not actually independent samples, but I want you to see the difference between doing them as independent sample and doing them as paired samples doing this hypothesis testing as paired samples.3211

Step 1, the hypothesis, the null hypothesis is that if I get these sample3224

and they are independent this difference of means on average is going to be 0.3231

The mu sub x bar - y bar is going to = 0.3240

The alternative hypothesis is that the mu sub x bar - y bar does not equal to 0.3244

Here I am going to put alpha =.05, two-tailed.3252

I am going to draw myself an SDOD.3261

Just to let you know it is the differences of means.3268

Here we know that this is going to be 0 and we probably should find out the standard error.3273

The standard error of this difference of means is going to be the square roots of the variance of x bar + the variance of y bar.3283

I am going to write this out to be s sub x2/ n sub x + s sub y2 /n sub y.3303

The variance of x and x bar, the variance of x /n, the variance of y/n.3315

We will probably need to find the degrees of freedom and that is going to be n sub x – 1 + n sub y -1.3321

Finally we will probably need to know the critical t but I will put that up here.3340

Let us look at this data, go to example 3.3345

Click on example 3 and take a look at this data.3353

Let us assume independent samples.3356

Here we are going to assume that this silence is just one group of scores and3359

this background noise is another group of scores and they are not paired.3368

They are actually paired.3372

This belongs to subject one, these 2 belongs to subject 3, this belongs to subject 5.3374

Here is the list order, it is A, B.3380

We get A list first then list B and here is the noise order.3384

They get it silent first then noisy.3388

This guy gets noisy first then silent.3390

All these orders are randomly assigned and the noise orders are randomly assigned as well.3393

For this exercise, we are going to assume we do not have any of this stuff.3406

We are going to assume this is gone and that this just a bunch of scores from one group of subjects3412

that listen to a list of words in silence and another group of subjects that listen to list of words in background noise.3418

We do the independent samples t test and we start with step 3.3427

We know we need to find the standard error, which is going to be the square root of the variance of x ÷ n(x) + the variance of y / n sub y.3433

All that added together and a square root.3473

We need to find the variance of x.3476

We need to find n sub x.3479

We also need to find the variance of y and n sub y before we can find standard error.3481

Variance is pretty easy.3488

We will just call silence x and the count of this is 24.3491

The count for y is going to be the same, but what is the variance of y?3504

The variance of y slightly different.3513

In order to find this guy, the standard error, we are going to put in square root of the variance of x ÷ 24 + the variance of y ÷ 24.3521

We get a standard error of 2.26 and standard error gives it just in terms of number of words accurately heard.3547

We also need to find the degrees of freedom.3564

In order to find degrees of freedom, we need the degrees of freedom for x + degrees of freedom for y.3567

The degrees of freedom for x is just going to be 24 - 1 and the degrees of freedom for y is also going to be 24 – 1.3574

The new degrees of freedom is 23 + 23 = 46.3586

Once we have that we can find our critical t.3593

Our critical t, we know that alpha is .05 so we are going to put in t in and3606

put in our two-tailed probability and the degrees of freedom 46.3615

We get a critical t of + or -2.01.3620

Our critical t is + or -2.01.3625

I will just leave that stuff on the Excel file.3631

Given all this now let us deal with the sample.3636

When we find the sample t what we are doing is finding the difference in means and then find the difference3641

between that difference and our expected difference 0 and divide all of that by standard error to find how many standard errors away we are.3653

Here I will put step 4, sample t.3666

In order to find sample t we need to find x bar - y bar - mu and all of that ÷ standard error.3672

Thankfully, we have a bunch of those things available to us quite easily.3697

We have x bar, we can get y bar, we can get standard error.3701

Let us find x bar, the average number of words heard accurately in silence and that is about 33 words.3708

The average number of words heard correctly with background noise, and that is 29 words.3723

Is the difference of about 4 words big enough to be statistically different?3732

We would take this - this and we know mu = 0 so I am going to ignore that / standard error found up here.3741

That would give us 1.75.3754

1.75 that is more extreme than + or -2.01.3758

1.75 we will actually say do not reject.3765

We should find the p value too.3769

This p value should be greater than .05.3773

We will put in t dist then our sample t, degrees of freedom which is 46 and we want a two tailed and we get .09.3777

.09 is greater than .05.3790

Step 5, fail to reject.3797

Now that we have all that, we want to know is it more sensitive?3809

Can we detect the statistical difference better if we used paired examples?3821

Let us start.3829

Here we would say p =.09 and 5 is failed to reject.3831

It is not outside of our rejection zone, it is inside our fail to reject zone.3846

Let us talk about the null hypotheses here.3855

What we are going to do is find the differences first then the mean of those differences.3860

We are saying if they are indeed not that different from each other that mean different should be 0.3866

The alternative is that the mean difference is not equal to 0.3871

Once again alpha = .05 two tailed and now we will draw our SDOD bar which means3877

it is a standard sampling distribution of means mean made of differences.3892

Here we want to put 0.3909

We probably also want to figure out standard error somewhere along the line,3919

which is going to be s sub d bar which is s sub d ÷ √n sub d.3924

We probably also want to find the degrees of freedom, which is going to be n sub d -1.3933

We probably also want to find the critical t.3941

Let us find out that.3945

Here I will start my paired samples section.3949

I will also start with step 3.3955

Let me move all of these over here.3957

Let us start here with step 33965

Let us find standard error and that is going to be s sub d not d bar ÷ √n sub d.3970

We can find s sub d very easily and we could also find n sub d.3986

First we need to create a column of d.3994

I will find the standard deviation of the d but I realized that I do not have any d.4002

The d look something like this silence - background noise.4008

This is how many more words, they are able to hear accurately in silence and background noise.4020

Here we see that some people hear a lot of words better in silence.4026

Some people here words better with a little bit of background noise.4032

Some people are exactly the same.4035

We could find a standard deviation of all these differences.4037

We could also find the mean of them4045

The n of them will be the same as 24 because there are 24 people that came from.4053

There is 24 differences.4062

We could find out standard error.4065

Standard deviation of d ÷ √24.4070

That is standard error, notice that is quite different from finding a standard error of independent samples.4076

Let us find degrees of freedom for d and that is going to be n sub d -1 and that is 24 -1.4086

Our critical t should be t inverse .05 two tailed=23 and we get 2.07.4101

So far it seems that our standard for how extreme it has to be is more far out.4127

That makes sense because the degrees of freedom is smaller than 46.4134

+ or -2.07.4139

Let us talk about our sample.4152

In order to find our sample t, we want to find the average of difference subtract from the hypothesized mu4154

and divide all of that by standard error to find out how many standard errors away our sample mean difference is.4169

We also want to find p value.4179

Here is step 4, our sample t would be d bar - mu ÷ standard error.4181

What is d bar and how would we find it?4196

Just use the average function and the average of our d like this d bar.4205

We can do d bar -0 / standard error= 2.97.4211

That is more extreme than 2.06.4226

Let us figure out why.4233

We might look at standard error, the standard error is much smaller and the steps are smaller.4235

How many steps we need to take to get all the way out to this d bar?4251

There is more of them than these the bigger steps.4257

These are almost twice as big.4261

These bigger steps, there is few of them that you need.4263

That is what the sample t I get is how many of these standard errors,4267

how many of these steps does it take to get all the way out to d bar or x bar – y bar?4273

We need almost 3 steps out.4279

What is our p value?4282

Our p value should be less than .05 that is going to be t dist.4287

Here is our t value I will put in our degrees of freedom and two tailed and its .007.4293

That certainly less than .05.4303

Step 5, here we reject whereas here we fail to reject.4306

Since there is this difference and we detected it with this one but not with this one,4313

we would say that this is the more sensitive test given that there is something to detect out there.4321

This is the difference if it does exist.4328

This one is a little coarser, there is a couple of reasons for that.4331

One of the reasons is because the standard error are usually larger than the standard error of differences.4336

Another issue is that x bar - y bar, the difference here if we look at x bar - y bar this difference is roughly around the same.4343

This difference is the same as this difference.4362

It is not that bad but it is that you are dividing by a smaller standard error here then you are here.4366

Here, the standard error is quite large.4373

The steps are quite large.4375

Here, the standard errors are small.4376

The steps are quite small.4379

It is because you are taking out some of the variation caused by having some people4380

just being able to hear a lot of words accurately all the time with noise.4385

Some people are very good at hearing anyway.4392

They might have over a low number of scores but with d bar you do not care about those individual differences.4397

You end up accounting for those by subtracting them out.4405

Here this is a more sensitive test.4409

Here we get p=.006 and we reject.4412

Which test is more sensitive?4425

Which test is able to detect the difference, if there is a difference?4431

Paired samples.4435

That principles are little more complicated to collect that data but it is worth it because it is a more sensitive test.4436

Thanks for using