Statistics Introduction to Hypothesis Testing

Hi and welcome to www.educator.com.0000

We are going to be talking about hypothesis testing today.0002

The first thing we need to do is situate ourselves where do hypothesis testing fit in with all of inferential statistics.0005

We are going to talk about how to create the hypothesis that we are going to test and that hypothesis is going to be about a population.0015

When we say about a population we mean about population parameters.0023

There is actually two parts to any hypothesis that we test.0028

There is the no hypothesis and the alternative hypothesis.0033

We are going to talk about how they fit together.0036

We are going to talk about potential errors in hypothesis testing because it is good to know going into it.0039

Finally, we are going to end with the steps of hypothesis testing and we are going to do the steps of hypothesis testing,0045

When sigma the population standard deviation is given and when it is not given.0051

And if you had just refresh yourself with the confidence interval lesson,0057

You can probably guess that when sigma is given we are going to be using z distributions or normal distributions.0063

When sigma is not given and we have to estimate the population standard deviation from the sample using s then we will use t-distributions.0072

In order to use the t distribution we need to figure out the degrees of freedom.0086

Let us go back and situate ourselves with all of inferential statistics.0094

Basically the idea of inferential statistics is that we use some known populations to figure out the sampling distribution.0101

The one that we are using a lot is the SDOM.0115

We are going to use the another one later.0121

We figure out sampling distributions and now we want to compare a sample from an unknown distribution.0123

We want to compare sample from that to the sampling distribution.0136

If the sampling distribution says the sample is very likely then we might say maybe the sample,0145

this unknown population is very similar to the known population.0154

But if the sampling distribution tells us the sample was very unlikely then we could rule out0159

the known population as a potential candidate for this unknown population.0169

In doing all of this in inferential statistics there are two issues that come up.0176

What happens when we do not know what the population looks like at all and0183

We want to try to figure out where the population mean or different parameters of the population might be.0188

In that case we use confidence intervals and when we use confidence intervals we try to figure out where mu is from x bar.0195

Another way of thinking about it is we try to figure out something about the population0211

From the sample information because we have that sample information.0216

Another technique that we could take is that we could use this idea and say how do we decide when a sample is unlikely?0220

How do we decide when to draw x?0235

When do we decide this side is weird?0238

In order to do that we now have to learn about hypothesis testing.0243

The goal of hypothesis testing is to be slightly different from confidence interval yet related.0248

It is the flip side of the coin.0254

Basically, you are going to try to figure out whether your x bar is unlikely given a hypothetical population.0256

In that case, what we are doing is we are setting up a population.0281

It is like the population is stable and we are going to compare the sample to it.0290

Here is our sample and here is our set standard.0296

Here the population is moving but this is the target and this is what we use to get that target.0305

Here this is already set and we are comparing this guy to this guy.0316

In this way you need both confidence intervals and hypothesis testing to give you the full story.0323

You might also hear that hypothesis testing another word or phrase for it will be a test of significance.0331

A lot of students misinterpret that to be a test of importance.0343

That is the modern way the word significance is used but that is not actually what we are talking about here.0348

When we call this at test of significance this is actually using the meaning of significance0354

from the early 20th century when this test was actually invented.0367

Back then significant adjustment prominence or standing out.0370

I like to think of it as being weird like how much does this sample stand out?0377

Is that significant?0386

Is it prominent and different or is it very, very similar?0387

Those are the ways you could think about it.0392

I do not want to think of it as a test of importance.0398

Now that we know why we need hypothesis testing, how do we hypothesize the population?0401

How do we make up a population?0411

Do we have to make up all the individual numbers of the population?0413

What do we got to do?0415

Here is the thing, we could assume things about population parameters and test those assumptions.0417

We do not have to stimulate every single member of the population we could just make some assumptions about parameters.0424

In order to set up a hypothetical population you set up a parameter.0431

For instance, you say mu is equal to something.0437

That is how you set up a population then check whether our sample is likely to have come from such a population.0440

In doing this we need to figure out how to we hypothesize rigorously so that we could get as much paying for our book from our hypothesis?0448

In order to do this we have two parts to a hypothesis and this is going to make our hypothesis better.0462

The first part of hypothesis is what we call the null hypothesis and null means 0 or not important.0472

The null hypothesis in this case is your hypothetical population.0487

We write the null hypothesis like this h sub 0 or h sub knot.0492

We might say mu= 0.0502

We have created a null hypothesis.0507

I just made up to 0 but there are better ways of doing this and we will talk about those later.0510

We could also write this in terms of standard deviation or other things but frequently0516

you will see the mean being the hypothesis of the population.0532

The alternative hypothesis is what do we learn if this is not true?0536

If we rule this out then what have we learned?0544

In that way these two make up the full hypothesis.0548

If we find this then we learn this.0554

If we do not find that we learn this other thing.0557

What we learn if this is not true is at least that mu does not equal 0.0560

This is called the alternative hypothesis and it helps us at least figure out something when we do not figure out that.0566

If we do not find this to be true at least we find this to be true.0575

If this is not true then we will always find this to be true.0580

These two hypotheses together this is more powerful than just having one hypothesis alone.0584

We will talk a little bit about why and it goes back to that idea of the test of significance.0597

Hypothesis testing or the test of significance is a test of weirdness.0607

It tests how weird the x bar is.0617

This is the question that it can answer is the x bar weird?0625

Is it different from the population?0633

But it actually tell is x bar very similar to the population?0636

That is not what number gives you but only tells you how weird it is.0642

It does not tell you how similar it is.0646

These are actually not flip sides of the same coin and because of that our goal here in all0648

of hypothesis testing is we find out the most when we reject the null hypotheses.0658

That is when we would find out the most.0668

This may not seem like we are finding out of luck because we ruled out 0.0671

There is an infinite number of mu that we need to test but actually in hypothesis testing0676

what you want to do is reject the no rather than accept or fail to reject the null.0682

Just because it is set up as a test of weirdness that is the only thing you can find out.0690

It is true that it would be nice if we can find out more than that but that is the limitation of this hypothesis testing.0695

It is a limitation that is also like the fact of life because even as the limitations this hypothesis testing still a powerful tool.0703

But it is good to keep in mind that this one is a limitation.0714

A little bit more about these two hypotheses.0716

These two hypotheses, the null and the alternative, sometimes you might see the alternative written as h sub 1.0722

They must be mutually exclusive.0729

This means if one is true the other cannot be true.0732

If the other is true, the first cannot be true.0736

You cannot have a null hypotheses and alternative hypotheses like mu=1 and mu=2.0739

They are not mutually exclusive.0748

If one is false, the other one does not have to be true.0751

It could be true but it does not have to be.0755

Whereas mu does not equal 1, mu = 1.0758

Those are mutually exclusive.0763

If you rule out one you absolutely know that the other one has to be true.0764

Together they must include all possible values of the parameter.0768

You can think of the parameters such as mu on a number line and you need to cover the entire number line.0774

You can have a null hypothesis like mu > 0.0782

You might say mu >0 but then your alternative hypotheses have to be mu < or = 0.0788

You color that in and color all of that in too because that is where you will cover the entire space, the parameter space.0800

If these are both true, here is what you get.0811

One of these two hypotheses must represent the true condition of the population.0815

You find out something that is true about the population and then as we said before,0821

typically in research your goal is to reject the null and find support for the alternative hypothesis.0827

You can actually prove the null hypothesis but you can reject the null hypotheses.0833

And the whole reason is because hypothesis testing is a test of significance or test of weirdness.0838

This x bar stands out.0848

You can only tell me whether it stands out a lot from the population or not.0851

They can tell me it is probably similar to the population.0856

You cannot tell me that part.0860

Let us talk about some errors that we could potentially make in hypothesis testing.0862

There are some foibles, you need to watch out for.0868

Well first, it helps to imagine that there are two potential realities and we do not know which one of them is true.0871

One is that the null hypothesis is true.0883

It is actually true.0887

We do not know yet, but it is true.0888

Other possible reality is that the null hypothesis is false.0892

Your sample did not come from the population.0898

Those are your two possible realities but only one can be true at any given time.0901

You cannot have both the null population being true and false at the same time.0907

You got to have one or the other.0915

These two boxes, this one and this one have to add up to 100%, but these two boxes , this one and this one have to add up to 1.0916

That is because we have a 100% possibility of this being true and 100% possibility of this being true.0934

If this is true then this is not true.0942

Given that this is reality but we do not know reality, what is the deal?0944

How do we put that together with hypothesis testing?0955

When we do have hypothesis testing we have 1 of 2 outcomes.0959

We could either reject the null successfully, that is what we wanted to do.0964

We could either reject the null or we can fail to reject the null.0968

We do not call this accept the alternative or accepting the null.0972

We call it failing to reject because that is how much we wish we could have rejected the null.0980

We failed to reject the null.0987

Let us think about these two decisions in conjunction with reality.0989

Here is the thing, when we reject the null hypothesis and say this sample did not come from the population.0997

If it did not come from that population we would be correct here.1006

This would be a correct decision.1011

If this is our decision and this is indeed the world we live in, this is a correct decision.1014

If we fail to reject the null however but the null is actually true we should not have rejected it1021

then this also represents a correct decision.1034

Good job not rejecting the null because it is right all along.1039

These two are ways that we could be correct.1044

That leaves us two ways that we could be incorrect.1048

One way is this, we could successfully reject the null but the null is actually true but1051

we said that it is false but the null is actually true.1063

This is an incorrect decision.1068

We call this a false alarm because we are rejecting that now.1074

It is false alarm we should have not rejected that null.1084

The probability of that false alarm is represented by the term alpha.1088

On the other hand, there is another way that we could be wrong and that way is this.1097

We could fail to reject the null.1107

We could say we may not be wrong.1109

We fail to reject it but the null is wrong.1114

This is also an incorrect decision.1121

This is not called a false alarm instead it is called a miss.1127

This is going to be called the beta rate.1134

Obviously the alpha and the beta have a probability of less than 1, but greater than 0.1143

What we want to do in hypothesis testing is reduce our chance of errors.1150

We can also figure out what is our probability of getting different kinds of correct decisions?1157

We know that this is one version of the world and that should add up to 100% this probability of failing to reject when we should have kept it around.1167

This probability is 1 – alpha.1183

This is what we call a correct failure.1188

It sounds odd but it sounds good that you have failed.1198

You failed to reject it and you should have failed to reject it.1203

It is like you failed to reject a date and you know that date was really good.1208

He is a good guy so you should have failed to reject him.1216

On the other hand, this is another possible set of what could be right in the world.1225

This should add up to 100%, so this should be 1 – beta.1232

That is our rate of correct decision where we successfully rejected the null and it is indeed false.1238

In dating it might be reject somebody who comes up to you and good job you should have rejected them.1245

They are a total loser.1253

That is what we call a hit.1255

It is like in a battleship when you hit it.1258

This is the hit rate, miss rate, false alarm rate, and the correct failure rate.1263

Let us talk about the steps of hypothesis testing.1272

Well there are going to be 5 steps.1281

The first step just starts out with setting up your hypothetical population.1284

This is the hypothetical population and you need to create both a null hypothesis and an alternative hypothesis then pick a significance level.1290

You can think of the word significant as a stand outness like how much it standout.1304

How much does it have to standout?1310

When it stands out a lot you have a very low false alarm rate.1313

If your x bar is out there and then you have a small chance of false alarming.1318

You are saying this really does not look like it belongs in the population because it is so out here.1326

And that is where your false alarm rate is low.1335

You want to set a low one.1338

If you want to be more conservative, you want to set an even lower false alarm rate.1340

For instance, alpha = .01 that would be even lower rate of false alarm.1344

Then you want to set a decision stage.1351

So far, we have not done anything except like setting things up yet and still we are setting things up.1355

We set up the decision stage and what you want to do is draw the SDOM, the sampling distribution.1361

We have the hypothetical population and we create a sampling distribution so that we can take our sample1368

and compare it to that sampling distribution.1375

You draw the SDOM and you identify the critical limits.1378

Here is my SDOM and you want to identify the extreme regions where you say if your x bar1383

is somewhere out here then you want to reject the null.1396

You want to say it is very, very unlikely to have come from this null population.1402

Then choose a test statistic because the test statistic will tell you how far out from the mean it is in terms of standard error.1407

How many jumps out you are?1419

This will be called choose a critical test statistics.1421

You are saying what are the extreme boundaries such that if x is outside those boundaries we reject it.1429

If it is inside the boundaries we do not reject.1440

And then we use the sample.1444

This is the first time we are doing anything with the sample.1447

We use the sample and the SDOM from here to compute the sample test statistic and p value.1450

And the p value is going to tell you given that x is out here how much of that curve does it actually cover?1458

What is the probability of false alarming there at that particular value?1468

And then you compare the sample to this SDOM population and you decide to reject the null or not?1476

One word about p value versus alpha.1487

The p value is going to be the probability of belonging to the null population given sample x bar.1494

What is the probability that this value belongs in here?1513

Alpha is what we call the critical limit.1519

This is what we are able to tolerate we just set it.1526

Alpha is often decided just by the scientific community.1532

In fact alpha is often set to something like .05 or .01 because that is commonly accepted in scientific communities.1536

We call that just being by tradition or convention.1546

It is not that we figured out the alpha level.1550

On the other hand we figure out the p value level given our sample x.1553

And what we want is for the p value to be lower than the critical limit.1559

Let us go through some examples.1566

Here is an example of single sample hypothesis testing, also called t tests of 1 mean or single mean t test.1572

This is also another term for it.1594

Let us talk about this when sigma is available.1597

The population standard deviation has been given to us.1601

Here it says that the average Www.www.facebook.com.com user has 230 friends, a sigma of 950, a random sample of college students n=39 showed that the sample mean was 393 friends.1605

Our college students like the average www.www.facebook.com.com user.1620

Let us try to think about this by using hypothesis testing.1624

The first thing is perhaps we should set up the best standard population as the average www.www.facebook.com.com user,1631

the real population of all Www.www.facebook.com.com users.1643

Our null hypothesis might be something like mu= 230.1648

That the null hypothesis is that our college students sample is just like everybody else.1655

The alternative hypothesis is that our samples are not similar to that population.1667

Let us set the significance level.1678

Here we could just use alpha = .05 by convention.1683

We could say that is traditional, we will use that too.1693

Let us set the decision stage.1698

Here we want to start off by drawing the SDOM and I like to label for myself that it is the SDOM1701

just so that I do not get confused and mistake it for the population or something like that.1711

We want to draw a critical limit.1717

If this is the only false alarm that we are willing to tolerate then we might say everything out here we reject.1721

Everything out here we reject.1730

That would mean that everything in here is 95% and out here these two regions together add up to 5%.1734

Because we are going to reject it there is still some probability that this sample belongs to the population.1745

But we are going to reject the null.1751

We need to split up 5% distributed to both sides so this would make this 2.5% and this would be also 2.5%.1754

That is the error that we are going to tolerate.1768

I will color n right now my rejection regions so that means if it out here in the extremes I am going to reject my null hypothesis.1771

And because we know that this SDOM comes from the population, that is how we are creating this SDOM.1783

We know that the mu of SDOM is exactly equal to the mu of the population so that will be 230.1792

Mu sub x bar = 230.1801

We can also figure out the sigma sub x bar and that would be just sigma ÷ √n Which is 950 ÷ √239.1805

You could just pull out a calculator to do this.1819

I am just going to use the blank Excel file and here is 950 ÷ √239= 61.5.1823

That is my standard error of this population.1839

And what I want to know is it is nice to have that but if it would also be nice to know what is the z score out here?1848

We use z score because we are using sigma.1856

What is the z score out here?1861

Actually I had just made you memorize it when we previously talked about confidence intervals so we know that is 1.96 and -1.96.1864

If you wanted to you could also figure it out by using either the table in the back of your book or Excel1876

so we could put in normsin because we have the probability.1885

I want the two tailed probability this is actually one tailed.1890

The one tailed probability is going to be .025 way down here.1902

This little bottom part down here it is covered .025 of this and Excel is telling me that the z score right there is about 1.96.1910

Now that we have all of that settled, we could start tinkering with our actual sample.1924

Let me draw some space here.1933

Let us talk about our sample.1938

When we talk about our sample we should figure out how far away is our sample mean?1942

We just do not want to know in terms of how far away they are in terms of friends but we want to know1955

how far away in terms of the standard deviation because only standard deviation will tell us what proportion of the curve is colored.1962

Even if we find out the actual raw distance away 163, we do not know where that is in relation to this curve.1971

It would be nice if we could find the z score of 393 then we will know where it is in relation to this curve.1983

That would be 393 – 230 so how far is it away from 230, all divided by the standard error 61.51990

because that will give me how many standard errors away we are.2002

Let me just calculate that.2007

That would be 393 - 230 and I need parentheses because I need it to do the subtraction before the division and that gives me 2.65.2011

My z score is 2.65.2032

Here this maybe 1 z score away, this is almost 2 z scores away and let us say this is 3 z scores away.2036

I know that my 393 is somewhere around here because it is around 2.65.2049

This area is very tiny, so I need to find the p value here.2061

What is the p value here?2070

What is the probability that x bar is greater than or equal to 393?2072

That equals the probability that z is greater than or equal to 2.65.2091

Not only that but remember we have a two tailed hypothesis.2100

We are interested in either being greater than or less than the mean.2106

We actually have to find this thing out and multiply it by 2.2112

What you can do is look this up in the back of your book and multiply it by 2 or Excel will actually calculate it for you2117

like you could put in normsdist and put in the negative side because normsdist gives it to me going from the negative side to positive side.2128

I am going to color this part first.2143

-2.65 and it should be a very tiny number that will be .004.2144

That is a tiny number and then we take that one side and we multiply it by 2 to give us our p value.2153

What we are really doing is we are coloring this base, pretend that is inside and also getting -2.652160

and coloring that space and adding those two together.2179

That will give us .008.2183

What about a single sample hypothesis test when sigma is not available?2188

Well this is the exact same problem in fact I have crossed this out so you can no longer use it.2201

It is no longer available to you.2208

Here what we have to do is estimate sigma and use s instead of sigma.2212

Let us go ahead and start off just hypothesis testing.2219

Our null hypothesis is mu=230 that are our sample of college students is just like everybody else.2222

Our alternative is that they are different from everybody else.2233

Different in some way, either have more friends or less friends.2239

We also need to pick a significance level.2244

How extreme does this x bar have to be?2248

We are going to pick alpha=.05 just by convention we do not figure it out or anything.2255

And then we need to set our decision stage.2260

Here we want to start off by drawing our SDOM helps to keep this in mind that this is a bunch of means, a bunch of x bars.2264

We can just use this information because this is our known population.2276

We are going to use that information to figure out our SDOM.2284

Here we run into the problem how can we figure out standard error?2288

Well, we cannot figure out sigma sub x bar but we can actually figure out s sub x bar.2294

That standard error using s instead of sigma.2302

That will be s(x) ÷ √n.2307

We have s for more sample, the standard deviation of our sample which is 447 ÷ v239.2316

And I will just pull out my Excel in order to calculate this.2326

447 ÷ v239 and I get 28.9.2346

I am actually going to draw in my rejection regions, anything more extreme is going to be rejected.2356

Fail to reject in the middle and this rejection region is .025 and this rejection region is .025 because2375

I need to split that significance level in 2.2389

What we do here is we want to figure out what is our actual t statistic?2393

How many standard errors we are when we talk about these borders?2404

What is our critical t?2408

That would be the t values here.2410

This is our raw values in terms of friends but we want to know it in terms of standard error.2413

Here are our t values so we cannot just put in 1.96 because that would be for z distributions.2418

We need a t distribution and in order to find a t distribution we need degrees of freedom.2426

The degrees of freedom is n-1 and that is 238 because 239 – 1.2434

You can either look this up in the back of your book or I am going to look this up on Excel.2443

Here I am going to use my t inverse and I put in my two tailed probability .05 and my degrees of freedom which is 238.2451

And I get 1.97.2465

1.97 and -1.97 because t distributions has many problems as they have they are perfectly symmetrical.2470

Those are critical t.2485

That is the boundary t values.2488

Now we have all of that, now we can start thinking about our sample.2491

Let us think about our samples t and p value.2499

The sample t would be the distance that our sample is away from our mean ÷ standard error because we want how many standard errors away we are.2505

393 - 230 ÷ standard error 28.9.2523

I will put that into my Excel 393 – 230 ÷ 28.9 = 5.6.2532

Let us find the p value there.2546

We know that it is far out here our t value so this is about 2, 4, 5.6.2552

It is way out here.2560

Imagine this going all the way out here.2562

That is where x bar landed.2565

Already we know that it is pretty far out but let us find the precise p value there.2569

In order to find the p value we want to use t dist because that is going to give us the probability.2577

We put in the x and that is Excel's word for t.2583

When you see x here in t distribution just put in your t value and it only accepts positive t values.2588

I will just point to this one, our degrees of freedom which is 238 and how many tails?2600

We have a two tailed hypothesis.2609

We get 4.8 × 10^-8 so that would be our p value.2612

Our probability of getting a t that is greater than or equal to 5.64 or t is less than or equal to -5.64 because it is two tailed equals 4.8 × 10^-8.2624

Imagine .07 × 48 and so that is the pretty number.2658

This number is so small that they cannot even show you the decimal places.2669

It is super close to there but not 0.2677

This is our p value, is the p value less than .05?2680

Indeed it is.2686

What do we do?2688

We reject the null hypothesis.2691

This is what we do when sigma is not available.2695

Just to recap about alpha versus p value.2702

P value is the probability of seeing that sample t or an even more extreme statistic given that the null hypothesis is true.2709

And we say extreme because they can be like way bigger or ways smaller either side right.2720

Alpha gives you the level of significance.2729

That level of extremeness that you have to reach in order to reject your null.2733

This is the set standard.2739

And this is the thing that you are going to compare to that set standard.2742

I want to talk briefly about one versus two-tailed hypotheses.2751

When we talk about a one tailed hypothesis, you might have something like mu is going to be greater than 0.2757

Or your alternative will be mu is less than 0.2768

If that is the case and your set alpha level is .05 then here is what you would do in your SDOM.2777

You will only use one side of it because you are not interested if your x values are way up here.2786

You only care if your x value is way smaller than your population.2798

In this case, you might set up this as your rejections zone and notice that it only on one side because one tailed and these are end tails.2805

That probability will be .05 and this failed to reject side will be .95.2817

This is a one tailed hypothesis.2830

Frequently we will be dealing with two tailed hypotheses.2833

In that case that might be that you do not really care.2838

We do not really care if mu is less than, way smaller or way bigger than what we expected.2845

We just care if it is extreme in some way, different in some way.2854

We do not really care which way and that would be mu = 0 and the alternatives is that mu do not = 0.2858

If we had something like alpha = .05 in a two-tailed hypotheses then we would split up2868

that rejection region into the two-tails so that will be .025 and .025.2879

We reject , we reject, but inside of these boundaries we fail to reject and this is 90.95%.2889

Whatever p value you find we want to compare it to the set alpha level.2906

Let us talk about some examples.2915

Your chemistry text book says that if you dissolve table salt and water the freezing point will be lower than it is for pure water 32°f.2920

To test this theory, your school does an experiment with 15 teams of students dissolved salt and water and put them in the freezer with the digital thermometer.2931

Periodically checking to observe the temperature at which the solution freezes.2940

The data is shown in the download below.2945

What can you conclude from this data?2948

If you look at your download and go to example 1, here are all my freezing temperatures that each of my teams got2951

and I think there are only 14 teams here.2963

Let us suggest that to be 14.2967

What should we do first?2969

Just to give you an example of what it is like to do one tailed hypothesis testing, let us have a one tailed test here.2973

Because it does say that putting the salt and water the freezing point should be lower2982

that automatically gives us a direction that we expect, the freezing point to go in.2990

What would our null or default hypothesis be?2999

The default hypothesis would be that it is not different from pure water.3004

They are the same.3010

It might be something like mu=32°f.3011

But do we care if our samples are all greater than 32°?3019

Maybe the freezing point is higher.3028

Do we really care about that?3032

No not really.3035

Null hypothesis is really that we do not care if it is anything higher than or equal to 32°.3037

What we eventually want to know is it lower like weird in this low direction.3051

The alternative hypothesis is that it is weird, but in a particular direction that it is too low way lower than 32°.3058

Our Alpha is going to be .05, but let us make it clear that it is one tailed.3071

Usually they do not say anything but most people assume two tails as the default.3079

Let us say one tailed.3086

Let us draw this SDOM for the decision stage and here is idea.3088

The default is that all the samples come from a population with 32° is the mean of this SDOM but3096

we want to know is it weird and a lot lower than that?3113

It is consistently lower than that.3126

That is our rejection region and that rejection region is going to be .05 because our fail to reject region is going to be .95.3128

Now that we have that it would be useful to know what our t statistic here.3144

This is raw in terms of degrees Fahrenheit.3150

We also want to know the t statistic.3156

Here at 0 what is the t statistics here that looks like boundary?3159

In order to know that we need to figure out a couple of things.3164

I will start with step 3, one of the things I want to know is that t statistics there.3168

In order to find that t statistics we need to know degrees of freedom for the sample and that is just account how many axis we have in our sample -1?3179

That is 13° of freedom.3193

What is the t value there?3196

We have the probabilities and we want to know the critical t or boundary t.3199

In order to know that we need to use t in here it asks for a two tailed probability.3212

We need a one tailed hypothesis so we have to turn that into a two tail probability.3221

If this was a two-tailed it would it be .1 and the degrees of freedom is 13.3228

It will only give you the positive side, but we could just turn it into -1 because it is perfectly symmetrical.3237

This critical t is -1.77.3248

Okay, now that we have that, we can start on step 4.3252

Step 4 deals with the sample t.3259

In order to find the sample t we probably need to find the mean of sample and that is average and we probably also need to know the standard error.3264

In order to find standard error what we need is s ÷ √n.3289

It is not like for Excel, this is just for me as I need to know s.3299

What is my s?3304

That would just be stdv in all of these.3308

Once I have that then I could calculate standard error s ÷ √n Which is 14.3314

We have a standard error, we have a mean, now we can find our sample t3327

and that is going to be the mean of the sample - the hypothesized mu 32 ÷ the standard error.3334

I get -3.7645.3347

We know that this is much more extreme on the negative side than -1.77.3354

We also need to find the p value.3363

What is the p value there?3366

We need to use pdist because we do not know the probability there.3370

We put in our t value but remember Excel only accept positive one and I am only going to put so two – is +.3376

The degrees of freedom, which is 13 up here and how many tails?3390

Just one.3398

That is going to be .001 p value.3399

Since I have ran out of room I will just write the p value here so p = .001.3407

Is that p value smaller than this alpha?3416

Yes, indeed.3420

What can we say?3421

We can reject the null.3424

What can I conclude from this data?3426

I can say that this data shows that it is very unlikely to come from the same population as pure water.3430

The freezing point of water will have a variation.3445

It will have some probability of not being exactly 32 and this deviation on the negative side is much greater than would be expected by chance.3449

Let us see.3461

Example 2, the heights of women in the United States are approximately normally distributed with a mean of 64.8 in.3465

The heights of 11 players on a recent roster of the WNBA team are these in inches.3472

Is there sufficient evidence to say that this sample is so much taller than the population that3479

this difference cannot reasonably be attributed to chance alone?3485

Let us do some hypothesis testing.3489

Here our null hypothesis is that our sample is just like regular women.3493

The mean is 64.8.3500

I am going to use a two tailed alternative here, is that they are not like this population.3504

We can probably guess by using common sense that they are on average taller, but we will do a two-tailed test.3514

It is actually more conservative.3522

It is safer to go with that two tailed test.3525

Here we will make alpha=.05 and it will be two-tailed.3527

Let us draw the SDOM here.3536

Here we might draw these boundaries and because it is two tailed this is .025 .025 and here it is .95.3542

All together it adds up to .1.3565

Now that we have this can we figure out the t?3568

In order to figure out the t, we need to have the degrees of freedom.3575

If you go to the download and go to example 2, I have listed this data here for you and we can actually find the degrees of freedom here.3579

Here I put step 3 so that we know where we are.3590

In step 3, we need degrees of freedom and that would be count of all of these guys -1.3596

We have 11 players 10° of freedom.3606

Let us find the critical t.3610

The critical t would be t inverse because we know the two tailed probability .05 and the degrees of freedom.3613

That gives us the positive critical t.3626

That is 2.23 and -2.23 those are our critical boundaries and anything outside of that, we reject the null.3629

Let us go to step 4.3640

In step 4 we can start dealing with the sample.3643

Let us figure out the sample t in order to do that we need the x bar - the mu ÷ standard error.3646

We need to know the samples average x bar.3656

We also need to know mu and we also need to know standard error.3663

Standard errors is going to be s ÷ √n.3669

I need to write these things down because it helps me figure out what we need.3674

It is like a shopping list.3679

Here I need s.3680

Now that I have written all these things down I can just calculate them.3684

I need the average and mu which I already know from the problem 64.8.3688

I need to get my standard error but before I do that I need to get s standard deviation3709

and 1 standard deviation I can take that and ÷ the square root of n which is 11.3718

That is my standard error and once I have all of these ingredients, I can assemble my t which is x bar – mu ÷ standard error.3730

I get 7.97 and that is way higher than 2.2.3746

I am pretty sure I can step 5, reject the null.3755

If I go back to my problem, then let me see is there is sufficient evidence to say that this sample is so much taller than the population,3763

that this difference cannot be reasonably attributed to chance alone.3776

I should say yes because when you are way out here, your probability that you belong to this chance distribution is small3780

that it is reasonable for us to say that the sample came from a different population.3793

Final example, select the best way to complete the sentence.3802

The probability that the null hypothesis is true, that is a false alarm rate.3810

It is when the null hypothesis is true, but also it is not just that.3824

It is not just the possibility that the null hypothesis is true it is that given that you have a particular sample it seems to leave some information.3835

It is not quite complete, but it is not entirely false.3850

It is just that it does not have the whole truth.3856

It does not have the condition.3859

Given that you have this particular sample value, the probability that the null hypothesis is false, that is not true.3861

Even if you just remember this.3870

Remember this column was null is true.3873

Alpha is the set one but the p ones are the ones in there.3877

That is just not true.3885

The probability that an alternative hypothesis is true.3889

Actually, we have not talked about that at all.3895

We only talked about having a very low possibility that the null hypothesis is true,3898

but we have not talked about increasing the probability that the alternative hypothesis is true.3905

Beside why would you reject the null when you have a really small t value?3910

A small possibility that the alternative hypothesis is true that does not make sense.3915

What about the probability of seeing a sample t as extreme as the one given that the null hypothesis is true.3921

This is our entire story I can process it now.3934

It is not just that the null hypothesis is true, but it also that when you have a certain sample, that also has to be part of the definition of p value.3938

The idea is if we have this t value and it is pretty extreme and the null hypothesis is true.3956

That is given.3967

Given that the null hypothesis is true, what is the possibility of seeing such extreme t value?3968

It is very small.3979

We are trying to lower our false alarm rate.3981

That is the end of one sample hypothesis testing.3986