Statistics Chi-Square Test of Homogeneity

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We are going to talk about the chi-square test of homogeneity.0002

Previously we talked about the chi-square goodness of fit test now were in a contrast that with this new test is still 0018.3 chi-square test but it is a test of homogeneity now.0005

We are going to try and figure out when do we use which test.0022

The test we are testing a new idea , we are not testing goodness of that would actually testing homogeneity similar.0027

We actually have slightly different null hypotheses and alternative null and alternative hypotheses .0035

We are going to talk about how those have changed then we are going to go over the chi-square statistic and also finding 0051.0 the expected values is going to be a little bit different in test of homogeneity .0041

Finally working to go through chi-square distributions as well as degrees of freedom and the conditions for the test of homogeneity,0055

one can you actually care conduct this test service statistically legally.0065

Okay so the first thing is what is the difference between the test of homogeneity and test of goodness of fit?0069

Well in the goodness of fit hypothesis testing we wanted to determine whether sample proportions are very different from hypothesized0082

population proportion one way you could think about this is that you have one sample and you are comparing it to some hypothetical population.0089

In test of homogeneity and I called it goodness of fit, it is about how well these two things fit together.0098

How well does the sample fit with the hypothesized proportion.0108

In test of homogeneity homogeneous means similar right, that they are made up of the same stuff.0112

In test of homogeneity we want to determine whether 2 populations that are sorted into categories share the same proportions or not.0120

And here you could also substitute this word population here because ultimately were using the sample as a proxy for the population.0130

So here we have 2 population and we want to know whether those two populations are similar in their proportions or not0142

right were not comparing them to some hypothesized population were comparing them to each other.0152

And so really you can think of this as an analogy you think of the their relationship by using an analogy from the0159

one sample to the independent samples t-test.0167

In the one sample t-test we had one sample and we compared it to the null hypothesis right?0170

That was when we would have null hypotheses such as new equals zero or new equals 200 or new equals -5 versus an independent sample.0176

We had 2 samples and we wanted to know how similar they were to each other right or how different0190

they were from each other and our null hypothesis was changed to something like use of X bar minus Y bar equals zero right,0198

that they are either made up of the same mean or different means.0208

And in a in a similar way the goodness of fit chi-square is really asking whether this proportion in my sample0213

is similar to the proportion in our population.0229

So that is how I am comparing , this is my null hypothesis in some ways .0232

In our inner test of homogeneity we have 2 sample 2 population 2 sample that come from 2 unknown population and we want to know0240

whether these have similar proportions to each other and so that is going to be our null hypothesis that these have the same proportion or have different one.0255

For null hypotheses is similar proportion.0267

And so in that way I hope you could see that goodness of fit in homogeneity their ideas that we have looked at before0275

comparing one sample to a hypothesized population or comparing two samples to each other but we have looked at it before0285

not with proportion but with means, right?0294

And now are looking at it with proportion okay since you are looking at proportion we should have hypotheses about0297

proportion so the null hypotheses with something like this the proportion of all the each category the proportion that0305

all into each category is the same for each population so however many categories you have so let us say we have0313

in a three categories.0322

If we believe that they are the same and they should roughly have the same proportion so these have similar proportion.0341

It does not actually matter what the proportions are it could be 90, 10 could be 10,10 it could be 75 20 like when the proportions0347

that were think there similar for each population and whatever 780 whatever category is 75% of the population0360

that category will also be 75% of the population.0368

The alternative hypothesis says that for at least one category the populations do not have the same proportion so just like before0371

were now talking about differences that the differences are really in the proportions the predicted the populations proportion.0383

So just to give you an example.0394

Here is the problem and let us try to change it into the null hypothesis as well as alternative hypothesis.0396

So according to a poll for and six Democrats said they were very satisfied with candidate A while 510 were unsatisfied0401

however 910 Republicans were satisfied with candidate a while 60 were not.0410

And in a chi-square test of homogeneity we could see whether the proportions of Democrats and Republicans that Democrats were satisfied are0415

similar to the proportions were Republican of Republicans were satisfied versus unsatisfied.0427

So let us draw this out first.0436

So here we have about 400 Democrats saying there satisfied while 500 saying unsatisfied.0439

Let put satisfied in blue and so that is a little bit less than half and the unsatisfied people are a little bit0451

more than half so this is the Democratic population that they look like.0460

The Republican population looks very different so here we see most of the Republicans being pretty satisfied and0467

only a very small minority being unsatisfied right.0479

And so the question is are these two are the two similar are the proportions that fall into each category0483

satisfied or unsatisfied the same for each population?0493

Are they different?0497

The null hypothesis would probably say something like this.0498

The proportion of satisfied and unsatisfied people like us are similar are the same for Dans as well as republicans.0501

The alternative hypothesis says for at least one category either satisfied or unsatisfied, Dans and Republicans do not have the same proportion.0531

Okay so note that in the case of 2, once category changes once the proportion of one category changes the other one automatically changes.0561

So if we somehow were able to change has satisfied the Democrats were with candidate A, we would also see the0584

proportion of unsatisfied people just automatically change.0592

So that is in the case of two categories but in the case of multiple categories maybe 2 might change but the others may0595

not change right so in that way this would be a more general way of saying alternative hypothesis.0606

Now let us talk about the chi-square statistic.0612

Now the nice thing about the chi-square statistic is that it is the same as the goodness of fit test.0616

We use the same idea so chi-square is going to be observed frequencies and the difference between that and0621

expected frequencies where over the proportion of expected frequency.0631

But there is just one subtle difference before it was for each category.0638

Now we have different categories in different population right so we not only have like category 1 and category 20643

category 3 so on and so forth but we also have population 1 and population 2 at least right?0651

And so we have multiple of observed frequencies and so what do we do right?0659

Well what we do here is that we consider each of these combination of which population your in and which category0668

are talking about each of these are going to be called cells.0681

And so we do this for each cell so I will go from one of to the number of cells.0686

And how do we get the number of cells?0694