For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

### Chi-Square Goodness-of-Fit Test

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Roadmap
- Where Does the Chi-Square Test Belong?
- A New Twist on HT: Goodness-of-Fit
- Hypotheses about Proportions
- Chi-Square Statistic
- Chi-Square Distributions
- Conditions for Chi-Square
- Example 1: Chi-Square Goodness-of-Fit Test
- Example 2: Chi-Square Goodness-of-Fit Test
- Example 3: Which of These Statements Describe Properties of the Chi-Square Goodness-of-Fit Test?

- Intro 0:00
- Roadmap 0:05
- Roadmap
- Where Does the Chi-Square Test Belong? 0:50
- Where Does the Chi-Square Test Belong?
- A New Twist on HT: Goodness-of-Fit 7:23
- HT in General
- Goodness-of-Fit HT
- Hypotheses about Proportions 12:17
- Null Hypothesis
- Alternative Hypothesis
- Example
- Chi-Square Statistic 17:52
- Chi-Square Statistic
- Chi-Square Distributions 24:31
- Chi-Square Distributions
- Conditions for Chi-Square 28:58
- Condition 1
- Condition 2
- Condition 3
- Condition 4
- Example 1: Chi-Square Goodness-of-Fit Test 32:23
- Example 2: Chi-Square Goodness-of-Fit Test 44:34
- Example 3: Which of These Statements Describe Properties of the Chi-Square Goodness-of-Fit Test? 56:06

### General Statistics Online Course

### Transcription: Chi-Square Goodness-of-Fit Test

*Hi, welcome to educator.com. *0000

*We are going to talk about the chi-square goodness of fit test. *0002

*So first, we are going to start with the bigger review of where the chi-square test actually fits in. *0005

*Amongst all the different inferential statistics we have been learning so far and then we are going to talk *0012

*about a new kind of hypothesis testing, the goodness of fit hypothesis test. *0018

*So it is going to be similar to hypothesis testing as we been doing so far but there is a slightly different logic behind it.*0023

*So because it is a slightly different logic there is a new all hypothesis as well as the alternative hypothesis. *0029

*Then we are going to introduce the chi-square distribution and the chi-square statistic. *0037

*And then we are going to talk about the conditions for chi-square test when do we actually do it. *0044

*So where does the chi-square test belong? *0049

*And it is been a while since we have looked at this if you are going in order with the videos but I think it is*0054

*pretty good to stop right now and sort of think where we come from? *0059

*Where are we now? *0063

*So the first thing we want to think about are the different independent variables that we been able to look at. *0065

*We been able to look at independent variables the predictor variables that are either categorical or continuous. *0072

*When the idea is categorical you have groups right? *0084

*Or different samples, right? *0095

*When the idea is continuous you do not have different groups you have a different levels that predict something. *0098

*So just to give you a idea of a categorical IV that would be something like experimental group versus the *0107

*control group or something like this categorical IV may be someone who gets a drug versus someone who *0116

*gets the placebo , a group that gets the drivers of the group that gets the placebo and example of the *0127

*continuous IV might be looking at how much you study predicting your score on a test , so how much you *0132

*study would be a continuous IV. *0140

*So that is one of the dimensions that we need to know, is your IV categorical or continuous. *0143

*You also need to know whether the DV is categorical or continuous so the DV is the thing that were *0150

*interested in measuring at the end of the day the things that we want to know that this thing change this is*0160

*the thing we want to predict right, and so far here is how would come. *0167

*At the very beginning we looked at continuous types of tests and those types of measures and those were *0177

*the regression, linear regression, as well as correlation. *0187

*Remember R and regression was that stuff about like Y equals the not + b sub 1 times X, so that was *0193

*regression and correlation way back in the day. *0210

*We have been covering a lot of this quadrant actually looking at t-tests and ANOVA right?*0215

* One important thing to know that t-tests and ANOVAs are both hypothesis tests, only so far have not *0224

*learned hypothesis testing with regression and correlation. *0238

*A lot of inferential statistics in college does not cover hypothesis testing of regression until you get to more advance levels of statistics. *0241

*So what do ANOVAs and t-tests sort of have in common? *0255

*Well they have in common that they are both categorical IV and continuous DV. *0261

*The IV is categorical and you only have one, one IV. *0269

*And your DV is continuous. *0277

*So that sort of what they have in common, what is different about them? *0282

*Well the difference is that the IV in t-tests has two levels in only two levels so there is only two groups or two samples. *0287

*In ANOVAs we could test for more than two samples, we can do that for 3 4 5 samples. *0297

*So that IV has greater than two levels and so that is where we been spending a lot of our time. *0302

*So for the most part continuous DV are really important because they tell us a lot, they tell us the find ways*0312

*that we could actually be different, that the data could actually be different.*0320

*So you are going to, it is more rare that you will use the categorical dependent variable, that is not going to*0327

*be as informative to us but it is still possible and that is where the chi-square is going to come in. *0334

*The chi-square is been coming right in this quadrant where we have categorical IV also a categorical DV so *0340

*for instance we might want to see something like if you are given a particular job or the placebo, do you *0347

*feel like you are getting better, yes or no right? *0357

*So that is a categorical DV, it is not like the score that we can find a mean and so this is where the chi-square tests come in. *0360

*And there is going to be 2 chi-square tests that we are going to look at. *0375

*The first one, we are going to cover today and it is called goodness of fit. *0379

*The next one is in the next lesson and it is called a test of homogeneity. *0382

*They are both chi-square test. *0386

*The other way you will see that what is written is chi-squared, so sometimes, do not think of, oh what is this doing here? *0387

*When it has this little curvy part here we need chi-square, the Greek letter chi, finally this is a test that*0398

*rarely is covered in inferential statistics but at more advanced levels of statistics he did cover it and it is called *0407

*the logistic test and logistic test takes you from continuous IV to categorical DV. *0415

*But that is rare design used in conducting science, it is not as informative as continuous to continuous or categorical to continues. *0424

*Alright so we are going to spend your time right in here. *0436

*So there is a new twist on hypothesis testing, it is not totally different, it is still very similar but there is there is a subtle difference. *0441

*Today we are going to start off with the chi-square goodness of fit test. *0454

*Basically let us think about hypothesis testing in general. *0457

*In general you want to determine whether a sample is very different from expected results that is the big idea of hypothesis testing *0462

*and expected results come from your hypothesized population.*0470

*If your sample is very different than we usually determine that with some sort of test statistic and looking*0474

*at how far it is on the on the tested statistics distribution right and we look at whether it is past that Alpha*0481

*cut off or the critical test statistic right and then we say, oh this sample is so different than would be *0489

*expected given that the null hypothesis is true that we are going to reject the null hypothesis. *0496

*That is usually hypothesis testing. It still takes that idea whether to look at whether a sample is very *0504

*different from expected results, but the question is how are we going to compare these two things? *0511

*We are not going to compare means anymore, we are not going to look at the distance between means, *0517

*nor are we going to look at the proportion of variances that is not what we are going to look at either. *0521

*Instead we are going to determine whether the sample proportions for some category are very different *0527

*from the hypothesized population proportion. *0539

*And the question will be how do we determine very different and here is what I mean by determine *0542

*whether the sample proportions are different from the hypothesized population proportion.*0549

*So here I am just going to draw for you sort of schematically what the hypothesized population proportions might look like. *0554

*So this is just sort about the idea, so you might think of the population as being like this and in the *0569

*population you might see a proportion of one third being blue, one third being red, and one third being yellow. *0577

*Now already it is hard to think about like you could already sort of see, well we cannot get the average of*0588

*blue red and yellow right like what would be the average of that, and how would you find the variability of *0597

*that so already we are starting to see why you cannot use t-tests or ANOVAs if you cannot find the mean or *0605

*variance you cannot use those test so is this is what our hypothesized population looks like and when we *0613

*get a sample we get a little sample from that population, we want to know whether our sample *0622

*proportions are very different from the hypothesized proportions or not, so let us say in our sample *0631

*proportion we get mostly blue, little bit of red, little bit of yellow so let say 60% blue 20% red 20% yellow. *0637

*Are those proportions different enough from our hypothesized proportion?*0650

*Another sample we might get is you know, half blue and half red and no yellow, is that really different from our hypothesized proportion? *0655

*Another sample we might get might be only like 110 blue and then 40% red and then the other half will be yellow. *0674

*So something like that we want to say if it is really different from these hypothesized population *0694

*proportion, and so that is what our new our new goal is. *0700

*How different are these proportions from these proportion and then the question becomes okay how to *0706

*determine whether something is very different? *0713

*Is this very different or just different? *0717

*How do we determine very different, that is going to be the key question here. *0724

*And that is why we are going to need the chi-square statistic and the chi-square distribution. *0728

*So we are changing our hypotheses a little bit now the null hypotheses is really about proportion and here is what we are talking about. *0733

*The null hypothesis now is that the proportions of the population are real population that we do not know? *0749

*Will this population be like the predicted or theorized proportion and so here we are asking is this unknown*0756

*population like or known population right and it should sound familiar as that sort of the fundamental basis of inferential statistics. *0772

*So that is our new null hypothesis. *0782

*That the proportions in the population are like the predicted will be like the predicted population proportion still be the same. *0785

*Remember sameness is always the hallmark of the null hypothesis alternatively if you want to say at least *0798

*one of the proportion in the population will be different than predicted so going back to our example, if our*0807

*population are hypothesized population is something like one third, one third, one third maybe what we *0816

*will find is something like in our sample will have one third blue but then some smaller proportion like 15% red and on the rest being yellow. *0830

*Now the one third should match up. *0856

*The one third matches up but what about these other two? *0860

*And so an alternative hypothesis at least one proportion in the population will be different from the predicted proportion, *0864

*there just has to be one guy that is different. *0875

*Suggest I give you an example, let us turn this problem into a null hypothesis in an alternative hypothesis.*0878

*So here it said according to early polls candidate A was supposed to win 63% of the votes and candidate B was supposed to win 37%. *0886

*When the votes are counted candidate a won 340 votes while B won 166 votes so here just to give you that *0898

*picture again the null hypothesis population was that candidate A color A in blue, candidate A should have *0908

*won 63% of the vote and candidate B all color in red should have won 37% of the vote so what would be our null hypothesis? *0918

*Our null hypothesis would be that our unknown population will be like this predicted the proportions of my unknown population*0933

*will have the same proportion as our predicted population. *0945

*So here we might see something like A's proportion of votes of the actual real votes should be like this, *0949

*the predicted population, and B’s proportion of votes should be like predicted population. *0982

*So let us say, A’s proportion the real proportion of votes should be like this, and so should B, B should be like this. *1009

*The other way we could say that is that the proportion of votes the real proportion of votes should be like *1017

*the predicted proportion of votes, and then you could just say for every single category for both A and B. *1025

*So what would be the alternative version of this? *1031

*The alternative would say at least one of the proportion one of the categories either A or B one of those *1035

*proportions will be different from the hypothesized proportion. *1043

*And in fact in this example if one of them is different the other will be different to because since we only*1048

*have two categories if we make one really different than the other one will automatically change. *1056

*But later on we might see example 3, 4, 5 category and so in those cases this will make more sense. *1061

*Okay so now let us talk about how to actually find out if out proportions are really off or not. *1070

*Are our proportion statistical outliers are they deviant, are they significant, do they stand out, that is what we want to know. *1080

*And in order to do that we have to use measure called the chi-square statistic instead of the T statistic *1092

*which looks at a distance away in terms of standard error instead of the S statistic which looks at the *1099

*proportion of the variance are interested in over the variance we cannot explain the chi-square does something different. *1106

*It is now looking at expected values what would we expect and what would we actually observe and so the *1113

*chi-square is going to look like this, so be careful that you do not, usually it is like a uppercase accident and*1124

*it is a little bit different than like a regular letter X, it is usually a little more curvy to let you know it is chi-square. *1134

*So the chi-square is really going to be interested in the difference between what we observe the actual *1142

*observed frequency or percentages minus the expected frequency. *1150

*So what were looking at observed versus expected this is what we see in our sample and this is what we *1157

*would predict given our hypothesized population so this is that predicted population part. *1170

*So were interested in the difference between those two frequencies. *1180

*Now although you could use proportions as well you can only do that if you have the same, if you have a *1185

*constant number of items so you probably are safer to go with frequencies because those are assertively *1200

*weeded proportion so you probably want to go with that. *1203

*So were interested in this difference but remember when we look at this different sometimes there can be *1207

*positive sometimes there can be negative and so we what we do here as is usual in statistics as we square *1214

*the whole thing, but we also want to know about this difference as a proportion of what was expected and we want to do this for every category. *1220

*For the number of categories and I goes from one to the number of categories and there is actually an I down here for everything. *1234

*So what this is saying is that for each category, each proportion that you are looking at so in our in our sort*1249

*of toy example with the red blue and yellow, in this example we would do this for blue we would do this *1259

*for red and we would do this for yellow so number of categories, so categories really speak to what are the proportions made of? *1275

*So in here we have three categories so we would do this three times and add those proportions up and we *1291

*want to eventually be able to find observed frequency and the expected frequency. *1315

*Now in the example that we saw with the voting of for candidate A and B, one of the things I hope you *1321

*noticed was that the observed frequencies were given is just number of votes how many people voted but *1330

*the expected frequencies would be expected hypothesized population, that was given as a percentage so *1336

*you cannot subtract votes from percentage, you have to translate them both into something that is the *1346

*same and so in that it is helpful to change the expected percentages into expected frequency and there is *1353

*going to be another reason for changing it into expected frequencies instead of changing the observed *1366

*frequencies into the observed proportion and I am going to that a little bit later. *1371

*So here is what I want you to think of this, is really the square difference between observed and expected *1377

*frequencies as a proportion of expected frequency and you want to do that and you want to sum that over all the categories. *1384

*Once you have that then you get your chi-square value, now let us think about this chi-square value. *1394

*If this difference is very large right so observed frequencies are just very different than expected one, is that difference is very large? *1400

*You are going to have a very large chi-square also if this difference is very small, they are really close to each other, then your chi-square is be very small. *1413

*So chi-square is giving us a measure of how far apart the observed and expected frequencies are, also I *1422

*want to see that the chi-square cannot be negative. *1434

*First of all because were squaring this difference right so the numerator cannot be negative not only that *1439

*the expected frequencies also cannot be negative because we are counting up how many things we have , *1445

*how many things we observed and so this also cannot be negative so this whole thing cannot be negative. *1451

*So already we see in our mind the chi-square distribution will probably be positive and positively skewed *1457

*because it stops at zero there is a wall at zero.*1465

*Okay so now let us actually talk and draw the chi-square distribution so imagine having some sort of data *1470

*set and taking from it over and over again samples so you take a sample and so have this big data set, you *1479

*take the sample and you calculate the chi-square statistic and you plot that. *1487

*And then you put that back in you take another sample and you take the chi-square plotted again and do *1493

*that over and over and over and over again. *1502

*You will never get a value that is below zero and you will get values that might be way higher than zero *1505

*sometimes but for the most part though be clustered over here so you will get a skewed distribution and *1514

*indeed the chi-square distribution is a skewed distribution. *1520

*Now here when we look at this you might think, hey, that looks sort of like the F distribution and you are *1527

*right overall and shape it looks just like the F distribution and in a lot of ways we could apply the reasoning*1536

*from the F distribution directly to the chi-square distribution. *1544

*For instant in the chi-square distribution, our alpha is automatically one tailed it is only on one side and so*1548

*when we say something like alpha equals .05 this is what we mean, we mean that we will reject the null *1556

*when we have a chi-square value that somewhere out here or here or here but we will fail to reject if we *1565

*get a chi-square value in here from our sample. *1573

*Now this chi-square distribution like the S and t-distribution, it is a family of distribution, not just one*1576

*distribution the only one that is just one distribution is the normal distribution. *1586

*The chi-square distribution again depends on degrees of freedom and the degrees of freedom that the chi-*1591

*square depends on is going to be the number of categories -1 . *1598

*So if you have a lot of categories the chi-square it will look distribution will look different if you have a small*1608

*number of the categories like 2, the chi-square distribution will look different. *1615

*So let us talk about what Alpha means here. *1619

*The alpha here is this set significance level we are going to say, we are going to use this as the boundary so*1623

*that if we have a chi-square from our sample that bigger than this boundary then we will reject the null. *1630

*What is the difference now with P value? *1643

*Now the P value said this is the probability so we might have a P value somewhere out here or we might *1647

*have a P value somewhere here, the P value is going to be very similar to other hypothesis test what the P *1656

*value means and other hypothesis test, basically is going to be the probability of getting a high square value*1669

*larger more extreme and in this case there is only one kind of extreme, positive larger than the one from our sample but under condition. *1681

*Remember in this world which one is true? *1700

*The null hypothesis is true. *1703

*So considering if the null hypothesis were true this would be the probability of getting such an extreme chi-*1712

*square value , one that is that large or larger, that is all we need. *1720

*So, in that way the P value is from our data while the alpha is not from our data it is it is just something we sat as the cut off. *1727

*So there are some conditions that we need to know before we use the chi-square. *1737

*When we use the chi-square we cannot just always use it, there are conditions that have to be met so one of the conditions of the chi-square is this. *1745

*Each outcome in the population falls exactly into one of a fixed number of categories, so every time you *1756

*have some sort of case from the population so let us say we are drying out votes. *1765

*Each vote has to fall into one of a fixed number of categories so if it is two candidates, always two *1773

*candidates for every single voter so we cannot compare voters that had two candidates versus voters who had three candidates. *1785

*Also these have to be mutually exclusive categories, one vote cannot go to two candidates at ones so they *1792

*have to be mutually exclusive, you got vote for A or vote for B. *1802

*And you cannot opt out either, or else nobody has to be one of the fixed numbers of categories ahead of time. *1807

*So the numbering is slightly off here but the second condition that must be met is that you must have a *1816

*random sample from your population, that is just like all kinds of hypothesis testing though. *1826

*Number 3, the expected frequency in each category so once you once you compute all the expected *1832

*frequency in order to compute your chi-square, that needs to be each cell each square needs to have an *1840

*expected frequency of five or greater, here is why. *1850

*You need a big enough sample, if you have to small of the sample, again expected frequencies less than five *1854

*also unique big enough proportions, so let us say you want to compare proportions that are like you know *1862

*like one candidate is going to be predicted to win 99.999% of the votes and the other candidate is only *1871

*supposed to win .001% of the vote and you only have five people in your sample. *1883

*And so you need to also have big enough proportion and these balance each other out. *1890

*If you have a large and a sample than your proportions can be smaller also, if you have large enough *1897

*proportions in your sample could be smaller. *1903

*And the final condition is not really condition it is just sort of something I wanted you to know at the rule. *1905

*The chi-square goodness of fit test so that is always been talking about so far.*1913

*This test actually applies to more than two categories. *1920

*You do not just have 2 categories, you have 3 or 4 or 5 or 6 but they do need to be mutually exclusive and *1927

*each outcome in the population must be able to fall into any one of those. *1935

*So those are the conditions. *1940

*So now let us move on to some examples. *1943

*So the first example is the problem that we already looked at so far according to early polls candidate A *1947

*was supposed to win 63% of the vote and B was supposed to win 37%. *1953

*When the votes are counted, A won 340 votes while B won 166 votes. *1958

*One of the things that I like to do just to help myself is when I think of the null hypothesis, when I think of*1967

*the null hypothesis, I sort of write it in a sentence that the proportion of votes, that is my population, *1975

*should be like predicted proportions, and the alternative is that at least one of the proportion of votes will not be like predicted population. *1990

*What I also like to do is I like to draw this out for myself, I like to draw out the predicted population so I will*2032

*color candidate A in blue so that will be about 63%, candidate B will be in red, 37%. *2040

*And so eventually I want to know whether this is reflected in my actual votes. *2053

*The significance level we can set it up .05 just set of convention and we know that it has to be one tailed *2059

*because this is definitely going to be a chi-square and we know it is a chi-square because it is about expected proportions. *2068

*So now let us set our decision stage. *2075

*Now our decision stage, it is helpful to draw that chi-square distribution and to sort of label it, for alpha*2081

*here this is our rejection region .05, now it would be nice to know what our critical chi-square is, and in *2100

*order to find that we need degrees of freedom and degrees of freedom is the number of categories, in this *2111

*case 2 -1 and that is 1° of freedom and it is because if you know let us say that candidate B won that is *2119

*supposed to win 37% of the votes you could actually figure out candidate A like you do not need me to tell *2131

*you what that is to figure it out and candidate A cannot vary, the proportion cannot very freely once you *2138

*know this one and that is why it is number of categories – 1.*2143

*So now that we have that you might be useful to look at either in the back of your book or use XL *2148

*spreadsheet Excel function in order to find our critical chi-square. *2156

*So in order to find chi-square there are two functions that you need to know just like T this and T, F this and F in, now there is chi-this. *2161

*Actually we need to use chi in right now because here we have the probability .05 and the degrees of *2182

*freedom one and that will give us our critical chi-square and that is 3.84. *2190

*So critical and so this is the boundary were looking for 3.84 so anything more extreme more positive than *2198

*3.84 and were going to reject our null hypothesis. *2208

*So now that our decision stage is set, now it is helpful to actually work with our population and remember *2214

*when we talk about our population, should have left myself some room, when we talk about our actual sample here is what we ended having. *2221

*We have observed frequencies already so for candidate A, I am going to write a column for observed in *2236

*candidate B so candidate A, we observed 340 votes so that is our observed frequency for candidate B, we see 166 votes. *2243

*Now one that helps is we know what the total number of votes was, so the total number of votes is going to be 340+166 and that is 506. *2261

*So 506 people actually voted in this so down here I am going to write total 506. *2274

*Now the question is what should our active frequencies have been? *2283

*So here I am going to write expected and I know that my proportion of expected should be 63%. *2291

*That means is that the total number of people who voted? *2298

*So here is our little sample of 506 people. *2302

*This is our 100% but here we have 506 people in our sample, we should expect 63% of 506 to have voted *2308

*for A, and so how do we find that? *2323

*Well we are going to multiply 63% to 506 to find out how many votes that little blue bit is and so that is *2328

*going to be.63×506 that total amount. *2341

*If we multiply 506 x 1 we would get 506 right?*2350

*So if we multiply by a little bit of a smaller proportion that we get just that chunk. 318.78 actually I am *2355

*going to put this here, let me actually draw this little table right in here because that can help us do our 3939.1 finder chi-square much more quickly. *2367

*And so observed expected frequency observed frequency at 340 and 166, okay. *2383

*So what are the other expected frequency for B, so in order to find this little bit we are going to multiply*2394

*.37×506, so .37x506 and that is 187.22. *2401

*And usually if you add this entire column that you should get roughly a similar total. *2414

*When you do it, when you do these by hand sometimes you might not get exactly the same number it *2422

*might be off by just a little bit because of a rounding error, if you round to the nearest 10th, round to the nearest integer, *2429

*you make it a little bit around it here but you should be off by much so that one way you could check to see what you did was right. *2438

*And so once we have this, so let me just copy these down right here so 318.78 and 187.22 for each of these *2445

*the total is 506, so here, one of things we see is that the expected value for A are a little bit lower and the*2463

*expected values for B are little bit higher, but is this difference in proportion is that significant is that *2476

*standing out enough, and in order to find that we need to find the chi-square, the sample chi-square. *2485

*Now, we completely run out of room here. *2493

*But I will just write the chi-square formula up here. *2497

*So the chi-square is going to be the sum over all the categories of the observed frequency minus the *2500

*expected square as a proportion of the expected frequency. *2510

*And so what I am going to do is calculate this for each category, A and B and then add them up.*2517

*So right here I am going to call this a column, O minus E squared all over B. *2525

*So I am going to do that for A and B and then sum them up. *2540

*So, my observed minus expected squared all divided by expected and so here I get this proportion and I am *2547

*just going to copy and paste that down here and then here I am just going to some them up and I get 3.817. *2565

*We are really close but no cigar so where were right underneath so our sample chi-square is just a smidge *2577

*smaller than our critical chi-square so here were not rejecting the null, we are going to fail to reject the*2589

*null, so let us find the P value so in order to find the P value you could use chi disc or alternatively look it up*2597

*in the back of your book, look for the chi-square distribution. *2609

*It should be behind your normal, your T, your F and then chi-square should come right behind it, it usually goes in that order , maybe a slightly different order.*2614

*And our degrees of freedom remain the same one and so all our P value is just over .05, if we round, .51 right? *2627

*So because of that we are not going to reject the null so we are going to say the proportions of votes are roughly similar to the predicted proportions. *2640

*Well, they are not significantly different at least, they are not super similar but we cannot make a decision*2657

*about that but we can say they are not that different from, that they are not extremely different at least. *2663

*Okay, example 2. A study ask college students could tell dog food apart from expensive liver pâté liverwurst and spam. *2669

*All blended to the same consistency chilled and garnished with herbs and a lemon wedge, just to make it pretty. *2684

*Students are asked to identify which was dog food. *2695

*Researchers wanted to test the probability model where the students are randomly guessing. *2698

*How would they cast their hypothesized model? *2703

*Okay so see the download that shows how many students picked that item to be dog food, so it seems that *2707

*college students have a bunch of different choices in dog food liver Patty, liverwurst and spam, and then *2714

*they need to identify which was dog food so out of those, which of those is dog food? *2723

*So it is sort of like a multiple-choice question. *2728

*So if you hit example 2 in the download that listed below, you will see the number of students is selected that particular item as dog food. *2732

*Now be careful because some people right here, remember, you will really get this problem on a test and you would not know that it is a chi-square problem. *2741

*Sometimes people might immediately just think I will find the means and so they just go ahead and find the *2751

*mean but then if you do find the mean, ask yourself, what does this mean? *2758

*What is the idea or the concept? *2763

*If we average this, we would find the average number of students that selected any of these items as dog *2768

*food and that sort of a mean that does not make any sense right? *2775

*And so before you know, go ahead and find the mean, ask yourself whether the mean is actually meaningful. *2779

*So here we know that the chi-square because the students are choosing something and it is a categorical choice. *2788

*They are not giving you an answer like 20 inches or 50° or I got 10 questions correct right? *2798

*They are actually just saying, that one is dog food and they have five different choices and they have *2804

*chosen one of them as dog food so out of five choices of probability model that are just guessing would *2813

*mean that 20% of the time they should pick pâté, once we dog food, 20% of the time don't expand to be *2821

*dog food 20% of the time to pick dog food to be dog food and so on and so forth. *2828

*So let us try that probability model and by model we also need null hypothesis. *2835

*Model or hypothesized population so step one. *2844

*So the null hypothesis is the idea that they will fit into this picture so this is the population, and it is out of*2848

*100% and they have five choices of pictures just lightly un even, it helps really draw this is as well as you can, just as then it will help you reason to. *2858

*That they will have a equal chance of guessing either one of these and there is two liver patties that is why there are 5 choices. *2878

*So liver pâté 1, spam was next, then actual dog food just in the data set, patty 2 and a liverwurst. *2885

*So these are the five choices and were saying look the students are just guessing they should have a 20% probability of each. *2909

*Is this the right proportion for this sample, is the sample going to serve match that or be very different from this. *2923

*The alternative is that at least one of the real proportion is different from predicted. *2938

*So once we have that, we can set our alpha to be .05 our decision stage, could draw there chi-square and *2954

*our degrees of freedom, we now have five categories and so our degrees of freedom is 5-1 which equals 4 *2970

*and it is because once we know four of this, that we could actually figure out the proportion for the fifth one just from knowing 4 of this. *2978

*So that one is no longer free to vary, it does not have freedom anymore. *2987

*So what is our critical chi-square? *2991

*Well, if you want to pull up your Excel data, here I am just in a start off with step three, in step three we are*2998

*critical chi-square in order to find that we can use chi-in, put in the probability that were interested in and our degrees of freedom which is 4. *3011

*And so our critical chi-square is 9.49. *3026

*Noticed that as degrees of freedom goes up, what is happening to the chi distribution is that it is getting *3035

*fatter it is getting more variable and because of that we need a more extreme chi-square value. *3053

*So that is sort of different than like T distributions or F distribution. *3059

*Those distributions got sharper when we increased our degrees of freedom , chi distributions were the opposite way. *3066

*Those district chi distributions are getting more variable as degrees of freedom goes up. *3075

*So once we have this now we could start working on our actual data, our actual samples. *3080

*So step four is we need to find a sample chi-square and in order to do that it helps to draw out that table so*3089

*the table might look something like this. *3102

*I will just copy this down here and this is the type of food, so that is the category and here we have our observed frequencies. *3106

*The actual number of students that pick that thing to be dog food. *3125

*So here we seen one student pick pâté, one to be dog food, 15 students picked liverwurst to be the dog food. *3130

*What are the expected frequencies? *3138

*Well in order to find expected frequencies we know that the expected proportions are going to be .2 all the way down.*3142

*20% 20% 20% 20% and here I am just going to total this up. *3153

*And I see that 34 students were asked this question. *3161

*Are expected frequencies should add up to about 34? *3170

*Are expected proportions adds up to one? *3175

*And that is why we cannot just directly compare these two things, they are not in the same sort of currency *3179

*yet, you sort of have to change this currency into frequency. *3184

*So how do we do that? *3189

*Well we imagine here are all 34 students take 20% of them, how many students will that be? *3192

*So that is 0.2×34, this times 34. *3199

*And I am just going to lockdown that 34 because that total sum would not change. *3207

*So, this is what we should expect that if they were indeed guessing, this is the expected frequencies that *3214

*we should see and if I just move that over here , we will see that that also at the column also add up to 34. *3226

*Now once we have that we can compute our actual chi-square because remember that observed frequency *3233

*minus expected square divided by expected as a proportion of expected. *3240

*So, that is the observed frequency minus expected frequency squared divided by the expected frequency. *3247

*And I could take that down for each row and then add those up and here I get my chi-square statistic for *3257

*my sample and so my sample chi-square is going to be 16.29, and that is the larger more extreme chi-*3268

*square than my critical chi-square, and let's also find P value here. *3281

*In order to find P value I could use chi-disc, here I put in my chi-square and my degrees of freedom which is 4. *3286

*And so that is .003 and that is certainly smaller than .05 and so in step five, we reject the null. *3297

*Now I just want to make a comment here. *3315

*Notice that here, after we do the chi-square although we reject the null just like in the ANOVA we do not *3318

*actually know which of the categories is the one that is really off. *3325

*This one here, we can sort of see, this one probably seems to be the most off but we are just eyeballing it,*3330

*were not using actual statistical principles. *3340

*So once you reject the null there is a post hoc test that you could do but we are not going to cover those here. *3343

*So it seems that students are not randomly guessing they actually have a preference for something as being dog food. *3349

*My guess is liverwurst. *3362

*So example 3 which of these statements describe properties of the chi-square goodness of fit test? *3365

*So if you switch the order of categories the value of the test statistic does not change, that is actually true it*3376

*does not matter whether candidate A got added before candidate B addition is totally order insensitive you *3383

*could add A or B or B on A, you can add pâté or liverwurst and dog food or dog food the liverwurst and *3391

*pate, it does not really matter so this is actually true, as a true property. *3398

*Observed frequencies are always whole members that is also actually true because when you observe of *3403

*the frequency, you are actually counting how many category numbers you have so counting is going to be made up of whole numbers. *3410

*Expected frequencies are always whole numbers, that is actually not true, expected frequencies are predicted frequencies. *3418

*It is not that at any one time you will have plenty student saying that liverwurst is dog food but it is that on*3427

*average that is what you would predict given a certain proportion and so this is actually not true, expected*3435

*frequencies do not have to be whole numbers because they are theoretical, they are not actually things that we counted up in real life. *3445

*A high value of chi-square indicates high level of agreement between observed frequencies and the expected frequencies. *3452

*Actually if you think about the chi-square statistic, this is actually the opposite of what is the real case. *3462

*If we had a high level of agreement this number would be very small and because this numerator is small *3472

*the chi-square would also be small, a high value of chi-square would actually mean that this is quite large *3479

*compared to this and so this is actually also wrong, the opposite. *3486

*So that is it for chi-square goodness of fit test, join us next time on educator.com for chi-square test of homogeneity.*3494

0 answers

Post by Lois Han on April 30, 2012

You are a breath of fresh air in my statistics life. Thank you so much!

0 answers

Post by Matt Lin on March 18, 2012

Why we are not reject the Null if sample chi-square is larger than critical chi-square?