For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

### Type I and Type II Errors

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
- Errors and Relationship to HT and the Sample Statistic?
- Instead of a Box…Distributions!
- Usually, Lots of Overlap between Null and Alternative Distributions
- How Distributions and 'Box' Fit Together
- Example 1: Types of Errors
- Example 2: Types of Errors
- Example 3: What is the Danger of the Type I Error?

- Intro 0:00
- Roadmap 0:18
- Roadmap
- Errors and Relationship to HT and the Sample Statistic? 1:11
- Errors and Relationship to HT and the Sample Statistic?
- Instead of a Box…Distributions! 7:00
- One Sample t-test: Friends on Facebook
- Two Sample t-test: Friends on Facebook
- Usually, Lots of Overlap between Null and Alternative Distributions 16:59
- Overlap between Null and Alternative Distributions
- How Distributions and 'Box' Fit Together 22:45
- How Distributions and 'Box' Fit Together
- Example 1: Types of Errors 25:54
- Example 2: Types of Errors 27:30
- Example 3: What is the Danger of the Type I Error? 29:38

### General Statistics Online Course

### Transcription: Type I and Type II Errors

*Hi and welcome to www.educator.com.*0000

*Today we are going to talk more in-depth about type 1 and type 2 errors.*0001

*If you want to know more about power and effect size it is good to go through this lesson*0006

*because it is going to help you understand some of the pictures that we are going to draw in the future.*0013

*Here is the roadmap for today.*0017

*We need to know about these type 1 and type 2 errors, but we also need to know when we make those errors in relationship to hypothesis testing.*0021

*So far we only used t test as our hypothesis test.*0033

*We have shown these errors and their relationship to hypothesis testing before as a box, but frequently in hypothesis testing we draw distributions.*0037

*The SDOM to be more specific.*0048

*What I want to show you how the errors fit on this distribution picture.*0051

*We are going to show you how the box and the distributions fit together because these two things actually relationship to each other.*0058

*They refer to the same concept.*0065

*There are just 2 different ways of showing you that same concept.*0067

*We go through hypothesis testing, but in the real world there is some reality that either the null hypotheses is just true or the null hypothesis is false.*0071

*Although we do not know this reality, all we know is the result of our hypothesis testing.*0086

*There are two kinds of ways we can make errors.*0092

*We can make an incorrect decision by false alarming.*0095

*We reject the null, but we should not have rejected the null.*0099

*That is called the false alarm or a type 1 error.*0106

*I used to get confused between which one is type 1 and type 2, these are arbitrate.*0110

*I like to think of this as the more serious error when you successfully reject the null hypothesis that is a more extreme thing that you do.*0116

*This is actually more dangerous than this miss.*0127

*That is not much of an error but actually false alarming.*0131

*That is how I remember the number 1 error you should look out for.*0136

*The type 1 error is often also called the likelihood of false alarming.*0142

*The probability of false alarming and that is referred to as alpha.*0151

*If the reality that we do not know is that this null hypothesis is true we have a probability of false alarming with the rate of alpha.*0157

*We have the probability of failing to reject when we should have rejected, a correct failure your probability is 1-alpha.*0171

*These two things add up to 1.*0186

*The probability of false alarming + the probability of making a correct failure =1.*0189

*On the flipside, let us say that null hypothesis is false that is not a true picture or model of the world.*0199

*Then we really should have reject it.*0210

*It is not true, we should reject it, that would be a correct decision and that is called the hit where we are rejecting the null when we should have rejected it.*0213

*That gives us the probability of hits.*0226

*We could be incorrect and fail to reject when we should have rejected that is also another incorrect decision.*0230

*That is the type 2 error.*0242

*It is a miss and the probability of miss is given as beta.*0244

*Beta + 1 –beta = 1.*0248

*The probability of misses + probability of hits =1.*0254

*In which of these boxes is the sample statistic statistically significant?*0262

*In which of these boxes is our p value less than .05 or whatever our alpha level is.*0274

*Let us think about that.*0280

*When we reject the null hypothesis that means our test statistic in this case t is extreme.*0282

*Our p value is significant and remember we mean significant as it stands out.*0294

*It is very weird.*0304

*In this case, these two quadrants up here is what we should worry about.*0307

*This is the decision we need to worry about when we reject the null hypothesis.*0315

*The other possibility is that when we reject the null hypotheses and our p is significant we made a correct decision.*0322

*These are our two choices if we know that p is less than alpha or if our test statistic is extreme.*0334

*Here p is not significant.*0343

*It is not too weird and because of that we will fail to reject and we can be correct in failing to reject*0349

*or when we fail to reject we could be wrong by making a type 2 error.*0358

*Here is what I want you to know.*0363

*Let us say we carry out hypothesis testing and I think I have a really low p value.*0365

*I am going to reject my null hypotheses.*0372

*Which error am I likely to make, a false alarm or a missed?*0375

*Since I rejected my null, the only error I can possibly make is this one where I reject the null and get wrong.*0381

*Let us say I go through my hypothesis testing and I get p=.4.*0397

*Let us say I do not reject my null.*0405

*What mistake or what error could I have possibly made?*0408

*The only error I can make is the missed error.*0411

*Here I fail to reject and I could be wrong in doing it.*0414

*Let us talk about distributions and how errors fit in here.*0418

*We have a one sample t test we set up some null population.*0426

*This is our null hypotheses population and our hypothesized mu might be 230.*0431

*We do not know whether our sample is part of this or it is part of some other population, not the null population.*0443

*We can hypothesize maybe it comes from some other population like this one.*0454

*When we set our alpha levels and create critical t and zones of rejection and all of that stuff what we are doing is recreating the line.*0459

*If our sample t is outside here then we are going to reject the null.*0476

*So far we have only colored in this part, but we really mean this part as well as all of this part.*0492

*That is our reject the null zone, this entire area.*0508

*In order to find out whether we should reject the null or not we also need to look past the raw score.*0514

*We need to look past the raw score and we need to look at it in terms of the critical t.*0528

*The critical t might be whatever like -2. Something .*0536

*We need to find out this t value and so I am just going to make one.*0544

*Let us say this t value is 5.5 and if our t value is sufficiently extreme then we reject are null hypothesis.*0549

*This would be our critical t and this is our sample x bar, but this is our sample t.*0560

*And that is how it looks out here.*0574

*Our possibility of making an error is this little gray spot that I have colored in red.*0577

*Just in case my sample really does come from these areas, I should not have rejected the null.*0587

*If it happens by chance rule 50 heads in a row it is very unlikely but it is still possible.*0596

*It is still possible that I got this x bar even though this is the true population distribution.*0613

*This is my possibility of making a type 1 error.*0621

*We actually have to add this side up to this side type 1 error.*0628

*We know that this is alpha=.05.*0640

*This part is 1 – alpha which is .95 and that is our possibility of not rejecting given that the null hypothesis is true.*0646

*That is the example of one sample hypothesis testing.*0660

*This is the same picture as before.*0666

*I just written it more neatly for you by typing it out and you can think of this test statistic as just t.*0669

*I have just written the generic word test statistic to think of this as critical t and sample t.*0676

*Here is the important thing to realize.*0682

*This gray distribution here represents an SDOM and that is why this is mu sub x bar and there is also an x bar here as a sample.*0685

*This SDOM actually represents the probability where the null hypothesis is true and that probability equal to 1.*0696

*Remember we talked about that before when we said the area underneath the normal distribution equal to 1.*0706

*This represents the possibility that this may not be true and that there exists some other population that our sample really came from.*0713

*We do not just know what that population is.*0727

*That is the probability that the null hypothesis is false.*0731

*That normal distribution also has an area =1.*0736

*What we can additionally find out is when we create the zones of rejection and we say anything outside of this critical t reject it.*0744

*We color in this area here.*0759

*What we are saying is this is the probability of rejecting given that the null hypothesis is true.*0761

*This is the area where we fail to reject.*0777

*This probability right here represents the conditional probability of failing to reject given that h knot or null hypothesis is true.*0784

*And that equals 1 – alpha because this one equal alpha.*0807

*Those are the important things to remember.*0815

*These are all conditional probabilities as we learned about previously in probability lessons.*0819

*Let us talk about a two sample t test.*0826

*The idea behind the two sample t test is almost exactly the same except there are just a couple of exceptions now.*0830

*Instead of a raw score we have difference of scores and we still have a test statistic.*0838

*Here our mean hypothesized difference between our non college sample and our college sample is going to be 0 because that means they are the same.*0846

*Remember, these are SDOD (Sampling distributions of differences of means).*0862

*This is 0 and this might be our actual sample difference x bar – y bar, the actual difference between the samples.*0875

*Same thing down here, we have this as our critical test statistic and this is our sample t.*0887

*We want to know whether our sample t is way far out, more extreme than our critical t.*0902

*Here this represents the probability that the null hypothesis, that there is no difference is true and that is =1.*0910

*Same thing here, the probability that the null hypothesis is false and actually there some other distribution we just do not know what that is.*0923

*We will draw it like a ghost with blue.*0933

*It is important to know that this mu is mu sub x bar - y bar because we are talking about SDOD.*0936

*That is why it is a difference of means.*0946

*Once we know this, now what we need to do is figure out what these probabilities mean.*0950

*Here, let me draw the cut off again, here we have our rejection zone and our fail to reject zone.*0958

*Once again we can find those conditional probabilities.*0977

*What is the probability of rejecting given this thing is true, inside of this space where the null hypothesis is true?*0981

*What is the probability of failing to reject given that the null hypothesis is true?*0992

*That is the conditions that we are working under.*0999

*It is still the same.*1005

*Here we see alpha and here we see 1 – alpha.*1008

*Ideally when we have these differences between distributions what we really would like is that*1018

*there was very little overlap between these two distributions.*1027

*The null distribution and the like real one that we do not know anything about.*1031

*It will be nice if there was very little overlap.*1036

*But in real life, there is usually a lot of overlap.*1038

*The real world is noisy and the real population might be very, very different.*1043

*The real population might be very similar to the null population.*1055

*If that is the case, there is some overlap between their distributions.*1071

*There are some chances that we might get a score over here and it could be part of the real population or part of the null population.*1077

*If this is the case and we need to understand these conditional probabilities in anyway.*1086

*Get ready here is the deal.*1098

*Instead of writing real population, I am going to say not null population because we do not know what it is.*1100

*It is just not the null population.*1112

*I am going to take this picture, this great curve and I will draw up here in two ways.*1115

*I am going to split it up into two parts.*1121

*One part is going to be this blue part, this fail to reject region and that is that whole part.*1123

*Here I am also going to draw the red part.*1144

*I just draw it separated from each other so that you can see.*1147

*Here we have this little part and that is red and it is red because we have rejected it.*1158

*This is the case where we are actually wrong.*1167

*This is the case where we are actually right.*1171

*Here we are wrong because we rejected the null hypotheses that we should not have rejected.*1174

*Here we are correct, because we fail to reject and truly we should not have rejected it.*1180

*Now that is the case if the null hypothesis population is true.*1185

*What happens in a case where it is not true?*1193

*The null hypothesis is false.*1200

*What happens here?*1203

*Here I am going to draw a different looking picture because I'm going to draw this curve but this curve split up.*1206

*Here I am going to split this curve up like this.*1218

*On this side of the line I am going to draw this little section and draw just this little section.*1227

*That part of it I have failed to reject.*1253

*That is wrong so I am going to color it in red because we should have rejected it but we fail to reject it.*1257

*On the other side, I am going to try the other part of this curve.*1274

*It is this part and here I am going to color that in blue because although we rejected it we should have rejected it.*1279

*Here we rejected the null hypothesis and you are right we should have rejected it because we are in this new unknown population.*1292

*You should have rejected it.*1308

*Let us look at the places where we are correct.*1310

*We are correct here and this is called a correct failure.*1314

*Here we are also correct and this is called a hit.*1319

*Here we are incorrect and that is called a false alarm.*1331

*Here we are also incorrect and this is called a miss.*1344

*It is a miss because we have failed to reject it.*1352

*We failed to hit the target when we should have hit the target.*1357

*Given that, let us see how the distributions and the box go together.*1361

*The false alarm is really that place.*1369

*Remember when the hypothesis is true I am going to draw it in black.*1373

*The correct decision is going to be this whole section where we fail to reject, but that is okay we are in this fail to reject zone.*1378

*You are good to go.*1393

*Here is the other part of this part.*1395

*Here this is an error because we have rejected when we should not have rejected because it is actually true.*1401

*This is our false alarm.*1416

*Now, in the case of a correct decision where you actually hit it, this means you rejected it and it is good*1418

*that you rejected it because actually a different population is true, not this null population.*1430

*That is going to be the area where you reject, all rejections are going one on the right side of this line.*1438

*You should have rejected it because you are in a different population.*1454

*You are not in the null population.*1461

*This is a good thing for you.*1462

*You should have rejected it.*1466

*The other part of that, the other piece of that is down here.*1469

*It is this little piece down here.*1474

*Here it is incorrect, because although you are part of a different population, not the null population, you did not reject it.*1477

*You fail to reject.*1491

*I want you to notice something here.*1493

*All the fail to reject are always on this side of the line because these are values that are less extreme than the mean.*1500

*And the rejection ones are all in this side of the line.*1508

*I could also drawn it two tailed and also showing you the side but I'm showing you one tailed.*1511

*It is all outside of the line, on the outer boundaries of this line, more extreme than the hypothesized mean.*1517

*This is less extreme than the hypothesized mean.*1524

*My hypothesized mean is somewhere here, less extreme than that.*1527

*It is relative to the hypothesized mean.*1533

*That is how these four pictures fit together.*1539

*When you see those two distributions drawn, do not get confused you already know it.*1543

*You just have to break it apart in slightly different ways.*1548

*Let us go on to some examples.*1553

*On the basis of results from a large sample of students from a university, a professor reports the mean high from my sample is not significantly below 60.*1556

*That means he did not reject.*1573

*This is fail to reject.*1576

*If he said significantly that would be rejecting the null.*1581

*Which type of error will this professor worry about?*1586

*He failed to reject, that is important to know.*1590

*What is the only error you can make if you fail to reject?*1593

*Well if you fail to reject, but you should have rejected it, the null hypotheses is false, what kind of error is that?*1596

*That is a missed and a type 2 error.*1617

*The error rates are given by alpha and beta and this is actually beta so these are wrong.*1624

*These are both correct rates instead of the error rates and this is nonsense having a non significant results are all error statistically.*1631

*It is never the case.*1642

*You are damned if you do and damned if you do not.*1643

*There is always a way you can make an error either type 1 or type 2.*1645

*Example 2, a researcher worries about trying incorrect conclusion.*1649

*The researcher plan to select a sample of size 20 and to use the .01 level of significance.*1655

*Here alpha is .01.*1662

*In a two tailed test of the null hypothesis the critical t should be + or - because it is a two tailed test.*1664

*It is + or -2.86.*1676

*If he obtains the t of 2.8 which type of error would he be worried about and why?*1681

*Well, you definitely know that he is not going to reject.*1695

*Fail to reject because this is less extreme than this.*1704

*This is less extreme so he fail to reject.*1717

*The only error you can have when you fail to reject is if you fail to reject given the null hypothesis is false.*1722

*What kind of error is that?*1729

*That is a missed or type 2.*1733

*What if he obtains a t of 2.869 which type of error would he be worried about?*1744

*That is more extreme than this.*1752

*In this case he would reject the null.*1754

*When is he wrong when he rejects?*1757

*When he should have not rejected it because the null hypothesis is actually true.*1760

*What kind of error is that?*1765

*That is a false alarm or type 1 error.*1768

*Example 3, what is the danger of the type 1 error?*1776

*This is a more conceptual question.*1782

*The danger is mistakenly concluding that there is no significant difference between the obtained mean and the hypothetical population mean.*1785

*When you make a type 1 error you have rejected the null but null hypothesis is true.*1794

*Mistakenly concluding that there is no significant difference but that is not true*1808

*because you concluded that there is a significant difference that is why you rejected the null.*1814

*Mistakenly concluding that there is a significant difference between the obtained mean and the hypothetical population mean.*1818

*That is true.*1826

*You mistakenly rejected the null and said there is a significant difference but you should not have done that.*1829

*Mistakenly being alarmed about a hypothesis when you should become.*1838

*That is non sense.*1843

*Mistakenly calculating the wrong test score.*1844

*These errors are not errors that you can actually avoid.*1847

*These are not errors because we were sloppy.*1851

*These are errors that are made because we do not know the real nature of the world.*1854

*This is actually not what we are talking about when we are talking about type 1 or 2 errors.*1860

*Mistakenly choosing the wrong population standard deviation to calculate standard error, that is not it either.*1865

*These two are just regular old mistakes or errors in calculation.*1872

*They are not type 1 and 2 errors of hypothesis testing.*1878

*That is it for type 1 and 2 errors.*1881

*Thank you for using www.educator.com.*1885

0 answers

Post by Windesson Almeida on August 5, 2012

nice..pretty teacher too