 Hello, SC0X learners and welcome to the second life event of SC0X 2019. I'm Ima Borrella, I'm the course lead for SC0X and I'm here with Andrea Moreno. Hi everyone, I am Andrea Moreno and I'm the SC0X teaching assistant. Thank you for being here. Oh thank you. It's great to have you. So today what we want to do is, let's start with maybe showing the agenda. Yeah, absolutely. So you know what to expect from the next 30 minutes. Okay, so first of all, we will check where we are in the course. So a little reminder of the SC0X contents and the pace you should follow because we're only three weeks until the exam right now. So we really need to accelerate the learning. Then we will give some advice on how to get ready for the final exam and after that we will dive into exam preparation. Today we will cover the topic of statistics. Particularly we will be talking about confidence intervals and hypothesis testing. We know many of you struggle with these concepts. So hopefully today's life event will help you to grasp these concepts and clarify some of your questions. Okay, so let's continue. So again, we are only three weeks until we open the final exam on June 19. And we can't stress enough the need for you to start working on modules three, four and five. These modules, particular module three and four, are longer than the other modules and the time you have to invest there is bigger. So please plan accordingly and allocate more hours in the next week. So you are ready for the final exam. Yeah, so what I would like to add is a lot of students, it makes sense that some students take more time than others. But for example, I would estimate I would say it's maybe a hundred hours that someone will take to pass SC0X. So just to make the calculation, if you are missing, for example, module four and five, you still have a big percentage like 50 percent of hours that you have to allocate to pass this course successfully. So please be mindful of that. Please allocate enough time. There are no extensions to the final, unfortunately. So we want you to be prepared, but you should start early. There's not much time left. And Ima is going to show you, I think, where we are right now. So we're May 29, right? Yes, so we're May, or today is May 30, actually. Oh, yeah. So even less than three weeks for the exam. Exactly. So please plan accordingly. So to prepare, what do we recommend you to do? So please, like, of course, review all the contents in all the modules, but something we are providing is a test, your knowledge section. It's maybe towards the end of the module. So maybe a little hidden for some of you. But this section is very important because it's like a mini exam for the module. If you can take these problems, try to solve them. And if you do well there, you can be pretty confident that you grasped and that you got all the key concepts for that module. If you don't perform very well there, maybe you need to go back to the module and review the contents again. So this is a tool you could use to assess how well you're learning the concepts for each module. Then we're also providing the exam preparation section that is towards the end of the course outline. And here we are posting problems from previous exams and solving them in step-by-step videos. So go there, take these problems, try them, and if you struggle to solve them, just check the videos. And then you will be pretty confident on how to solve that type of problem. And finally, we will also be providing a practice exam. This practice exam will become available one week before the final exam opens. It will be very similar to the final exam. We will try to be a similar level of challenge, similar length. And it will also be timed to four hours. So when you take it, you will get a pretty similar experience to taking the real exam that will open on June 19th. So this is for you to reduce the stress a little bit and really be exposed to an exam environment before you have to take the real exam. Yeah, so as you can see here in the calendar, it will start like Inma said on the 14th. And it will end on the 18th or like the 19th before the final opens. Something to clarify is that we will close the practice exam. You will no longer be able to access the practice exam on the 19th when the final exam opens. Because in previous occasions, we have the case where people will take the practice exam instead of the real exam. So we don't want that to happen. So let's say you're planning to take the final exam on the 24th. You won't be able to take the practice exam on the 24th before your real exam. The exam will close on the 18th. So plan accordingly again. Make sure you take the practice exam on time and then allocate whenever you want in that other week for the final. But be mindful that the practice will not be opened while the real exam is open. Yeah, something to comment to is that regarding the exam preparation section, we have posted already two problems. One about probability, one about statistics. But next week we will be posting a last problem in that section that will be solving a linear programming problem using Excel. So just keep an eye out for that. And we will have a last life event on June 13th. Yeah, so we have two questions regarding the exam. So there is a question that says the final exam starts June 16th and closes June 26th. Does it mean I am to select a convenient date within this range for the four hours for the final exam? Yes, so it starts on June 19th, not June 16th, until June 26th. So it will be seven days. Yes, it will be available during that time. But once you decide when to take it, once you start taking the exam, you will have four hours to complete it. And you will only have one chance. You cannot take the exam on June 19th and then again on June 21st. You will have one chance, four hours time slot, but you can decide which four hours within that period. Very important note is that the deadline is on June 26th at 1500 UTC. At that time, the exam closes. So if you want to have the full four hours, you should start four hours before the deadline. Because at the deadline, the exam closes and whatever you submitted is submitted and whatever is not, is not. So just be mindful of that. We have another question that says, will the exam questions be at the same level of difficulty as the prep questions? As the exam prep questions? It's general, but I think we could maybe talk about the practice questions that we have in the course on the exam prep or the practice exam. Yes. So the exam prep section actually have problems that were part of the final exam in previous rounds of SC0x. So you can expect the level of difficulty of these questions is similar, will be similar to the level of difficulty of the questions in the final exam this run. The practice problems have different levels of difficulty. So we cannot say that all the practice problems will have like show the same level of difficulty that you will find in the final exam because it varies. And I think in the practice exam, also the goal is that you are more comfortable with being on the time exam, where to click when to start, how to end, how to submit questions, just to make sure to have the experience. But the reality for having the level of difficulty, I think you can take that experience from the exam prep because like Ingma said, those are really problems that were taken in previous rounds during the exam. Any other questions? No. Make sure to pause your questions. You have anywhere monitoring the YouTube channel right now. So try to put them there. If it's outside the topic, we're moving forward in the solution of a different problem. We will make sure to answer them afterwards, but try to put them and we will try to answer them now. So now we'll dive into the exam prep. That's I know why most of you are here. So today we're going to talk about statistics. As I mentioned earlier, problem two in the exam prep section is already posted. In this, you will have the problem. You have the step-by-step solution in three videos, one by question. And now we are in the life event, where we resolve or answer questions about the problem two specifically and about the statistics concepts in general. So let's start with it. Question one of the problem, they give us one sample with an N of 136, certain MENA, certain standard deviation. And they ask us to create a 90% prediction interval and a 90% confidence interval. So let's have a quick reminder of what a prediction interval and a confidence interval is and how are they different. So for a prediction interval, this is something we already did in the probability section. We just want to find some bounds, some values, such as if we randomly choose a data point from the population, it will fall between them 3% of the time. In the case of the problem, they are asking us for like two values that give me a 90% confidence that any random value will fall between them 90% of the time. And then a confidence interval is something different. Actually, for a confidence interval, what we want to find is the bounds from the sample mean for the sample mean, such as the population, the true mean, the population mean, will fall between them as 3% of the time. In the case of the problem, 90% of the time. So prediction interval is just like random variable. We want to know if it falls between two values, a certain percentage of the time. And the confidence interval, we're talking about comparing the, like finding some values that are a confidence interval for the population mean, the true mean, based on sample values. So what is- Yeah, so something to add, I think we will look at later, but as you may have seen, watching the video, for example, like Inma said, you have a 90% interval, right? So you have 10% that you are leaving outside the tails of your interval, your confidence interval. But be mindful, here, for example, in this graph, we have a two tails. So to the right, to the left. So it's 10% that we're trying to leave outside. But you can't leave, you have to leave half and half. So it's 5% to the left, half, 5% to the right. That's why we divide alpha by two. So you're leaving the, like the summation is 10%. If you leave 10 here, and then 10 here, you're actually doing an 80% confidence interval. So you always have to be mindful of that. Now let's say you have just a one tail, we're just doing the left fail. So forget about this. So then you will be using the 10% here. So you actually have the 90% confidence interval. Great, that's a very common mistake. So keep that in mind. You have to be mindful that when you put it on Excel also, Excel doesn't know if this, in the normal distribution, for example, it doesn't know if you're doing a two tail or a one tail. So unless you specify correctly your alpha, it will bring the wrong answer, basically the wrong value for it. So how do we solve the problem? Actually for the prediction interval, we have like the distribution, the sample distribution with a mean of 16.25 and a standard deviation of 8.83. And we have to find these Y values that give us a 90% confidence that any random variable will fall between them. So this is how we calculate them. This formula, this is the formula we use. We use the mean of the sample, that's Tx. And then we use the standard deviation of the sample, that's Ts. But we are missing these values, C. How do we calculate C? Well, since we have a big enough sample, we can assume a normal distribution. So we can calculate the C by using the unit normal distribution. So the C in this case will be equal to the Z that we've been working with in the probability section 2. And we can calculate it with using the norm that is that in function in Excel. And as Andrea mentioned earlier, what is very important is that you take into account that to find this value using this formula, we have to put here like all the area under the curve from the start of the curve until the here. So we will have to put like this 90% value plus this alpha divided by 2 value. So in this case will be 90% plus 10 divided by 2, 5%. So all these will be 95%. We can think about it like that, or we can think about it like 1, that is the whole area under the curve, minus this alpha divided by 2. That's what I put here in this formula. If you don't do the 1 minus this, you will get the negative of Z. That's if you put here only the alpha divided by 2. But if you do like 90% plus 5%, you will get the same value. So if you calculate that with the values that are provided in the problem, you will get these two values. So the prediction interval for an 80% confidence will go from 1.7 to 30.8. On the other hand, when we want to calculate the 90% confidence interval, as we mentioned, this is different. We want to find these boundaries for the real means, the population mean, based on the values from the sample, the sample mean and the standard deviation, the sample standard deviation. So we use this other formula. It looks very similar to the prediction interval, but now we are dividing Z times S by the square root of N. N is the size of my sample. The C, we can use exactly the same C that we used before, because again, the N is big enough, we can assume a normal distribution, and the confidence interval in this case is 90% again. So it's exactly the same value that we used in the previous, to solve the previous question. In this case, when we calculate these values, we get for the confidence interval is between 50 and 17.5. As we see, this is way smaller than the prediction interval. So confidence interval prediction interval, very different. Prediction interval is usually wider, it's always wider, and the confidence interval is narrower. Great, so that's all about prediction and confidence intervals. If you have any questions, please let us know through the chat. Now we're going to jump into hypothesis testing. Hypothesis testing is a very, very important statistical method, and it's widely used in supply chain management and also in research. So questions two and three of the problem actually are about hypothesis testing. The first question is about, so they, in the problem, they tell us that they have introduced, this company have introduced new evaluation methods for selecting suppliers. So they want to see if this introduction of the new evaluation methods had had any impact in suppliers delivery date. So they give us a sample of this time difference in deliveries after introducing the new methods, and then they tell us the previous average time between promised delivery date and real delivery date. And they want us to see if there's a difference or not. So of course, this is a hypothesis test, and that's how we will solve it. What is a hypothesis test? Well, when we have to choose between two mutually exclusive and collectively exhausted alternatives, we have a hypothesis test. In a hypothesis test, we will always have two hypotheses, the new hypothesis and the alternative hypothesis. Keep in mind that for us, the new hypothesis will always be the situation where things don't change. So, and then the alternative hypothesis will be other possible outcomes. We will see this in more detail in the problem. Keep in mind that the new hypothesis cannot be rejected unless data argues overwhelmingly otherwise. So we'll usually stick to a new hypothesis unless data tells us otherwise. And the result of a hypothesis test will always be reject or fail to reject the new hypothesis. We don't talk in accept terms, this reject or fail to reject. And then the alternative hypothesis, I think, I'm not sure. I don't think you mentioned what the alternative. So you have like other possible outcomes. So in this thing, we're looking at the delays or the differences in the delivery dates we can have. It's bigger, so it takes more time or it takes less time, as we're going to see in the next examples. Yeah, we'll see like how to determine if it's, what is the alternative hypothesis in the exam? Okay. So how to do the hypothesis test? When you have a problem in the exam or in real life where you actually want to test something and want to see it like, you did something, I want to see something change, then you start by formulating your hypothesis. So as mentioned earlier, you will have to formulate your null hypothesis and your alternative hypothesis. As Andrea mentioned, the null hypothesis will be like the one where nothing changes and the alternative hypothesis will be the one that something naturally changed. Second step, select your test statistics. How to select the test statistics? Well, it depends on the size of the sample. If you have a sample of more than 30 observations, then you can assume an order distribution. If you have a sample of less or equal to 30 observations, then use the t distribution. Then you have to determine the type of test. Is it a one-tail test or a two-tail test? We will use one-tail test when we want to check if the difference is, so if the alternative hypothesis is something changed, like something is greater than before or something is smaller than before. So we're looking to one direction. We will use a two-tail test when we're just checking if something changed. It doesn't matter the direction. If it's greater, it's smaller, like we just want to see if it's different or not. Then the fourth step is to pick a significance level and a critical value. So what do we want to be our confidence level? So we want to be confident as 90%, 95%, 98%, we need to decide that, and that will give us the alpha value. The alpha value will be one minus that confidence level. So if it's our confidence level is 95%, the alpha value will be 5%. And this will give us also the critical value, z or t, using this alpha value in the distribution that we selected before. Then we have to calculate the test statistics. We can calculate it with this formula. Remember that the x and the s are the sample values and the mu will be the mean or the value we want to compare our new sample with. And then finally, the sixth step is once we've done all these calculations, we can compare the test statistics to the critical value or compare the p-value to the alpha value to decide if we reject or fail to reject the null hypothesis. So now we want to clarify how to do this last step. So what we can do to actually assess the results of my hypothesis test is to look at these test statistics or to look at the p-value. So what do we have? So we have this level of confidence, in this bluish area, that this is one tail test. So this bluish area gives me my confidence level, let's say it's 90%. So this alpha value will be 10%. And my critical value will be a certain value. It will depend. This is a distribution or a normal distribution, but I can easily calculate it by knowing my alpha value. So I have my alpha value and my critical value given by my confidence level and by the distribution that I'm using for the test. Then I calculate my test statistics that will be based on the data from my sample. Once I have my test statistics, I can calculate it, sorry, I can compare it with my critical value. And if the test statistics is greater than the critical value, then we can reject the null hypothesis. And the reason why we can reject it is because it's outside the boundary, right? So we're putting a boundary here. We're saying 90% confidence, like this is our 90% confidence interval from the left to this point. And so our test statistic for our new value is right here. So it's outside. And that's why we can reject it. When if our test statistic is inside, we can't be sure. We can't be 90% sure. And that's why we fail to reject. Exactly. And let's think about the p-value now because I know a lot of people are confused with the p-value. The p-value is just, let's say, the alpha value of my test statistics. I don't want to confuse you with this, but if you look at this, what we do is just calculate this area under the curve from the test statistics to the right. So this value, this area is the p-value. Therefore, we can say that if the p-value is smaller than the alpha value, then we can reject the null hypothesis too. We're just doing the same thing. We're comparing, instead of comparing my critical value of my test statistics, now I'm comparing the alpha value with the p-value. That is, these are areas instead of values in my x-axis. So if this area is smaller than this one, then I can reject the null hypothesis. Again, for the same reason that Andrea highlighted, because we are outside this 90% confidence interval that we selected. Exactly. So I think it's important, the way I always look at it, this like Inma said, these are values. So 1.28 and then we're hearing 1.50 or something. So we're outside. Well, these are percentage. So this is 0.5, 0.05. And let's say you got a p-value of 0.02, then you know it's smaller. So these are percentages. These are values. Yeah. And just keep this in mind. I think the image helps, but sometimes I guess people get confused with these two. Please check it out. That is a greater than the critical value we can reject. The p-value is smaller than the alpha value we can reject. So think of this image and it will help you to get the intuition behind this. For me specifically, and this is unsolicited advice, but for me it's very helpful to always write the graph. So I always put my numbers. I always put my mean, my intervals. So that really helps me to see, am I doing a two-tail to the right, like a two-tail or a one-tail to the right, a one-tail to the left? It really is helpful to understand what I'm doing. And like we saw in the graph, now we're like, okay, my test statistic, it's outside. So I have here and my new value, it's outside. So it makes sense to reject it, because here is what my value should be. And if I'm outside, I can reject it. So I suggest that if you are struggling with this, make the graph, make it more manageable. And it will be more intuitive, because if you just throw the formulas and we change the problem a bit, you're going to find it challenging. Yeah, great advice. Thank you. So let's see how the problem looks like with this in mind, what we just mentioned. So the question two, we mentioned it earlier, they were asking us how the introduction of new evaluation methods had an impact. If it had an impact or not, in the supplier's delivery date. So we have all this data, and we have to formulate the hypothesis test. So as I mentioned, no hypothesis, nothing changes. So my null hypothesis in this case will be like actually the new evaluation methods had no effect. The delivery date is, the delay in delivery dates is similar before and after the implementation of the new evaluation methods. While my alternative hypothesis will be that something changed. So the current situation is different than the previous situation. It does, we're not looking for direction. Here, we just want to see if something changed or not. Therefore, what we want to do is a two-tail test. We want to see if something changed. Therefore, this is two-tail. And we will use for our, we will use a normal distribution because we can assume that due to the size of our sample. The size is 138 observations. Therefore, we can use a normal distribution. So we're not going to go through the solution because it's already posted on the video. We just want to highlight how we look at it. If it's a two-tail, a one-tail, why we're saying a two-tail? Like Inma mentioned, two-tail because we're looking at differences. Any difference. We don't care bigger or smaller, just a difference, then it's a two-tail. And then the second question is also a hypothesis test. In this case, the problem, the question from the problem is, are there more delays now than before introducing the new evaluation methods? So now we have another sample. The sample is size 12. And the hypotheses that we need to formulate are different because now the null hypothesis is, okay, so the delays are less or equal than before. And the alternative hypothesis is that the delays are greater than before. That's what we want to test. So in this case, if the sample is smaller, we will use a t-distribution. And since there is directionality in the test, we want to see if actually the delays are greater. Then we will use a one-tail test. One-tail test to the right. And so this is what we wanted to highlight. We wanted to highlight how to think about the hypothesis, the null hypothesis and the alternative hypothesis, how to think about which distribution to select. Depending on the size of the sample, remember smaller than 30 t-distribution, greater than 30 normal distribution. And also when to use a two-tail test, a one-tail test, two-tail, when there's no directionality, we just want to compare if everything remains the same or if something changed. And then we will use a one-tail test when there is directionality. We want to check if something is greater than or smaller than before. Perfect. I don't know if we have any questions from the students. Yeah, so let's see. So it says, any possibility of having a PPT or PDF format of this explanation, especially for the curves? Yes, sir. We could share the presentation in a PDF format in the exam prep section. So right in the same section where we share the link to the life event, we will share the PPT. Perfect. There are other questions about the exam. I don't know if you want to maybe spend a few minutes on that. Sure, definitely. Let's do it. Okay. So it says, Marita Cornejo says, do we get immediate results of the questions after answered? No. We made this very clear in the instructions, but I think we should reinforce this idea again. You will not receive any feedback for your answers. So you will submit your answers. You will know that they are submitted, but you won't get any green or red checks telling you if you're right or wrong. So it will be like a real exam at university. Yeah. Alucin Camara says, any special software or platform, apart from the edX practice platform that we have been using to solve the problems. You can use any software we want, any software of your choice. You can use Excel, we can use SAS, you can use other software that you might like. You can use anything. It's an open book exam. You can also use slides. You can check the videos from the course, your own notes. The only thing that is forbidden, strictly forbidden, is to collaborate with others. So please read the edX on our code, but we do not allow anyone to talk to others or interact with others or try to solve the program exam with others. And we will take measures to avoid this from happening. Yeah. Maybe highlighting that during the exam, if you have any trouble, you could, since you are not allowed to talk to others and to use the discussion forums to talk about the exam, that's forbidden too, you could reach out directly to us through the SC0x help email. So it's sc0xhelp at MIT.edu and Andrea and myself will be there to answer your questions. But we won't provide tips on how to solve the problems because you have to solve them yourself. Exactly. We can provide clarifications. Maybe the question is not clear to you. If you think there is some mistake, we're always open to hear them. We will double check that our answers are right. Although we do beta test very extensively every exam before we release it. But if you have any burning questions, just send it to us. Don't share it with other people because that against the honor code. So we have Sivan that says, I'm worried about the difficulty of the exam. I'm not starting from external material. Should I worry? You shouldn't need any external or non-material for the exam. If you just go through the course and check all the contents with the practice problems, you should be more than prepared to take the final exam. The level of difficulty should be, we create the exam accordingly to the course, to the contents that we deliver in the course. So don't worry about that. I encourage you to take the, as I mentioned earlier, the test your knowledge section in each module because they are a little bit harder than the practice problems usually, more similar to exam problems. I encourage you to take the problems in the exam prep section and also the, how did we call it, the prep exam, the mock exam that we will provide one week. The practice. The practice standard we will provide one week before we open the real exam. And you will get a sense of the difficulty of the final exam. But be confident and I'm sure you will be able to do it. There was another question regarding the attempts. So I can maybe answer that. We have normally three attempts for QQs or PPs, right? Then for test your knowledge, we only have two attempts. The reason for that, we want you to be able to get the right answer within two attempts. Why? It's because on the final exam, you get two attempts. That doesn't mean you will get a green or red tick after the first attempt as in the problems. You would only have two attempts in case you made a mistake. You submit it in the wrong box and you're like, oh my God, I wanted to change this. You get a second attempt. But we are not going to let you know if your first answer is right or wrong, which is similar to when you take an exam, right? You can erase it. You can change your answer. But nobody's telling you if your first attempt is right or wrong. We have another question. I think it's interesting. I joined recently in the course. What happens if I fail the final exam? What are the next steps? You joined recently a course. I hope that you're allocating a lot of hours to get ready for the exam. If you dedicate enough time every week, you can still make it. But of course, you need to dedicate a lot of hours because it's the course. It has a lot of contents. And it also depends how familiar you are with the techniques and the contents that we're providing in the course. This being said, if you take the final exam and fail, then if you still want to get the certificate from the CCRX course, you will have to enroll as verified in the next run of a CCRX. Perfect. If you fail, yeah. Okay. How many questions will be there? Any specific timing per question or the whole thing is four hours? Yes. So the whole thing is four hours. There will be four problems. And we usually, we create them so they don't take more than two hours to solve for our beta testing team. So you should have more than enough timing for hours to complete the whole exam. But don't worry about the time. The process will be four problems and the exam will be comprehensive, meaning that it will cover all the topics from the whole course. And if you want more information, all the instructions about the final exam are in the first module of the course, in the course overview module. If you go to the subsection that is called Certificate and Assessment, I have listed there all these clarifications about the final exam. But please, if you have any other questions, just send us an email to CCRX Help or ask us in the forums and we will be happy to clarify. Great. I also noticed a lot of students in my experience that they are not maybe comfortable with these two options that we have in the exam, which are save and submit, just in practice problems. So saving, you just save locally. You're not sending it. You're not using your attempts. We're not grading this question. You're just saving it so it doesn't go anywhere. But if you don't submit your questions, they are not going to be graded. If you end your exam with saved questions, they are not going to be graded. That's very, very important. The other tip I have for you is you have four tabs, right? So you can go from one problem to the other. You can go backwards, forwards as you see fit. However, if let's say you're in problem one, you enter your answers. You don't save them. You don't submit them. And then you go to question two and then you go back to question one. The answers will be deleted because you didn't save them. You just put it there in your computer. They're not in our system. So if you just want to, like I don't know, maybe your strategy is put all the answers and then submit them, put them, but save them. And at the end, make sure to submit them. If the four hours pass and you don't submit them, they are not going to be submitted automatically. So please make sure to submit them. We had a lot of extreme cases of people that did the entire exam just saving their answers and they were not graded. Yeah, since you have two attempts, maybe the safest choice is to submit once you've solved the problem once and then go back. If you have time, review it and if you change your mind and want to change the numbers, you still have a second attempt to do that. So yes, yeah. Perfect. No live events for module one or two are they exempted from the exam? No, no. No modules are exempted from the exam. The exam will cover all the modules. We just have a limited time for live events because we have a limited time here to prepare for them. So we chose the topics where we know people struggle the most. So that's why we had a live event about probability, a live event about statistics and a live event about linear programming. The thing is, even in the statistics one or the probability one, we didn't cover the whole thing today. We didn't talk about TI square tests. We didn't talk about the central limit theorem. We didn't talk about regression. And these are very important topics too. So please don't think that only the things we cover in the live events are the things that will go into the exam. The exam will be comprehensive. And any topic from the whole course can be part of the exam. And actually we'll try that they are in there. Yep. And I think we have a final question. It says, what is the pass mark? And if I fail the exam, will I have to pay $200 again to retake it? Yeah, so the pass mark is 60%. You need to score 60% or above in the final exam. Yes, if you fail and you want to get the certificate, you'll have to pay the $200 again in a next course. So, but don't think about failing. I think you all can make it. We still have three weeks to go. And if you are completing the modules, you will be able to pass the exam. You should be ready. I think the first slide is helpful. Try to locate as much time as you can. If you have committed to it, make it through it. And we're here to help in anything that you may need. And I hope you found the explanation about confidence intervals and about hypothesis testing helpful. If you have any suggestions or comments or feedback, please share them with us. We're always trying to make things better. We're always trying to provide a live event that is helpful to you. And if you have any comments, we would be happy to hear them. Yeah, you can use the forum in the live event section to actually leave your feedback or let us know what you would like to have for the next live event that could be more useful for you to prepare for the final exam. But yeah, definitely we hope this was helpful. We know these concepts are a struggle for a lot of people. They are kind of confusing and they are sometimes hard to explain too. So I hope we were clear enough and we will keep working in the forums to answer all the questions that you may still have about these topics. If you are on your way to the entire MicroMasters credential, the good news is that this is the most math-intensive course. So things will get better. Things will be less math, less statistic-intensive. So if you make it through this, we're pretty sure you will get the credential smoothly. Agreed. Okay. Thank you so much. Thank you for joining us for this live event and see you in a couple of weeks for the last live event. Thank you everyone. Bye.