A final exam in elementary statistics averages 73 points with a standard deviation of 7.8 points. If we have 10 students that are randomly selected, is it possible to apply the central limit theorem? Sample: All right, so randomness is clearly expressed there, so again, as long as you have that, we're good to go. So now let's check the normality condition. Again, you only need one or the other of these, so the first one is population shape. Looking back at the prompt, were we even given anything about the population shape? Looking at the prompt, we weren't. So in this case, if your shape isn't even given, you don't get to use this option. You have to be explicitly given it's normal. So if either the shape is not given like what we have here in example two, or if the shape is given but the shape is not normal, meaning skewed, both of those situations are examples of where the population shape condition doesn't work. But what's the issue with my current sample size of 10 students? What's the problem with that sample size? What did we need that sample size to satisfy? It needed to be greater than 25, but 10 is not. 10 is not greater than 25. Uh-oh, double sad face. And so in situations where both fail, in situations where you both don't have the right shape as well as too small of a sample, unfortunately, it means then that the normal condition fails. And here's the cool thing: we get the same concept again where if even one condition fails, if even one condition fails in the central limit theorem, you once again cannot apply the central limit theorem. So ultimately it's the same concept like we saw all the way back in Chapter 7, that if the central limit theorem condition fails and more often than not the condition that will fail is the normal condition, we will honestly have an exact situation like this: you cannot apply the central limit theorem. The lucky for us someone ends up coming around and ends up giving me a little bit more information and in this case, they ended up giving me the little bit more information telling me that, okay, I gathered all of those stat scores, I graphed them out on my graphing calculator, and I saw the graph ended up being normal in shape. So you, utilizing this same random sample of 10 students, so we're still using the same random sample of 10 students, I want you to see that this new piece of information explicitly being given that the data I'm looking at is normal in shape, literally that is what I'm talking about when I say you need to be given the shape is normal. All right, and that once you're given that shape is normal, that's literally all you need to satisfy that second normal condition. And what I want you to note is I didn't even check the sample size here. Why? Because honestly, once your population shape is normal, it means the world is your oyster because if your population shape is normal, what that inherently means is then your sample should be normal in shape. Therefore, if your sample is normal, it almost doesn't matter what size your sample is because that sample should always be normal in shape. So bottom line is that honestly it's the best when the population is given to be normal because then it doesn't matter how big of a sample you have. And then lastly, checking large population, again we're looking at students so taking 10 times my sample size 10 times 10 is 100. Are there definitely over 100 students? Are there definitely over a 100 students taking statistics? Yeah, absolutely. So in this case, in this case, all the conditions hold when looking at the central limit theorem conditions every single condition. So we then can go to the second part of the central limit theorem, the results. The results for my sampling distribution, we again we can identify the shape of that sampling distribution. Can you guys remind me always always what will be the shape of the sampling distribution? Normal. I mean guys, we literally spent an entire chapter almost in Chapter 6 discussing normal data because ultimately normal data is super important when understanding sampling distributions. Center, when it comes to understanding the sampling distribution, its center is going to be equal to the population mean. So going back to the very, very, very, very beginning of the problem, what is the population mean here? Yeah, it's going to be that average of 73 points. It's that average of 73 points. So in this case, 73 points. Lastly, we need to calculate spread, the standard error formula of taking my, again, population standard deviation and dividing it by the square root of my sample size, dividing it by the square root of my sample size. So why you guys give me a hand here and I'm going to zoom out a little bit what is going to be my population standard deviation? What is Sigma here? Yeah, it's that given standard deviation of 7.8 points and we're going to divide this by the sample size. What is my sample size here? Yeah, sample size of 10. Find that standard error and I want you to give it to me with numbers as well as the proper units. What does this number even represent? It's going to be 2.47 points. Ultimately, the statistical inference here, what we expect when it comes to say the next time I give this final exam is I expect the mean final exam score to be that 73 points that I found from my center, give or take, give or take emphasizing there might be a little error and that that error is the spread I had just found 2.47. Central limit theorem, really the whole point was to get to this problem right here. The whole point of even discussing "what is a sampling distribution?", and really teaching you guys these formulas, was really to get us to a place where we could combine this all together to give us the central limit theorem. Now I'm going to perfectly level with you guys: these results of the central limit theorem - these formulas for center and spread - you're only going to see them here in Section 9.1 and 9.2. All right. So practice them. Get to know them. They will be on the exam, but ultimately, the main purpose was to make sure you guys understood part one of the central limit theorem. The main point, and probably the most important takeaway from today's class, is: Do you know how to show all three conditions of the central limit theorem? Why? Because the central limit theorem conditions, when you are looking at sample means, is going to come up over and over again for all of Chapter 9. You guys have already seen this in Chapter 7 and 8. I taught you guys the central limit theorem conditions for proportions, and notice how those conditions came up again in Chapter 7, again in Chapter 7, again in Chapter 8, again in Chapter 8. And so, in the same way, really, the main point was to introduce you guys to these three conditions when we're now looking at sample means because these three conditions are literally the three conditions we're going to use again and again and again and again throughout all of Chapter 9 so that we can do things like run confidence intervals and run hypotheses testing. Make sure you understand how to satisfy these three conditions.