So, then can we go to the second part of the Central Limit Theorem? We can ultimately look at the results of the Central Limit Theorem. Now, when it comes to the results of the Central Limit Theorem, they are all about the sampling distribution. So, again, sampling distribution would be if I took 200 drivers over here and found the proportion that texts while driving. I took another 200 drivers over here, found the proportion that texts were driving. I took another 200 drivers over there, took the proportion of them driving, and I did that over and over and over again. I found a bazillion samples of 200 drivers and found each proportion that texts while driving. And if I was to graph all those proportions, if I was to graph all those proportions, we would first identify that the shape is normal. Why? Because that is the result of the Central Limit Theorem. The first result of the Central Limit Theorem is that the shape of the sampling distribution will be normal. It was not proven to you guys. It's something that they have used pretty intensive calculus to prove. But ultimately, that's the first thing and always the same answer you will have. Always, always, always when it comes to the sampling distribution, the shape will always be normal. Why? Why is it so important that the sampling distribution have a shape that's normal? Well, let's go down to the application. At the end of the day, when it comes to statistics, more often than not, people aren't walking around asking, "What are the results of the Central Limit Theorem?" I mean, if they did, you would have heard of the Central Limit Theorem by this point. Practically, when it comes to the real world, the question people always ask is, "What is the probability? What is the chance something will happen?" That is the practical application question people are always asking. What is the chance more than 30% of the drivers on the road are texting while driving? Because that's terrifying. That's a lot of people texting. That's a lot of accidents that can happen. And so, ultimately, when it comes to this application of finding probability, we need to go back and ask ourselves, "Well, how did we find probability in the past?" And let's go back and remember, we found probability in the past, back in section 6.2. Because back in section 6.2, we found that probability is equal to normalcdf. We found back in section 6.2, probability is equal to normalcdf. And if you're like, "What does that mean?" You gotta go back to section 6.2 and learn that material. Do you understand probability is found by using normalcdf? But here's the thing: to be able to even use normalcdf to find probability, you literally need the shape of your data to be normal. And bam! That is what that first result of the Central Limit Theorem does for us. That first result of the Central Limit Theorem tells us, "Yes, my shape is normal." So, therefore, I, in fact, can find probability using normalcdf. And so, while your response is always, "It's always going to be the shape is normal," I want you to see that even though that response is going to always be the correct answer, I want you to understand the power of that response. The power of that normal shape means I can use normalcdf to find probability. Mic drop. Because now we literally then can go back to, "How did I utilize normalcdf?" Remember that in normalcdf, you then had four inputs: lower bound, upper bound, mean, and standard deviation. And so, the next question then is, "Well, then what is the mean? What is the standard deviation? What is the mean that I'll put in the middle of that normal curve? What is the standard deviation that I'm going to use to move from tick mark to tick mark?" And that's where we go back to the Central Limit Theorem results. That mean, that standard deviation, then can be found using the center and spread results of the Central Limit Theorem. So, let's go back a couple of pages and see what that is ultimately. We said that the center would equal the population proportion and the spread would equal that standard error formula. So, let's apply it. Let's apply it now. Ultimately, first center, the center, the mean. The mean of all of those sample proportions is the population proportion. But what's the issue with that? Well, guys, I'm color-coding on purpose. I am color-coding our lecture notes on purpose because I'm really trying to help you see visually where different pieces of data weave throughout the problem. And what I want you to know is there's nothing green in the example one. Notice in the example one, I was given my sample size. I was given my samples number of successes, but nowhere was I given the population proportion. Notice that the population proportion is unknown. Notice how nowhere were we given that population proportion. And so, at that point, we would be like, "Shoot, I'm in a pickle." Literally, this value P, this value P, which the Central Limit Theorem result says should be my center, is unknown. So, Shannon, what do I do? And that's where we go back and remember. We go back and remember because all the conditions hold. We made a good sample and that a good sample means that I can replace my population proportion with my sample proportion. I can replace P with P-hat. I can replace P with P-hat. I can replace the population proportion, which is unknown, with now the sample proportion. Why is that so powerful, guys? Well, it's because we can literally construct this sample proportion. Remember, proportions are fractions where you put your number of successes on the top and the total number in your sample on the bottom. Why is this so powerful? It's because literally, we were given those two numbers. What fraction then is the sample proportion? Yeah, it's going to be the 48 people who said I text while driving out of the 200. Why is this so powerful? It's because my center then went from some unknown number to then some known number. And while this is my sample because we identified it as a good sample, what that means is that this proportion will be a fairly good representation of my population. 48 divided by 200 then is 0.24. That 0.24 then is going to be the mean, the center of my normal curve. And so, I want you to see that by identifying this center, it's giving us the center of this normal curve. It's quite literally giving us the third input into normalcdf. And so, all we have left to do then is find the spread. And remember, the spread is the standard error formula of the square root of P * (1 - P) / n. And again, I want to make a point: don't memorize this formula. It's an ugly formula. Write it on your note sheet. Don't memorize it. Now, again, the pickle about this formula is it involves this unknown P again. But again, using what we just highlighted in yellow above because we have a good sample, I then can replace P with P-hat. I then can replace P with P-hat. So, in this case, I can take the square root using my P-hat value of 0.24. 0.24 * (1 - 0.24), all divided by that sample size of 200. But what you should get is 0.033. What is the spread? 0.03? Well, that's then the standard deviation. That is then the standard deviation of this normal curve, meaning that 0.03 is then representing how much we will increase and decrease to get from tick mark to tick mark. And so, from there, we then can draw out each of the tick marks.