According to the CDC, at the end of 2011, a proportion of 0.04 people have HIV. You want to sample a thousand random people. I want us to check the conditions of the central limit theorem. So let's check my first condition: random sample. Randomly selected sample, do we have that in this sample of a thousand people? Yeah, literally, you just look for the word "random", that's enough to be like, "Yep, condition one satisfied." Second one. Second condition that we need to check is the condition of large sample. We need to check the condition of large sample. Now again, to check the condition of large sample, you need to check the number of successes and the number of failures. So contextually, we sampled a thousand people, we asked them a question. What was considered the successful response here? What is considered the success here? And again, I need to emphasize when you're answering what is success, it's words, it's having HIV. And so when we say find the number of successes, we want then the number of people that had HIV in this sample of a thousand. Except what I want you guys to note here is that the number of successes is not given in this problem. What is given: the sample size, sampled a thousand people; what is given: the proportion, the proportion of people that have HIV. And so keep in mind we can use these two values by multiplying them together to ultimately find the number of successes. We can multiply that sample size of a thousand people times the proportion of 0.04 to find the number of successes, meaning number of people that have HIV. Can you guys calculate that for me and tell me what are my number of successes? We look at this sample of a thousand and we would say, "Dang, a thousand seems like a huge number that definitely has to be a large sample." But remember, in the central limit theorem and frankly what makes sound statistics, what makes a large sample is you need your number of successes to be greater than what? What do we need our number of successes to be greater than? And is that the case here? Is the number of successes greater than 10? Here, no. No, in this case, the number of successes are not greater than 10. And so what does that emphasize? It emphasizes that my sample is too small. Even though this sample of a thousand, a thousand people seems like a lot of people, the issue is that there were still not enough successes. There were still not enough successes in my collection. And so what does that mean? If even one of the conditions fail, in this case, large sample, in this case, if even one condition fails, unfortunately, we will need to throw the central limit theorem out the window. In this case, because my number of successes did not satisfy being greater than 10, we then have to note: the large sample condition failed. The large sample condition failed. And ultimately, if even one condition fails, I mean, if two conditions or even all three conditions fail, this is bad, but if even one condition fails, then that means we cannot apply the central limit theorem. So what does this mean? It means then that the conditions for the central limit theorem of clutch, it's literally the first domino to fall if we want to do any central limit theorem problem, that if all the conditions hold, you get the green light to go to part two and write down the results. But if even one condition fails, it stops the problem, and we cannot go any further. Notice there's no point in checking failures, there's no point in checking large populations, 'cause my sample was not large enough.