We are going to do some practical examples from Section 7.4. I mean, really, what we're going to do are the types of questions that are real-life applications of how we use confidence intervals. Then, on a practical sense, it's what the majority of your homework questions will look like, as well as being practically the type of questions I'm going to ask you on the exam. How do we make a confidence interval? How can I practically find my confidence interval using my graphing calculator? Ultimately, I want you guys in this class to only find your confidence interval using your TI84 or 83 calculator. All right, don't do it by hand. So, what are the three steps when it comes to finding this confidence interval? Well, ultimately, it's going to have three main steps. The first main step is actually a step we have already talked about back in Section 7.3. Remember back in Section 7.3, we verified the conditions of the Central Limit Theorem to, in essence, determine if we have a good sample? Meaning if the sample we collected is, in fact, a good representation of the population. And remember, we verified that sample by checking if it was collected randomly, by checking if it was large enough, and by checking if the population is large enough. So, that skill we learned in Section 7.3 is going to come back and help us now in Section 7.4 because, once again, as step one, it's ensuring to us that we, in fact, collected a good sample. It's ensuring to us that all the work we're going to do next is, in fact, a legitimately good result. So, after we've done step one, you then can go on to step two. Well, ultimately, step two is going to then be doing the calculator work. It's going to be doing the calculator work, and I am going to talk about all of the different buttons you need to press, all of the different numbers you need to plug in, so that ultimately the outputs of that calculator work will be the confidence interval. So that we then can do step three: take that confidence interval and ultimately write it as a complete sentence. And so, what I want you to see is that ultimately as we talk about confidence intervals, there will be three main steps. All right, and in general, what we're going to talk about for 7.4 and 7.5 are going to be confidence intervals, and these will always be the three steps. And as I mentioned to you guys already when it comes to these lecture notes, so I'm giving you a practical usage of this packet again, there's going to be Parts in the packet that's black and those are the, the, the prompts to the problem. Literally in black is what you will see on the exam. Those will be the exam questions in black or what you will see in the homework. In black are the practical applications and given bits of information you'll see when we need to use statistics. And that in purple will then be the scaffolding you will need to complete these problems. It will be the outlines to complete these problems. And again, before we even get into the nitty-gritties of this particular problem, I want you to see that in this scaffolding in purple, we literally have the three steps of the confidence interval. Notice how we have step one where Step One is emphasizing checking the conditions. Notice we are going to have step two where step two is about using the appropriate calculator function to find that confidence interval. And then I want you to note we'll have step three where step three, once again, is making an interpretation. And as you guys can see, I provided a template in purple because the interpretation will always follow this purple template. And so before we even get into the weeds and the details of this problem, I want you to see I provided a template for you so that you can ultimately use this every single time we are asked to do a confidence interval problem. All right, now let's get into it. In 200,000, the GSS, which is a Surveying Company, asked: "Are you willing to pay much higher prices in order to protect the environment?" And of 1,154 randomly selected respondents, 518 said they were willing to pay higher prices. Find and interpret a 95% confidence interval for the population proportion of adult Americans willing to do so at the time of the survey. So, step one. Step one is you need to check the conditions for the Central Limit Theorem to hold. And guys, literally, step one, these conditions are the exact conditions from Section 7.3. It's exactly the same. It's exactly checking if the sample is random. Exactly checking both things to make a large sample. Checking you have a large population. And so this will be a really good review of last class. Again, random selection. Random selection is asking if the sample you collected was, in fact, collected randomly. Reading the prompt, was the sample collected randomly? Yeah, absolutely, guys, all you got to do is literally look for the word random, and we're good to go. All right, we're good to go. That's all you need. Second, you need to check if you have a large sample. If you have a large sample. Now, remember when it comes to checking if you have a large sample, there are two numbers you need to find and check. You need to check both the number of successes and the number of failures. And so, first things first, we got to figure out what is considered the success here. So looking at my question, "Are you willing to pay much higher prices in order to protect the environment?" Remember, success is the yes response. Yes, yes I am willing to pay much higher prices is to protect the environment. And so that now we know the successful answer was saying, "I am willing." What is the number of successes here? Practically, it's a number. How many people said they are willing to pay higher prices to protect the environment? 518 said, "Absolutely, I am willing to pay higher prices." Remember, failure is simply the not response. So those who say no, I am not willing. And remember that the number of failures is pretty much taking everyone who was surveyed. So, that's n. That is the 1,154 people surveyed. And we note the fact that, well, 518 said, "Yeah, yeah, I'm willing to pay higher prices." So subtracting, not including, ignoring those 518 people who said yes, what's left then are those who said no. How many people are not willing? What are the number of failures here? Perfect, 636. Again, you need to check both of these numbers. And not only do you need to check both of these numbers, you need to check both of these numbers are greater than a specific value. And I'm totally forgetting it, guys, can you help me? What do both of these numbers need to be greater than? What does both 518 and 636 need to be greater than? Perfect, perfect, they both need to be greater than 10, which they are, 500, 600, definitely bigger than 10. And when both are satisfied, only then can you say yes, I have a large sample. Lastly, you need to check large population. So what is my population here? Who am I studying here? Perfect, the adult Americans. And so in this case, you need to ask the question: Are there definitely over 10 times the sample size of that? So, we have already identified our sample size is going to be that 1,154. So you take 10 times the amount of that, so 11,540, and you just need to ask if this statement is correct: there are definitely more than 11,540 adult Americans. Is that statement an accurate statement? Yeah, absolutely. Again, the point of us checking every one of these conditions and getting a yes, yes, yes is ultimately to emphasize that this is a good sample, meaning this sample of 1,154 respondents and 518 saying, "I am willing," it is emphasizing from all of these conditions that this sample was a good sample.