Step two is actually finding that confidence interval. Finding that confidence interval using your calculator. Please, please, please don't do the crazy huge formula to calculate the confidence interval by hand. Please, please, please don't do that. Alright, rather, I want you to use your graphing calculator. So to get to the steps of doing that, we go back a couple of pages and we can see the steps here for calculating using our graphing calculator. What's the first two steps? Well, it's already steps you should be comfortable with from section 7.4. You go to "Stat" and you go to "Test". You go to "Stat", toggle over to the right-hand column of "Test", and notice this is where we went to get to "1-PropZInt". That's what we did in section 7.4, except "1" emphasized one population. Well now we're looking at two populations. So we're now going to choose option "B", which is "2-PropZInt". We're now going to pick option "B", which is now "2-PropZInt". Why is "2-PropZInt" so important? Well, for starters, two is really the big one, guys. Two is emphasizing we're looking at two populations. And so that's why we need two. Because we're looking at two populations. "Prop" again is emphasizing the fact we're looking at a proportion, meaning we're looking at percentages. And then "Int" - well "Int" is literally to emphasize we are looking at a confident interval. All right, and so when it comes to the calculator function of "2-PropZInt", I want to emphasize that the name of it is lending itself to what we want to do. We're looking at two populations, we're studying proportions, we're doing a confidence interval. And the beautiful thing is that when you select the correct calculator function, it's going to tell you exactly what you need to plug in: X1, X2, N1, N2. It's going to tell you everything you need to plug in. And what is incredible, guys, is that literally every single one of these values is going to have been given to you already. So let's zoom out a little bit and let's see what we have here. In this case, let's go back and remember that X represents number of successes and that the "one" means from group one. So X1 means successes from group one. The number of successes from group one was literally the first value we calculated. So quite literally, guys, you are going to take the number from above that we used to check the number of successes is greater than 10, and that is our first input in the calculator. N1, "N" means sample size, "one" means from group one. It's the 3,000 that we used to calculate that number of successes. Notice the "one" emphasizes all of our information coming from group one. So inherently then, X2 and N2 are going to come from group two. What's X2 here? Yeah, number of successes from group two. It's going to be that 930. And lastly, N2, we can already see that here, it's the 3,000 that was surveyed in 2004. Lastly, confidence level. The "C" like the letter "C" Level, it's given to us here. What is the confidence level here? 95% confidence or 0.95. Typing those values into my calculator, we have 1,410 from X1, 3,000 for N1, 930 for X2, 3,000 for N2, with a confidence level of 0.95. So let's hit calculate. And if you hit calculate again, some of you might be given lower and upper explicitly or some of you might be given it as two numbers inside parentheses. But again, notice, you'll get the same two numbers: 0.1357 to 0.1843. That's step two. Step two is simply finding that confidence interval. But step three then is interpreting it. Step three is going to be interpreting this confidence interval. And I'm going to tell you guys right now, step three will be the hardest part of section 7.5. And that's why we're going to do three examples together to make sure we really clear up any confusion there might be. So what we're going to do is look at this confidence interval. Let's just look at it on a number line. We're looking at a confidence interval from 0.1357 to 0.1843. Again, what does this interval represent? Remember, this interval, any number between 0.1357 to 0.1843, is representing possible values for the difference of P1 minus P2. All right, I need to emphasize this. Confidence interval is representing possible values used for the difference of P1 minus P2. I need to emphasize that 0.13 is not P1. 0.18 is not P2. Rather, any number in this interval is representing what the difference of these two proportions can be. And so, how do we interpret this? What we need to do is ask the question: Is this entire interval entirely positive? Is this entire interval entirely negative? Or is zero somewhere in this interval? Is this entire interval entirely positive? Entirely negative? Or does it contain zero? How can you tell? How can you answer this question? Well, you want to look at your lower and upper bound. We can see here, my lower bound is a positive number. We can see here, my upper bound is a positive number. If both these numbers are positive, it means they are both greater than zero, which means that this entire interval is positive. It means that this entire interval is positive. And why is it important that this entire confidence interval is positive? Well, again, this interval, this yellow interval, means that any possible value in this interval will represent this difference of P1 minus P2. And so if every single value in this interval is positive, what that means is that this difference between P1 minus P2 must be positive. It's telling us that the difference between P1 minus P2 is positive. Why is this so important? Well, let's just think about two random numbers. Let's just think about two random numbers. I want to take their difference, again. Again, taking the difference, just like we're doing here. Difference, difference. And I now know their difference is positive, like positive 3. Well, can you guys give me just some pairs of numbers where when I subtract them, it will give me 3? 8 and 5. 8 minus 5, it's going to give me positive 3. Can I get another example? 4 minus 1. Where in both these examples, whether we did 4 minus 1, 8 minus 5, I want you to see it was the bigger number. It's the bigger number that must come first when you are looking at subtraction and you get a positive number. Notice, it's the bigger number that needs to come first. How can we show this a little more algebraically? Well, again, if P1 minus P2 is positive, meaning greater than zero, well, we can solve this inequality by adding P2 on both sides. And ultimately, what I want you to see here is that P1 is larger. What I want you to see here is that ultimately when the entire confidence interval is positive, what that's emphasizing is that the proportion of group one of group one is larger. All right, what I want you guys to see here is this: We find this confidence interval in step two, and honestly, all we care about is if the entire interval is positive, if the entire interval is negative, or if it includes zero. And in particular, if the entire interval is positive, what that's emphasizing is that the proportion of group one—meaning that first number, P1, the proportion of group one—is larger. And that, guys, is the first of three interpretations we're going to see here today. If you guys go back a couple of pages, if you go back a couple of pages to step three, I want you guys to see example one. Is this first option. If the entire confidence interval is positive, it means P1 is significantly larger. It means that if the entire confidence interval is positive, it means the proportion is larger for population one. I want you guys to see that we just experienced firsthand in a practical example, the first of three possible interpretation options we'll have in section 7.5. And it's that if the entire interval is positive, it means then P1, population one's proportion, is larger. And so I gave you guys then a template emphasizing this again. You list your confidence level. We are 95% confident that the proportion of—well, the characteristic we are studying here, in this case, we are studying people who strongly favor gay marriage. All right, we are 95% confident that the proportion of people that strongly favor gay marriage is between 13.6% to 18.4%. Where are these percentages coming from? They're literally the confidence interval percentages. And we have that this percentage is going to be larger for group one. So those in group one are ultimately those who are in 2012. Then in group two are the people in 2004. So what's the big takeaway? It's the green boxes. All right, the big takeaway is that if your entire interval is positive, it means that the proportion for group one is larger. So the larger proportion will come from, however, you're describing your group.