The San Mateo County Clerk wishes to improve voter registration. One method under consideration is to send reminders in the mail to all citizens in the county who are eligible to register. As part of a pilot study, 1,250 potential voters were selected and divided into two groups. Group one is 625 voters who received no reminder, and of those, 295 registered. In group two, the 625 voters who were sent a reminder, and now 350 are registered to vote. So, in this case, who are my two groups? My two groups are voters who got no reminder, and my group two is then the voters who were sent a reminder. Those are my two groups, and ultimately, my success here is registering to vote. Now, ultimately, we just did Step Zero, and for this particular problem, we are just assuming all the conditions hold. But we can really see that fast. We can see that they selected people randomly; condition one done. You either get a reminder or you don't, so Independence is done. We can see we have large samples here because we can see our number of successes are both greater than 10, and if you subtract the sample size from successes also greater than 10. And that we're looking at voters in San Mateo County, which consists of like tens of thousands of people; it's a huge population. So those conditions are good. So, what I really want to focus on is steps two and three: how do we find the confidence interval and ultimately how do we interpret it? So I'll give you guys a moment to do that. I'll give you guys a moment to identify what is X1, N1, X2, N2, plug that into 2-PropZInt. I want you guys to be able to dissect the numbers given and plopping that into the calculator. Again, the beautiful part about this particular problem is literally every number that you need to plug in 2-PropZInt is given in the prompt. It literally gave you the number of successes and the sample size, and again, all four of those values is what we plug into 2-PropZInt, and we ultimately get that we are looking at the interval from -0.143 to -0.033. So, I want you guys to see a practical example now of a confidence interval where the lower bound is a negative number (-0.143), and the upper bound is also a negative number (-0.033). And so in situations where your confidence interval has a negative lower bound and a negative upper bound, what then can we say about the entire confidence interval? Is it going to be entirely positive or entirely negative or containing zero? In situations where my lower bound is negative and my upper bound is negative, what then can we say about my entire confidence interval? Yeah, ultimately when two numbers are both negative, what that's emphasizing is that they are all less than zero. And being less than zero means everything is negative. So, what that's emphasizing is that every possible value for P1 minus P2 is negative. The idea here is if the entire interval is negative, if the lower bound is negative and the upper bound is negative, what that's emphasizing is that the entire interval is negative, and therefore P1 minus P2 must be negative. And again, just going back to just basic algebra, basic algebra of us subtracting, again, that's what we're doing. We are subtracting two numbers. Basic algebra. If we are subtracting two numbers and the result is a negative number, we know that when you're doing that subtraction, it's the second number that needs to be bigger. So, for instance, 3 - 5 gives me -2, or if I do say 4 - 6, we see here that we get -2. In either situation, 4 - 6, 3 - 5, I want you to see it is the second number that is bigger. And so, proving this a little more algebraically, again, the fact that P1 minus P2 is negative is going to then emphasize that P1 minus P2 is less than zero. So adding P2 to both sides of this inequality, we can see here in this inequality P1 is smaller than P2, or more importantly, we see here that the larger proportion is coming from group two. The proportion of group two is larger. And that, guys, is the final relationship that we will have when the entire confidence interval is negative. It means then the proportion of group two is larger. That is the final relationship that we will have. So, let me scroll back up and emphasize that all the way back at the beginning of section 7.5, all the way back to our overall steps of running confidence intervals. I want you to see in example three, we're seeing that final option. If the entire confidence interval is negative, it means P2 is significantly larger than group two. And so in the template, in the template, what we then have here is that if the entire confidence interval is negative, the proportion will be larger for population two. And so what I want you guys to see here is that in these last three examples, and yes, I'm color-coding it so you can see the differences, I want you to see we've covered each one of these scenarios in these last three examples. And so let's do the template then for this final example. All right, in this final example, the big thing I want to emphasize is that the larger group is going to be the group two. Now, again, who is group two here? Group two is ultimately voters who were sent a reminder. So, I want to emphasize before we even write anything else in this template, notice that the larger group, the larger proportion group, will be the voters who are sent a reminder. So, let's fill in the rest. We are 95% confident that the proportion of voters who will register to vote, all right, remember that was my characteristic of interest, that's what we said here in orange. Orange was ultimately the characteristic of registering to vote is between something and something. We'll get there in just a moment, larger for those voters who were sent a reminder than if the voter did not receive a reminder. So, you're like, well, Shannon, you left something blank here. You left the percentages blank here. What percentages then do I write down? Well, the thing is those negative signs in the lower and upper value of my confidence interval, those negative signs were in essence already used by identifying the large proportion came from group two. More so, remember, percentages can only be positive values. So, because of that, we're just going to take the value of these decimals, not the negative sign, just the value, and those values are what we will use as my percentages. All right, why? Namely because percentages can't be negative. We learned that in chapter five. Percentages cannot be negative. And so, because of that, because of that, we use the positive of that. And let me just emphasize that now. All right, only when the entire confidence interval is negative, which is exactly what we have here in this scenario, only when the entire confidence interval is negative in the interpretation will you include the positive values. Will you only include the positive values. All right, when the entire confidence interval is negative, the percentages in your interpretation need to be positive: 3% positive, 14%.