Hello and welcome back. My name is Mark, and today I'm going to be showing you how to use the Seeing Theory App. Seeing Theory was created by Daniel Kunin, an undergraduate at Brown University. His goal was to make statistics more accessible through the use of interactive visualizations. Specifically today, we'll be working with the interactive visualization he has created for confidence intervals. To get started, either click on the link provided or head to your web browser and type in Seeing Theory Brown University. So, whether you use the link or use your web browser, you should see a page that looks like this, students.brown.edu/seeing-theory. Once you're here, go ahead and click on chapters on the top left corner, and then to navigate to the chapter we want, go ahead and click on Frequentist inference. You can see the six chapters listed near the top of the page, go ahead and click on that fourth one. Once you're here, you're going to click on Interval Estimation, this middle white box and that should take you to the applet that we desire. On the left-hand side, are going to be a couple of inputs that we're going to change throughout today, and on the right will be the interactive visualization where that takes place. So, first things first, we need to choose a distribution in which we're going to sample from. They've listed a couple of different ones here, we're going to start with a normal distribution. Now, you can see up on the right-hand side, we've got a normal distribution and the population parameter for that distribution is right down the center as notated by this dotted line. Now, we can choose a sample size and a confidence level for confidence intervals that we'll be creating, and we can go ahead and start sampling. For the first go around, we're just going to stay with a sample size of five and a confidence level of 90 percent. So, what does this 90 percent mean? Ninety percent confidence level means that if we were to take repeated samples of the same size, here our sample size is five, then we would expect each of the out of the resulting intervals that we have, 90 percent of them should contain that true population parameter that we're after. So, let's go ahead and start sampling. So, you can see that we're taking samples of five from this distribution up here, and they're dropping down and creating an interval. We've got our best estimate and the center, plus or minus some margin of error. The app goes ahead and colors the green one if they contain or overlap with that true population parameter, and it changes red if it does not. If you can see on the left side here, we've got a nice bar chart that shows just what percentage of these intervals have captured that true population parameter and which ones have not. Again, over time since we set this to a 90 percent confidence level, we should see that this bar chart should level out to about 90 percent and 10 percent for those that contain the true parameter, and those that do not. So, now let's see how these intervals are affected by changing the sample size and the confidence level. First, let's change the sample size. So, if we slide this n equals five all the way to the right to its max value of 30, we'll see that the widths of our interval's got much narrower than they previously we're. So, now I'm looking at this, again if we watch over time since we didn't change our confidence level at all, we should still see about 90 percent and 10 percent for these two bars, and they're fairly close now 91 percent and eight percent. So, if we kept watching this in the long run over time, we should see this level out to about 90 percent and 10 percent. Finally, let's go ahead and change our confidence level, if we slide this bar a little bit to the left let's say 0.6 or so, this is where the app can get a little bit touchy, so 60 percent. Now, we should still be getting, we still have a sample size of 30. But if we look at how many of these are green and how many of these are read, only about 60 percent of these new intervals being created should contain that true population proportion. So, this gives you a good idea of really what confidence level is. A confidence interval is just a single interval here as it moves down the screen. The confidence level however saying that overtime if you took repeated samples of the same sample size, we would expect to see 60 percent of these resulting intervals containing the true population proportion. That's pretty much roughly what we're seeing here. Now, feel free to go ahead change the distribution, the sample size, or the confidence level. Play with this on your own. But this will give you a good grasp of confidence intervals, confidence levels, and the inputs that changed them.