This module looks at confidence intervals. We'll discuss the interpretation and how to construct confidence intervals in some standard situations. It's important to have a solid grasp on these concepts because most statistic analyses provide some kind of confidence interval. Let's look again at the US Presidential approval ratings. There are about 140 million likely voters. And let's suppose for a minute that 60% of them approve of the way the president is handling his job. Now we do a poll. Let's say, we poll 1,000 of them. The approval percentage we get in the poll should be somewhat similar to 60%, but it will be off a little bit. How much will it be off? Well, we already know that the sample percentage is likely off by about one standard error. Oftentimes, one would like to have a statement that is somewhat more precise. This is done by a confidence interval. We already know that according to the central limit theorem, the sample percentage follows the normal curve with expected value equal to mu equal the population percentage and standard error equal to the population standard deviation sigma divided by square root sample size. In this case, sigma turns out to be 0.49. To compute that, recall that we are looking at the case where we are counting things. Namely, we are counting voters who approve of the president. So, we introduce labels where each voter who approves gets a 1 and every other voter gets a 0. And there are 140 million voters. So, we end up with 140 million labels. And from that, we can compute the standard deviation sigma as 0.48. So, what this means is that there is a population percentage mu, which is 60%. And then, if we sample 1,000 voters, the sample percentage will follow the normal curve. And the empirical rule says, if we go two standard errors each way, then there's a 95% chance that the sample percentage will be somewhere in that range. So, we know that there's a 95% chance that the sample percentage is no more than two standard errors away from the population percentage mu. But saying that the sample percentage is no more than two standard errors away from the population percentage mu is the same as saying that the population percentage mu is no more than two standard errors away from the sample percentage. What this means is, we can take our sample percentage, for example, it may turn out to be 58%, and then we go two standard errors in each direction. And that gives us a range of plausible values for the population percentage. Remember, when we do a poll, we do not know the population percentage. The whole idea is to take a sample and get a sample percentage. And hopefully, the population percentage will be close to the sample percentage. So, what we do is, we take the sample percentage of 58%, go two standard errors in each direction, which takes us to 54.8% and 61.2%. And the interval between these two numbers is called a 95% confidence interval for the unknown population percentage. So, why is this called a confidence interval? Where does the name confidence come from? And why don't we simply say probability? Well, the population percentage mu is a fixed number. And that number is either inside the interval or outside the interval. So once we write down the confidence interval, there are no more chances involved. And for that reason, we talk about confidence. The randomness in this procedure comes really through the sampling. If we sampled another 1,000 voters, we will get a slightly different interval. So, the interpretation of a 95% confidence interval is, that if we do many polls and for each poll we do a confidence interval, then 95% of these intervals trap the population percentage and therefore, 5% percent will miss it. The number 95% is called the confidence level that comes with a confidence interval. Here's a picture that explains the situation. There are some population percentage mu. Let's say it's 60%. And then we take a sample of size 1,000, and we compute a confidence interval. For example, that confidence interval might be from 55% down here to 65% up here. And you see, in this case, the confidence interval traps the population percentage mu. Then, next week, Gallup might do another poll, and it comes up with another confidence interval. And that might run from here to there. And somebody else might do a third confidence interval, and it may be from here to here. And another poll may result in a confidence interval from here to here. And now you see, this one misses the population percentage. So, among this large number of 95% confidence intervals, most cover the population percentage, but some don't, like this one will not and this one will not. The way a confidence interval is reported is oftentimes something like, "I'm 95% confident that the President's approval rating is between 54.8% and 61.2%." And the interpretation of this is that 95% of the time, I'm correct when making such a statement, and 5% of the time, I may be wrong. So, keep in mind that the randomness is in the sampling and not in the population percentage.
