Welcome. Now that we've seen how to make confidence intervals, we want to talk a little bit more about how to correctly interpret them, what they are used for, and how they help us. So, we're going to talk about how to interpret confidence intervals, little bit about what that confidence level really means, and then also understand how we might change up our interval if we want to instead be 99 percent confident, instead of 95 percent. To get us going, let's look at a background to have a little context. In a sample of 659 parents with toddlers, 540, about 85 percent, stated they use a car seat for all travel with their toddler. From these results, a confidence interval was provided, going from about 82.3 percent up to 87.7 percent. So, what does this confidence interval tell us? How might we interpret this interval? To better understand that, it might be first good to ask a question. What do we make confidence intervals for? So, think about it. We make a confidence interval for a, is it a parameter or a statistic? We make a confidence interval for a parameter. In our example, we would say that this interval is a confidence interval for a population proportion of all parents with toddlers who report they use a car seat regularly. For a population proportion, now we use our sample proportion to construct this interval but we don't refer to this interval as being a confidence interval for a sample proportion. It's a range of values to help us estimate what we think that population rate might be with a high level of confidence. What we have here is a pretty good statement about the interval reporting it and then some good context around it, but we can improve upon that in a couple ways. We want to make it more of an interpretation that conveys that its coming from data and make sure we always are talking about what confidence level we were using. So, here's a couple of good statements. We estimate with 95 percent confidence that population proportion of parents with toddlers who report they use a car seat for all travel somewhere between that 0.8227 up to that 0.8773. Another good way of saying based on our sample of 659 parents with toddlers. With 95 percent confidence, we'd estimate between 82.3 and up to 87.7 percent of all parents report they use a car seat for all travel. Both of these statements well-written do include the confidence level that was used, that phrase with 95 percent confidence. Both of these statements convey that we are using the data to estimate the population parameter. In this case, estimating a population proportion. In the second phrase, it says based on our sample we would estimate the rate to be for all such parents or for that population of all such parents with toddlers. Finally, that last statement nicely includes the sample size which is often left out in a very important feature for us to truly understand the results of a study. So, we have this range of reasonable values for which we would estimate could possibly be that population proportion. In fact, 83 percent falls in this interval, so it would be a reasonable value for that population rate, we could not refute it. Our data would not reject 83 percent could be that true rate, 86 percent could be that true rate. We get a range of reasonable values for that parameter based on our one sample from that population of interest. So, how do we better understand what that with 95 percent confidence phrase really means? Well, let's think about that a moment. Does the interval we have here, at 82 to that 87, does it contain the sample proportion of parents with toddlers who reported they use a car seat for all travel? Does it contain the sample proportion? Think about that. Is the sample proportion in this interval? Of course it is. Most certainly is in the interval because we centered the interval at that best guess, that 85 percent, that was our best estimate. We went from that best estimate each way a margin of error. So, every interval is going to contain that corresponding sample statistic, that best estimate would be right in the middle. But what would your answer be to this question? Does our confidence interval contain the population proportion of all parents with toddlers who report they use a car seat for all travel? What did you think? We would have to answer this as we don't know. In order for me to be able to say whether this answer would be a yes, it's in there or no, it's not, I would have to know the value of that population proportion. That population proportion is a fixed value but it still unknown. I don't know what it is. That's why we designed a study to be able to gather data that's representative, to be able to estimate its value. So, I cannot tell you whether this particular already made interval has the true proportion in it or not. So, that leads us to one way that might often be incorrectly interpreting that confidence level. We don't want to think of it as a chance or probability that there's a 95 percent chance the population proportion is in our interval that we just made. That intervals are already computed, its fixed. That population proportion, the true rate for our population is a fixed number two. We're trying to learn what it is but it ends up either being in between these two values or it's not. In some sense, the probability would either be one or zero. We just don't know which of those two answers it is. The confidence level refers instead then to the confidence we have in the process we use to make our interval. Not as a probability after our interval was made but how confident we are about the statistical procedure that was used. So, let's think about that. Let's take a population for which we know already. The population proportion who would say yes is 50 percent, usually that's not null, but we need to know what the right answer is so we can evaluate how well the process is working. Let's go to that population and take a sample of 25. Let's take our best estimate from that sample, the sample of proportion and let's make an interval around it. Let's do that a lot of times. Let's say 100 times. Take 100 samples each of the same size from the same population, create a confidence interval each time. What might we see? Well, here the very first interval, the first line at the top of this image, is showing us that the estimate was a little bit below the true rate of 50 percent but when we went around that best estimate each way, the interval still captured, had the true proportion of 0.5 in it. So, it was colored blue. It's a good interval. The next interval also included the true proportion. The estimate was initially a little bit high but the margin of error each way allowed us to have an interval that in the end, once it was observed, did have the true proportion in it. But a little further down, we see another interval, it's got a different color. It was an estimate from that sample that ended up being a bit too high, so much so that the interval around it did not capture that 50 percent, that true rate, 96 of our intervals here did contain the true proportion, four of them did not. Each particular interval either did or did not. Do this again another 100 times and the results will vary, different intervals based on different samples but here 95, all those 100 generated intervals, did end up containing the true parameter and only five did not. Each interval, either does or does not contain the true value but in the long run we expect most of them to. So, this 95 percent confidence level is what we would expect in the long run. Regarding that statistical procedure, about 95 percent of the intervals that are made with this method we used, are expected to contain the true value. That's why we're confident in the one that we did. We're going to do this once. Take one sample, representative if we can, large sample size, produce our interval and we're confident in that result because this process guarantees that you would expect 95 percent of those intervals made with that method you're using to contain that true value. Our last idea is to look at how you might change up that confidence level. What if you want to be 99 percent confident or maybe only 90 percent confident? Well, if you remember how we construct our interval, best guess plus or minus a few estimated standard errors, that few is the thing that reflects your confidence. So, if you want to be more confident, the multiplier we need to use then has to be adjusted. Instead of being just 1.96 or about two for 95 percent, you might have to make it now, two-and-a-half, 2.576 if we want to be 99 percent confident. The more confident we want to be, the multiplier is getting larger, the interval's going to be wider. So, it's a trade off. Don't need to be as confident, your interval can be a little more precise. Want to be more confident, you need that wider net to hopefully produce an interval that's going to have that true value in it. We go back to our parents with toddlers. We would see that the 90 percent interval has a smaller margin of error and as we become more confident indeed the interval is going to be wider. Here's the image that shows 90 percent intervals being generated and the 99 percent intervals, 99 percent confidence intervals are generally wider than the corresponding 90 percent intervals based on the same data. So, we have seen now how to interpret our confidence intervals better. We better understand what that confidence level really means, and we do have some options for the desired confidence level that we would like to use.
