Greetings, and welcome to lecture set seven. In this set of lectures, we're going to build on the momentum we got in lecture set six looking at sampling distributions and culminating with the central limit theorem, to use the results from the central limit theorem to make a statement of an unknown population truth like a mean, proportion, or incidence rate that we can't observe directly based on the estimate we get from a single sample incorporating some measure of uncertainty about that estimate into our statement that will ultimately result in something called a Confidence Interval. So in the first section here, we're going to look at doing Confidence Intervals for Population Means. But first, we'll review the basic results from the central limit theorem and talk about how they can help us in this process of getting an interval for the unknown truth. So upon completion of this lecture section, you will be able to explain how the central limit theorem sets the groundwork for computing a confidence interval for an unknown population parameter. By parameter I mean an underlying mean, proportion, or incidence rate that we can't observe directly, we can only estimate based on the results from a sample from the population. Going and then show how to estimate 95 percent confidence intervals for a population mean based on the results of a single sample from the population. We're going to show how to estimate other level confidence intervals for a population mean based on the results from a single sample from the population as well. So let's just recall the central limit theorem, or CLT for those in the know. The CLT states that if we were, it tells us what would have happened without us having to do it, if we were to take all possible random samples of the same size n, and n here could be any fixed number, 25, 100, 1000 et cetera. But if we took all possible random samples of the same size, we're taking from the same population. A summary statistic were computed whether it be a mean, proportion, or incidence rate depending on the data we have, if a summary statistic were computed on each of these infinite samples we took, and then we did a histogram of this ostensibly infinite number of summary statistic values and smooth histogram, the distribution these values would be approximately normal, and centered at their mean, and median, and mode. Furthermore, the mean, median, and mode of this distribution would be the true value of the underlined parameter we're trying to estimate from our sample. So again each point in the smooth histogram, if I were to draw bars, we smooth over, each point in this picture is a single sample summary statistic. A single sample mean, a single sample proportion et cetera, and there's variation in estimates from sample to sample, but on average they equal the underlying true value of what we're estimating. Furthermore, because this is a normal distribution, approximately normal, most of the estimates we could get from most of the samples we could obtain just by random chance from the population of interest will fall within plus or minus two standard errors of the truth. Now we standard errors and measure the variation in the summary statistics from sample to sample and you may be thinking, and we talked about this before, but just to remind you might say well John how can we estimate the variation in the statistics from sample to sample if we only ultimately have one sample? Recall the central limit theorem also gives us the opportunity to do that using the information taken from our single-sample as well. So you might be saying okay well we can estimate the variability in these estimates from sample to sample, we know on average they equal the truth. We know because the distribution of these estimates is normal around the truth that most of our estimates fall within plus or minus two standard errors of the unknown truth. But how's that help us? Because if we knew the truth, we wouldn't care about a range of values for imperfect estimates of the truth that fell within plus or minus two standard errors. I certainly agree with you, but we're not going to know the truth. We're only going to take one sample from any population we want to study, and we're going get one sample statistic estimate from that one sample. Where will that sample statistic estimate fall under this theoretical curve describing the distribution of all possible values for sample statistics? Well, it will fall somewhere under this curve. It could fall right in the middle. We could hit the truth right on the nose. It may way, our sample statistics may be much larger than the underlying truth, or it may be smaller et cetera. We'll never know exactly where it falls, but here's the thing for most of the samples we can take randomly, the estimate we get will fall within this plus or minus two standard error range from the truth. So if we take our sample estimate whether it be a sample mean, sample proportion, or sample incidence rate, and add subtract two standard errors which we will have to estimate from our samples as well, then this interval we create has a high likelihood of containing the unknown truth. This interval is called a 95 percent confidence interval. So let's operationalize this with some examples looking at creating confidence intervals for a population mean. Let's start with a population blood pressure mean for a clinical population of men, we're going to use the results from a single sample of 113 to then make this interval. So we've looked at the sample before, and I'll feel him on the highlights. There were 113 men in the sample. The sample mean which was our best estimate for the underlying population mean we can observe directly is 123.6 millimeters of mercury. The sample standard deviation which is an estimate of the underlying variability blood pressures from man to man in the entire population is 12.9 millimeters of mercury. We can estimate how variable sample means based on random samples of 113 men are from sample to sample based only the result we have from the single sample because you may recall that the theoretical standard deviation of sample means across multiple random samples as a function of the variability of individual values in the population sigma divided by the square root of the sample size each mean is based on. We don't know sigma, so we substitute with our best estimate which is our sample standard deviation 12.9 millimeters of mercury, and divide by the square root of the sample size which is square root of 113. If you do this, this is the standard error is approximately equal to 1.2 millimeters of mercury. So again just to reiterate because this is important, the standard error estimates how far on average single systolic blood pressure means based on 113 randomly sampled men from this clinical population, how far they will fall on average from the true population mean blood pressure. So just again it's important to restate this a couple times to let it sink in. This standard error quantifies the variability in sample means based on random samples of 113 men across the samples. The CLT tells us the theoretical distribution of all possible sample means based on random samples of N equals 113 is approximately normal, so we can get our estimated confidence interval by taking the mean of 123.6 millimeters of mercury plus or minus two times 1.2 millimeters of mercury, 123.6 plus or minus 2.4 which gives us 121.6 millimeters of mercury to 126 millimeters of mercury. So, this gives us a range of plausible values for the underlying true mean among all men in this population. We'll come back and talk more about the interpretation later in this lecture set but first we'll just dole out some examples to get the estimation process down. Let's look at another example, our heritage health length of stay large sample, 12,928 observations where we had the mean length of stay for all these persons who had at least one day of inpatient visits in 2011. The mean for the samples 4.3 days, the standard deviation of the individual hospital length of stay in the samples 4.9. We can estimate the standard error of mean, estimates from this population based on samples ran, all random samples of 12,928. It's quite small because our sample is so large, it's 0.04 days. Again, the standard error quantifies how far the length of stay means based on 12,928 from the insurance population, will fall from the true mean length of stay where we actually sample from the population and infinite number of times. Each time taking a very large sample of 12,928 suggests that the actual estimates of the true population mean or not that variable around the truth on average, the standard error was 0.04 days. So, if we do this, we can get a confidence interval for the true underlying population mean by taking our estimate plus or minus two estimated standard errors four point three days, plus or minus 2.04 days or point zero eight days, which gives a confidence interval of 4.22 days to 4.38 days. So, a pretty tight confidence interval narrowing in on the underlying truth because we had such a large sample and the standard error estimate was small. One more example here, weight change in diet type says a low carbohydrate is compared with a low fat diet in severe obesity. This study was done when low-carb diets were initially becoming invoke. This was one of the first studies to demonstrate that they can have an impact on weight change. One hundred and thirty two severely obese subjects were randomized to one of two diet groups. So, low carbohydrate or a low fat diet group. Each of the subjects in east the randomization groups was followed for a six month period and of interest was the average weight change after six months compared to baseline, which was six months prior at the time of randomization. Here summary statistics on both groups are 64 subjects randomized to the low-carb group, they lost 5.7 kilograms on average and there was a lot of variation though in the individual weight changes among the 64 persons. The 68 people randomized to the low-fat diet group, they lost 1.8 kilograms on average. Remember this is after minus before and it's negative, indicates a weight loss and there was also a fair amount but not as much was with the low-carb group. A fair amount of variation in the individual weight changes among those 68 subjects. Here the confidence intervals for the true weight change where everyone given the low-carb diet or were everyone given a low fat diet? The entire population of severely obese people that is. So, you can see the confidence interval in the weight change in the low-carb diet group, the average weight changed, the observed change was negative 5.7 kilograms, it ranges from negative 7.8 to negative 3.5. So, you can see that all possibilities for the truth, are indicative of all weight loss, all possibilities are negative. Similarly, with the low fat diet group, if we do the computations, they are all possibilities for the true mean, change are negative as well. So, both groups even after accounting for sampling variability seems that they would lose weight. What we're getting at towards here is comparing the results between the groups. I'll just put this out there. Notice that while both confidence intervals only contain negative values for both groups, the confidence intervals for the two groups do not overlap. Just put that out there and we'll come back to that shortly in our next lecture set when we start comparing results between two or more populations via two or more samples. So, I want to throw that note on the level of confidence 95 percent confidence intervals or the industry standard and research but it's certainly possible to estimate intervals with different levels of confidence. Again, just using cutoffs under the standard normal curve. So, if we wanted to have less confidence in our resulting interval, we could do a 90 percent confidence interval and we'd only need to add and subtract 1.65 standard errors as opposed to two. So, our interval will be narrower but it wouldn't cover the truth as often as frequently 90 percent instead of 95 percent. We wanted to have higher level of confidence up to 99 percent. We need to add and subtract 2.5 standard errors to get the coverage. So, in order to get 40 percent more coverage than a 95 percent confidence interval, we need to add and subtract a lot more in terms of standard errors to get that and that's because of the decrease in proportional observations, the further we go from the center. So, in summary, it's actually pretty straight forward, analytically to compute a confidence interval for a single population mean based on the results from a single sample. We can just take our sample mean estimate and subtract two estimated standard errors, estimated standard error is also based on results from our single-sample, the observed sample standard deviation and the sample size. So, s over the square root of n is our estimate. The standard error of the sample mean quantifies the theoretical variation in sample means across random samples from the same size. We can estimate this without taking multiple random samples of the same size and that's the beauty of the CLT. If we wanted to change the level of confidence, we could adjust the number of standard errors added and subtracted from the sample mean. In this course, we will strictly use 95 percent confidence intervals. So, hopefully, you see the pattern that we're going to be working out, we'll see very similar results for proportions and incidence rates and then we'll start to struggle with perhaps the harder idea, what are we actually getting from this confidence interval and what does it tell us? What does it mean to give a range of possibilities for an unknown truth. So, stay tuned more to come.
