Okay, so we may have decided that a result is significant or, in our case we've decided it wasn't significant. But we don't really know how large the effect typically will be, what sort of values will our statistic take on these repeated experiments. We did the experiment many, many times, what's the sort of range of values it'll typically take. So the classical method for completing this confidence interval is to take the critical value we found from the, that we looked up in that table, from t-statistic and multiply by the standard error of the t-statistic to compute these marginals. So the difference in means minus this marginal and the difference of mean plus this marginal give us this range. And it can be the, different significance levels will give you different size of confidence intervals, right? So we picked a signifiance level of 0.5%, 0.05, which means it's, two and a half percent to 97.5%. So for the resampling method though, it's utterly straightforward. Just literally run this experiment over and over and over again using these bootstrap samples that I'll talk about in a second. And just compute the fifth and 95th percentiles of all the different results you get. If you do 10,000 repetitions of the experiment, what's the fifth percent, so now you have 10,000 difference of means, take the fifth percentile and the 95th percentile and that's the range. So it just directly computes it. The only trick is whether we can trust this bootstrap sampling process or not. And so that's been worked out. And it's a rigorous way of doing it with a couple of caveats that we'll talk about in a second. So given a dataset of size N, draw N samples. So, if you have 137 patients, you're gonna create another sample of size of 137. But you're gonna sample with replacement, so you may get duplicates in the result. We repeat this lots of times, 1,000, 10,000, many, many times, which is easy with computers now. So these are, you now have 1,000 or 10,000 sample data sets that you know were all drawn from the same original population. And so you can compute these 1,000 sample statistics. And you can interpret these as a repeated experiment, which is exactly what the classical methods call for. You're exactly capturing the goal of the theory developed in the classical methods. Okay, the key assumption here is that all the samples need to be independent. But you don't have to assume very much else. So we don't really know the distribution from which the data was drawn. But, the data we've already collected estimates its own distribution, so drawing random samples from that distribution gives you an estimate. So why is with replacement most replacement? Well, one is if you're doing sample size of, a sample of size N from a population of size N, and you do it without replacement, you're gonna get the same thing every time. But even if you were allowed to do different sizes, which does come up at some times, you're gonna wanna assume independence. Remember, we said assuming independence comes up over and over and over again. If you don't do with replacement, then now the samples will be dependent, right? There'll be a lower probability, namely zero, of ever drawing that, you draw a value 95 in the future, you'll never have a chance of drawing that again if you don't do it with replacements, so now they're no longer independent. Okay, so just to flash an example of this on the screen. Imagine we have the data set, one, two, three, four, five, six. We compute the mean of that is 3.5. Then we draw a bootstrap sample. The first value in our bootstrap sample is four, then we draw three, then we draw four again, because it's with replacement and so on. And we get a mean of 3.33 and then we can repeat that sampling and mean computation procedure over and over and over again until we have a set of observations of the mean. And we can do this with difference of means or any other kind of statistic you may want to derive, with a couple of exceptions we'll mention in a second. So, with our own data set we can do this as well, but it's important that we take a bootstrap sample of the test cohort and then a bootstrap sample of the standard cohort. Okay, and if we do that we can paint out a histogram that might look like this where you can see that the 0 value, so where the mean is 0 is squarely within the confidence interval. These two red lines are the two and a half percentile in the 97.5 percentile to give a signifiance level 0.05. And our mean, if you remember our derived sample mean, or our mean from our original sample, was 12.4, so somewhere in here. But there's, confirming what we found earlier, there is no reason to reject the null hypothesis here. Zero is square root within the confidence interval. It would vary, not unlikely at all that it would happen by chance that we would see that value. Okay. So you can do this with other statistics as well, not just confidence intervals. You can say do it when you're doing linear regression. You take a bunch of points and you compute the linear regression, and draw bootstrap samples from those set of points. Drawing with replacement to derive 1,000 samples or 10 thousand samples. Recompute the linear regression each time and you'll get a bunch of little lines that all have the same similar slope, but not exactly the same slope. And now by reasoning about that range of slopes that you got from all those samples, you can construct a confidence interval over the slope of the true line. So this works for any parameter you may want to derive. Again, with a couple of exceptions.