We have assumed that we can work with standardized data. Now, we're going to make an additional assumption, which is a parametric assumption. We assume that X and Y are Gaussian in shape. In other words, individual values are drawn from probability distribution function that is normal in distribution. So then we can write that there's the Gaussian distribution for the values of x with mean 0 and standard deviation 1. And similarly, there's a Gaussian distribution for y variables with mean 0 and standard deviation 1. So here is a question. We would like to know, what is the standard deviation of our errors? We know that it, too is a Gaussian. We know that the mean of errors is 0, but we don't know what the value is for the variants of the error itself. And what we'd like to do is, we'd like to relate the variance of this error directly to the correlation, and directly to the slope of the best fit line. And that is exactly what we are going to do next. So how are we going to do that? Well, we know that our Y values are made up of the sum of our X values with our model [BLANK AUDIO] and now our model consists of beta, the linear association, because alpha is equal to 0, plus an error term. Root means square error. Or, We can write it like so. So our error term looks like this. We know that the mean of the errors is 0. The mean of residuals is 0 for the best fit line. And the variance is as so. We know that the variance of something multiplied by beta is going to be multiplied by beta squared. So our variance here is, again, mean 0 and variance beta squared sigma x squared. And we know that our value for y is [BLANK AUDIO] equal to 1. So we know that this is equal to 1 for our standardized problem. And this is equal to 1 for our standardized problem. What that tells us is the following. By our basic Gaussian addition rules, 1, that's the standard deviation of y and the variance of y = sigma squared + [BLANK AUDIO] the variance of the error. But we know that for our standardized data, beta is equal to the correlation. So we know that 1 = R squared + the error squared. So this tells us the relationship between correlation or more precisely, the coefficient of determination. Coefficient of determination is a fancy name for our correlation value squared, so we know that R squared equals 1 minus our error squared. And we know that our error squared = 1- R squared. What this means is that there is a direct relationship between the standard deviation of our errors and the correlation. And the reason why this is important and valuable is that when we use our linear regression model to make a forecast, so we are forecasting Y as beta X sub i. There's always going to be some error built into our forecast, and we would like to know what that error is, so that we can give not only a point forecast, but actually a forecast that's in the form of a probability distribution, where the mean is our estimate for Y sub i, and we have error around that point. And the standard deviation is equal to the standard deviation of our residuals. And this allows us to be very precise, for example, to say we have a maximum likelihood point, but what we're really saying is perhaps we have a 95% confidence that the value will be between these different Z-values. Let's suppose we had a relationship where there is no association whatsoever. Correlation is equal to 0. In that case, the line, there really is no line, the line is flat. The standard deviation of y = 1, and the standard deviation of our error = 1. At the other extreme, would be our perfect model where R = 1, and in this case, there would be no error at all. And this would mean that every single value actually was on this line. So you can think about our confidence intervals shrinking to 0, from 1 to 0 as R changes. So if you think about the picture like this, if we plot R here, and we plot the error here, the relationship is R equals the square root of 1 minus sigma squared. So that is gonna look something like this. Be sure and note that the error itself is not a Z-score. We know that it's always going to be less than or equal to 1. It's going to be equal to [BLANK AUDIO] beta squared in our standardized model. So we've got an error around our ideal point. And if, let's say we want a 90% confidence interval, so we have 5% above and 5% below. Then, we're going to be at Z-scores 1.64 and minus 1.64. So if we have an error of let's say, 0.51 which is what we would have if we had a correlation of R = 0.7 And if we have a standard deviation of error that is equal to 0.51. We would take our best estimate for Y. Which is beta x sub i. And we would add 1.64 times 0.51. And that would put us out here, and we would subtract, so we'd have (-1.64) (0.51). And that would put us out here. And that would give us our 90% confidence interval.