So given a data table and asked to create a scatter plot and determine the linear regression and then the our value that goes with that. So this this scatter plot was created. Using Excel on Excel has a nice feature that with a few clicks of the mouse, you can add a trend line, and it will do. The regression for you will even allow you to choose which shape function. It doesn't have to be a line. Furthermore, if you want the R squared value displayed on the graph, it's another click. Same with the equation. So for us to determine our we're going to take advantage of that R squared R is equal to the square root of R squared, and it's either positive or negative. In this case, it will be positive because there's a positive correlation as X increases. Why increases another way to look at? It is the trend is sloping upwards. So if we take 0.942 and take the square root of it, then we're going to get seat 0.9 for to take the square root, we get 0.9706 so the R values 0.9706 That looks like a pretty good fit. There are other places where we would be pretty happy with that. We look at the data, I want her. I'm sure you noticed that. Really? It's got some curvature to it. I wants to be some function that looks more like that, which is not a line. So sometimes with data, you want to try a different shape, a question we are going to start off trying. What does a quadratic dio of X squared function parabola and what we see here is, all of a sudden, all of the data points are quite near to the curve, and our our our value shoots up 2.9805 That's much better. Um, see what happens if we go degree three. If we make it a cubic, we could raise it one more degree. Looks like it gets even better a lot of times with these higher degree polynomial ills. For a small region of the data, you can get a fantastic fit like this, Um, and it does look like every single dot lies on the curb. So it's tempting to choose that one. Let's see what happens if we raise it one more degree? So now we're looking at an extra. The Four Excel believes that the R squared value is one for this particular, um, fit. And if you verify that on another graphing utility, you'll see it's something like 0.9998 which is essentially one. And again. Every single dot seems to be right on the line. It was a slightly complicated shape, but it does seem to fit very well. Let's look at natural log. This one has also a very, very high correlation. 0.9988 That's extremely close to one, and furthermore, this shape is a little simpler, and it very much seems to represent the data. So even though degree for appears to have a better R or R squared, the natural log just looks like a better fit. All right, so finally, let's take a look at a power function. This one Excel didn't do such a great job. The the exponents is pretty near one, which makes this look very much like a lot, and we can clearly see that this is a curved kind of function. So the power best fit really wasn't our option. And in fact, we could see the arse words the lowest of the six things we tried. So when we're coming down to pick, which is the best model when could either choose one of the higher degree polynomial is because the R squared is better. Or we could choose the natural log because not only is the r squared very close to one, but also the shape just appears to represent the data better, and I'm inclined to choose natural Log. One could probably debate that for a while, but this just appears toe match the trend very well. And in either case, we have a fairly high degree of certainty. If we were using the equation to predict a Y value for one of the next that we didn't already have, we would we would have some high degree of confidence that we would get an accurate result. The, uh, further away you get from the range of exes that you had actual data for this, you know, between one and 16 the last good your model is. My gut tells me the natural logs going to do a better job with it. So that's what we're gonna choose