Today, we're going to be learning about how to draw a scatter plot, how to come up with the correlation coefficient, interpret the slope, find the domain and understand how this all works. And so today we're going to start off by understanding how to draw a scatter plot. And so we have our X and Y axis. You're going to make a scale of your X and Y axis in this question, our independent variable is our internet ads and our dependent variable is going to be based off of our magazines. So up here we will write magazines for our Y. Value and so our internet and our magazines, we use our table and we're going to create a scatter plot by plotting these points. So 8.2 and 14, bus 10.7 and 13.9, 14 and 12.5, 15.5 12 really 17.2 and 12 22 and 10.7. So you start to see a little bit of a pattern in this scatter plot. And finally, and so you start to see a pattern of the scatter plot based off of the independent verily internet and the dependent burial. The magazine. The magazine depends on what's going on with the internet in the context of this question. Understanding us advertising spending. And so the question is, does this appear to be linear? And so we look at the graph and it does appear to be linear with a negative slope. Right. And so we're now going to use a graphing utility to find the line of best fit. That would model the relationship between this. And so now that we've drawn it on paper, we're going to take our graphing calculator and we're going to use this data to fill in our information. And so we're going to go to stat. We're going to go to edit And we're going to put in our information of our percentages. And so we're going to put in 8.2, 10.7 14, 17.2 all the way down as our X. Values our independent variable. And then for L. two, we're gonna put our dependent variable, which is our magazine based off of these percentages. And we're going to keep everything as a whole number as a percentage. Understanding that they are percentages. We want to make sure all those accurate everything lines up nice and neatly and so we want to look and see now what is the line of best fit. And so you take your regression. And since it looks like a line, we're going to go to stat, we're going to go over to couch and we're going to go down to linear regression. Once you're in linear regression, you're going to put in your data lines, which is going to be L one comma L. Two. If you put it under L. Two and L. Three, that's what you would put uh in this little spot here. If you have an 84 plus, you can probably just go right to calculate. Just make sure your X. and Y is lined up and you're going to get a whole bunch of information in that information would be the line of best fit. And so your line of best fit based off of your graphing calculator is going to be why equals And your slope is going to be negative .24. And so we're gonna use negative .24. Which makes sense because the slope is looking downwards And then our y intercept going to be 16 two. And so this would be the line of best fit. That we would draw in our graphing calculator. And so if you take a look at your graphing calculator here, you would look at your stop plot and you would see you're a graph shown here below. And so when you see your graph, which in this case is going to be Y equals negative 0.24 X plus 16.2, you are then going to take go to stat. You're going to go over to help One of our stat again and you're going to take a look at your r value. And if you see your r value is negative .99, that is your correlation coefficient. That means we have a very very strong correlation coefficient because correlation coefficients the strongest one would be negative one or positive one, negative being that it's going in a downwards direction. And so we would label this as our correlation coefficient, which we now know is very strong, which means that it's going to be very close to a linear function. We use the right type of regression and so we have our line of best fit and we now have our correlation coefficient. And so that would answer questions be and questions see the only part about question be that we now have to add in is to write it as a function. And so instead of saying why equals, we're going to write it as F of X. F of X is in function notation. And so it's going to be F of X equals negative 0.24 X Plus 16.2, which is going to be our line of best fit. So this would be part B. And now this would be part six. And so the next part of this says, does the correlation coefficient support your conclusion from part A. And so. Exactly right. This will conclude it because we said that it looked linear and now this confirms that it is a linear graph because it has a really strong correlation coefficient. And so to interpret the slope and see how close we are from the slope, we can take two of these points and find the slope to interpret it. And so what we're gonna do is we're gonna pick out two points, I'll pick up one of these, you can pick any two points you want I'll pick out 17.2, common 12. And so we're going to do 17 points two comma 12. And then we're gonna pick out another point, such as let's pick 10.7 and 13.9. And so since it looks downward and our slope was negative .24, we see that it is going downwards. And so basically as your internet increases, as the price of internet increases, we know that magazines would decrease. Which makes sense right? The more you are using things on the internet, unless you're probably going to read a magazine. And so to prove that we would just do our slope formula, which is why to minus Y one Over X 2 -11, which in this case is 13 -12 Over 10.7 -17.2. And we're hoping to get a negative slope here to prove that we're doing this correctly. Well 13 -12 is one, 10.7 -17.2. And so you take your calculator, You would do 10.7 -17.2 And you get negative 6.5 and you would divide these values. And when you divide these values, you're going to get one Divided by negative 6.5 And you're gonna get negative .15, right? Which is very close to negative 2.4. Right? And so we're right in that ballpark of understanding our line of best fit the domain of this function. We know domain are going to be your X values. And so your domain of this function, you're going to use interval notation and we're going to talk about our X values. And so our X value, we look at our smallest X value here which is 8.2% And our largest x value is 35.1. So we say that all values From 8.2 to 35.1 is our domain. So this is what we call interval notation. You could also say 8.2 is less than or equal to X, which is less than or equal to 35.1. And then finally they want to know to predict the percentage of your spending on magazine ads when Internet ads are 26%. So we want to know what our why value is Given 26% as our x value of this function. And so the last part which is part F Is going to be to plug in f. 26. So when we plug in f of 26, We are going to set that equal to negative .24, which is our regression Times 26, which is our new percentage to figure out how much magazines would cost. Plus our 16.2. And so we say f of 26 or f of 26 would be what percentage from magazines. And so we take our calculator and we would do Negative .24 times 26 Plus 16.2 grounded. And you would get 9.96 person