All right, So in this problem, we have Wei have data, pretend students. And for each student, they give us their own height and their ideal partners height. The first thing that has asked of us is to find a regression line on the correlation coefficient and car observations on it. Silk. Um, first, we're going to do that. A regression line. It looks something like this. It's, ah, equation for a line and due to find it. So if we need the value for M and B, that's the slope and the y intercept for the line. Um, to do that, we need to build this table. Over here. Come I used Excel Rivers on the first two columns that they we have here is the data that they give us this excellent light. Next we build the column that for each row they tell us the multiplication x y then iss X squared and y squared and the last hero we have because some of each Carlin So these values are the ones ever can use to find them and be so the's values I compute them over here and thes to equations are the definition for the Slope and slope and B. Why intercept of the regression line. So, um so our first thing to do is to plug in those values that we found on through the table. Um, and and the only one that's not here. And it's actually the number off, uh, the amount of data and we have We have ten entries and will be equal to ten. Okay, now, already so. And is ten. Uh, the some ex wise is forty six thousand, four hundred and ninety three. We said tracked first s o X. That is six hundred and eighty two times the sum of why that is also six hundred and eighty two. And that goes over, uh, n that's him. And next, we have the sum of the squares minus the square of the sun. So these were two different numbers. This sum of squares is going first, Queer X, and then we add all the the fact of Andries de So they're talking about this number right here. Thiss iss forty six thousand seven hundred and thirty. Huh? Uh, minus than this square. Uh, this number that's the summer exists. That is six hundred and eighty two and then this number he's cleared. So now we have this expression for them. It's all constants. So weakness. Either you do it by hand or use a calculator. I've found that is approximately equal to minus zero point zero night. All right, so that would be the slope. And now, to find the why intercept. Can you see the formula? Any definition, also from now on. So we stirred with self wise that it's six hundred and eighty two, six hundred eighty two minus, um And this is the one. The value that we just found that it's minus zero point. I scare nothing, turns the sum of excess POTUS also six hundred and eighty two, and that is over. And who has a value of ten Gandhi throughout constants. You can place it into the calculator and get seventy two. No, I'm sorry. Seventy four kind. Twenty eight. Keep. Okay. All right. So this will be the y intercept for the regression line. We can write out the equation for the line explicitly accusing this slope interests of former Emma. So we stopped. We found out this line. ISS What cause minus serial point. Syria, a nine thanks, plus seventy four turned twenty eight. And so this is good. Lined with the I'm sorry. The question for the line. Uh huh. So what's characteristic above this of the slope? Here it is. Almost zero. So that will be almost a horizontal line around this value. Okay, But we're not done. We are also asked to find the correlation coefficient, Uh, correlation coefficient, which we call our It is defined like this. And we're using the same values that we found on this table to calculate that so we can go ahead, answer plugging in these values. Uh, and it's tend the son of X y. It's forty six thousand four hundred ninety three good. Minus the sum of the outfits that is six hundred and eighty two times flies. That is also six hundred and eighty two. All this goes over over firstly, people, we start with square root. And that's ten again, where you were using Sum of squares and square Cem. So we begin with some squares. Hey, good. Education. Forty six thousand seven hundred and thirty minus minus. I said, uh, the sum of X, that is six hundred and eighty two. And this number is quick. That is our first school route. This is multiplied by another square room beginning. Also end. That's ten and some of squares minus the square of some. So ten times sum of squares for why isthe forty six thousand six hundred and seventy. Linus, Um, B x minus scripts. Um, so some is six hundred eighty two, and then this number squared so we can Oh, this is some new thinking square. Now we have this really big, but but just complete. No, it's all Constance. Which means we can either put that into the calculator, do it by hand, and I got that Our ISS negative. Zero point one two three. All right, So, um, so what if this number, this coefficient mean means first, our is closed to either one positive one or negative fun? That means that, uh, this line is a very good fit for the data. Yes, but if it is close to zero, it means it's not a very good fit. All of this not model. It is not a good model for the data, so this isthe very close to Ciro. It's definitely closer to Ciro them to one minus one. So what this is telling it. This line that's almost horizontal. It it's not a very good fit, huh? Model these trends for that. The ideal partner set. So that is a first for second part, huh? It is revealed to us that the first five data pairs are for female students and a second five for male students. So we're asked applying again. They're regression line on the correlation coefficient, but for each set of data separately. So we're going into the first fight first, uh, that isthe course months, two girls. So since we are doing the exact same thing to regression line and Correlation coefficient, we need a similar table. So now it's just the vice. But first, first five data points. Um, the approach is going to be the same. So we're not going to go into too much detail. They already know how to build this Temple Peace, bro. Oh, this really is the sum for each column. And then we're going to apply again. The for the definition. Put on your red definition for slough. Nope. And for eggs for slope. And why intercept and the court the coefficient. But worthies you've got so since We already did it once. I'm just gonna give you what I what? I thought so for slope. I got syrup point sixty seven, and with wind yourself got twenty seven point nine, we can write explicitly. The the regression, the question for the regression line. So M iss serial point sixty seven spoke close X. That's wind yourself. That is twenty seven point night. All right, so we have this new line that fits the data for the female students and the correlation coefficient can Using the definition is this number, and that is a lot of a lot closer to one. So that means this is a much better data for the The much better fete for the data. You see, the first back puts. All right, uh, now we're going to do the same thing for boys. We construct the people. This last role is this Last root is the use a little color? Uh, peace are the sums for each column, and these are the values were getting use to find the slope and intercept for the regression line. Um, I already use these definitions, and I found that the slope was sirrah point pretty tree and the one interested just thirty four point syrup. Or so we can provide the agression line that fits the the data with points the ISS. Why equals Sierra Point forty three times X plus? Why intercept That's twenty four points zero four and the correlation coefficient that we found us this. It's not as good as the one for the girls, but it's still a lot closer to one. So that means it's good enough fit for the data with Putz. So and that's part asked us to plot all the data and one graph using different types of points to distinguish the data for the males and the females? No. So, um, since I heard a lot of points already planted a few I'm using red for girls and green for voice. So we're missing just the last two points X represents. The height s is acceptable than white is the ideal partners. So good luck. It's less too. Censuses X will find it on this. Access seventy three will be around here and, of course, plunging y value. That would be the second coordinate is sixty six. It would be somewhere around here, so you go up go to the site and it will be somewhere around here. You. And for this, for the last point, seventy five here should be around up way between these two and sixty six. It should be on the same height. Cell, same Highness. This one a little bit to our right. There you go. Probably a lot less close. So, uh, so that would be Yeah. So here on the girl's data are over here. Um, voice data is a lot closer and under the gross data. So it as this It says that using this plant and the result of propriety explain why we found, in part a sow here for the girls. They have, ah, higher slope, Hunter Slope. And the white intercept is also lower. And here it's, uh well, it's slower at lower slope and higher Who? Why intercept? So this regression for the girls will be more unless around here, shit And for the boys, it'LL be around here. That would explain why the slope for the girls this a lot at a steeper but the voices throughout lower and the Y intercept for the grossest all the way over here. What? The voice is only, um, up here. Uh, so as further what we observed in print, eh? It was almost a core central line. So if we let's get these out of the way. If we were to take all of this into one data set the line, don't go through, it would be around here. It's kind of like averaging out both data sets. And that's why it would be almost home was the horizontal line. Hey.