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Given are five observations for two variables, $x$ and $y$$$\frac{x_{i}}{y_{i}} \left| \begin{array}{ccccc}{3} & {12} & {6} & {20} & {14} \\ \hline y_{i} & {55} & {40} & {55} & {10} & {15}\end{array}\right.$$$$\begin{array}{l}{\text { a. Develop a scatter diagram for these data }} \\ {\text { b. What does the scatter diagram developed in part (a) indicate about the relationship }} \\ {\text { between the two variables? }} \\ {\text { c. Try to approximate the relationship between } x \text { and } y \text { by drawing a straight line }} \\ {\text { through the data. }}\end{array}$$$$\begin{array}{l}{\text { d. Develop the estimated regression equation by computing the values of } b_{0} \text { and } b_{1} \text { using }} \\ {\text { equations }(12.6) \text { and }(12.7) .} \\ {\text { e. Use the estimated regression equation to predict the value of } y \text { when } x=10 \text { . }}\end{array}$$

a. See scatter diagramb. Positive linear relationshipc. See scatter diagramd. $\hat{y}=68-3 x$e. 38

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

Temple University

University of North Carolina at Chapel Hill

Idaho State University

Lectures

0:00

04:51

Given are five observation…

06:28

01:37

03:12

Five observations taken fo…

So in this problem, we're giving the following data set. And what we want to first do is come up with scatter diagram for this data. So I did this and Excel and I found the following plot. And what is the relationship that we've seen between the two variables on the horizontal axis, we have the X and on the vertical, we have the why. And what I see is as the X increases the wide decreases. So, um, that would be the relationship between these two. Another way of saying that is by calling it by its official name, a negative linear relationship. Um, and what this linear part tells us is that it's changing at a constant rate. So if we were to draw a line of best fit through our data points, we would get something that looks like this. Now assume that this is a straight line. Um, and you would see that as X increases. So at five X would be over here and at 10 we were here. The change here is the same change here is the same as a change from 10 to 15. If my drawing was accurate, um, so that is the basic idea of a linear relationship. And because we see that as our dependent variable R X increases our why decreases, We call it a negative linear relationship. Now, we also answered part. See, by doing this and by drawing in this line of best fit, and, um, now we have to come up with an estimated regression equation. Now, the regression equation in general is always Why are a linear regression equation is always why equals basis of zero plus betas of one times X And what data Sub one represents is our slope and baby zero represents our there. Now, if you remember back in algebra one, we had something that looked like y equals mx plus b m was a slope in B was over. Why? Intercept is basically the same idea. Our slope represents our model. Um, what we mean by model is something to represent our data up that is kind of accurate, and by error, we don't mean something that is wrong. It just means something that our model can't capture. So now we have to come up with beta someone beta subzero. And the formula for beta sub one is equal to the sum of each individual data point minus the mean for our exes times Which individual? Why value minus the Why me? Over the sun of squared. Extra useable. So what that means is the sum of the difference between each individual X value and the ex meat. Sorry, this isn't the residual, but this is our This is our status of one formula. And now for our beta subzero, that is simply going to be equal to our mean of why, minus the basis of one that we found times our mean for X. So let's first find a beta sub one beta someone. And to do this, we have to first come up with an ex bar. Let's go back to our data set and our ex bar you do is in a different color. Our export is going to be equal to the sum of all ex data points over the number of values that we have in our data set. So this is equal to three plus 12 plus six plus 28 was 14 all over five and we get a value of 11 and now we're gonna do the same thing for y bar would be 55 plus 40 post 55 plus 10 plus 15 divided by five. So it's going to be ableto 1 75 over five, which is equal to 35 now, using this weekend, come up with the sum of our differences. So in order to find, uh, our X our exit by minus X bar, we're going to be taking the difference between each individual data point and our mean for exes. So, um, I'm just gonna show you guys one example. So our first value is three, and our mean is 11 so this would be equal to negative eight, so that would be the first value. And now we're going to take the sum of each of our differences throughout our entire data set for the X column and for the White column, it would be the same thing. It would be 55 minus our Why, I mean are why bar of 35 which would lead to a value of 20. And we would sum up all the values in this row. Um, the differences between each of these values and the mean and this rope. So once we do that we would get in our formula. The sum of would be equal to you. Um, the sum of the ex differences timesthe wide differences over the sum of ex differences squared. Right. So this would be equal to negative 540 over 180 which is equal to negative three. This is the value for abated. One value now for beta subzero is equal to our what I mean, plus our basis of one times x mean and we got that or why, I mean is equal to 35 plus. And we just figured out a beta sub one value to be negative three and our ex bar to be 11. So this is equal two. Um, 35. Oh, sorry. This should be a minus. Yeah, it's why bar minus beta sub one times export. So minus. So 35 minus negative. 33 would be equal to 68. So we get that our ah, our linear equation is equal to I had our estimated y is equal to 68 minus three x. So this is our answer to party. And now in part E, we have to use this aggression equation to come up with some value of why when X is equal to 10. So if we have, why hat when excess 10 that would be equal to 68 minus three times 10 which is equal to 38.

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