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The following data are the monthly salaries $y$ and the grade point averages $x$ for studentswho obtained a bachelor's degree in business administration with a major in informationsystems. The estimated regression equation for these data is $\hat{y}=1790.5+581.1 x .$$$\begin{array}{l}{\text { a. Compute SST, SSR, and SSE. }} \\ {\text { b. Compute the coefficient of determination } r^{2} \text { . Comment on the goodness of fit. }} \\ {\text { c. What is the value of the sample correlation coefficient? }}\end{array}$$

a. $S S T=335000, S S R=249881.1378, S S E=85135.1378$b. $0.7459,74.59 \%$ of the variability between $x$ and $y$ is explained by theregression equation.c. $r=0.8637$

Intro Stats / AP Statistics

Chapter 12

Simple Linear Regression

Linear Regression and Correlation

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Given the following information, we have to come up with some of squares of the total sum of squares of the regression sum of squares of the errors. The first thing we'll find is the sum of squares of the air and sum of squares of the air is simply that each individual why value each individual dependent variable minus the expected value given our X squared. So I'll give you guys an example of what they should be. So are why hat is equal to as a general formula 17. 90.5 plus 581.1 x And given that our first X is 2.6, we would expect our why hat at 2.6 to be 3301 2.6 to be equal to 3301 0.36 And now each individual why value minus the white hat would be equal to 3330 minus 303,301 0.36 Yes, OK, And now, because we're interested in looking at the difference squared, you would square this value and this would be equal Thio negative 1.36 and we would square this value. How can we get Ah, approximately 1.849? Um and we would do the same thing for each of the values. And then we would take eventually the sum of these values. So here's the table that we get. And now once we take the sum of everything in this column over here, we would get a sum of squares due to error of 8000 500 135 0.137 gate 8501 185,135 85010.1378 Now we have to find the sum of squares of the total. That's simply each individual. Why value minus our ah, why bar which is our means of the Y variables square? So we would take, uh, first, we have to find our wide bar, which is equal to the sum of all the values, all the Y values divided by the number of y value. So they'd be 3 300 plus 3600 plus 4000 plus 3500 plus 3900 plus 3600 divided by six and eventually we get a why and bar of, um one second, make sure this value is right. We get a wide bar of 3650 3650 and now you take the difference between this value and our Y bar sitting square. So we'll take our first value by one minus. Why excuse why Bar squared is equal to 3300 minus 36 50 squared, which is equal to negative 350 squared is equal to 122,000 500. And once we do that, for all the values, we would get the following table and then we have to find the sum of squares you of the total, which is some of the values in this column, and we would eventually get a total of 335,000. And now we have to find the sum of squares your aggression, which is sum of squares of the total minus the sum of squares of the air equal to 335,000 minus ah, 85,000 100. Yeah. 85,135 0.1378 this Will you lead lead to a sum of squares due to a regression of, um, 249,000 249,864 0.862 And now we have to find a coefficient of determination. So the coefficient of determination is simply the sum of squares to you to the regression over the sum of squares of the total. So that is equal to the big number that we just got to 49864.862 Divided by the sum of squares, it was total of 335,000. 35,000 is equal to 0.7 459 And now we have to come up with a coefficient of determination. The coefficient of determination is just the sign of our, um our beta value, our beta value times the square of our coefficient of determination. So our beta value is positive. So this is equal to positive. 0.7459 equal to positive 0.8636 though, and going back. One question this over here means that approximately 74.59% of the variability between X and Y is explained by our, um, regression equation.

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