In Exercises 57 - 60, find the least squares regression line...

In Exercises 57 - 60, find the least squares regression line $ y = ax + b $ for the points $ \left(x_1,y_1\right) $, $ \left(x_2,y_2\right) $, . . . ,$ \left(x_n,y_n\right) $ by solving the system for $ a $ and $ b $. $ nb + \left(\sum_{i=1}^{n}x_i\right) a = \left(\sum_{i=1}^{n}y_i\right) $ $ \left(\sum_{i=1}^{n}x_i\right)b + \left(\sum_{i=1}^{n}x_i^2\right)a = \left(\sum_{i=1}^{n}x_i y_i\right) $ Then use a graphing utility to confirm the result. (If you are unfamiliar with summation notation, look at the discussion in Section 9.1 or in Appendix B at the website for this text atacademic.cengage.com.) $ (0, 8), (1, 6), (2, 4), (3, 2) $

In Exercises 57 - 60, find the least squares regression line $ y = ax + b $ for the points

$ \left(x_1,y_1\right) $, $ \left(x_2,y_2\right) $, . . . ,$ \left(x_n,y_n\right) $

by solving the system for $ a $ and $ b $.

$ nb + \left(\sum_{i=1}^{n}x_i\right) a = \left(\sum_{i=1}^{n}y_i\right) $

$ \left(\sum_{i=1}^{n}x_i\right)b + \left(\sum_{i=1}^{n}x_i^2\right)a = \left(\sum_{i=1}^{n}x_i y_i\right) $

Then use a graphing utility to confirm the result. (If you are unfamiliar with summation notation, look at the discussion in Section 9.1 or in Appendix B at the website for this text atacademic.cengage.com.)

$ (0, 8), (1, 6), (2, 4), (3, 2) $

Transcript

00:01 Okay, so in this problem, they want us to use a formula that's just above the problem set.

00:07 Okay, so the problem set we're dealing with, we have points 08, 16, 24, and 3, 2.

00:20 Okay, so all these are points.

00:25 Okay, and it gives us a formula to use n, where n's the number of points times b, plus sigma x, which is the sum of the x values times a, equals sigma y, which is the sum of the y values.

00:39 And our second equation is our sum of the x values times b plus the sum of the x values squared times a, and that's equal to the sum of the product of the x and y values.

00:54 So to determine these things, let's just make the list of all of our coefficients that we do know.

00:59 So end, the number of points we have is four.

01:04 The sum of our x values is six.

01:09 Some of our y values is 20.

01:13 Okay, so the sum of our x values squared.

01:18 Okay, well, that's going to be 0 plus 1 plus 4.

01:25 1 plus 4 is 5.

01:26 5 and 9 is going to be 14.

01:30 And the sum of our, the product of our x and y values is going to be 0 plus 6, plus 8, plus 6, which is 20.

01:38 Again.

01:41 Okay, so i'm going to use these coefficients on the next slide and we will solve our system of equations.

01:47 Okay, so we're going to have 4b plus 6a equals 20.