The question is attached to this post. Please give full solution and explanation to the answer. On the "Rate Your Professor.com" website, you can rate your professors’ “Easiness” and “Quality” on a scale of 1-5, with higher numbers signifying easier (or higher quality) professors. At right is a scatterplot containing the ratings of all the math professors at SPSCC who had been rated at least 20 times as of Spring 2016, when I collected this data. The black dot, for example, is me. (My average E&Q scores were 2.6 and 3.3 respectively.)The line running through the data is called a “least-squares line” (also called a “regression line” or “best-fit line”). No doubt you’ve seen such lines before. But where do they come from? The slope and y-intercept of such a line are precisely determined by the coordinates of the various data points. In this extra credit problem, you’ll learn how.Suppose we have $ n $ pairs of data: $ (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) $.The line we seek will ultimately have the form $ y = mx + b $. Our task is to determine the numerical values of $ m $ and $ b $.When we consider any specific line as a candidate for the “best fit,” we judge its quality as follows: The vertical distances between the data points and the line are called residuals. The “best-fit” line is the one that minimizes the sum of the squared residuals. (Hence the name “least-squares” line.) In other words, we want to choose $ m $ and $ b $ so that they will minimize the function\[S(m, b) = \sum_{i=1}^{n} (mx_i + b - y_i)^2.\]### Your problems (You’ll need to write up your work on the blank pages that follow):**EC1.** Prove that $ m $ and $ b $ can be found by solving this system of linear equations:\[m \sum x_i^2 + b \sum x_i = \sum x_i y_i \quad \text{and} \quad m \sum x_i + nb = \sum y_i,\]where the sums all run from \( i = 1 \

EC1 From the given information, m∑xi2+b∑xi=∑xiyim∑xi+nb=∑yi Consider, m∑xi+nb=∑yi…

On the "Rate Your Professor.com website, you can rate your professors "Easiness" and "Quality" on a scale of 1-5, with higher numbers signifying easier (or higher quality) professors. At right is a scatterplot containing the ratings of all the math professors at SPSCC who had been rated at least 20 times as of Spring 2016, when I collected this data. The black dot, for example, is me. (My average E&Q scores were 2.6 and 3.3 respectively.) The line running through the data is called a "least-squares line" (also called a "regression line" or "best-fit line"). No doubt you've seen such

On the "Rate Your Professor.com website, you can rate your professors "Easiness" and "Quality" on a scale of 1-5, with higher numbers signifying easier (or higher quality) professors. At right is a scatterplot containing the ratings of all the math professors at SPSCC who had been rated at least 20 times as of Spring 2016, when I collected this data. The black dot, for example, is me. (My average E&Q scores were 2.6 and 3.3 respectively.) The line running through the data is called a "least-squares line" (also called a "regression line" or "best-fit line"). No doubt you've seen such

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

The question is attached to this post. Please give full solution and explanation to the answer.

On the "Rate Your Professor.com" website, you can rate your professors’ “Easiness” and “Quality” on a scale of 1-5, with higher numbers signifying easier (or higher quality) professors. At right is a scatterplot containing the ratings of all the math professors at SPSCC who had been rated at least 20 times as of Spring 2016, when I collected this data. The black dot, for example, is me. (My average E&Q scores were 2.6 and 3.3 respectively.)

The line running through the data is called a “least-squares line” (also called a “regression line” or “best-fit line”). No doubt you’ve seen such lines before. But where do they come from? The slope and y-intercept of such a line are precisely determined by the coordinates of the various data points. In this extra credit problem, you’ll learn how.

Suppose we have $ n $ pairs of data: $ (x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n) $.
The line we seek will ultimately have the form $ y = mx + b $. Our task is to determine the numerical values of $ m $ and $ b $.

When we consider any specific line as a candidate for the “best fit,” we judge its quality as follows: The vertical distances between the data points and the line are called residuals. The “best-fit” line is the one that minimizes the sum of the squared residuals. (Hence the name “least-squares” line.) In other words, we want to choose $ m $ and $ b $ so that they will minimize the function

\[
S(m, b) = \sum_{i=1}^{n} (mx_i + b - y_i)^2.
\]

### Your problems (You’ll need to write up your work on the blank pages that follow):

**EC1.** Prove that $ m $ and $ b $ can be found by solving this system of linear equations:

\[
m \sum x_i^2 + b \sum x_i = \sum x_i y_i \quad \text{and} \quad m \sum x_i + nb = \sum y_i,
\]

where the sums all run from \( i = 1 \

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