[Statistics] Suppose that we have n points (1, yı), -... (In: Yn) in the plane, and we wish to draw the most reasonable straight line through them. It is not likely that the line will be able to pass exactly through every point, but we would like it to be as "close as possible". Let y = a + ßx be the linear function that we use to go through these points, for a and 3 to be determined. There are various criteria we could use to determine what the "best" line is, but the most common is the least-squares line of best fit, also called the regression line. For this line, we seek to minimize the sum of the of the vertical deviations of the points from the line. (a) In everything that follows, remember that the r, and y, are the constants, and a and 3 are the variables. We seek the values of a and B (in terms of the r; and the y) which minimize f(a, B) =
[Statistics] Suppose that we have n points (1, yı), -... (In: Yn) in the plane, and we wish to draw the most reasonable straight line through them. It is not likely that the line will be able to pass exactly through every point, but we would like it to be as "close as possible". Let y = a + ßx be the linear function that we use to go through these points, for a and 3 to be determined. There are various criteria we could use to determine what the "best" line is, but the most common is the least-squares line of best fit, also called the regression line. For this line, we seek to minimize the sum of the of the vertical deviations of the points from the line. (a) In everything that follows, remember that the r, and y, are the constants, and a and 3 are the variables. We seek the values of a and B (in terms of the r; and the y) which minimize f(a, B) =
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 27EQ
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![4. [Statistics] Suppose that we have n points (21, y1), ... (In, Yn) in the plane, and we wish to draw the
most reasonable straight line through them. It is not likely that the line will be able to pass exactly through
every point, but we would like it to be as "close as possible". Let
y = a + Br
be the linear function that we use to go through these points, for a and 3 to be determined. There are
various criteria we could use to determine what the "best" line is, but the most common is the least-squares
line of best fit, also called the regression line. For this line, we seek to minimize the sum of the squares
of the vertical deviations of the points from the line.
(a) In everything that follows, remember that the r; and y4 are the constants, and a and 3 are the
variables. We seek the values of a and B (in terms of the r; and the y:) which minimize
f(a, 8) =
(b) which we do by solving the following system
af
da
af
(c) to get the following values of a and 3:
To keep the expressions from getting too messy, you can leave the expression for a in terms of 3 (so
that if you ever need to solve for a, just plug the whole B expression into it).
B =
a =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F98bf22a5-dede-49bc-a067-b329e3bb80ae%2F6ab85413-bd0d-4710-a918-0278dd6f1648%2F3dvxtxu_processed.png&w=3840&q=75)
Transcribed Image Text:4. [Statistics] Suppose that we have n points (21, y1), ... (In, Yn) in the plane, and we wish to draw the
most reasonable straight line through them. It is not likely that the line will be able to pass exactly through
every point, but we would like it to be as "close as possible". Let
y = a + Br
be the linear function that we use to go through these points, for a and 3 to be determined. There are
various criteria we could use to determine what the "best" line is, but the most common is the least-squares
line of best fit, also called the regression line. For this line, we seek to minimize the sum of the squares
of the vertical deviations of the points from the line.
(a) In everything that follows, remember that the r; and y4 are the constants, and a and 3 are the
variables. We seek the values of a and B (in terms of the r; and the y:) which minimize
f(a, 8) =
(b) which we do by solving the following system
af
da
af
(c) to get the following values of a and 3:
To keep the expressions from getting too messy, you can leave the expression for a in terms of 3 (so
that if you ever need to solve for a, just plug the whole B expression into it).
B =
a =
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