Explanation Given: The Least Squares Line- The least squares line or regression line is the line that best fits a set of points in the plane. The line that best fits the n data points ( x 1 , y 1 ) , ( x 2 , y 2 ) , … , ( x n , y n ) is the line y = a x + b , here the coefficients a and b satisfy the following pair of linear equations. (The notation ∑ k = 1 n x k stands for the sum of all the x 's .) ( ∑ k = 1 n x k ) a + n b = ∑ k = 1 n y k ( ∑ k = 1 n x k 2 ) a + ( ∑ k = 1 n x k ) b = ∑ k = 1 n x k y k Approach: Consider the equation of line, y = a x + b Here the coefficients a and b satisfy the following pair of linear equations, ( ∑ k = 1 n x k ) a + n b = ∑ k = 1 n y k ( ∑ k = 1 n x k 2 ) a + ( ∑ k = 1 n x k ) b = ∑ k = 1 n x k y k Plot the equation of line along with the given set of points on a graph and on TI-83 plus calculator as follows. • First enter data into TI-83 plus as follows. • Press on STAT and then, press on EDIT . • Enter the data. • Press on STAT , press on CALC and select linear regression, press VARS . • Select Y-VARS, and Function. • Select Y 1 press ENTER . • Press Y= . • Press the STAT PLOT and adjust the same. • Adjust the window settings. • Press GRAPH button to display the final graph. Calculation: Given, Data points- ( 1 , 3 ) = ( x 1 , y 1 ) ( 2 , 5 ) = ( x 2 , y 2 ) ( 3 , 6 ) = ( x 3 , y 3 ) ( 5 , 6 ) = ( x 4 , y 4 ) , ( 7 , 9 ) = ( x 5 , y 5 ) , i.e. n = k = 5 Solve for the coefficients a and b in y = a x + b , ( ∑ k = 1 n x k ) a + n b = ∑ k = 1 n y k ( ∑ k = 1 5 x k ) a + 5 b = ∑ k = 1 5 y k ( ∑ k = 1 n x k 2 ) a + ( ∑ k = 1 n x k ) b = ∑ k = 1 n x k y k ( ∑ k = 1 5 x k 2 ) a + ( ∑ k = 1 5 x k ) b = ∑ k = 1 5 x k y k ⋯ ( 1 ) ∑ k = 1 5 x k = x 1 + x 2 + x 3 + x 4 + x 5 = 1 + 2 + 3 + 5 + 7 = 18 ∑ k = 1 5 y k = y 1 + y 2 + y 3 + y 4 + y 5 = 3 + 5 + 6 + 6 + 9 = 29 ∑ k = 1 5 x k 2 = x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 = 1 2 + 2 2 + 3 2 + 5 2 + 7 2 = 1 + 4 + 9 + 25 + 49 = 88

4th Edition

ISBN: 9781305719781

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: PEARSON

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Chapter 10.1, Problem 77E

To determine

To find:

The least square line for the following data points:

(1,3), (2,5), (3,6), (5,6), (7,9)

Sketch the points and the line to confirm that the line fits these points well.

If the calculator computes regression lines, see whether it gives the same line as the formulas.

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The least square line for the following data points: ( 1 , 3 ) , ( 2 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) , ( 7 , 9 ) Sketch the points and the line to confirm that the line fits these points well. If the calculator computes regression lines, see whether it gives the same line as the formulas.

Solution Summary: The author explains that the least squares line or regression line is the line that best fits a set of points in the plane. If the calculator computes regression lines, it gives the same line as the formulas.

Concept explainers

Videos

To find:

The least square line for the following data points:

(1,3), (2,5), (3,6), (5,6), (7,9)

Sketch the points and the line to confirm that the line fits these points well.

If the calculator computes regression lines, see whether it gives the same line as the formulas.

Chapter 10 Solutions

The least square line for the following data points: ( 1 , 3 ) , ( 2 , 5 ) , ( 3 , 6 ) , ( 5 , 6 ) , ( 7 , 9 ) Sketch the points and the line to confirm that the line fits these points well. If the calculator computes regression lines, see whether it gives the same line as the formulas.

Solution Summary: The author explains that the least squares line or regression line is the line that best fits a set of points in the plane. If the calculator computes regression lines, it gives the same line as the formulas.

Concept explainers

Videos

To find: The least square line for the following data points: (1,3), (2,5), (3,6), (5,6), (7,9) Sketch the points and the line to confirm that the line fits these points well. If the calculator computes regression lines, see whether it gives the same line as the formulas.

Chapter 10 Solutions

To find:

The least square line for the following data points:

(1,3), (2,5), (3,6), (5,6), (7,9)

Sketch the points and the line to confirm that the line fits these points well.

If the calculator computes regression lines, see whether it gives the same line as the formulas.