(A) Explanation Formulae used: “For a set of points ( x 1 , y 1 ) , ( x 2 , y 2 ) , ⋯ , ( x n , y n ) , the coefficients of the least squares line y = a x + b are the solutions of the system of normal equations, ( ∑ k = 1 n x k 2 ) a + ( ∑ k = 1 n x k ) b = ∑ k = 1 n x k y k ( ∑ k = 1 n x k ) a + n b = ∑ k = 1 n y k And are given by the formulas, a = n ( ∑ k = 1 n x k y k ) − ( ∑ k = 1 n x k ) ( ∑ k = 1 n y k ) n ( ∑ k = 1 n x k 2 ) − ( ∑ k = 1 n x k ) 2 and b = ( ∑ k = 1 n y k ) − a ( ∑ k = 1 n x k ) n .” Given that the value of n = 5 . The x coordinates of the data points are 2 , − 1 , 0 , 1 , 2 . Use the given data and construct the table as follows. x k y k x k y k x k 2 − 2 y 1 − 2 y 1 4 − 1 y 2 − y 2 1 0 y 3 0 0 1 y 4 y 4 1 2 y 5 2 y 5 4 ∑ k = 1 5 x k = 0 ∑ k = 1 5 y k = y 1 + y 2 + y 3 + y 4 + y 5 ∑ k = 1 5 x k y k = − 2 y 1 − y 2 + y 4 + 2 y 5 ∑ k = 1 5 x k 2 = 10 Table 1 From the above Table 1, it is observed that the values of ∑ k = 1 5 x k = 0 , ∑ k = 1 5 y k = y 1 + y 2 + y 3 + y 4 + y 5 , ∑ k = 1 5 x k y k = − 2 y 1 − y 2 + y 4 + 2 y 5 and ∑ k = 1 5 x k 2 = 10 (B) To determine To show: Part (A) holds whenever the average of the x coordinates of the given data points is 0.

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

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Question

Chapter 7.5, Problem 25E

(A)

To determine

To show: The value of a=∑xkyk∑xk2 for the given system and b equals the average of the values of yk.

(B)

To determine

To show: Part (A) holds whenever the average of the x coordinates of the given data points is 0.

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Chapter 7.5, Problem 24E

Chapter 7.5, Problem 26E

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Chapter 7 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

Ch. 7.1 - Prob. 1MP Ch. 7.1 - Matched Problem 2 Find f(2, 2, 3) for f in Example...Ch. 7.1 - Prob. 3MP Ch. 7.1 - Matched Problem 4 For the box in Example 4, find...Ch. 7.1 - Matched Problem 5 Refer to Example 5. Find the...Ch. 7.1 - Matched Problem 6 Find the coordinates of the...Ch. 7.1 - Matched Problem 7 (A) Describe the cross sections...Ch. 7.1 - Explore and Discuss 1 Imagine that you are facing...Ch. 7.1 - In Problems 18, find the indicated value of the...Ch. 7.1 - Prob. 2E

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The value of a = ∑ x k y k ∑ x k 2 for the given system and b equals the average of the values of y k .

Solution Summary: The author explains that the values of a, b, and y are the solutions of the system of normal equations.

To show: The value of a=∑xkyk∑xk2 for the given system and b equals the average of the values of yk.

To show: Part (A) holds whenever the average of the x coordinates of the given data points is 0.

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Chapter 7 Solutions