Explanation Given: The table shows the average SAT verbal scores of high school seniors during the 5 years since the district implemented its program. Year x 1 2 3 4 5 Average Score y 436 438 428 430 426 Concept: For n data points, P 1 ( x 1 , y 1 ) , P 2 ( x 2 , y 2 ) , P 3 ( x 3 , y 3 ) , ……, P n ( x n , y n ) The least square (regression) line for the data is given by the linear equation (function), y = f ( x ) = m x + b Where, the constant m and b satisfy the normal equations, n b + ( x 1 + x 2 + ⋅ ⋅ ⋅ + x n ) m = y 1 + y 2 + ⋅ ⋅ ⋅ + y n ( x 1 + x 2 + ⋅ ⋅ ⋅ + x n ) b + ( x 1 2 + x 2 2 + ⋅ ⋅ ⋅ + x n 2 ) m = x 1 y 1 + x 2 y 2 + ⋅ ⋅ ⋅ + x n y n Calculation: Consider the least square (regression) line, y = m x + b … (1) Consider the normal equations, n b + ( x 1 + x 2 + ⋅ ⋅ ⋅ + x n ) m = y 1 + y 2 + ⋅ ⋅ ⋅ + y n … (2) ( x 1 + x 2 + ⋅ ⋅ ⋅ + x n ) b + ( x 1 2 + x 2 2 + ⋅ ⋅ ⋅ + x n 2 ) m = x 1 y 1 + x 2 y 2 + ⋅ ⋅ ⋅ + x n y n … (3) Consider the table below, x y x 2 x y 1 436 1 436 2 438 4 876 3 428 9 1284 4 430 16 1720 5 426 25 2130 ∑ x = 15 ∑ y = 2158 ∑ x 2 = 55 ∑ x y = 6446 Table (1) Here, x i y i is the product of the observations x i and y i , substitute 1 for x 1 and 436 for y 1 in x 1 y 1 can be calculated as, To determine (b) To plot: A scatter diagram for the given data and graph the least squares line. To determine (c) To estimate: The average SAT verbal score of high school seniors 3 years from now.

WebAssign Printed Access Card for Tan's Finite Mathematics for the Managerial, Life, and Social Sciences, 12th Edition, Single-Term

12th Edition

ISBN: 9781337652766

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 1.5, Problem 9E

To determine

(a)

To find:

The equation of the least-square line for the given data.

To determine

(b)

To plot:

A scatter diagram for the given data and graph the least squares line.

To determine

(c)

To estimate:

The average SAT verbal score of high school seniors 3 years from now.

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Chapter 1.5, Problem 8E

Chapter 1.5, Problem 10E

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Chapter 1 Solutions

WebAssign Printed Access Card for Tan's Finite Mathematics for the Managerial, Life, and Social Sciences, 12th Edition, Single-Term

Ch. 1.1 - What can you say about the signs of a and b if the...Ch. 1.1 - Refer to the accompanying figure. a. Given the...Ch. 1.1 - In Exercises 16, refer to the accompanying figure...Ch. 1.1 - In Exercises 16, refer to the accompanying figure...Ch. 1.1 - In Exercises 16, refer to the accompanying figure...Ch. 1.1 - In Exercises 16, refer to the accompanying figure...Ch. 1.1 - In Exercises 16, refer to the accompanying figure...Ch. 1.1 - In Exercises 16, refer to the accompanying figure...Ch. 1.1 - In Exercises 712, refer to the accompanying...Ch. 1.1 - Prob. 8E

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(a) To find: The equation of the least-square line for the given data.

Solution Summary: The author explains the equation of the least-square line for the given data: y=(-2.8)x+440.

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(a) To find: The equation of the least-square line for the given data.

(b) To plot: A scatter diagram for the given data and graph the least squares line.

(c) To estimate: The average SAT verbal score of high school seniors 3 years from now.

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Chapter 1 Solutions

(a)

To find:

The equation of the least-square line for the given data.

(b)

To plot:

A scatter diagram for the given data and graph the least squares line.

(c)

To estimate:

The average SAT verbal score of high school seniors 3 years from now.